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Similarly, simulations of processes on long timescales (beyond about 1 microsecond) are prohibitively expensive, because they require so many time steps. In these cases, one can sometimes tackle the problem by using reduced representations, which are also called coarse-grained models.
Examples for coarse graining (CG) methods are discontinuous molecular dynamics (CG-DMD) and Go-models. Coarse-graining is done sometimes taking larger pseudo-atoms. Such united atom approximations have been used in MD simulations of biological membranes. Implementation of such approach on systems where electrical properties are of interest can be challenging owing to the difficulty of using a proper charge distribution on the pseudo-atoms. The aliphatic tails of lipids are represented by a few pseudo-atoms by gathering 2 to 4 methylene groups into each pseudo-atom.
The parameterization of these very coarse-grained models must be done empirically, by matching the behavior of the model to appropriate experimental data or all-atom simulations. Ideally, these parameters should account for both enthalpic and entropic contributions to free energy in an implicit way. When coarse-graining is done at higher levels, the accuracy of the dynamic description may be less reliable.
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Ideally, these parameters should account for both enthalpic and entropic contributions to free energy in an implicit way. When coarse-graining is done at higher levels, the accuracy of the dynamic description may be less reliable. But very coarse-grained models have been used successfully to examine a wide range of questions in structural biology, liquid crystal organization, and polymer glasses.
## Examples of applications
of coarse-graining:
- protein folding and protein structure prediction studies are often carried out using one, or a few, pseudo-atoms per amino acid;
- liquid crystal phase transitions have been examined in confined geometries and/or during flow using the Gay-Berne potential, which describes anisotropic species;
- Polymer glasses during deformation have been studied using simple harmonic or FENE springs to connect spheres described by the Lennard-Jones potential;
- DNA supercoiling has been investigated using 1–3 pseudo-atoms per basepair, and at even lower resolution;
- Packaging of double-helical DNA into bacteriophage has been investigated with models where one pseudo-atom represents one turn (about 10 basepairs) of the double helix;
- RNA structure in the ribosome and other large systems has been modeled with one pseudo-atom per nucleotide.
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But very coarse-grained models have been used successfully to examine a wide range of questions in structural biology, liquid crystal organization, and polymer glasses.
## Examples of applications
of coarse-graining:
- protein folding and protein structure prediction studies are often carried out using one, or a few, pseudo-atoms per amino acid;
- liquid crystal phase transitions have been examined in confined geometries and/or during flow using the Gay-Berne potential, which describes anisotropic species;
- Polymer glasses during deformation have been studied using simple harmonic or FENE springs to connect spheres described by the Lennard-Jones potential;
- DNA supercoiling has been investigated using 1–3 pseudo-atoms per basepair, and at even lower resolution;
- Packaging of double-helical DNA into bacteriophage has been investigated with models where one pseudo-atom represents one turn (about 10 basepairs) of the double helix;
- RNA structure in the ribosome and other large systems has been modeled with one pseudo-atom per nucleotide.
The simplest form of coarse-graining is the united atom (sometimes called extended atom) and was used in most early MD simulations of proteins, lipids, and nucleic acids. For example, instead of treating all four atoms of a CH3 methyl group explicitly (or all three atoms of CH2 methylene group), one represents the whole group with one pseudo-atom.
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The simplest form of coarse-graining is the united atom (sometimes called extended atom) and was used in most early MD simulations of proteins, lipids, and nucleic acids. For example, instead of treating all four atoms of a CH3 methyl group explicitly (or all three atoms of CH2 methylene group), one represents the whole group with one pseudo-atom. It must, of course, be properly parameterized so that its van der Waals interactions with other groups have the proper distance-dependence. Similar considerations apply to the bonds, angles, and torsions in which the pseudo-atom participates. In this kind of united atom representation, one typically eliminates all explicit hydrogen atoms except those that have the capability to participate in hydrogen bonds (polar hydrogens). An example of this is the CHARMM 19 force-field.
The polar hydrogens are usually retained in the model, because proper treatment of hydrogen bonds requires a reasonably accurate description of the directionality and the electrostatic interactions between the donor and acceptor groups. A hydroxyl group, for example, can be both a hydrogen bond donor, and a hydrogen bond acceptor, and it would be impossible to treat this with one OH pseudo-atom. About half the atoms in a protein or nucleic acid are non-polar hydrogens, so the use of united atoms can provide a substantial savings in computer time.
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A hydroxyl group, for example, can be both a hydrogen bond donor, and a hydrogen bond acceptor, and it would be impossible to treat this with one OH pseudo-atom. About half the atoms in a protein or nucleic acid are non-polar hydrogens, so the use of united atoms can provide a substantial savings in computer time.
### Machine Learning Force Fields
Machine Learning Force Fields] (MLFFs) represent one approach to modeling interatomic interactions in molecular dynamics simulations. MLFFs can achieve accuracy close to that of ab initio methods. Once trained, MLFFs are much faster than direct quantum mechanical calculations. MLFFs address the limitations of traditional force fields by learning complex potential energy surfaces directly from high-level quantum mechanical data. Several software packages now support MLFFs, including VASP and open-source libraries like DeePMD-kit and SchNetPack.
## Incorporating solvent effects
In many simulations of a solute-solvent system the main focus is on the behavior of the solute with little interest of the solvent behavior particularly in those solvent molecules residing in regions far from the solute molecule. Solvents may influence the dynamic behavior of solutes via random collisions and by imposing a frictional drag on the motion of the solute through the solvent.
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## Incorporating solvent effects
In many simulations of a solute-solvent system the main focus is on the behavior of the solute with little interest of the solvent behavior particularly in those solvent molecules residing in regions far from the solute molecule. Solvents may influence the dynamic behavior of solutes via random collisions and by imposing a frictional drag on the motion of the solute through the solvent. The use of non-rectangular periodic boundary conditions, stochastic boundaries and solvent shells can all help reduce the number of solvent molecules required and enable a larger proportion of the computing time to be spent instead on simulating the solute. It is also possible to incorporate the effects of a solvent without needing any explicit solvent molecules present. One example of this approach is to use a potential mean force (PMF) which describes how the free energy changes as a particular coordinate is varied. The free energy change described by PMF contains the averaged effects of the solvent.
Without incorporating the effects of solvent simulations of macromolecules (such as proteins) may yield unrealistic behavior and even small molecules may adopt more compact conformations due to favourable van der Waals forces and electrostatic interactions which would be dampened in the presence of a solvent.
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The free energy change described by PMF contains the averaged effects of the solvent.
Without incorporating the effects of solvent simulations of macromolecules (such as proteins) may yield unrealistic behavior and even small molecules may adopt more compact conformations due to favourable van der Waals forces and electrostatic interactions which would be dampened in the presence of a solvent.
## Long-range forces
A long range interaction is an interaction in which the spatial interaction falls off no faster than
$$
r^{-d}
$$
where _ BLOCK1_ is the dimensionality of the system. Examples include charge-charge interactions between ions and dipole-dipole interactions between molecules. Modelling these forces presents quite a challenge as they are significant over a distance which may be larger than half the box length with simulations of many thousands of particles. Though one solution would be to significantly increase the size of the box length, this brute force approach is less than ideal as the simulation would become computationally very expensive. Spherically truncating the potential is also out of the question as unrealistic behaviour may be observed when the distance is close to the cut off distance.
## Steered molecular dynamics (SMD)
Steered molecular dynamics (SMD) simulations, or force probe simulations, apply forces to a protein in order to manipulate its structure by pulling it along desired degrees of freedom.
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## Steered molecular dynamics (SMD)
Steered molecular dynamics (SMD) simulations, or force probe simulations, apply forces to a protein in order to manipulate its structure by pulling it along desired degrees of freedom. These experiments can be used to reveal structural changes in a protein at the atomic level. SMD is often used to simulate events such as mechanical unfolding or stretching.
There are two typical protocols of SMD: one in which pulling velocity is held constant, and one in which applied force is constant. Typically, part of the studied system (e.g., an atom in a protein) is restrained by a harmonic potential. Forces are then applied to specific atoms at either a constant velocity or a constant force. Umbrella sampling is used to move the system along the desired reaction coordinate by varying, for example, the forces, distances, and angles manipulated in the simulation. Through umbrella sampling, all of the system's configurations—both high-energy and low-energy—are adequately sampled. Then, each configuration's change in free energy can be calculated as the potential of mean force. A popular method of computing PMF is through the weighted histogram analysis method (WHAM), which analyzes a series of umbrella sampling simulations.
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Then, each configuration's change in free energy can be calculated as the potential of mean force. A popular method of computing PMF is through the weighted histogram analysis method (WHAM), which analyzes a series of umbrella sampling simulations.
A lot of important applications of SMD are in the field of drug discovery and biomolecular sciences. For e.g. SMD was used to investigate the stability of Alzheimer's protofibrils, to study the protein ligand interaction in cyclin-dependent kinase 5 and even to show the effect of electric field on thrombin (protein) and aptamer (nucleotide) complex among many other interesting studies.
Examples of applications
Molecular dynamics is used in many fields of science.
- First MD simulation of a simplified biological folding process was published in 1975. Its simulation published in Nature paved the way for the vast area of modern computational protein-folding.
- First MD simulation of a biological process was published in 1976. Its simulation published in Nature paved the way for understanding protein motion as essential in function and not just accessory.
- MD is the standard method to treat collision cascades in the heat spike regime, i.e., the effects that energetic neutron and ion irradiation have on solids and solid surfaces.
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- First MD simulation of a biological process was published in 1976. Its simulation published in Nature paved the way for understanding protein motion as essential in function and not just accessory.
- MD is the standard method to treat collision cascades in the heat spike regime, i.e., the effects that energetic neutron and ion irradiation have on solids and solid surfaces.
The following biophysical examples illustrate notable efforts to produce simulations of a systems of very large size (a complete virus) or very long simulation times (up to 1.112 milliseconds):
- MD simulation of the full satellite tobacco mosaic virus (STMV) (2006, Size: 1 million atoms, Simulation time: 50 ns, program: NAMD) This virus is a small, icosahedral plant virus that worsens the symptoms of infection by Tobacco Mosaic Virus (TMV). Molecular dynamics simulations were used to probe the mechanisms of viral assembly. The entire STMV particle consists of 60 identical copies of one protein that make up the viral capsid (coating), and a 1063 nucleotide single stranded RNA genome. One key finding is that the capsid is very unstable when there is no RNA inside. The simulation would take one 2006 desktop computer around 35 years to complete. It was thus done in many processors in parallel with continuous communication between them.
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The simulation would take one 2006 desktop computer around 35 years to complete. It was thus done in many processors in parallel with continuous communication between them.
- Folding simulations of the Villin Headpiece in all-atom detail (2006, Size: 20,000 atoms; Simulation time: 500 μs= 500,000 ns, Program: Folding@home) This simulation was run in 200,000 CPU's of participating personal computers around the world. These computers had the Folding@home program installed, a large-scale distributed computing effort coordinated by Vijay Pande at Stanford University. The kinetic properties of the Villin Headpiece protein were probed by using many independent, short trajectories run by CPU's without continuous real-time communication. One method employed was the Pfold value analysis, which measures the probability of folding before unfolding of a specific starting conformation. Pfold gives information about transition state structures and an ordering of conformations along the folding pathway. Each trajectory in a Pfold calculation can be relatively short, but many independent trajectories are needed.
- Long continuous-trajectory simulations have been performed on Anton, a massively parallel supercomputer designed and built around custom application-specific integrated circuits (ASICs) and interconnects by D. E. Shaw Research.
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Pfold gives information about transition state structures and an ordering of conformations along the folding pathway. Each trajectory in a Pfold calculation can be relatively short, but many independent trajectories are needed.
- Long continuous-trajectory simulations have been performed on Anton, a massively parallel supercomputer designed and built around custom application-specific integrated circuits (ASICs) and interconnects by D. E. Shaw Research. The longest published result of a simulation performed using Anton is a 1.112-millisecond simulation of NTL9 at 355 K; a second, independent 1.073-millisecond simulation of this configuration was also performed (and many other simulations of over 250 μs continuous chemical time). In How Fast-Folding Proteins Fold, researchers Kresten Lindorff-Larsen, Stefano Piana, Ron O. Dror, and David E. Shaw discuss "the results of atomic-level molecular dynamics simulations, over periods ranging between 100 μs and 1 ms, that reveal a set of common principles underlying the folding of 12 structurally diverse proteins." Examination of these diverse long trajectories, enabled by specialized, custom hardware, allow them to conclude that "In most cases, folding follows a single dominant route in which elements of the native structure appear in an order highly correlated with their propensity to form in the unfolded state."
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In How Fast-Folding Proteins Fold, researchers Kresten Lindorff-Larsen, Stefano Piana, Ron O. Dror, and David E. Shaw discuss "the results of atomic-level molecular dynamics simulations, over periods ranging between 100 μs and 1 ms, that reveal a set of common principles underlying the folding of 12 structurally diverse proteins." Examination of these diverse long trajectories, enabled by specialized, custom hardware, allow them to conclude that "In most cases, folding follows a single dominant route in which elements of the native structure appear in an order highly correlated with their propensity to form in the unfolded state." In a separate study, Anton was used to conduct a 1.013-millisecond simulation of the native-state dynamics of bovine pancreatic trypsin inhibitor (BPTI) at 300 K.
Another important application of MD method benefits from its ability of 3-dimensional characterization and analysis of microstructural evolution at atomic scale.
- MD simulations are used in characterization of grain size evolution, for example, when describing wear and friction of nanocrystalline Al and Al(Zr) materials. Dislocations evolution and grain size evolution are analyzed during the friction process in this simulation. Since MD method provided the full information of the microstructure, the grain size evolution was calculated in 3D using the Polyhedral Template Matching, Grain Segmentation, and Graph clustering methods.
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Dislocations evolution and grain size evolution are analyzed during the friction process in this simulation. Since MD method provided the full information of the microstructure, the grain size evolution was calculated in 3D using the Polyhedral Template Matching, Grain Segmentation, and Graph clustering methods. In such simulation, MD method provided an accurate measurement of grain size. Making use of these information, the actual grain structures were extracted, measured, and presented. Compared to the traditional method of using SEM with a single 2-dimensional slice of the material, MD provides a 3-dimensional and accurate way to characterize the microstructural evolution at atomic scale.
## Molecular dynamics algorithms
- Screened Coulomb potentials implicit solvent model
### Integrators
- Symplectic integrator
- Verlet–Stoermer integration
- Runge–Kutta integration
- Beeman's algorithm
- Constraint algorithms (for constrained systems)
### Short-range interaction algorithms
- Cell lists
- Verlet list
- Bonded interactions
### Long-range interaction algorithms
- Ewald summation
- Particle mesh Ewald summation (PME)
- Particle–particle-particle–mesh (P3M)
- Shifted force method
### Parallelization strategies
- Domain decomposition method (Distribution of system data for parallel computing)
### Ab-initio molecular dynamics
- Car–Parrinello molecular dynamics
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### Parallelization strategies
- Domain decomposition method (Distribution of system data for parallel computing)
### Ab-initio molecular dynamics
- Car–Parrinello molecular dynamics
## Specialized hardware for MD simulations
- Anton – A specialized, massively parallel supercomputer designed to execute MD simulations
- MDGRAPE – A special purpose system built for molecular dynamics simulations, especially protein structure prediction
## Graphics card as a hardware for MD simulations
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Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. Magnetostatics is even a good approximation when the currents are not static – as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory.
## Applications
### Magnetostatics as a special case of Maxwell's equations
Starting from Maxwell's equations and assuming that charges are either fixed or move as a steady current
$$
\mathbf{J}
$$
, the equations separate into two equations for the electric field (see electrostatics) and two for the magnetic field. The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.
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The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.
Name Form Differential Integral Gauss's law for magnetism Ampère's law
Where ∇ with the dot denotes divergence, and B is the magnetic flux density, the first integral is over a surface
$$
S
$$
with oriented surface element
$$
d\mathbf{S}
$$
. Where ∇ with the cross denotes curl, J is the current density and is the magnetic field intensity, the second integral is a line integral around a closed loop
$$
C
$$
with line element
$$
\mathbf{l}
$$
. The current going through the loop is
$$
I_\text{enc}
$$
.
The quality of this approximation may be guessed by comparing the above equations with the full version of Maxwell's equations and considering the importance of the terms that have been removed. Of particular significance is the comparison of the
$$
\mathbf{J}
$$
term against the
$$
\partial \mathbf{D} / \partial t
$$
term. If the
$$
\mathbf{J}
$$
term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.
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Of particular significance is the comparison of the
$$
\mathbf{J}
$$
term against the
$$
\partial \mathbf{D} / \partial t
$$
term. If the
$$
\mathbf{J}
$$
term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.
### Re-introducing Faraday's law
A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term
$$
\partial \mathbf{B} / \partial t
$$
. Plugging this result into Faraday's Law finds a value for
$$
\mathbf{E}
$$
(which had previously been ignored). This method is not a true solution of Maxwell's equations but can provide a good approximation for slowly changing fields.
## Solving for the magnetic field
### Current sources
If all currents in a system are known (i.e., if a complete description of the current density
$$
\mathbf{J}(\mathbf{r})
$$
is available)
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## Solving for the magnetic field
### Current sources
If all currents in a system are known (i.e., if a complete description of the current density
$$
\mathbf{J}(\mathbf{r})
$$
is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation:
$$
\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int{\frac{\mathbf{J}(\mathbf{r}') \times \left(\mathbf{r} - \mathbf{r}'\right)}{|\mathbf{r} - \mathbf{r}'|^3} \mathrm{d}^3\mathbf{r}'}
$$
This technique works well for problems where the medium is a vacuum or air or some similar material with a relative permeability of 1. This includes air-core inductors and air-core transformers. One advantage of this technique is that, if a coil has a complex geometry, it can be divided into sections and the integral evaluated for each section. Since this equation is primarily used to solve linear problems, the contributions can be added. For a very difficult geometry, numerical integration may be used.
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Since this equation is primarily used to solve linear problems, the contributions can be added. For a very difficult geometry, numerical integration may be used.
For problems where the dominant magnetic material is a highly permeable magnetic core with relatively small air gaps, a magnetic circuit approach is useful. When the air gaps are large in comparison to the magnetic circuit length, fringing becomes significant and usually requires a finite element calculation. The finite element calculation uses a modified form of the magnetostatic equations above in order to calculate magnetic potential. The value of
$$
\mathbf{B}
$$
can be found from the magnetic potential.
The magnetic field can be derived from the vector potential. Since the divergence of the magnetic flux density is always zero,
$$
\mathbf{B} = \nabla \times \mathbf{A},
$$
and the relation of the vector potential to current is:
$$
\mathbf{A}(\mathbf{r}) = \frac{\mu_{0}}{4\pi} \int{ \frac{\mathbf{J(\mathbf{r}')} } {|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3\mathbf{r}'}.
$$
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The magnetic field can be derived from the vector potential. Since the divergence of the magnetic flux density is always zero,
$$
\mathbf{B} = \nabla \times \mathbf{A},
$$
and the relation of the vector potential to current is:
$$
\mathbf{A}(\mathbf{r}) = \frac{\mu_{0}}{4\pi} \int{ \frac{\mathbf{J(\mathbf{r}')} } {|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3\mathbf{r}'}.
$$
### Magnetization
Strongly magnetic materials (i.e., ferromagnetic, ferrimagnetic or paramagnetic) have a magnetization that is primarily due to electron spin. In such materials the magnetization must be explicitly included using the relation
$$
\mathbf{B} = \mu_0(\mathbf{M}+\mathbf{H}).
$$
Except in the case of conductors, electric currents can be ignored.
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### Magnetization
Strongly magnetic materials (i.e., ferromagnetic, ferrimagnetic or paramagnetic) have a magnetization that is primarily due to electron spin. In such materials the magnetization must be explicitly included using the relation
$$
\mathbf{B} = \mu_0(\mathbf{M}+\mathbf{H}).
$$
Except in the case of conductors, electric currents can be ignored. Then Ampère's law is simply
$$
\nabla\times\mathbf{H} = 0.
$$
This has the general solution
$$
\mathbf{H} = -\nabla \Phi_M,
$$
where
$$
\Phi_M
$$
is a scalar potential. Substituting this in Gauss's law gives
$$
\nabla^2 \Phi_M = \nabla\cdot\mathbf{M}.
$$
Thus, the divergence of the magnetization,
$$
\nabla\cdot\mathbf{M},
$$
has a role analogous to the electric charge in electrostatics and is often referred to as an effective charge density
$$
\rho_M
$$
.
The vector potential method can also be employed with an effective current density
$$
\mathbf{J_M} = \nabla \times \mathbf{M}.
$$
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A gravitational lens is matter, such as a cluster of galaxies or a point particle, that bends light from a distant source as it travels toward an observer. The amount of gravitational lensing is described by Albert Einstein's general theory of relativity. If light is treated as corpuscles travelling at the speed of light, Newtonian physics also predicts the bending of light, but only half of that predicted by general relativity. Extract of page 21 Extract of page 165
Orest Khvolson (1924) and Frantisek Link (1936) are generally credited with being the first to discuss the effect in print, but it is more commonly associated with Einstein, who made unpublished calculations on it in 1912 and published an article on the subject in 1936.
In 1937, Fritz Zwicky posited that galaxy clusters could act as gravitational lenses, a claim confirmed in 1979 by observation of the Twin QSO SBS 0957+561.
## Description
Unlike an optical lens, a point-like gravitational lens produces a maximum deflection of light that passes closest to its center, and a minimum deflection of light that travels furthest from its center. Consequently, a gravitational lens has no single focal point, but a focal line.
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## Description
Unlike an optical lens, a point-like gravitational lens produces a maximum deflection of light that passes closest to its center, and a minimum deflection of light that travels furthest from its center. Consequently, a gravitational lens has no single focal point, but a focal line. The term "lens" in the context of gravitational light deflection was first used by O. J. Lodge, who remarked that it is "not permissible to say that the solar gravitational field acts like a lens, for it has no focal length". If the (light) source, the massive lensing object, and the observer lie in a straight line, the original light source will appear as a ring around the massive lensing object (provided the lens has circular symmetry). If there is any misalignment, the observer will see an arc segment instead.
This phenomenon was first mentioned in 1924 by the St. Petersburg physicist Orest Khvolson, and quantified by Albert Einstein in 1936. It is usually referred to in the literature as an Einstein ring, since Khvolson did not concern himself with the flux or radius of the ring image. More commonly, where the lensing mass is complex (such as a galaxy group or cluster) and does not cause a spherical distortion of spacetime, the source will resemble partial arcs scattered around the lens.
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It is usually referred to in the literature as an Einstein ring, since Khvolson did not concern himself with the flux or radius of the ring image. More commonly, where the lensing mass is complex (such as a galaxy group or cluster) and does not cause a spherical distortion of spacetime, the source will resemble partial arcs scattered around the lens. The observer may then see multiple distorted images of the same source; the number and shape of these depending upon the relative positions of the source, lens, and observer, and the shape of the gravitational well of the lensing object.
There are three classes of gravitational lensing:
Strong lensing Where there are easily visible distortions such as the formation of Einstein rings, arcs, and multiple images. Despite being considered "strong", the effect is in general relatively small, such that even a galaxy with a mass more than 100 billion times that of the Sun will produce multiple images separated by only a few arcseconds. Galaxy clusters can produce separations of several arcminutes. In both cases the galaxies and sources are quite distant, many hundreds of megaparsecs away from the Milky Way Galaxy.
Weak lensing Where the distortions of background sources are much smaller and can only be detected by analyzing large numbers of sources in a statistical way to find coherent distortions of only a few percent.
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In both cases the galaxies and sources are quite distant, many hundreds of megaparsecs away from the Milky Way Galaxy.
Weak lensing Where the distortions of background sources are much smaller and can only be detected by analyzing large numbers of sources in a statistical way to find coherent distortions of only a few percent. The lensing shows up statistically as a preferred stretching of the background objects perpendicular to the direction to the centre of the lens. By measuring the shapes and orientations of large numbers of distant galaxies, their orientations can be averaged to measure the shear of the lensing field in any region. This, in turn, can be used to reconstruct the mass distribution in the area: in particular, the background distribution of dark matter can be reconstructed. Since galaxies are intrinsically elliptical and the weak gravitational lensing signal is small, a very large number of galaxies must be used in these surveys. These weak lensing surveys must carefully avoid a number of important sources of systematic error: the intrinsic shape of galaxies, the tendency of a camera's point spread function to distort the shape of a galaxy and the tendency of atmospheric seeing to distort images must be understood and carefully accounted for.
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Since galaxies are intrinsically elliptical and the weak gravitational lensing signal is small, a very large number of galaxies must be used in these surveys. These weak lensing surveys must carefully avoid a number of important sources of systematic error: the intrinsic shape of galaxies, the tendency of a camera's point spread function to distort the shape of a galaxy and the tendency of atmospheric seeing to distort images must be understood and carefully accounted for. The results of these surveys are important for cosmological parameter estimation, to better understand and improve upon the Lambda-CDM model, and to provide a consistency check on other cosmological observations. They may also provide an important future constraint on dark energy.
Microlensing Where no distortion in shape can be seen but the amount of light received from a background object changes in time. The lensing object may be stars in the Milky Way in one typical case, with the background source being stars in a remote galaxy, or, in another case, an even more distant quasar. In extreme cases, a star in a distant galaxy can act as a microlens and magnify another star much farther away. The first example of this was the star MACS J1149 Lensed Star 1 (also known as Icarus), thanks to the boost in flux due to the microlensing effect.
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The first example of this was the star MACS J1149 Lensed Star 1 (also known as Icarus), thanks to the boost in flux due to the microlensing effect.
Gravitational lenses act equally on all kinds of electromagnetic radiation, not just visible light, and also in non-electromagnetic radiation, like gravitational waves. Weak lensing effects are being studied for the cosmic microwave background as well as galaxy surveys. Strong lenses have been observed in radio and x-ray regimes as well. If a strong lens produces multiple images, there will be a relative time delay between two paths: that is, in one image the lensed object will be observed before the other image.
## History
Henry Cavendish in 1784 (in an unpublished manuscript) and Johann Georg von Soldner in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object as had already been supposed by Isaac Newton in 1704 in his Queries No.1 in his book Opticks. The same value as Soldner's was calculated by Einstein in 1911 based on the equivalence principle alone. However, Einstein noted in 1915, in the process of completing general relativity, that his (and thus Soldner's) 1911-result is only half of the correct value. Einstein became the first to calculate the correct value for light bending.
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However, Einstein noted in 1915, in the process of completing general relativity, that his (and thus Soldner's) 1911-result is only half of the correct value. Einstein became the first to calculate the correct value for light bending.
The first observation of light deflection was performed by noting the change in position of stars as they passed near the Sun on the celestial sphere. The observations were performed in 1919 by Arthur Eddington, Frank Watson Dyson, and their collaborators during the total solar eclipse on May 29. The solar eclipse allowed the stars near the Sun to be observed. Observations were made simultaneously in the cities of Sobral, Ceará, Brazil and in São Tomé and Príncipe on the west coast of Africa. The observations demonstrated that the light from stars passing close to the Sun was slightly bent, so that stars appeared slightly out of position.
The result was considered spectacular news and made the front page of most major newspapers. It made Einstein and his theory of general relativity world-famous. When asked by his assistant what his reaction would have been if general relativity had not been confirmed by Eddington and Dyson in 1919, Einstein said "Then I would feel sorry for the dear Lord. The theory is correct anyway." In 1912, Einstein had speculated that an observer could see multiple images of a single light source, if the light were deflected around a mass.
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The theory is correct anyway." In 1912, Einstein had speculated that an observer could see multiple images of a single light source, if the light were deflected around a mass. This effect would make the mass act as a kind of gravitational lens. However, as he only considered the effect of deflection around a single star, he seemed to conclude that the phenomenon was unlikely to be observed for the foreseeable future since the necessary alignments between stars and observer would be highly improbable. Several other physicists speculated about gravitational lensing as well, but all reached the same conclusion that it would be nearly impossible to observe.
Although Einstein made unpublished calculations on the subject, the first discussion of the gravitational lens in print was by Khvolson, in a short article discussing the "halo effect" of gravitation when the source, lens, and observer are in near-perfect alignment, now referred to as the Einstein ring.
In 1936, after some urging by Rudi W. Mandl, Einstein reluctantly published the short article "Lens-Like Action of a Star By the Deviation of Light In the Gravitational Field" in the journal Science.
In 1937, Fritz Zwicky first considered the case where the newly discovered galaxies (which were called 'nebulae' at the time) could act as both source and lens, and that, because of the mass and sizes involved, the effect was much more likely to be observed.
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In 1936, after some urging by Rudi W. Mandl, Einstein reluctantly published the short article "Lens-Like Action of a Star By the Deviation of Light In the Gravitational Field" in the journal Science.
In 1937, Fritz Zwicky first considered the case where the newly discovered galaxies (which were called 'nebulae' at the time) could act as both source and lens, and that, because of the mass and sizes involved, the effect was much more likely to be observed.
In 1963 Yu. G. Klimov, S. Liebes, and Sjur Refsdal recognized independently that quasars are an ideal light source for the gravitational lens effect.
It was not until 1979 that the first gravitational lens would be discovered. It became known as the "Twin QSO" since it initially looked like two identical quasistellar objects. (It is officially named SBS 0957+561.) This gravitational lens was discovered by Dennis Walsh, Bob Carswell, and Ray Weymann using the Kitt Peak National Observatory 2.1 meter telescope.
In the 1980s, astronomers realized that the combination of CCD imagers and computers would allow the brightness of millions of stars to be measured each night. In a dense field, such as the galactic center or the Magellanic clouds, many microlensing events per year could potentially be found.
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In the 1980s, astronomers realized that the combination of CCD imagers and computers would allow the brightness of millions of stars to be measured each night. In a dense field, such as the galactic center or the Magellanic clouds, many microlensing events per year could potentially be found. This led to efforts such as Optical Gravitational Lensing Experiment, or OGLE, that have characterized hundreds of such events, including those of OGLE-2016-BLG-1190Lb and OGLE-2016-BLG-1195Lb.
## Approximate Newtonian description
Newton wondered whether light, in the form of corpuscles, would be bent due to gravity. The Newtonian prediction for light deflection refers to the amount of deflection a corpuscle would feel under the effect of gravity, and therefore one should read "Newtonian" in this context as the referring to the following calculations and not a belief that Newton held in the validity of these calculations.
For a gravitational point-mass lens of mass
$$
M
$$
, a corpuscle of mass
$$
m
$$
feels a force
$$
\vec F = -\frac{GMm}{r^2} \hat r,
$$
where _ BLOCK3_ is the lens-corpuscle separation.
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For a gravitational point-mass lens of mass
$$
M
$$
, a corpuscle of mass
$$
m
$$
feels a force
$$
\vec F = -\frac{GMm}{r^2} \hat r,
$$
where _ BLOCK3_ is the lens-corpuscle separation. If we equate this force with Newton's second law, we can solve for the acceleration that the light undergoes:
$$
\vec a = -\frac{GM}{r^2} \hat r.
$$
The light interacts with the lens from initial time
$$
t = 0
$$
to
$$
t
$$
, and the velocity boost the corpuscle receives is
$$
\Delta \vec v = -\int_0^t dt'\, \frac{GM}{r(t')^2} \hat r(t').
$$
If one assumes that initially the light is far enough from the lens to neglect gravity, the perpendicular distance between the light's initial trajectory and the lens is b (the impact parameter), and the parallel distance is
$$
r_\parallel
$$
, such that
$$
r^2 = b^2 + r_\parallel^2
$$
.
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BLOCK3_ is the lens-corpuscle separation. If we equate this force with Newton's second law, we can solve for the acceleration that the light undergoes:
$$
\vec a = -\frac{GM}{r^2} \hat r.
$$
The light interacts with the lens from initial time
$$
t = 0
$$
to
$$
t
$$
, and the velocity boost the corpuscle receives is
$$
\Delta \vec v = -\int_0^t dt'\, \frac{GM}{r(t')^2} \hat r(t').
$$
If one assumes that initially the light is far enough from the lens to neglect gravity, the perpendicular distance between the light's initial trajectory and the lens is b (the impact parameter), and the parallel distance is
$$
r_\parallel
$$
, such that
$$
r^2 = b^2 + r_\parallel^2
$$
. We additionally assume a constant speed of light along the parallel direction,
$$
dr_\parallel \approx c\,dt
$$
, and that the light is only being deflected a small amount. After plugging these assumptions into the above equation and further simplifying, one can solve for the velocity boost in the perpendicular direction.
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We additionally assume a constant speed of light along the parallel direction,
$$
dr_\parallel \approx c\,dt
$$
, and that the light is only being deflected a small amount. After plugging these assumptions into the above equation and further simplifying, one can solve for the velocity boost in the perpendicular direction. The angle of deflection between the corpuscle’s initial and final trajectories is therefore (see, e.g., M. Meneghetti 2021)
$$
\theta = \frac{2GM}{c^2 r}.
$$
Although this result appears to be half the prediction from general relativity, classical physics predicts that the speed of light
$$
c
$$
is observer-dependent (see, e.g., L. Susskind and A. Friedman 2018) which was superseded by a universal speed of light in special relativity.
## Explanation in terms of spacetime curvature
In general relativity, light follows the curvature of spacetime, hence when light passes around a massive object, it is bent. This means that the light from an object on the other side will be bent towards an observer's eye, just like an ordinary lens. In general relativity the path of light depends on the shape of space (i.e. the metric).
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This means that the light from an object on the other side will be bent towards an observer's eye, just like an ordinary lens. In general relativity the path of light depends on the shape of space (i.e. the metric). The gravitational attraction can be viewed as the motion of undisturbed objects in a background curved geometry or alternatively as the response of objects to a force in a flat geometry. The angle of deflection is
$$
\theta = \frac{4GM}{c^2 r}
$$
toward the mass M at a distance r from the affected radiation, where G is the universal constant of gravitation, and c is the speed of light in vacuum.
Since the Schwarzschild radius
$$
r_\text{s}
$$
is defined as
$$
r_\text{s} = 2Gm/c^2
$$
, and escape velocity
$$
v_\text{e}
$$
is defined as
$$
v_\text{e} = \sqrt{2Gm/r} = \beta_\text{e} c
$$
, this can also be expressed in simple form as
$$
\theta = 2 \frac{r_\text{s}}{r} = 2 \left(\frac{v_\text{e}}{c}\right)^2 = 2\beta_\text{e}^2.
$$
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The angle of deflection is
$$
\theta = \frac{4GM}{c^2 r}
$$
toward the mass M at a distance r from the affected radiation, where G is the universal constant of gravitation, and c is the speed of light in vacuum.
Since the Schwarzschild radius
$$
r_\text{s}
$$
is defined as
$$
r_\text{s} = 2Gm/c^2
$$
, and escape velocity
$$
v_\text{e}
$$
is defined as
$$
v_\text{e} = \sqrt{2Gm/r} = \beta_\text{e} c
$$
, this can also be expressed in simple form as
$$
\theta = 2 \frac{r_\text{s}}{r} = 2 \left(\frac{v_\text{e}}{c}\right)^2 = 2\beta_\text{e}^2.
$$
## Search for gravitational lenses
Most of the gravitational lenses in the past have been discovered accidentally. A search for gravitational lenses in the northern hemisphere (Cosmic Lens All Sky Survey, CLASS), done in radio frequencies using the Very Large Array (VLA) in New Mexico, led to the discovery of 22 new lensing systems, a major milestone.
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## Search for gravitational lenses
Most of the gravitational lenses in the past have been discovered accidentally. A search for gravitational lenses in the northern hemisphere (Cosmic Lens All Sky Survey, CLASS), done in radio frequencies using the Very Large Array (VLA) in New Mexico, led to the discovery of 22 new lensing systems, a major milestone. This has opened a whole new avenue for research ranging from finding very distant objects to finding values for cosmological parameters so we can understand the universe better.
A similar search in the southern hemisphere would be a very good step towards complementing the northern hemisphere search as well as obtaining other objectives for study. If such a search is done using well-calibrated and well-parameterized instruments and data, a result similar to the northern survey can be expected. The use of the Australia Telescope 20 GHz (AT20G) Survey data collected using the Australia Telescope Compact Array (ATCA) stands to be such a collection of data. As the data were collected using the same instrument maintaining a very stringent quality of data we should expect to obtain good results from the search. The AT20G survey is a blind survey at 20 GHz frequency in the radio domain of the electromagnetic spectrum.
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As the data were collected using the same instrument maintaining a very stringent quality of data we should expect to obtain good results from the search. The AT20G survey is a blind survey at 20 GHz frequency in the radio domain of the electromagnetic spectrum. Due to the high frequency used, the chances of finding gravitational lenses increases as the relative number of compact core objects (e.g. quasars) are higher (Sadler et al. 2006). This is important as the lensing is easier to detect and identify in simple objects compared to objects with complexity in them. This search involves the use of interferometric methods to identify candidates and follow them up at higher resolution to identify them. Full detail of the project is currently under works for publication.
Microlensing techniques have been used to search for planets outside the Solar System. A statistical analysis of specific cases of observed microlensing over the time period of 2002 to 2007 found that most stars in the Milky Way galaxy hosted at least one orbiting planet within 0.5 to 10 AU.
In 2009, weak gravitational lensing was used to extend the mass-X-ray-luminosity relation to older and smaller structures than was previously possible to improve measurements of distant galaxies.
As of 2013 the most distant gravitational lens galaxy, J1000+0221, had been found using NASA's Hubble Space Telescope.
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In 2009, weak gravitational lensing was used to extend the mass-X-ray-luminosity relation to older and smaller structures than was previously possible to improve measurements of distant galaxies.
As of 2013 the most distant gravitational lens galaxy, J1000+0221, had been found using NASA's Hubble Space Telescope. While it remains the most distant quad-image lensing galaxy known, an even more distant two-image lensing galaxy was subsequently discovered by an international team of astronomers using a combination of Hubble Space Telescope and Keck telescope imaging and spectroscopy. The discovery and analysis of the IRC 0218 lens was published in the Astrophysical Journal Letters on June 23, 2014.
Research published September 30, 2013 in the online edition of Physical Review Letters, led by McGill University in Montreal, Québec, Canada, has discovered the B-modes, that are formed due to gravitational lensing effect, using National Science Foundation's South Pole Telescope and with help from the Herschel space observatory. This discovery would open the possibilities of testing the theories of how our universe originated.
## Solar gravitational lens
Albert Einstein predicted in 1936 that rays of light from the same direction that skirt the edges of the Sun would converge to a focal point approximately 542 AU from the Sun.
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This discovery would open the possibilities of testing the theories of how our universe originated.
## Solar gravitational lens
Albert Einstein predicted in 1936 that rays of light from the same direction that skirt the edges of the Sun would converge to a focal point approximately 542 AU from the Sun. Thus, a probe positioned at this distance (or greater) from the Sun could use the Sun as a gravitational lens for magnifying distant objects on the opposite side of the Sun. A probe's location could shift around as needed to select different targets relative to the Sun.
This distance is far beyond the progress and equipment capabilities of space probes such as Voyager 1, and beyond the known planets and dwarf planets, though over thousands of years 90377 Sedna will move farther away on its highly elliptical orbit. The high gain for potentially detecting signals through this lens, such as microwaves at the 21-cm hydrogen line, led to the suggestion by Frank Drake in the early days of SETI that a probe could be sent to this distance. A multipurpose probe SETISAIL and later FOCAL was proposed to the ESA in 1993, but is expected to be a difficult task. If a probe does pass 542 AU, magnification capabilities of the lens will continue to act at farther distances, as the rays that come to a focus at larger distances pass further away from the distortions of the Sun's corona.
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A multipurpose probe SETISAIL and later FOCAL was proposed to the ESA in 1993, but is expected to be a difficult task. If a probe does pass 542 AU, magnification capabilities of the lens will continue to act at farther distances, as the rays that come to a focus at larger distances pass further away from the distortions of the Sun's corona. A critique of the concept was given by Landis, who discussed issues including interference of the solar corona, the high magnification of the target, which will make the design of the mission focal plane difficult, and an analysis of the inherent spherical aberration of the lens.
In 2020, NASA physicist Slava Turyshev presented his idea of Direct Multipixel Imaging and Spectroscopy of an Exoplanet with a Solar Gravitational Lens Mission. The lens could reconstruct the exoplanet image with ~25 km-scale surface resolution, enough to see surface features and signs of habitability.
## Measuring weak lensing
Kaiser, Squires and Broadhurst (1995), Luppino & Kaiser (1997) and Hoekstra et al. (1998) prescribed a method to invert the effects of the point spread function (PSF) smearing and shearing, recovering a shear estimator uncontaminated by the systematic distortion of the PSF. This method (KSB+) is the most widely used method in weak lensing shear measurements.
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## Measuring weak lensing
Kaiser, Squires and Broadhurst (1995), Luppino & Kaiser (1997) and Hoekstra et al. (1998) prescribed a method to invert the effects of the point spread function (PSF) smearing and shearing, recovering a shear estimator uncontaminated by the systematic distortion of the PSF. This method (KSB+) is the most widely used method in weak lensing shear measurements.
Galaxies have random rotations and inclinations. As a result, the shear effects in weak lensing need to be determined by statistically preferred orientations. The primary source of error in lensing measurement is due to the convolution of the PSF with the lensed image. The KSB method measures the ellipticity of a galaxy image. The shear is proportional to the ellipticity. The objects in lensed images are parameterized according to their weighted quadrupole moments. For a perfect ellipse, the weighted quadrupole moments are related to the weighted ellipticity. KSB calculate how a weighted ellipticity measure is related to the shear and use the same formalism to remove the effects of the PSF.
KSB's primary advantages are its mathematical ease and relatively simple implementation. However, KSB is based on a key assumption that the PSF is circular with an anisotropic distortion.
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In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point is a pole of a function if it is a zero of the function and is holomorphic (i.e. complex differentiable) in some neighbourhood of .
A function is meromorphic in an open set if for every point of there is a neighborhood of in which at least one of and is holomorphic.
If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between zeros and poles, that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros.
## Definitions
A function of a complex variable is holomorphic in an open domain if it is differentiable with respect to at every point of . Equivalently, it is holomorphic if it is analytic, that is, if its Taylor series exists at every point of , and converges to the function in some neighbourhood of the point.
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## Definitions
A function of a complex variable is holomorphic in an open domain if it is differentiable with respect to at every point of . Equivalently, it is holomorphic if it is analytic, that is, if its Taylor series exists at every point of , and converges to the function in some neighbourhood of the point. A function is meromorphic in if every point of has a neighbourhood such that at least one of and is holomorphic in it.
A zero of a meromorphic function is a complex number such that . A pole of is a zero of .
If is a function that is meromorphic in a neighbourhood of a point
$$
z_0
$$
of the complex plane, then there exists an integer such that _ BLOCK1_is holomorphic and nonzero in a neighbourhood of
$$
z_0
$$
(this is a consequence of the analytic property).
If , then
$$
z_0
$$
is a pole of order (or multiplicity) of . If , then
$$
z_0
$$
is a zero of order
$$
|n|
$$
of . Simple zero and simple pole are terms used for zeroes and poles of order _ BLOCK6_ Degree is sometimes used synonymously to order.
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BLOCK6_ Degree is sometimes used synonymously to order.
This characterization of zeros and poles implies that zeros and poles are isolated, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole.
Because of the order of zeros and poles being defined as a non-negative number and the symmetry between them, it is often useful to consider a pole of order as a zero of order and a zero of order as a pole of order . In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0.
A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function (see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at . Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along .
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The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at . Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along .
In a neighbourhood of a point
$$
z_0,
$$
a nonzero meromorphic function is the sum of a Laurent series with at most finite principal part (the terms with negative index values):
$$
f(z) = \sum_{k\geq -n} a_k (z - z_0)^k,
$$
where is an integer, and
$$
a_{-n}\neq 0.
$$
Again, if (the sum starts with
$$
a_{-|n|} (z - z_0)^{-|n|}
$$
, the principal part has terms), one has a pole of order , and if (the sum starts with
$$
a_{|n|} (z - z_0)^{|n|}
$$
, there is no principal part), one has a zero of order
$$
|n|
$$
.
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In a neighbourhood of a point
$$
z_0,
$$
a nonzero meromorphic function is the sum of a Laurent series with at most finite principal part (the terms with negative index values):
$$
f(z) = \sum_{k\geq -n} a_k (z - z_0)^k,
$$
where is an integer, and
$$
a_{-n}\neq 0.
$$
Again, if (the sum starts with
$$
a_{-|n|} (z - z_0)^{-|n|}
$$
, the principal part has terms), one has a pole of order , and if (the sum starts with
$$
a_{|n|} (z - z_0)^{|n|}
$$
, there is no principal part), one has a zero of order
$$
|n|
$$
.
## At infinity
A function
$$
z \mapsto f(z)
$$
is meromorphic at infinity if it is meromorphic in some neighbourhood of infinity (that is outside some disk), and there is an integer such that
$$
\lim_{z\to \infty}\frac{f(z)}{z^n}
$$
exists and is a nonzero complex number.
In this case, the point at infinity is a pole of order if , and a zero of order
$$
|n|
$$
if .
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## At infinity
A function
$$
z \mapsto f(z)
$$
is meromorphic at infinity if it is meromorphic in some neighbourhood of infinity (that is outside some disk), and there is an integer such that
$$
\lim_{z\to \infty}\frac{f(z)}{z^n}
$$
exists and is a nonzero complex number.
In this case, the point at infinity is a pole of order if , and a zero of order
$$
|n|
$$
if .
For example, a polynomial of degree has a pole of degree at infinity.
The complex plane extended by a point at infinity is called the Riemann sphere.
If is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros.
Every rational function is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator.
## Examples
- The function
$$
f(z) = \frac{3}{z}
$$
is meromorphic on the whole Riemann sphere.
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Every rational function is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator.
## Examples
- The function
$$
f(z) = \frac{3}{z}
$$
is meromorphic on the whole Riemann sphere. It has a pole of order 1 or simple pole at
$$
z= 0,
$$
and a simple zero at infinity.
- The function
$$
f(z) = \frac{z+2}{(z-5)^2(z+7)^3}
$$
is meromorphic on the whole Riemann sphere. It has a pole of order 2 at
$$
z=5,
$$
and a pole of order 3 at
$$
z = -7
$$
. It has a simple zero at
$$
z=-2,
$$
and a quadruple zero at infinity.
- The function
$$
f(z) = \frac{z-4}{e^z-1}
$$
is meromorphic in the whole complex plane, but not at infinity. It has poles of order 1 at
$$
z=2\pi ni\text{ for } n\in\mathbb Z
$$
. This can be seen by writing the Taylor series of
$$
e^z
$$
around the origin.
-
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It has poles of order 1 at
$$
z=2\pi ni\text{ for } n\in\mathbb Z
$$
. This can be seen by writing the Taylor series of
$$
e^z
$$
around the origin.
- The function
$$
f(z) = z
$$
has a single pole at infinity of order 1, and a single zero at the origin.
All above examples except for the third are rational functions. For a general discussion of zeros and poles of such functions, see .
## Function on a curve
The concept of zeros and poles extends naturally to functions on a complex curve, that is complex analytic manifold of dimension one (over the complex numbers). The simplest examples of such curves are the complex plane and the Riemann surface. This extension is done by transferring structures and properties through charts, which are analytic isomorphisms.
More precisely, let be a function from a complex curve to the complex numbers. This function is holomorphic (resp. meromorphic) in a neighbourhood of a point of if there is a chart
$$
\phi
$$
such that
$$
f \circ \phi^{-1}
$$
is holomorphic (resp. meromorphic) in a neighbourhood of
$$
\phi(z).
$$
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meromorphic) in a neighbourhood of a point of if there is a chart
$$
\phi
$$
such that
$$
f \circ \phi^{-1}
$$
is holomorphic (resp. meromorphic) in a neighbourhood of
$$
\phi(z).
$$
Then, is a pole or a zero of order if the same is true for
$$
\phi(z).
$$
If the curve is compact, and the function is meromorphic on the whole curve, then the number of zeros and poles is finite, and the sum of the orders of the poles equals the sum of the orders of the zeros. This is one of the basic facts that are involved in Riemann–Roch theorem.
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The first law of thermodynamics is a formulation of the law of conservation of energy in the context of thermodynamic processes. For a thermodynamic process affecting a thermodynamic system without transfer of matter, the law distinguishes two principal forms of energy transfer, heat and thermodynamic work. The law also defines the internal energy of a system, an extensive property for taking account of the balance of heat transfer, thermodynamic work, and matter transfer, into and out of the system. Energy cannot be created or destroyed, but it can be transformed from one form to another. In an externally isolated system, with internal changes, the sum of all forms of energy is constant.
An equivalent statement is that perpetual motion machines of the first kind are impossible; work done by a system on its surroundings requires that the system's internal energy be consumed, so that the amount of internal energy lost by that work must be resupplied as heat by an external energy source or as work by an external machine acting on the system to sustain the work of the system continuously.
## Definition
For thermodynamic processes of energy transfer without transfer of matter, the first law of thermodynamics is often expressed by the algebraic sum of contributions to the internal energy,
$$
U,
$$
from all work,
$$
W,
$$
done on or by the system, and the quantity of heat,
$$
Q,
$$
supplied to the system.
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An equivalent statement is that perpetual motion machines of the first kind are impossible; work done by a system on its surroundings requires that the system's internal energy be consumed, so that the amount of internal energy lost by that work must be resupplied as heat by an external energy source or as work by an external machine acting on the system to sustain the work of the system continuously.
## Definition
For thermodynamic processes of energy transfer without transfer of matter, the first law of thermodynamics is often expressed by the algebraic sum of contributions to the internal energy,
$$
U,
$$
from all work,
$$
W,
$$
done on or by the system, and the quantity of heat,
$$
Q,
$$
supplied to the system. With the sign convention of Rudolf Clausius, that heat supplied to the system is positive, but work done by the system is subtracted, a change in the internal energy,
$$
\Delta U,
$$
is written
$$
\Delta U = Q - W.
$$
Modern formulations, such as by Max Planck, and by IUPAC, often replace the subtraction with addition, and consider all net energy transfers to the system as positive and all net energy transfers from the system as negative, irrespective of the use of the system, for example as an engine.
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## Definition
For thermodynamic processes of energy transfer without transfer of matter, the first law of thermodynamics is often expressed by the algebraic sum of contributions to the internal energy,
$$
U,
$$
from all work,
$$
W,
$$
done on or by the system, and the quantity of heat,
$$
Q,
$$
supplied to the system. With the sign convention of Rudolf Clausius, that heat supplied to the system is positive, but work done by the system is subtracted, a change in the internal energy,
$$
\Delta U,
$$
is written
$$
\Delta U = Q - W.
$$
Modern formulations, such as by Max Planck, and by IUPAC, often replace the subtraction with addition, and consider all net energy transfers to the system as positive and all net energy transfers from the system as negative, irrespective of the use of the system, for example as an engine.
When a system expands in an isobaric process, the thermodynamic work,
$$
W,
$$
done by the system on the surroundings is the product,
$$
P~ \Delta V,
$$
of system pressure,
$$
P,
$$
and system volume change,
$$
\Delta V,
$$
whereas _ BLOCK9_ is said to be the thermodynamic work done on the system by the surroundings.
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whereas _ BLOCK9_ is said to be the thermodynamic work done on the system by the surroundings. The change in internal energy of the system is:
$$
\Delta U = Q - P~ \Delta V,
$$
where
$$
Q
$$
denotes the quantity of heat supplied to the system from its surroundings.
Work and heat express physical processes of supply or removal of energy, while the internal energy
$$
U
$$
is a mathematical abstraction that keeps account of the changes of energy that befall the system. The term
$$
Q
$$
is the quantity of energy added or removed as heat in the thermodynamic sense, not referring to a form of energy within the system. Likewise,
$$
W
$$
denotes the quantity of energy gained or lost through thermodynamic work. Internal energy is a property of the system, while work and heat describe the process, not the system. Thus, a given internal energy change,
$$
\Delta U,
$$
can be achieved by different combinations of heat and work. Heat and work are said to be path dependent, while change in internal energy depends only on the initial and final states of the system, not on the path between.
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Thus, a given internal energy change,
$$
\Delta U,
$$
can be achieved by different combinations of heat and work. Heat and work are said to be path dependent, while change in internal energy depends only on the initial and final states of the system, not on the path between. Thermodynamic work is measured by change in the system, and, because of friction, is not necessarily the same as work measured by forces and distances in the surroundings, though, ideally, such can sometimes be arranged; this distinction is noted in the term 'isochoric work', at constant system volume, with
$$
\Delta V = 0,
$$
which is not a form of thermodynamic work.
For thermodynamic processes of energy transfer with transfer of matter, the extensive character of internal energy can be stated: for the otherwise isolated combination of two thermodynamic systems with internal energies
$$
U_1
$$
and
$$
U_2
$$
into a single system with internal energy
$$
U_0,
$$
$$
U_0=U_1+U_2.
$$
## History
In the first half of the eighteenth century, French philosopher and mathematician Émilie du Châtelet made notable contributions to the emerging theoretical framework of energy, for example by emphasising Leibniz's concept of ' vis viva ', mv2, as distinct from Newton's momentum, mv.
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## History
In the first half of the eighteenth century, French philosopher and mathematician Émilie du Châtelet made notable contributions to the emerging theoretical framework of energy, for example by emphasising Leibniz's concept of ' vis viva ', mv2, as distinct from Newton's momentum, mv.
Empirical developments of the early ideas, in the century following, wrestled with contravening concepts such as the caloric theory of heat.
In the few years of his life (1796–1832) after the 1824 publication of his book Reflections on the Motive Power of Fire, Sadi Carnot came to understand that the caloric theory of heat was restricted to mere calorimetry, and that heat and "motive power" are interconvertible. This is known only from his posthumously published notes. He wrote:
At that time, the concept of mechanical work had not been formulated. Carnot was aware that heat could be produced by friction and by percussion, as forms of dissipation of "motive power". As late as 1847, Lord Kelvin believed in the caloric theory of heat, being unaware of Carnot's notes.
In 1840, Germain Hess stated a conservation law (Hess's law) for the heat of reaction during chemical transformations.
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As late as 1847, Lord Kelvin believed in the caloric theory of heat, being unaware of Carnot's notes.
In 1840, Germain Hess stated a conservation law (Hess's law) for the heat of reaction during chemical transformations. This law was later recognized as a consequence of the first law of thermodynamics, but Hess's statement was not explicitly concerned with the relation between energy exchanges by heat and work.
In 1842, Julius Robert von Mayer made a statement that was rendered by Clifford Truesdell (1980) as "in a process at constant pressure, the heat used to produce expansion is universally interconvertible with work", but this is not a general statement of the first law, for it does not express the concept of the thermodynamic state variable, the internal energy. Mayer, Robert (1841). "Remarks on the Forces of Nature". Quoted in Lehninger, A. (1971). Bioenergetics – the Molecular Basis of Biological Energy Transformations, 2nd ed. London: The Benjamin/Cummings Publishing Company. Also in 1842, Mayer measured a temperature rise caused by friction in a body of paper pulp. This was near the time of the 1842–1845 work of James Prescott Joule, measuring the mechanical equivalent of heat.
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Also in 1842, Mayer measured a temperature rise caused by friction in a body of paper pulp. This was near the time of the 1842–1845 work of James Prescott Joule, measuring the mechanical equivalent of heat. In 1845, Joule published a paper entitled The Mechanical Equivalent of Heat, in which he specified a numerical value for the amount of mechanical work required to "produce a unit of heat", based on heat production by friction in the passage of electricity through a resistor and in the rotation of a paddle in a vat of water.
The first full statements of the law came in 1850 from Rudolf Clausius, and from William Rankine. Some scholars consider Rankine's statement less distinct than that of Clausius.
### Original statements: the "thermodynamic approach"
The original 19th-century statements of the first law appeared in a conceptual framework in which transfer of energy as heat was taken as a primitive notion, defined by calorimetry. It was presupposed as logically prior to the theoretical development of thermodynamics. Jointly primitive with this notion of heat were the notions of empirical temperature and thermal equilibrium. This framework also took as primitive the notion of transfer of energy as work. This framework did not presume a concept of energy in general, but regarded it as derived or synthesized from the prior notions of heat and work.
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This framework also took as primitive the notion of transfer of energy as work. This framework did not presume a concept of energy in general, but regarded it as derived or synthesized from the prior notions of heat and work. By one author, this framework has been called the "thermodynamic" approach.
The first explicit statement of the first law of thermodynamics, by Rudolf Clausius in 1850, referred to cyclic thermodynamic processes, and to the existence of a function of state of the system, the internal energy. He expressed it in terms of a differential equation for the increments of a thermodynamic process. This equation may be described as follows:
Reflecting the experimental work of Mayer and of Joule, Clausius wrote:
Because of its definition in terms of increments, the value of the internal energy of a system is not uniquely defined. It is defined only up to an arbitrary additive constant of integration, which can be adjusted to give arbitrary reference zero levels. This non-uniqueness is in keeping with the abstract mathematical nature of the internal energy. The internal energy is customarily stated relative to a conventionally chosen standard reference state of the system.
The concept of internal energy is considered by Bailyn to be of "enormous interest". Its quantity cannot be immediately measured, but can only be inferred, by differencing actual immediate measurements.
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The concept of internal energy is considered by Bailyn to be of "enormous interest". Its quantity cannot be immediately measured, but can only be inferred, by differencing actual immediate measurements. Bailyn likens it to the energy states of an atom, that were revealed by Bohr's energy relation . In each case, an unmeasurable quantity (the internal energy, the atomic energy level) is revealed by considering the difference of measured quantities (increments of internal energy, quantities of emitted or absorbed radiative energy).
### Conceptual revision: the "mechanical approach"
In 1907, George H. Bryan wrote about systems between which there is no transfer of matter (closed systems): "Definition. When energy flows from one system or part of a system to another otherwise than by the performance of mechanical work, the energy so transferred is called heat." This definition may be regarded as expressing a conceptual revision, as follows. This reinterpretation was systematically expounded in 1909 by Constantin Carathéodory, whose attention had been drawn to it by Max Born. Largely through Born's influence, this revised conceptual approach to the definition of heat came to be preferred by many twentieth-century writers. It might be called the "mechanical approach".
Energy can also be transferred from one thermodynamic system to another in association with transfer of matter.
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It might be called the "mechanical approach".
Energy can also be transferred from one thermodynamic system to another in association with transfer of matter. Born points out that in general such energy transfer is not resolvable uniquely into work and heat moieties. In general, when there is transfer of energy associated with matter transfer, work and heat transfers can be distinguished only when they pass through walls physically separate from those for matter transfer.
The "mechanical" approach postulates the law of conservation of energy. It also postulates that energy can be transferred from one thermodynamic system to another adiabatically as work, and that energy can be held as the internal energy of a thermodynamic system. It also postulates that energy can be transferred from one thermodynamic system to another by a path that is non-adiabatic, and is unaccompanied by matter transfer. Initially, it "cleverly" (according to Martin Bailyn) refrains from labelling as 'heat' such non-adiabatic, unaccompanied transfer of energy. It rests on the primitive notion of walls, especially adiabatic walls and non-adiabatic walls, defined as follows. Temporarily, only for purpose of this definition, one can prohibit transfer of energy as work across a wall of interest.
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It rests on the primitive notion of walls, especially adiabatic walls and non-adiabatic walls, defined as follows. Temporarily, only for purpose of this definition, one can prohibit transfer of energy as work across a wall of interest. Then walls of interest fall into two classes, (a) those such that arbitrary systems separated by them remain independently in their own previously established respective states of internal thermodynamic equilibrium; they are defined as adiabatic; and (b) those without such independence; they are defined as non-adiabatic.
This approach derives the notions of transfer of energy as heat, and of temperature, as theoretical developments, not taking them as primitives. It regards calorimetry as a derived theory. It has an early origin in the nineteenth century, for example in the work of Hermann von Helmholtz, but also in the work of many others.
## Conceptually revised statement, according to the mechanical approach
The revised statement of the first law postulates that a change in the internal energy of a system due to any arbitrary process, that takes the system from a given initial thermodynamic state to a given final equilibrium thermodynamic state, can be determined through the physical existence, for those given states, of a reference process that occurs purely through stages of adiabatic work.
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It has an early origin in the nineteenth century, for example in the work of Hermann von Helmholtz, but also in the work of many others.
## Conceptually revised statement, according to the mechanical approach
The revised statement of the first law postulates that a change in the internal energy of a system due to any arbitrary process, that takes the system from a given initial thermodynamic state to a given final equilibrium thermodynamic state, can be determined through the physical existence, for those given states, of a reference process that occurs purely through stages of adiabatic work.
The revised statement is then
For a closed system, in any arbitrary process of interest that takes it from an initial to a final state of internal thermodynamic equilibrium, the change of internal energy is the same as that for a reference adiabatic work process that links those two states. This is so regardless of the path of the process of interest, and regardless of whether it is an adiabatic or a non-adiabatic process. The reference adiabatic work process may be chosen arbitrarily from amongst the class of all such processes.
This statement is much less close to the empirical basis than are the original statements, but is often regarded as conceptually parsimonious in that it rests only on the concepts of adiabatic work and of non-adiabatic processes, not on the concepts of transfer of energy as heat and of empirical temperature that are presupposed by the original statements.
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The reference adiabatic work process may be chosen arbitrarily from amongst the class of all such processes.
This statement is much less close to the empirical basis than are the original statements, but is often regarded as conceptually parsimonious in that it rests only on the concepts of adiabatic work and of non-adiabatic processes, not on the concepts of transfer of energy as heat and of empirical temperature that are presupposed by the original statements. Largely through the influence of Max Born, it is often regarded as theoretically preferable because of this conceptual parsimony. Born particularly observes that the revised approach avoids thinking in terms of what he calls the "imported engineering" concept of heat engines.
Basing his thinking on the mechanical approach, Born in 1921, and again in 1949, proposed to revise the definition of heat. In particular, he referred to the work of Constantin Carathéodory, who had in 1909 stated the first law without defining quantity of heat. Born's definition was specifically for transfers of energy without transfer of matter, and it has been widely followed in textbooks (examples:). Born observes that a transfer of matter between two systems is accompanied by a transfer of internal energy that cannot be resolved into heat and work components.
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Born's definition was specifically for transfers of energy without transfer of matter, and it has been widely followed in textbooks (examples:). Born observes that a transfer of matter between two systems is accompanied by a transfer of internal energy that cannot be resolved into heat and work components. There can be pathways to other systems, spatially separate from that of the matter transfer, that allow heat and work transfer independent of and simultaneous with the matter transfer. Energy is conserved in such transfers.
## Description
### Cyclic processes
The first law of thermodynamics for a closed system was expressed in two ways by Clausius. One way referred to cyclic processes and the inputs and outputs of the system, but did not refer to increments in the internal state of the system. The other way referred to an incremental change in the internal state of the system, and did not expect the process to be cyclic.
A cyclic process is one that can be repeated indefinitely often, returning the system to its initial state. Of particular interest for single cycle of a cyclic process are the net work done, and the net heat taken in (or 'consumed', in Clausius' statement), by the system.
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A cyclic process is one that can be repeated indefinitely often, returning the system to its initial state. Of particular interest for single cycle of a cyclic process are the net work done, and the net heat taken in (or 'consumed', in Clausius' statement), by the system.
In a cyclic process in which the system does net work on its surroundings, it is observed to be physically necessary not only that heat be taken into the system, but also, importantly, that some heat leave the system. The difference is the heat converted by the cycle into work. In each repetition of a cyclic process, the net work done by the system, measured in mechanical units, is proportional to the heat consumed, measured in calorimetric units.
The constant of proportionality is universal and independent of the system and in 1845 and 1847 was measured by James Joule, who described it as the mechanical equivalent of heat.
## Various statements of the law for closed systems
The law is of great importance and generality and is consequently thought of from several points of view. Most careful textbook statements of the law express it for closed systems. It is stated in several ways, sometimes even by the same author. Münster, A. (1970).
For the thermodynamics of closed systems, the distinction between transfers of energy as work and as heat is central and is within the scope of the present article.
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Münster, A. (1970).
For the thermodynamics of closed systems, the distinction between transfers of energy as work and as heat is central and is within the scope of the present article. For the thermodynamics of open systems, such a distinction is beyond the scope of the present article, but some limited comments are made on it in the section below headed '
## First law of thermodynamics for open systems
'.
There are two main ways of stating a law of thermodynamics, physically or mathematically. They should be logically coherent and consistent with one another.
An example of a physical statement is that of Planck (1897/1903):
It is in no way possible, either by mechanical, thermal, chemical, or other devices, to obtain perpetual motion, i.e. it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing.
This physical statement is restricted neither to closed systems nor to systems with states that are strictly defined only for thermodynamic equilibrium; it has meaning also for open systems and for systems with states that are not in thermodynamic equilibrium.
An example of a mathematical statement is that of Crawford (1963):
For a given system we let large-scale mechanical energy, large-scale potential energy, and total energy.
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An example of a mathematical statement is that of Crawford (1963):
For a given system we let large-scale mechanical energy, large-scale potential energy, and total energy. The first two quantities are specifiable in terms of appropriate mechanical variables, and by definition
$$
E^{\mathrm{tot}}=E^{\mathrm{kin}}+E^{\mathrm{pot}}+U\,\,.
$$
For any finite process, whether reversible or irreversible,
$$
\Delta E^{\mathrm{tot}}=\Delta E^{\mathrm{kin}}+\Delta E^{\mathrm{pot}}+\Delta U\,\,.
$$
The first law in a form that involves the principle of conservation of energy more generally is
$$
\Delta E^{\mathrm{tot}}=Q+W\,\,.
$$
Here and are heat and work added, with no restrictions as to whether the process is reversible, quasistatic, or irreversible.[Warner, Am. J. Phys., 29, 124 (1961)]
This statement by Crawford, for , uses the sign convention of IUPAC, not that of Clausius. Though it does not explicitly say so, this statement refers to closed systems.
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J. Phys., 29, 124 (1961)]
This statement by Crawford, for , uses the sign convention of IUPAC, not that of Clausius. Though it does not explicitly say so, this statement refers to closed systems. Internal energy is evaluated for bodies in states of thermodynamic equilibrium, which possess well-defined temperatures, relative to a reference state.
The history of statements of the law for closed systems has two main periods, before and after the work of George H. Bryan (1907), of Carathéodory (1909), and the approval of Carathéodory's work given by Born (1921). The earlier traditional versions of the law for closed systems are nowadays often considered to be out of date.
Carathéodory's celebrated presentation of equilibrium thermodynamics refers to closed systems, which are allowed to contain several phases connected by internal walls of various kinds of impermeability and permeability (explicitly including walls that are permeable only to heat). Carathéodory's 1909 version of the first law of thermodynamics was stated in an axiom which refrained from defining or mentioning temperature or quantity of heat transferred.
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Carathéodory's celebrated presentation of equilibrium thermodynamics refers to closed systems, which are allowed to contain several phases connected by internal walls of various kinds of impermeability and permeability (explicitly including walls that are permeable only to heat). Carathéodory's 1909 version of the first law of thermodynamics was stated in an axiom which refrained from defining or mentioning temperature or quantity of heat transferred. That axiom stated that the internal energy of a phase in equilibrium is a function of state, that the sum of the internal energies of the phases is the total internal energy of the system, and that the value of the total internal energy of the system is changed by the amount of work done adiabatically on it, considering work as a form of energy. That article considered this statement to be an expression of the law of conservation of energy for such systems. This version is nowadays widely accepted as authoritative, but is stated in slightly varied ways by different authors.
Such statements of the first law for closed systems assert the existence of internal energy as a function of state defined in terms of adiabatic work. Thus heat is not defined calorimetrically or as due to temperature difference. It is defined as a residual difference between change of internal energy and work done on the system, when that work does not account for the whole of the change of internal energy and the system is not adiabatically isolated.
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Thus heat is not defined calorimetrically or as due to temperature difference. It is defined as a residual difference between change of internal energy and work done on the system, when that work does not account for the whole of the change of internal energy and the system is not adiabatically isolated. Reif, F. (1965), p. 122.
The 1909 Carathéodory statement of the law in axiomatic form does not mention heat or temperature, but the equilibrium states to which it refers are explicitly defined by variable sets that necessarily include "non-deformation variables", such as pressures, which, within reasonable restrictions, can be rightly interpreted as empirical temperatures, and the walls connecting the phases of the system are explicitly defined as possibly impermeable to heat or permeable only to heat.
According to A. Münster (1970), "A somewhat unsatisfactory aspect of Carathéodory's theory is that a consequence of the Second Law must be considered at this point [in the statement of the first law], i.e. that it is not always possible to reach any state 2 from any other state 1 by means of an adiabatic process." Münster instances that no adiabatic process can reduce the internal energy of a system at constant volume. Carathéodory's paper asserts that its statement of the first law corresponds exactly to Joule's experimental arrangement, regarded as an instance of adiabatic work.
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Münster instances that no adiabatic process can reduce the internal energy of a system at constant volume. Carathéodory's paper asserts that its statement of the first law corresponds exactly to Joule's experimental arrangement, regarded as an instance of adiabatic work. It does not point out that Joule's experimental arrangement performed essentially irreversible work, through friction of paddles in a liquid, or passage of electric current through a resistance inside the system, driven by motion of a coil and inductive heating, or by an external current source, which can access the system only by the passage of electrons, and so is not strictly adiabatic, because electrons are a form of matter, which cannot penetrate adiabatic walls. The paper goes on to base its main argument on the possibility of quasi-static adiabatic work, which is essentially reversible. The paper asserts that it will avoid reference to Carnot cycles, and then proceeds to base its argument on cycles of forward and backward quasi-static adiabatic stages, with isothermal stages of zero magnitude.
Sometimes the concept of internal energy is not made explicit in the statement. Reif, F. (1965), p. 82.
Sometimes the existence of the internal energy is made explicit but work is not explicitly mentioned in the statement of the first postulate of thermodynamics.
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Reif, F. (1965), p. 82.
Sometimes the existence of the internal energy is made explicit but work is not explicitly mentioned in the statement of the first postulate of thermodynamics. Heat supplied is then defined as the residual change in internal energy after work has been taken into account, in a non-adiabatic process.
A respected modern author states the first law of thermodynamics as "Heat is a form of energy", which explicitly mentions neither internal energy nor adiabatic work. Heat is defined as energy transferred by thermal contact with a reservoir, which has a temperature, and is generally so large that addition and removal of heat do not alter its temperature. A current student text on chemistry defines heat thus: "heat is the exchange of thermal energy between a system and its surroundings caused by a temperature difference." The author then explains how heat is defined or measured by calorimetry, in terms of heat capacity, specific heat capacity, molar heat capacity, and temperature.
A respected text disregards the Carathéodory's exclusion of mention of heat from the statement of the first law for closed systems, and admits heat calorimetrically defined along with work and internal energy. Another respected text defines heat exchange as determined by temperature difference, but also mentions that the Born (1921) version is "completely rigorous".
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A respected text disregards the Carathéodory's exclusion of mention of heat from the statement of the first law for closed systems, and admits heat calorimetrically defined along with work and internal energy. Another respected text defines heat exchange as determined by temperature difference, but also mentions that the Born (1921) version is "completely rigorous". These versions follow the traditional approach that is now considered out of date, exemplified by that of Planck (1897/1903).
## Evidence for the first law of thermodynamics for closed systems
The first law of thermodynamics for closed systems was originally induced from empirically observed evidence, including calorimetric evidence. It is nowadays, however, taken to provide the definition of heat via the law of conservation of energy and the definition of work in terms of changes in the external parameters of a system. The original discovery of the law was gradual over a period of perhaps half a century or more, and some early studies were in terms of cyclic processes.
The following is an account in terms of changes of state of a closed system through compound processes that are not necessarily cyclic. This account first considers processes for which the first law is easily verified because of their simplicity, namely adiabatic processes (in which there is no transfer as heat) and adynamic processes (in which there is no transfer as work).
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The following is an account in terms of changes of state of a closed system through compound processes that are not necessarily cyclic. This account first considers processes for which the first law is easily verified because of their simplicity, namely adiabatic processes (in which there is no transfer as heat) and adynamic processes (in which there is no transfer as work).
### Adiabatic processes
In an adiabatic process, there is transfer of energy as work but not as heat. For all adiabatic process that takes a system from a given initial state to a given final state, irrespective of how the work is done, the respective eventual total quantities of energy transferred as work are one and the same, determined just by the given initial and final states. The work done on the system is defined and measured by changes in mechanical or quasi-mechanical variables external to the system. Physically, adiabatic transfer of energy as work requires the existence of adiabatic enclosures.
For instance, in Joule's experiment, the initial system is a tank of water with a paddle wheel inside. If we isolate the tank thermally, and move the paddle wheel with a pulley and a weight, we can relate the increase in temperature with the distance descended by the mass. Next, the system is returned to its initial state, isolated again, and the same amount of work is done on the tank using different devices (an electric motor, a chemical battery, a spring,...).
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If we isolate the tank thermally, and move the paddle wheel with a pulley and a weight, we can relate the increase in temperature with the distance descended by the mass. Next, the system is returned to its initial state, isolated again, and the same amount of work is done on the tank using different devices (an electric motor, a chemical battery, a spring,...). In every case, the amount of work can be measured independently. The return to the initial state is not conducted by doing adiabatic work on the system. The evidence shows that the final state of the water (in particular, its temperature and volume) is the same in every case. It is irrelevant if the work is electrical, mechanical, chemical,... or if done suddenly or slowly, as long as it is performed in an adiabatic way, that is to say, without heat transfer into or out of the system.
Evidence of this kind shows that to increase the temperature of the water in the tank, the qualitative kind of adiabatically performed work does not matter. No qualitative kind of adiabatic work has ever been observed to decrease the temperature of the water in the tank.
A change from one state to another, for example an increase of both temperature and volume, may be conducted in several stages, for example by externally supplied electrical work on a resistor in the body, and adiabatic expansion allowing the body to do work on the surroundings.
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No qualitative kind of adiabatic work has ever been observed to decrease the temperature of the water in the tank.
A change from one state to another, for example an increase of both temperature and volume, may be conducted in several stages, for example by externally supplied electrical work on a resistor in the body, and adiabatic expansion allowing the body to do work on the surroundings. It needs to be shown that the time order of the stages, and their relative magnitudes, does not affect the amount of adiabatic work that needs to be done for the change of state. According to one respected scholar: "Unfortunately, it does not seem that experiments of this kind have ever been carried out carefully. ... We must therefore admit that the statement which we have enunciated here, and which is equivalent to the first law of thermodynamics, is not well founded on direct experimental evidence." Another expression of this view is "no systematic precise experiments to verify this generalization directly have ever been attempted".
This kind of evidence, of independence of sequence of stages, combined with the above-mentioned evidence, of independence of qualitative kind of work, would show the existence of an important state variable that corresponds with adiabatic work, but not that such a state variable represented a conserved quantity. For the latter, another step of evidence is needed, which may be related to the concept of reversibility, as mentioned below.
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This kind of evidence, of independence of sequence of stages, combined with the above-mentioned evidence, of independence of qualitative kind of work, would show the existence of an important state variable that corresponds with adiabatic work, but not that such a state variable represented a conserved quantity. For the latter, another step of evidence is needed, which may be related to the concept of reversibility, as mentioned below.
That important state variable was first recognized and denoted
$$
U
$$
by Clausius in 1850, but he did not then name it, and he defined it in terms not only of work but also of heat transfer in the same process. It was also independently recognized in 1850 by Rankine, who also denoted it
$$
U
$$
; and in 1851 by Kelvin who then called it "mechanical energy", and later "intrinsic energy". In 1865, after some hesitation, Clausius began calling his state function
$$
U
$$
"energy". In 1882 it was named as the internal energy by Helmholtz. If only adiabatic processes were of interest, and heat could be ignored, the concept of internal energy would hardly arise or be needed.
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In 1882 it was named as the internal energy by Helmholtz. If only adiabatic processes were of interest, and heat could be ignored, the concept of internal energy would hardly arise or be needed. The relevant physics would be largely covered by the concept of potential energy, as was intended in the 1847 paper of Helmholtz on the principle of conservation of energy, though that did not deal with forces that cannot be described by a potential, and thus did not fully justify the principle. Moreover, that paper was critical of the early work of Joule that had by then been performed. A great merit of the internal energy concept is that it frees thermodynamics from a restriction to cyclic processes, and allows a treatment in terms of thermodynamic states.
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Moreover, that paper was critical of the early work of Joule that had by then been performed. A great merit of the internal energy concept is that it frees thermodynamics from a restriction to cyclic processes, and allows a treatment in terms of thermodynamic states.
In an adiabatic process, adiabatic work takes the system either from a reference state
$$
O
$$
with internal energy
$$
U(O)
$$
to an arbitrary one
$$
A
$$
with internal energy
$$
U(A)
$$
, or from the state
$$
A
$$
to the state
$$
O
$$
:
$$
U(A)=U(O) - W^\mathrm{adiabatic}_{O\to A}\,\, \mathrm{or}\,\,U(O)=U(A) - W^\mathrm{adiabatic}_{A\to O}\,.
$$
Except under the special, and strictly speaking, fictional, condition of reversibility, only one of the processes
$$
\mathrm{adiabatic},\,O\to A
$$
or
$$
\mathrm{adiabatic},\,{A\to O}\,
$$
is empirically feasible by a simple application of externally supplied work. The reason for this is given as the second law of thermodynamics and is not considered in the present article.
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In an adiabatic process, adiabatic work takes the system either from a reference state
$$
O
$$
with internal energy
$$
U(O)
$$
to an arbitrary one
$$
A
$$
with internal energy
$$
U(A)
$$
, or from the state
$$
A
$$
to the state
$$
O
$$
:
$$
U(A)=U(O) - W^\mathrm{adiabatic}_{O\to A}\,\, \mathrm{or}\,\,U(O)=U(A) - W^\mathrm{adiabatic}_{A\to O}\,.
$$
Except under the special, and strictly speaking, fictional, condition of reversibility, only one of the processes
$$
\mathrm{adiabatic},\,O\to A
$$
or
$$
\mathrm{adiabatic},\,{A\to O}\,
$$
is empirically feasible by a simple application of externally supplied work. The reason for this is given as the second law of thermodynamics and is not considered in the present article.
The fact of such irreversibility may be dealt with in two main ways, according to different points of view:
Since the work of Bryan (1907), the most accepted way to deal with it nowadays, followed by Carathéodory,Buchdahl, H. A. (1966), p. 43.
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The reason for this is given as the second law of thermodynamics and is not considered in the present article.
The fact of such irreversibility may be dealt with in two main ways, according to different points of view:
Since the work of Bryan (1907), the most accepted way to deal with it nowadays, followed by Carathéodory,Buchdahl, H. A. (1966), p. 43. is to rely on the previously established concept of quasi-static processes,Maxwell, J. C. (1871). Theory of Heat, Longmans, Green, and Co., London, p. 150.Planck, M. (1897/1903), Section 71, p. 52.Bailyn, M. (1994), p. 95. as follows. Actual physical processes of transfer of energy as work are always at least to some degree irreversible. The irreversibility is often due to mechanisms known as dissipative, that transform bulk kinetic energy into internal energy. Examples are friction and viscosity. If the process is performed more slowly, the frictional or viscous dissipation is less. In the limit of infinitely slow performance, the dissipation tends to zero and then the limiting process, though fictional rather than actual, is notionally reversible, and is called quasi-static.
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If the process is performed more slowly, the frictional or viscous dissipation is less. In the limit of infinitely slow performance, the dissipation tends to zero and then the limiting process, though fictional rather than actual, is notionally reversible, and is called quasi-static. Throughout the course of the fictional limiting quasi-static process, the internal intensive variables of the system are equal to the external intensive variables, those that describe the reactive forces exerted by the surroundings. Adkins, C. J. (1968/1983), p. 35. This can be taken to justify the formula
Another way to deal with it is to allow that experiments with processes of heat transfer to or from the system may be used to justify the formula () above. Moreover, it deals to some extent with the problem of lack of direct experimental evidence that the time order of stages of a process does not matter in the determination of internal energy. This way does not provide theoretical purity in terms of adiabatic work processes, but is empirically feasible, and is in accord with experiments actually done, such as the Joule experiments mentioned just above, and with older traditions.
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Moreover, it deals to some extent with the problem of lack of direct experimental evidence that the time order of stages of a process does not matter in the determination of internal energy. This way does not provide theoretical purity in terms of adiabatic work processes, but is empirically feasible, and is in accord with experiments actually done, such as the Joule experiments mentioned just above, and with older traditions.
The formula () above allows that to go by processes of quasi-static adiabatic work from the state
$$
A
$$
to the state
$$
B
$$
we can take a path that goes through the reference state
$$
O
$$
, since the quasi-static adiabatic work is independent of the path
$$
-W^\mathrm{adiabatic,\,quasi-static}_{A\to B}=-W^\mathrm{adiabatic,\,quasi-static}_{A\to O}-W^\mathrm{adiabatic,\,quasi-static}_{O\to B} = W^\mathrm{adiabatic,\,quasi-static}_{O\to A}- W^\mathrm{adiabatic,\,quasi-static}_{O\to B} = -U(A) + U(B) = \Delta U
$$
This kind of empirical evidence, coupled with theory of this kind, largely justifies the following statement:
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This way does not provide theoretical purity in terms of adiabatic work processes, but is empirically feasible, and is in accord with experiments actually done, such as the Joule experiments mentioned just above, and with older traditions.
The formula () above allows that to go by processes of quasi-static adiabatic work from the state
$$
A
$$
to the state
$$
B
$$
we can take a path that goes through the reference state
$$
O
$$
, since the quasi-static adiabatic work is independent of the path
$$
-W^\mathrm{adiabatic,\,quasi-static}_{A\to B}=-W^\mathrm{adiabatic,\,quasi-static}_{A\to O}-W^\mathrm{adiabatic,\,quasi-static}_{O\to B} = W^\mathrm{adiabatic,\,quasi-static}_{O\to A}- W^\mathrm{adiabatic,\,quasi-static}_{O\to B} = -U(A) + U(B) = \Delta U
$$
This kind of empirical evidence, coupled with theory of this kind, largely justifies the following statement:
For all adiabatic processes between two specified states of a closed system of any nature, the net work done is the same regardless the details of the process, and determines a state function called internal energy,
$$
U
$$
.
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https://en.wikipedia.org/wiki/First_law_of_thermodynamics
|
The formula () above allows that to go by processes of quasi-static adiabatic work from the state
$$
A
$$
to the state
$$
B
$$
we can take a path that goes through the reference state
$$
O
$$
, since the quasi-static adiabatic work is independent of the path
$$
-W^\mathrm{adiabatic,\,quasi-static}_{A\to B}=-W^\mathrm{adiabatic,\,quasi-static}_{A\to O}-W^\mathrm{adiabatic,\,quasi-static}_{O\to B} = W^\mathrm{adiabatic,\,quasi-static}_{O\to A}- W^\mathrm{adiabatic,\,quasi-static}_{O\to B} = -U(A) + U(B) = \Delta U
$$
This kind of empirical evidence, coupled with theory of this kind, largely justifies the following statement:
For all adiabatic processes between two specified states of a closed system of any nature, the net work done is the same regardless the details of the process, and determines a state function called internal energy,
$$
U
$$
.
### Adynamic processes
A complementary observable aspect of the first law is about heat transfer.
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For all adiabatic processes between two specified states of a closed system of any nature, the net work done is the same regardless the details of the process, and determines a state function called internal energy,
$$
U
$$
.
### Adynamic processes
A complementary observable aspect of the first law is about heat transfer. Adynamic transfer of energy as heat can be measured empirically by changes in the surroundings of the system of interest by calorimetry. This again requires the existence of adiabatic enclosure of the entire process, system and surroundings, though the separating wall between the surroundings and the system is thermally conductive or radiatively permeable, not adiabatic. A calorimeter can rely on measurement of sensible heat, which requires the existence of thermometers and measurement of temperature change in bodies of known sensible heat capacity under specified conditions; or it can rely on the measurement of latent heat, through measurement of masses of material that change phase, at temperatures fixed by the occurrence of phase changes under specified conditions in bodies of known latent heat of phase change. The calorimeter can be calibrated by transferring an externally determined amount of heat into it, for instance from a resistive electrical heater inside the calorimeter through which a precisely known electric current is passed at a precisely known voltage for a precisely measured period of time.
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A calorimeter can rely on measurement of sensible heat, which requires the existence of thermometers and measurement of temperature change in bodies of known sensible heat capacity under specified conditions; or it can rely on the measurement of latent heat, through measurement of masses of material that change phase, at temperatures fixed by the occurrence of phase changes under specified conditions in bodies of known latent heat of phase change. The calorimeter can be calibrated by transferring an externally determined amount of heat into it, for instance from a resistive electrical heater inside the calorimeter through which a precisely known electric current is passed at a precisely known voltage for a precisely measured period of time. The calibration allows comparison of calorimetric measurement of quantity of heat transferred with quantity of energy transferred as (surroundings-based) work. According to one textbook, "The most common device for measuring
$$
\Delta U
$$
is an adiabatic bomb calorimeter." According to another textbook, "Calorimetry is widely used in present day laboratories." According to one opinion, "Most thermodynamic data come from calorimetry...".
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According to another textbook, "Calorimetry is widely used in present day laboratories." According to one opinion, "Most thermodynamic data come from calorimetry...".
When the system evolves with transfer of energy as heat, without energy being transferred as work, in an adynamic process, the heat transferred to the system is equal to the increase in its internal energy:
$$
Q^\mathrm{adynamic}_{A\to B}=\Delta U\,.
$$
### General case for reversible processes
Heat transfer is practically reversible when it is driven by practically negligibly small temperature gradients. Work transfer is practically reversible when it occurs so slowly that there are no frictional effects within the system; frictional effects outside the system should also be zero if the process is to be reversible in the strict thermodynamic sense.
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### General case for reversible processes
Heat transfer is practically reversible when it is driven by practically negligibly small temperature gradients. Work transfer is practically reversible when it occurs so slowly that there are no frictional effects within the system; frictional effects outside the system should also be zero if the process is to be reversible in the strict thermodynamic sense. For a particular reversible process in general, the work done reversibly on the system,
$$
W^{\mathrm{path}\,P_0,\, \mathrm{reversible}}_{A\to B}
$$
, and the heat transferred reversibly to the system,
$$
Q^{\mathrm{path}\,P_0,\, \mathrm{reversible}}_{A\to B}
$$
are not required to occur respectively adiabatically or adynamically, but they must belong to the same particular process defined by its particular reversible path,
$$
P_0
$$
, through the space of thermodynamic states. Then the work and heat transfers can occur and be calculated simultaneously.
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For a particular reversible process in general, the work done reversibly on the system,
$$
W^{\mathrm{path}\,P_0,\, \mathrm{reversible}}_{A\to B}
$$
, and the heat transferred reversibly to the system,
$$
Q^{\mathrm{path}\,P_0,\, \mathrm{reversible}}_{A\to B}
$$
are not required to occur respectively adiabatically or adynamically, but they must belong to the same particular process defined by its particular reversible path,
$$
P_0
$$
, through the space of thermodynamic states. Then the work and heat transfers can occur and be calculated simultaneously.
Putting the two complementary aspects together, the first law for a particular reversible process can be written
$$
-W^{\mathrm{path}\,P_0,\, \mathrm{reversible}}_{A\to B} + Q^{\mathrm{path}\,P_0,\, \mathrm{reversible}}_{A\to B} = \Delta U\, .
$$
This combined statement is the expression the first law of thermodynamics for reversible processes for closed systems.
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Then the work and heat transfers can occur and be calculated simultaneously.
Putting the two complementary aspects together, the first law for a particular reversible process can be written
$$
-W^{\mathrm{path}\,P_0,\, \mathrm{reversible}}_{A\to B} + Q^{\mathrm{path}\,P_0,\, \mathrm{reversible}}_{A\to B} = \Delta U\, .
$$
This combined statement is the expression the first law of thermodynamics for reversible processes for closed systems.
In particular, if no work is done on a thermally isolated closed system we have
$$
\Delta U = 0\,
$$
.
This is one aspect of the law of conservation of energy and can be stated:
The internal energy of an isolated system remains constant.
### General case for irreversible processes
If, in a process of change of state of a closed system, the energy transfer is not under a practically zero temperature gradient, practically frictionless, and with nearly balanced forces, then the process is irreversible. Then the heat and work transfers may be difficult to calculate with high accuracy, although the simple equations for reversible processes still hold to a good approximation in the absence of composition changes.
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### General case for irreversible processes
If, in a process of change of state of a closed system, the energy transfer is not under a practically zero temperature gradient, practically frictionless, and with nearly balanced forces, then the process is irreversible. Then the heat and work transfers may be difficult to calculate with high accuracy, although the simple equations for reversible processes still hold to a good approximation in the absence of composition changes. Importantly, the first law still holds and provides a check on the measurements and calculations of the work done irreversibly on the system,
$$
W^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B}
$$
, and the heat transferred irreversibly to the system,
$$
Q^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B}
$$
, which belong to the same particular process defined by its particular irreversible path,
$$
P_1
$$
, through the space of thermodynamic states.
$$
-W^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B} + Q^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B} = \Delta U\, .
$$
This means that the internal energy
$$
U
$$
is a function of state and that the internal energy change
$$
\Delta U
$$
between two states is a function only of the two states.
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Then the heat and work transfers may be difficult to calculate with high accuracy, although the simple equations for reversible processes still hold to a good approximation in the absence of composition changes. Importantly, the first law still holds and provides a check on the measurements and calculations of the work done irreversibly on the system,
$$
W^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B}
$$
, and the heat transferred irreversibly to the system,
$$
Q^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B}
$$
, which belong to the same particular process defined by its particular irreversible path,
$$
P_1
$$
, through the space of thermodynamic states.
$$
-W^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B} + Q^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B} = \Delta U\, .
$$
This means that the internal energy
$$
U
$$
is a function of state and that the internal energy change
$$
\Delta U
$$
between two states is a function only of the two states.
### Overview of the weight of evidence for the law
The first law of thermodynamics is so general that its predictions cannot all be directly tested.
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https://en.wikipedia.org/wiki/First_law_of_thermodynamics
|
Importantly, the first law still holds and provides a check on the measurements and calculations of the work done irreversibly on the system,
$$
W^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B}
$$
, and the heat transferred irreversibly to the system,
$$
Q^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B}
$$
, which belong to the same particular process defined by its particular irreversible path,
$$
P_1
$$
, through the space of thermodynamic states.
$$
-W^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B} + Q^{\mathrm{path}\,P_1,\, \mathrm{irreversible}}_{A\to B} = \Delta U\, .
$$
This means that the internal energy
$$
U
$$
is a function of state and that the internal energy change
$$
\Delta U
$$
between two states is a function only of the two states.
### Overview of the weight of evidence for the law
The first law of thermodynamics is so general that its predictions cannot all be directly tested. In many properly conducted experiments it has been precisely supported, and never violated.
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https://en.wikipedia.org/wiki/First_law_of_thermodynamics
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### Overview of the weight of evidence for the law
The first law of thermodynamics is so general that its predictions cannot all be directly tested. In many properly conducted experiments it has been precisely supported, and never violated. Indeed, within its scope of applicability, the law is so reliably established, that, nowadays, rather than experiment being considered as testing the accuracy of the law, it is more practical and realistic to think of the law as testing the accuracy of experiment. An experimental result that seems to violate the law may be assumed to be inaccurate or wrongly conceived, for example due to failure to account for an important physical factor. Thus, some may regard it as a principle more abstract than a law.
## State functional formulation for infinitesimal processes
When the heat and work transfers in the equations above are infinitesimal in magnitude, they are often denoted by , rather than exact differentials denoted by , as a reminder that heat and work do not describe the state of any system. The integral of an inexact differential depends upon the particular path taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential is always zero.
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The integral of an inexact differential depends upon the particular path taken through the space of thermodynamic parameters while the integral of an exact differential depends only upon the initial and final states. If the initial and final states are the same, then the integral of an inexact differential may or may not be zero, but the integral of an exact differential is always zero. The path taken by a thermodynamic system through a chemical or physical change is known as a thermodynamic process.
The first law for a closed homogeneous system may be stated in terms that include concepts that are established in the second law. The internal energy may then be expressed as a function of the system's defining state variables , entropy, and , volume: . In these terms, , the system's temperature, and , its pressure, are partial derivatives of with respect to and . These variables are important throughout thermodynamics, though not necessary for the statement of the first law. Rigorously, they are defined only when the system is in its own state of internal thermodynamic equilibrium. For some purposes, the concepts provide good approximations for scenarios sufficiently near to the system's internal thermodynamic equilibrium.
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Rigorously, they are defined only when the system is in its own state of internal thermodynamic equilibrium. For some purposes, the concepts provide good approximations for scenarios sufficiently near to the system's internal thermodynamic equilibrium.
The first law requires that:
$$
dU=\delta Q-\delta W \, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text {(closed system, general process, quasi-static or irreversible).}
$$
Then, for the fictive case of a reversible process, can be written in terms of exact differentials. One may imagine reversible changes, such that there is at each instant negligible departure from thermodynamic equilibrium within the system and between system and surroundings. Then, mechanical work is given by and the quantity of heat added can be expressed as .
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