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For an ideal gas is
$$
\Omega (U)\propto\mathcal{F}U^{\frac{\mathcal{F}}{2}-1}\delta U
$$
.
In this description, the particles are distinguishable. If the position and momentum of two particles are exchanged, the new state will be represented by a different point in phase space. In this case a single point will represent a microstate. If a subset of M particles are indistinguishable from each other, then the M! possible permutations or possible exchanges of these particles will be counted as part of a single microstate. The set of possible microstates are also reflected in the constraints upon the thermodynamic system.
For example, in the case of a simple gas of N particles with total energy U contained in a cube of volume V, in which a sample of the gas cannot be distinguished from any other sample by experimental means, a microstate will consist of the above-mentioned N! points in phase space, and the set of microstates will be constrained to have all position coordinates to lie inside the box, and the momenta to lie on a hyperspherical surface in momentum coordinates of radius U.
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The set of possible microstates are also reflected in the constraints upon the thermodynamic system.
For example, in the case of a simple gas of N particles with total energy U contained in a cube of volume V, in which a sample of the gas cannot be distinguished from any other sample by experimental means, a microstate will consist of the above-mentioned N! points in phase space, and the set of microstates will be constrained to have all position coordinates to lie inside the box, and the momenta to lie on a hyperspherical surface in momentum coordinates of radius U. If on the other hand, the system consists of a mixture of two different gases, samples of which can be distinguished from each other, say A and B, then the number of microstates is increased, since two points in which an A and B particle are exchanged in phase space are no longer part of the same microstate. Two particles that are identical may nevertheless be distinguishable based on, for example, their location. (See configurational entropy.) If the box contains identical particles, and is at equilibrium, and a partition is inserted, dividing the volume in half, particles in one box are now distinguishable from those in the second box.
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(See configurational entropy.) If the box contains identical particles, and is at equilibrium, and a partition is inserted, dividing the volume in half, particles in one box are now distinguishable from those in the second box. In phase space, the N/2 particles in each box are now restricted to a volume V/2, and their energy restricted to U/2, and the number of points describing a single microstate will change: the phase space description is not the same.
This has implications in both the Gibbs paradox and correct Boltzmann counting. With regard to Boltzmann counting, it is the multiplicity of points in phase space which effectively reduces the number of microstates and renders the entropy extensive. With regard to Gibbs paradox, the important result is that the increase in the number of microstates (and thus the increase in entropy) resulting from the insertion of the partition is exactly matched by the decrease in the number of microstates (and thus the decrease in entropy) resulting from the reduction in volume available to each particle, yielding a net entropy change of zero.
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Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects may result in displacements, which are changes of an object's position relative to its environment.
Theoretical expositions of this branch of physics has its origins in Ancient Greece, for instance, in the writings of Aristotle and ArchimedesRana, N.C., and Joag, P.S.
### Classical
Mechanics. West Petal Nagar, New Delhi. Tata McGraw-Hill, 1991, pg 6. (see
## History
of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo Galilei, Johannes Kepler, Christiaan Huygens, and Isaac Newton laid the foundation for what is now known as classical mechanics.
As a branch of classical physics, mechanics deals with bodies that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as the physical science that deals with the motion of and forces on bodies not in the quantum realm.
History
### Antiquity
The ancient Greek philosophers were among the first to propose that abstract principles govern nature.
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History
### Antiquity
The ancient Greek philosophers were among the first to propose that abstract principles govern nature. The main theory of mechanics in antiquity was Aristotelian mechanics, though an alternative theory is exposed in the pseudo-Aristotelian Mechanical Problems, often attributed to one of his successors.
There is another tradition that goes back to the ancient Greeks where mathematics is used more extensively to analyze bodies statically or dynamically, an approach that may have been stimulated by prior work of the Pythagorean Archytas. Examples of this tradition include pseudo-Euclid (On the Balance), Archimedes (On the Equilibrium of Planes, On Floating Bodies), Hero (Mechanica), and Pappus (Collection, Book VIII)."A Tiny Taste of the History of Mechanics". The University of Texas at Austin.
### Medieval age
In the Middle Ages, Aristotle's theories were criticized and modified by a number of figures, beginning with John Philoponus in the 6th century. A central problem was that of projectile motion, which was discussed by Hipparchus and Philoponus.
Persian Islamic polymath Ibn Sīnā published his theory of motion in The Book of Healing (1020).
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A central problem was that of projectile motion, which was discussed by Hipparchus and Philoponus.
Persian Islamic polymath Ibn Sīnā published his theory of motion in The Book of Healing (1020). He said that an impetus is imparted to a projectile by the thrower, and viewed it as persistent, requiring external forces such as air resistance to dissipate it. Ibn Sina made distinction between 'force' and 'inclination' (called "mayl"), and argued that an object gained mayl when the object is in opposition to its natural motion. So he concluded that continuation of motion is attributed to the inclination that is transferred to the object, and that object will be in motion until the mayl is spent. He also claimed that a projectile in a vacuum would not stop unless it is acted upon, consistent with Newton's first law of motion.
On the question of a body subject to a constant (uniform) force, the 12th-century Jewish-Arab scholar Hibat Allah Abu'l-Barakat al-Baghdaadi (born Nathanel, Iraqi, of Baghdad) stated that constant force imparts constant acceleration.
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He also claimed that a projectile in a vacuum would not stop unless it is acted upon, consistent with Newton's first law of motion.
On the question of a body subject to a constant (uniform) force, the 12th-century Jewish-Arab scholar Hibat Allah Abu'l-Barakat al-Baghdaadi (born Nathanel, Iraqi, of Baghdad) stated that constant force imparts constant acceleration. According to Shlomo Pines, al-Baghdaadi's theory of motion was "the oldest negation of Aristotle's fundamental dynamic law [namely, that a constant force produces a uniform motion], [and is thus an] anticipation in a vague fashion of the fundamental law of classical mechanics [namely, that a force applied continuously produces acceleration]."
Influenced by earlier writers such as Ibn Sina and al-Baghdaadi, the 14th-century French priest Jean Buridan developed the theory of impetus, which later developed into the modern theories of inertia, velocity, acceleration and momentum. This work and others was developed in 14th-century England by the Oxford Calculators such as Thomas Bradwardine, who studied and formulated various laws regarding falling bodies. The concept that the main properties of a body are uniformly accelerated motion (as of falling bodies) was worked out by the 14th-century Oxford Calculators.
###
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The concept that the main properties of a body are uniformly accelerated motion (as of falling bodies) was worked out by the 14th-century Oxford Calculators.
### Early modern age
Two central figures in the early modern age are Galileo Galilei and Isaac Newton. Galileo's final statement of his mechanics, particularly of falling bodies, is his Two New Sciences (1638). Newton's 1687 Philosophiæ Naturalis Principia Mathematica provided a detailed mathematical account of mechanics, using the newly developed mathematics of calculus and providing the basis of Newtonian mechanics.
There is some dispute over priority of various ideas: Newton's Principia is certainly the seminal work and has been tremendously influential, and many of the mathematics results therein could not have been stated earlier without the development of the calculus. However, many of the ideas, particularly as pertain to inertia and falling bodies, had been developed by prior scholars such as Christiaan Huygens and the less-known medieval predecessors. Precise credit is at times difficult or contentious because scientific language and standards of proof changed, so whether medieval statements are equivalent to modern statements or sufficient proof, or instead similar to modern statements and hypotheses is often debatable.
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However, many of the ideas, particularly as pertain to inertia and falling bodies, had been developed by prior scholars such as Christiaan Huygens and the less-known medieval predecessors. Precise credit is at times difficult or contentious because scientific language and standards of proof changed, so whether medieval statements are equivalent to modern statements or sufficient proof, or instead similar to modern statements and hypotheses is often debatable.
### Modern age
Two main modern developments in mechanics are general relativity of Einstein, and quantum mechanics, both developed in the 20th century based in part on earlier 19th-century ideas. The development in the modern continuum mechanics, particularly in the areas of elasticity, plasticity, fluid dynamics, electrodynamics, and thermodynamics of deformable media, started in the second half of the 20th century.
## Types of mechanical bodies
The often-used term body needs to stand for a wide assortment of objects, including particles, projectiles, spacecraft, stars, parts of machinery, parts of solids, parts of fluids (gases and liquids), etc.
Other distinctions between the various sub-disciplines of mechanics concern the nature of the bodies being described. Particles are bodies with little (known) internal structure, treated as mathematical points in classical mechanics.
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Other distinctions between the various sub-disciplines of mechanics concern the nature of the bodies being described. Particles are bodies with little (known) internal structure, treated as mathematical points in classical mechanics. Rigid bodies have size and shape, but retain a simplicity close to that of the particle, adding just a few so-called degrees of freedom, such as orientation in space.
Otherwise, bodies may be semi-rigid, i.e. elastic, or non-rigid, i.e. fluid. These subjects have both classical and quantum divisions of study.
For instance, the motion of a spacecraft, regarding its orbit and attitude (rotation), is described by the relativistic theory of classical mechanics, while the analogous movements of an atomic nucleus are described by quantum mechanics.
## Sub-disciplines
The following are the three main designations consisting of various subjects that are studied in mechanics.
Note that there is also the "theory of fields" which constitutes a separate discipline in physics, formally treated as distinct from mechanics, whether it be classical fields or quantum fields. But in actual practice, subjects belonging to mechanics and fields are closely interwoven. Thus, for instance, forces that act on particles are frequently derived from fields (electromagnetic or gravitational), and particles generate fields by acting as sources.
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But in actual practice, subjects belonging to mechanics and fields are closely interwoven. Thus, for instance, forces that act on particles are frequently derived from fields (electromagnetic or gravitational), and particles generate fields by acting as sources. In fact, in quantum mechanics, particles themselves are fields, as described theoretically by the wave function.
Classical
The following are described as forming classical mechanics:
- Newtonian mechanics, the original theory of motion (kinematics) and forces (dynamics)
- Analytical mechanics is a reformulation of Newtonian mechanics with an emphasis on system energy, rather than on forces. There are two main branches of analytical mechanics:
- Hamiltonian mechanics, a theoretical formalism, based on the principle of conservation of energy
- Lagrangian mechanics, another theoretical formalism, based on the principle of the least action
- Classical statistical mechanics generalizes ordinary classical mechanics to consider systems in an unknown state; often used to derive thermodynamic properties.
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Classical
The following are described as forming classical mechanics:
- Newtonian mechanics, the original theory of motion (kinematics) and forces (dynamics)
- Analytical mechanics is a reformulation of Newtonian mechanics with an emphasis on system energy, rather than on forces. There are two main branches of analytical mechanics:
- Hamiltonian mechanics, a theoretical formalism, based on the principle of conservation of energy
- Lagrangian mechanics, another theoretical formalism, based on the principle of the least action
- Classical statistical mechanics generalizes ordinary classical mechanics to consider systems in an unknown state; often used to derive thermodynamic properties.
- Celestial mechanics, the motion of bodies in space: planets, comets, stars, galaxies, etc.
- Astrodynamics, spacecraft navigation, etc.
- Solid mechanics, elasticity, plasticity, or viscoelasticity exhibited by deformable solids
- Fracture mechanics
- Acoustics, sound (density, variation, propagation) in solids, fluids and gases
- Statics, semi-rigid bodies in mechanical equilibrium
- Fluid mechanics, the motion of fluids
- Soil mechanics, mechanical behavior of soils
- Continuum mechanics, mechanics of continua (both solid and fluid)
- Hydraulics, mechanical properties of liquids
- Fluid statics, liquids in equilibrium
- Applied mechanics (also known as engineering mechanics)
- Biomechanics, solids, fluids, etc. in biology
- Biophysics, physical processes in living organisms
-
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There are two main branches of analytical mechanics:
- Hamiltonian mechanics, a theoretical formalism, based on the principle of conservation of energy
- Lagrangian mechanics, another theoretical formalism, based on the principle of the least action
- Classical statistical mechanics generalizes ordinary classical mechanics to consider systems in an unknown state; often used to derive thermodynamic properties.
- Celestial mechanics, the motion of bodies in space: planets, comets, stars, galaxies, etc.
- Astrodynamics, spacecraft navigation, etc.
- Solid mechanics, elasticity, plasticity, or viscoelasticity exhibited by deformable solids
- Fracture mechanics
- Acoustics, sound (density, variation, propagation) in solids, fluids and gases
- Statics, semi-rigid bodies in mechanical equilibrium
- Fluid mechanics, the motion of fluids
- Soil mechanics, mechanical behavior of soils
- Continuum mechanics, mechanics of continua (both solid and fluid)
- Hydraulics, mechanical properties of liquids
- Fluid statics, liquids in equilibrium
- Applied mechanics (also known as engineering mechanics)
- Biomechanics, solids, fluids, etc. in biology
- Biophysics, physical processes in living organisms
-
### Relativistic
or Einsteinian mechanics
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- Celestial mechanics, the motion of bodies in space: planets, comets, stars, galaxies, etc.
- Astrodynamics, spacecraft navigation, etc.
- Solid mechanics, elasticity, plasticity, or viscoelasticity exhibited by deformable solids
- Fracture mechanics
- Acoustics, sound (density, variation, propagation) in solids, fluids and gases
- Statics, semi-rigid bodies in mechanical equilibrium
- Fluid mechanics, the motion of fluids
- Soil mechanics, mechanical behavior of soils
- Continuum mechanics, mechanics of continua (both solid and fluid)
- Hydraulics, mechanical properties of liquids
- Fluid statics, liquids in equilibrium
- Applied mechanics (also known as engineering mechanics)
- Biomechanics, solids, fluids, etc. in biology
- Biophysics, physical processes in living organisms
-
### Relativistic
or Einsteinian mechanics
### Quantum
The following are categorized as being part of quantum mechanics:
- Schrödinger wave mechanics, used to describe the movements of the wavefunction of a single particle.
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### Relativistic
or Einsteinian mechanics
### Quantum
The following are categorized as being part of quantum mechanics:
- Schrödinger wave mechanics, used to describe the movements of the wavefunction of a single particle.
- Matrix mechanics is an alternative formulation that allows considering systems with a finite-dimensional state space.
- Quantum statistical mechanics generalizes ordinary quantum mechanics to consider systems in an unknown state; often used to derive thermodynamic properties.
- Particle physics, the motion, structure, and behavior of fundamental particles
- Nuclear physics, the motion, structure, and reactions of nuclei
- Condensed matter physics, quantum gases, solids, liquids, etc.
Historically, classical mechanics had been around for nearly a quarter millennium before quantum mechanics developed. Classical mechanics originated with Isaac Newton's laws of motion in Philosophiæ Naturalis Principia Mathematica, developed over the seventeenth century. Quantum mechanics developed later, over the nineteenth century, precipitated by Planck's postulate and Albert Einstein's explanation of the photoelectric effect. Both fields are commonly held to constitute the most certain knowledge that exists about physical nature.
Classical mechanics has especially often been viewed as a model for other so-called exact sciences.
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Both fields are commonly held to constitute the most certain knowledge that exists about physical nature.
Classical mechanics has especially often been viewed as a model for other so-called exact sciences. Essential in this respect is the extensive use of mathematics in theories, as well as the decisive role played by experiment in generating and testing them.
Quantum mechanics is of a bigger scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the correspondence principle, there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The correspondence principle states that the behavior of systems described by quantum theories reproduces classical physics in the limit of large quantum numbers, i.e. if quantum mechanics is applied to large systems (for e.g. a baseball), the result would almost be the same if classical mechanics had been applied. Quantum mechanics has superseded classical mechanics at the foundation level and is indispensable for the explanation and prediction of processes at the molecular, atomic, and sub-atomic level. However, for macroscopic processes classical mechanics is able to solve problems which are unmanageably difficult (mainly due to computational limits) in quantum mechanics and hence remains useful and well used.
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Quantum mechanics has superseded classical mechanics at the foundation level and is indispensable for the explanation and prediction of processes at the molecular, atomic, and sub-atomic level. However, for macroscopic processes classical mechanics is able to solve problems which are unmanageably difficult (mainly due to computational limits) in quantum mechanics and hence remains useful and well used.
Modern descriptions of such behavior begin with a careful definition of such quantities as displacement (distance moved), time, velocity, acceleration, mass, and force. Until about 400 years ago, however, motion was explained from a very different point of view. For example, following the ideas of Greek philosopher and scientist Aristotle, scientists reasoned that a cannonball falls down because its natural position is in the Earth; the Sun, the Moon, and the stars travel in circles around the Earth because it is the nature of heavenly objects to travel in perfect circles.
Often cited as father to modern science, Galileo brought together the ideas of other great thinkers of his time and began to calculate motion in terms of distance travelled from some starting position and the time that it took. He showed that the speed of falling objects increases steadily during the time of their fall. This acceleration is the same for heavy objects as for light ones, provided air friction (air resistance) is discounted.
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He showed that the speed of falling objects increases steadily during the time of their fall. This acceleration is the same for heavy objects as for light ones, provided air friction (air resistance) is discounted. The English mathematician and physicist Isaac Newton improved this analysis by defining force and mass and relating these to acceleration. For objects traveling at speeds close to the speed of light, Newton's laws were superseded by Albert Einstein's theory of relativity. [A sentence illustrating the computational complication of Einstein's theory of relativity.] For atomic and subatomic particles, Newton's laws were superseded by quantum theory. For everyday phenomena, however, Newton's three laws of motion remain the cornerstone of dynamics, which is the study of what causes motion.
Relativistic
Akin to the distinction between quantum and classical mechanics, Albert Einstein's general and special theories of relativity have expanded the scope of Newton and Galileo's formulation of mechanics. The differences between relativistic and Newtonian mechanics become significant and even dominant as the velocity of a body approaches the speed of light. For instance, in Newtonian mechanics, the kinetic energy of a free particle is , whereas in relativistic mechanics, it is (where is the Lorentz factor; this formula reduces to the Newtonian expression in the low energy limit).
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The differences between relativistic and Newtonian mechanics become significant and even dominant as the velocity of a body approaches the speed of light. For instance, in Newtonian mechanics, the kinetic energy of a free particle is , whereas in relativistic mechanics, it is (where is the Lorentz factor; this formula reduces to the Newtonian expression in the low energy limit).
For high-energy processes, quantum mechanics must be adjusted to account for special relativity; this has led to the development of quantum field theory.
## Professional organizations
- Applied Mechanics Division, American Society of Mechanical Engineers
- Fluid Dynamics Division, American Physical Society
- Society for Experimental Mechanics
- Institution of Mechanical Engineers is the United Kingdom's qualifying body for mechanical engineers and has been the home of Mechanical Engineers for over 150 years.
- International Union of Theoretical and Applied Mechanics
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In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.
Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions
$$
f
$$
and
$$
g
$$
on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the
## Poisson bracket
of
$$
f
$$
and
$$
g
$$
.
## Definition
Suppose that
$$
(M,\omega)
$$
is a symplectic manifold.
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## Poisson bracket
of
$$
f
$$
and
$$
g
$$
.
## Definition
Suppose that
$$
(M,\omega)
$$
is a symplectic manifold. Since the symplectic form
$$
\omega
$$
is nondegenerate, it sets up a fiberwise-linear isomorphism
$$
\omega: TM \to T^*M,
$$
between the tangent bundle
$$
TM
$$
and the cotangent bundle
$$
T^*M
$$
, with the inverse
$$
\Omega: T^* M \to T M, \quad \Omega = \omega^{-1}.
$$
Therefore, one-forms on a symplectic manifold _ BLOCK6_ may be identified with vector fields and every differentiable function
$$
H:M\rightarrow\mathbb{R}
$$
determines a unique vector field
$$
X_H
$$
, called the Hamiltonian vector field with the Hamiltonian
$$
H
$$
, by defining for every vector field
$$
Y
$$
on
$$
M
$$
,
$$
\mathrm{d}H(Y) = \omega(X_H,Y).
$$
Note: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature.
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BLOCK6_ may be identified with vector fields and every differentiable function
$$
H:M\rightarrow\mathbb{R}
$$
determines a unique vector field
$$
X_H
$$
, called the Hamiltonian vector field with the Hamiltonian
$$
H
$$
, by defining for every vector field
$$
Y
$$
on
$$
M
$$
,
$$
\mathrm{d}H(Y) = \omega(X_H,Y).
$$
Note: Some authors define the Hamiltonian vector field with the opposite sign. One has to be mindful of varying conventions in physical and mathematical literature.
## Examples
Suppose that
$$
M
$$
is a
$$
2n
$$
-dimensional symplectic manifold. Then locally, one may choose canonical coordinates
$$
(q^1,\cdots,q^n,p_1,\cdots,p_n)
$$
on
$$
M
$$
, in which the symplectic form is expressed as:
$$
\omega = \sum_i \mathrm{d}q^i \wedge \mathrm{d}p_i,
$$
where
$$
\operatorname{d}
$$
denotes the exterior derivative and
$$
\wedge
$$
denotes the exterior product.
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## Examples
Suppose that
$$
M
$$
is a
$$
2n
$$
-dimensional symplectic manifold. Then locally, one may choose canonical coordinates
$$
(q^1,\cdots,q^n,p_1,\cdots,p_n)
$$
on
$$
M
$$
, in which the symplectic form is expressed as:
$$
\omega = \sum_i \mathrm{d}q^i \wedge \mathrm{d}p_i,
$$
where
$$
\operatorname{d}
$$
denotes the exterior derivative and
$$
\wedge
$$
denotes the exterior product. Then the Hamiltonian vector field with Hamiltonian
$$
H
$$
takes the form:
$$
\Chi_H = \left( \frac{\partial H}{\partial p_i},
- \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,
$$
where _ BLOCK9_ is a
$$
2n\times 2n
$$
square
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Then the Hamiltonian vector field with Hamiltonian
$$
H
$$
takes the form:
$$
\Chi_H = \left( \frac{\partial H}{\partial p_i},
- \frac{\partial H}{\partial q^i} \right) = \Omega\,\mathrm{d}H,
$$
where _ BLOCK9_ is a
$$
2n\times 2n
$$
square matrix
$$
\Omega =
\begin{bmatrix}
0 & I_n \\
-I_n & 0 \\
\end{bmatrix},
$$
and
$$
\mathrm{d}H = \begin{bmatrix}
\frac{\partial H}{\partial q^i} \\
\frac{\partial H}{\partial p_i}
\end{bmatrix}.
$$
The matrix
$$
\Omega
$$
is frequently denoted with
$$
\mathbf{J}
$$
.
Suppose that
$$
M=\mathbb{R}^{2n}
$$
is the
$$
2n
$$
-dimensional symplectic vector space with (global) canonical coordinates.
- If
$$
H = p_i
$$
then
$$
X_H = \partial/\partial q^i;
$$
-
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Suppose that
$$
M=\mathbb{R}^{2n}
$$
is the
$$
2n
$$
-dimensional symplectic vector space with (global) canonical coordinates.
- If
$$
H = p_i
$$
then
$$
X_H = \partial/\partial q^i;
$$
- if
$$
H = q_i
$$
then
$$
X_H = -\partial/\partial p^i;
$$
- if
$$
H = \frac{1}{2} \sum (p_i)^2
$$
then
$$
X_H = \sum p_i\partial/\partial q^i;
$$
- if
$$
H = \frac{1}{2} \sum a_{ij} q^i q^j, a_{ij} = a_{ji}
$$
then
$$
X_H = -\sum a_{ij} q_i\partial/\partial p^j.
$$
## Properties
- The assignment
$$
f\mapsto X_f
$$
is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
- Suppose that
$$
(q^1,\cdots,q^n,p_1,\cdots,p_n)
$$
are canonical coordinates on
$$
M
$$
(see above).
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## Properties
- The assignment
$$
f\mapsto X_f
$$
is linear, so that the sum of two Hamiltonian functions transforms into the sum of the corresponding Hamiltonian vector fields.
- Suppose that
$$
(q^1,\cdots,q^n,p_1,\cdots,p_n)
$$
are canonical coordinates on
$$
M
$$
(see above). Then a curve
$$
\gamma(t)=(q(t),p(t))
$$
is an integral curve of the Hamiltonian vector field
$$
X_H
$$
if and only if it is a solution of Hamilton's equations:
$$
\begin{align}
\dot{q}^i & = \frac{\partial H}{\partial p_i} \\
\dot{p}_i & =-\frac{\partial H}{\partial q^i}.
\end{align}
$$
- The Hamiltonian
$$
H
$$
is constant along the integral curves, because
$$
\langle dH, \dot{\gamma}\rangle = \omega(X_H(\gamma),X_H(\gamma)) = 0
$$
. That is,
$$
H(\gamma(t))
$$
is actually independent of
$$
t
$$
.
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if and only if it is a solution of Hamilton's equations:
$$
\begin{align}
\dot{q}^i & = \frac{\partial H}{\partial p_i} \\
\dot{p}_i & =-\frac{\partial H}{\partial q^i}.
\end{align}
$$
- The Hamiltonian
$$
H
$$
is constant along the integral curves, because
$$
\langle dH, \dot{\gamma}\rangle = \omega(X_H(\gamma),X_H(\gamma)) = 0
$$
. That is,
$$
H(\gamma(t))
$$
is actually independent of
$$
t
$$
. This property corresponds to the conservation of energy in Hamiltonian mechanics.
- More generally, if two functions
$$
F
$$
and
$$
H
$$
have a zero Poisson bracket (cf. below), then
$$
F
$$
is constant along the integral curves of
$$
H
$$
, and similarly,
$$
H
$$
is constant along the integral curves of
$$
F
$$
. This fact is the abstract mathematical principle behind Noether's theorem.
- The symplectic form
$$
\omega
$$
is preserved by the Hamiltonian flow.
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This fact is the abstract mathematical principle behind Noether's theorem.
- The symplectic form
$$
\omega
$$
is preserved by the Hamiltonian flow. Equivalently, the Lie derivative
$$
\mathcal{L}_{X_H} \omega=0
$$
.
Poisson bracket
The notion of a Hamiltonian vector field leads to a skew-symmetric bilinear operation on the differentiable functions on a symplectic manifold
$$
M
$$
, the Poisson bracket, defined by the formula
$$
\{f,g\} = \omega(X_g, X_f)= dg(X_f) = \mathcal{L}_{X_f} g
$$
where
$$
\mathcal{L}_X
$$
denotes the Lie derivative along a vector field
$$
X
$$
. Moreover, one can check that the following identity holds:
$$
X_{\{f,g\}}= -[X_f,X_g]
$$
,
where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians
$$
f
$$
and
$$
g
$$
.
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|
Poisson bracket
The notion of a Hamiltonian vector field leads to a skew-symmetric bilinear operation on the differentiable functions on a symplectic manifold
$$
M
$$
, the Poisson bracket, defined by the formula
$$
\{f,g\} = \omega(X_g, X_f)= dg(X_f) = \mathcal{L}_{X_f} g
$$
where
$$
\mathcal{L}_X
$$
denotes the Lie derivative along a vector field
$$
X
$$
. Moreover, one can check that the following identity holds:
$$
X_{\{f,g\}}= -[X_f,X_g]
$$
,
where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians
$$
f
$$
and
$$
g
$$
. As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity:
$$
\{\{f,g\},h\} + \{\{g,h\},f\} + \{\{h,f\},g\}=0
$$
,
which means that the vector space of differentiable functions on
$$
M
$$
, endowed with the Poisson bracket, has the structure of a Lie algebra over
$$
\mathbb{R}
$$
, and the assignment
$$
f\mapsto X_f
$$
is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if
$$
M
$$
is connected).
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|
Moreover, one can check that the following identity holds:
$$
X_{\{f,g\}}= -[X_f,X_g]
$$
,
where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians
$$
f
$$
and
$$
g
$$
. As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity:
$$
\{\{f,g\},h\} + \{\{g,h\},f\} + \{\{h,f\},g\}=0
$$
,
which means that the vector space of differentiable functions on
$$
M
$$
, endowed with the Poisson bracket, has the structure of a Lie algebra over
$$
\mathbb{R}
$$
, and the assignment
$$
f\mapsto X_f
$$
is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if
$$
M
$$
is connected).
## Remarks
## Notes
## Works cited
- See section 3.2.
-
-
-
-
## External links
- Hamiltonian vector field on nLab
Category:Hamiltonian mechanics
Category:Symplectic geometry
Category:William Rowan Hamilton
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The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions – excluding gravity) in the universe and classifying all known elementary particles. It was developed in stages throughout the latter half of the 20th century, through the work of many scientists worldwide, with the current formulation being finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, proof of the top quark (1995), the tau neutrino (2000), and the
### Higgs boson
(2012) have added further credence to the Standard Model. In addition, the Standard Model has predicted various properties of weak neutral currents and the W and Z bosons with great accuracy.
Although the Standard Model is believed to be theoretically self-consistent and has demonstrated some success in providing experimental predictions, it leaves some physical phenomena unexplained and so falls short of being a complete theory of fundamental interactions. For example, it does not fully explain why there is more matter than anti-matter, incorporate the full theory of gravitation as described by general relativity, or account for the universe's accelerating expansion as possibly described by dark energy. The model does not contain any viable dark matter particle that possesses all of the required properties deduced from observational cosmology.
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For example, it does not fully explain why there is more matter than anti-matter, incorporate the full theory of gravitation as described by general relativity, or account for the universe's accelerating expansion as possibly described by dark energy. The model does not contain any viable dark matter particle that possesses all of the required properties deduced from observational cosmology. It also does not incorporate neutrino oscillations and their non-zero masses.
The development of the Standard Model was driven by theoretical and experimental particle physicists alike. The Standard Model is a paradigm of a quantum field theory for theorists, exhibiting a wide range of phenomena, including spontaneous symmetry breaking, anomalies, and non-perturbative behavior. It is used as a basis for building more exotic models that incorporate hypothetical particles, extra dimensions, and elaborate symmetries (such as supersymmetry) to explain experimental results at variance with the Standard Model, such as the existence of dark matter and neutrino oscillations.
## Historical background
In 1928, Paul Dirac introduced the Dirac equation, which implied the existence of antimatter.
In 1954, Yang Chen-Ning and Robert Mills extended the concept of gauge theory for abelian groups, e.g. quantum electrodynamics, to nonabelian groups to provide an explanation for strong interactions.
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## Historical background
In 1928, Paul Dirac introduced the Dirac equation, which implied the existence of antimatter.
In 1954, Yang Chen-Ning and Robert Mills extended the concept of gauge theory for abelian groups, e.g. quantum electrodynamics, to nonabelian groups to provide an explanation for strong interactions. In 1957, Chien-Shiung Wu demonstrated parity was not conserved in the weak interaction.
In 1961, Sheldon Glashow combined the electromagnetic and weak interactions. In 1964, Murray Gell-Mann and George Zweig introduced quarks and that same year Oscar W. Greenberg implicitly introduced color charge of quarks. In 1967 Steven Weinberg and Abdus Salam incorporated the Higgs mechanism into Glashow's electroweak interaction, giving it its modern form.
In 1970, Sheldon Glashow, John Iliopoulos, and Luciano Maiani introduced the GIM mechanism, predicting the charm quark. In 1973 Gross and Wilczek and Politzer independently discovered that non-Abelian gauge theories, like the color theory of the strong force, have asymptotic freedom. In 1976, Martin Perl discovered the tau lepton at the SLAC. In 1977, a team led by Leon Lederman at Fermilab discovered the bottom quark.
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In 1976, Martin Perl discovered the tau lepton at the SLAC. In 1977, a team led by Leon Lederman at Fermilab discovered the bottom quark.
The Higgs mechanism is believed to give rise to the masses of all the elementary particles in the Standard Model. This includes the masses of the W and Z bosons, and the masses of the fermions, i.e. the quarks and leptons.
After the neutral weak currents caused by Z boson exchange were discovered at CERN in 1973, the electroweak theory became widely accepted and Glashow, Salam, and Weinberg shared the 1979 Nobel Prize in Physics for discovering it. The W± and Z0 bosons were discovered experimentally in 1983; and the ratio of their masses was found to be as the Standard Model predicted.
The theory of the strong interaction (i.e. quantum chromodynamics, QCD), to which many contributed, acquired its modern form in 1973–74 when asymptotic freedom was proposed (a development that made QCD the main focus of theoretical research) and experiments confirmed that the hadrons were composed of fractionally charged quarks.
The term "Standard Model" was introduced by Abraham Pais and Sam Treiman in 1975, with reference to the electroweak theory with four quarks.
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The theory of the strong interaction (i.e. quantum chromodynamics, QCD), to which many contributed, acquired its modern form in 1973–74 when asymptotic freedom was proposed (a development that made QCD the main focus of theoretical research) and experiments confirmed that the hadrons were composed of fractionally charged quarks.
The term "Standard Model" was introduced by Abraham Pais and Sam Treiman in 1975, with reference to the electroweak theory with four quarks. Steven Weinberg has since claimed priority, explaining that he chose the term Standard Model out of a sense of modesty and used it in 1973 during a talk in Aix-en-Provence in France.
## Particle content
The Standard Model includes members of several classes of elementary particles, which in turn can be distinguished by other characteristics, such as color charge.
All particles can be summarized as follows:
### Fermions
The Standard Model includes 12 elementary particles of spin , known as fermions. Fermions respect the Pauli exclusion principle, meaning that two identical fermions cannot simultaneously occupy the same quantum state in the same atom. Each fermion has a corresponding antiparticle, which are particles that have corresponding properties with the exception of opposite charges. Fermions are classified based on how they interact, which is determined by the charges they carry, into two groups: quarks and leptons.
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Each fermion has a corresponding antiparticle, which are particles that have corresponding properties with the exception of opposite charges. Fermions are classified based on how they interact, which is determined by the charges they carry, into two groups: quarks and leptons. Within each group, pairs of particles that exhibit similar physical behaviors are then grouped into generations (see the table). Each member of a generation has a greater mass than the corresponding particle of generations prior. Thus, there are three generations of quarks and leptons. As first-generation particles do not decay, they comprise all of ordinary (baryonic) matter. Specifically, all atoms consist of electrons orbiting around the atomic nucleus, ultimately constituted of up and down quarks. On the other hand, second- and third-generation charged particles decay with very short half-lives and can only be observed in high-energy environments. Neutrinos of all generations also do not decay, and pervade the universe, but rarely interact with baryonic matter.
There are six quarks: up, down, charm, strange, top, and bottom. Quarks carry color charge, and hence interact via the strong interaction.
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There are six quarks: up, down, charm, strange, top, and bottom. Quarks carry color charge, and hence interact via the strong interaction. The color confinement phenomenon results in quarks being strongly bound together such that they form color-neutral composite particles called hadrons; quarks cannot individually exist and must always bind with other quarks. Hadrons can contain either a quark-antiquark pair (mesons) or three quarks (baryons). The lightest baryons are the nucleons: the proton and neutron. Quarks also carry electric charge and weak isospin, and thus interact with other fermions through electromagnetism and weak interaction. The six leptons consist of the electron, electron neutrino, muon, muon neutrino, tau, and tau neutrino. The leptons do not carry color charge, and do not respond to strong interaction. The charged leptons carry an electric charge of −1 e, while the three neutrinos carry zero electric charge. Thus, the neutrinos' motions are influenced by only the weak interaction and gravity, making them difficult to observe.
### Gauge bosons
The Standard Model includes 4 kinds of gauge bosons of spin 1, with bosons being quantum particles containing an integer spin.
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Thus, the neutrinos' motions are influenced by only the weak interaction and gravity, making them difficult to observe.
### Gauge bosons
The Standard Model includes 4 kinds of gauge bosons of spin 1, with bosons being quantum particles containing an integer spin. The gauge bosons are defined as force carriers, as they are responsible for mediating the fundamental interactions. The Standard Model explains the four fundamental forces as arising from the interactions, with fermions exchanging virtual force carrier particles, thus mediating the forces. At a macroscopic scale, this manifests as a force. As a result, they do not follow the Pauli exclusion principle that constrains fermions; bosons do not have a theoretical limit on their spatial density. The types of gauge bosons are described below.
-
### Electromagnetism
: Photons mediate the electromagnetic force, responsible for interactions between electrically charged particles. The photon is massless and is described by the theory of quantum electrodynamics (QED).
- Strong Interactions: Gluons mediate the strong interactions, which binds quarks to each other by influencing the color charge, with the interactions being described in the theory of quantum chromodynamics (QCD).
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### Electromagnetism
: Photons mediate the electromagnetic force, responsible for interactions between electrically charged particles. The photon is massless and is described by the theory of quantum electrodynamics (QED).
- Strong Interactions: Gluons mediate the strong interactions, which binds quarks to each other by influencing the color charge, with the interactions being described in the theory of quantum chromodynamics (QCD). They have no mass, and there are eight distinct gluons, with each being denoted through a color-anticolor charge combination (e.g. red–antigreen). As gluons have an effective color charge, they can also interact amongst themselves.
- Weak Interactions: The , , and gauge bosons mediate the weak interactions between all fermions, being responsible for radioactivity. They contain mass, with the having more mass than the . The weak interactions involving the act only on left-handed particles and right-handed antiparticles respectively. The carries an electric charge of +1 and −1 and couples to the electromagnetic interaction. The electrically neutral boson interacts with both left-handed particles and right-handed antiparticles. These three gauge bosons along with the photons are grouped together, as collectively mediating the electroweak interaction.
-
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The electrically neutral boson interacts with both left-handed particles and right-handed antiparticles. These three gauge bosons along with the photons are grouped together, as collectively mediating the electroweak interaction.
-
### Gravity
: It is currently unexplained in the Standard Model, as the hypothetical mediating particle graviton has been proposed, but not observed. This is due to the incompatibility of quantum mechanics and Einstein's theory of general relativity, regarded as being the best explanation for gravity. In general relativity, gravity is explained as being the geometric curving of spacetime.
The Feynman diagram calculations, which are a graphical representation of the perturbation theory approximation, invoke "force mediating particles", and when applied to analyze high-energy scattering experiments are in reasonable agreement with the data. However, perturbation theory (and with it the concept of a "force-mediating particle") fails in other situations. These include low-energy quantum chromodynamics, bound states, and solitons. The interactions between all the particles described by the Standard Model are summarized by the diagrams on the right of this section.
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These include low-energy quantum chromodynamics, bound states, and solitons. The interactions between all the particles described by the Standard Model are summarized by the diagrams on the right of this section.
Higgs boson
The Higgs particle is a massive scalar elementary particle theorized by Peter Higgs (and others) in 1964, when he showed that Goldstone's 1962 theorem (generic continuous symmetry, which is spontaneously broken) provides a third polarisation of a massive vector field. Hence, Goldstone's original scalar doublet, the massive spin-zero particle, was proposed as the Higgs boson, and is a key building block in the Standard Model. It has no intrinsic spin, and for that reason is classified as a boson with spin-0.
The Higgs boson plays a unique role in the Standard Model, by explaining why the other elementary particles, except the photon and gluon, are massive. In particular, the Higgs boson explains why the photon has no mass, while the W and Z bosons are very heavy. Elementary-particle masses and the differences between electromagnetism (mediated by the photon) and the weak force (mediated by the W and Z bosons) are critical to many aspects of the structure of microscopic (and hence macroscopic) matter.
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In particular, the Higgs boson explains why the photon has no mass, while the W and Z bosons are very heavy. Elementary-particle masses and the differences between electromagnetism (mediated by the photon) and the weak force (mediated by the W and Z bosons) are critical to many aspects of the structure of microscopic (and hence macroscopic) matter. In electroweak theory, the Higgs boson generates the masses of the leptons (electron, muon, and tau) and quarks. As the Higgs boson is massive, it must interact with itself.
Because the Higgs boson is a very massive particle and also decays almost immediately when created, only a very high-energy particle accelerator can observe and record it. Experiments to confirm and determine the nature of the Higgs boson using the Large Hadron Collider (LHC) at CERN began in early 2010 and were performed at Fermilab's Tevatron until its closure in late 2011. Mathematical consistency of the Standard Model requires that any mechanism capable of generating the masses of elementary particles must become visible at energies above ; therefore, the LHC (designed to collide two proton beams) was built to answer the question of whether the Higgs boson actually exists.
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Experiments to confirm and determine the nature of the Higgs boson using the Large Hadron Collider (LHC) at CERN began in early 2010 and were performed at Fermilab's Tevatron until its closure in late 2011. Mathematical consistency of the Standard Model requires that any mechanism capable of generating the masses of elementary particles must become visible at energies above ; therefore, the LHC (designed to collide two proton beams) was built to answer the question of whether the Higgs boson actually exists.
On 4 July 2012, two of the experiments at the LHC (ATLAS and CMS) both reported independently that they had found a new particle with a mass of about (about 133 proton masses, on the order of ), which is "consistent with the Higgs boson". On 13 March 2013, it was confirmed to be the searched-for Higgs boson.
## Theoretical aspects
### Construction of the Standard Model Lagrangian
Parameters of the Standard Model # Symbol Description Renormalization scheme (point) Value1meElectron mass0.511 MeV2mμMuon mass105.7 MeV3mτTau
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## Theoretical aspects
### Construction of the Standard Model Lagrangian
Parameters of the Standard Model # Symbol Description Renormalization scheme (point) Value1meElectron mass0.511 MeV2mμMuon mass105.7 MeV3mτTau mass1.78 GeV4muUp quark massμ = 2 GeV1.9 MeV5mdDown quark massμ = 2 GeV4.4 MeV6msStrange quark massμ = 2 GeV87 MeV7mcCharm quark massμ = mc1.32 GeV8mbBottom quark massμ = mb4.24 GeV9mtTop quark mass On shell scheme173.5 GeV10θ12CKM 12-mixing angle13.1°11θ23CKM 23-mixing angle2.4°12θ13CKM 13-mixing angle0.2°13δCKM CP violation Phase0.99514g1 or gU(1) gauge couplingμ = mZ0.35715g2 or gSU(2) gauge couplingμ = mZ0.65216g3 or gsSU(3) gauge couplingμ = mZ1.22117θQCDQCD vacuum angle~018vHiggs vacuum expectation value246 GeV19mHHiggs mass
Technically, quantum field theory provides the mathematical framework for the Standard Model, in which a Lagrangian controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical field that pervades space-time.
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mass1.78 GeV4muUp quark massμ = 2 GeV1.9 MeV5mdDown quark massμ = 2 GeV4.4 MeV6msStrange quark massμ = 2 GeV87 MeV7mcCharm quark massμ = mc1.32 GeV8mbBottom quark massμ = mb4.24 GeV9mtTop quark mass On shell scheme173.5 GeV10θ12CKM 12-mixing angle13.1°11θ23CKM 23-mixing angle2.4°12θ13CKM 13-mixing angle0.2°13δCKM CP violation Phase0.99514g1 or gU(1) gauge couplingμ = mZ0.35715g2 or gSU(2) gauge couplingμ = mZ0.65216g3 or gsSU(3) gauge couplingμ = mZ1.22117θQCDQCD vacuum angle~018vHiggs vacuum expectation value246 GeV19mHHiggs mass
Technically, quantum field theory provides the mathematical framework for the Standard Model, in which a Lagrangian controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical field that pervades space-time.
The construction of the Standard Model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.
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Each kind of particle is described in terms of a dynamical field that pervades space-time.
The construction of the Standard Model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.
The global Poincaré symmetry is postulated for all relativistic quantum field theories. It consists of the familiar translational symmetry, rotational symmetry and the inertial reference frame invariance central to the theory of special relativity. The local SU(3) × SU(2) × U(1) gauge symmetry is an internal symmetry that essentially defines the Standard Model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different representations of the various symmetry groups of the Standard Model (see table). Upon writing the most general Lagrangian, one finds that the dynamics depends on 19 parameters, whose numerical values are established by experiment. The parameters are summarized in the table (made visible by clicking "show") above.
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Upon writing the most general Lagrangian, one finds that the dynamics depends on 19 parameters, whose numerical values are established by experiment. The parameters are summarized in the table (made visible by clicking "show") above.
#### Quantum chromodynamics sector
The quantum chromodynamics (QCD) sector defines the interactions between quarks and gluons, which is a Yang–Mills gauge theory with SU(3) symmetry, generated by
$$
T^a = \lambda^a/2
$$
. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by
$$
\mathcal{L}_\text{QCD} = \overline{\psi} i\gamma^\mu D_{\mu} \psi - \frac{1}{4} G^a_{\mu\nu} G^{\mu\nu}_a,
$$
where
$$
\psi
$$
is a three component column vector of Dirac spinors, each element of which refers to a quark field with a specific color charge (i.e. red, blue, and green) and summation over flavor (i.e. up, down, strange, etc.) is implied.
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Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by
$$
\mathcal{L}_\text{QCD} = \overline{\psi} i\gamma^\mu D_{\mu} \psi - \frac{1}{4} G^a_{\mu\nu} G^{\mu\nu}_a,
$$
where
$$
\psi
$$
is a three component column vector of Dirac spinors, each element of which refers to a quark field with a specific color charge (i.e. red, blue, and green) and summation over flavor (i.e. up, down, strange, etc.) is implied.
The gauge covariant derivative of QCD is defined by
$$
D_{\mu} \equiv \partial_\mu - i g_\text{s}\frac{1}{2}\lambda^a G_\mu^a
$$
, where
- are the Dirac matrices,
- is the 8-component (
$$
a = 1, 2, \dots, 8
$$
) SU(3) gauge field,
- are the 3 × 3 Gell-Mann matrices, generators of the SU(3) color group,
- represents the gluon field strength tensor, and
- is the strong coupling constant.
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The Dirac Lagrangian of the quarks coupled to the gluon fields is given by
$$
\mathcal{L}_\text{QCD} = \overline{\psi} i\gamma^\mu D_{\mu} \psi - \frac{1}{4} G^a_{\mu\nu} G^{\mu\nu}_a,
$$
where
$$
\psi
$$
is a three component column vector of Dirac spinors, each element of which refers to a quark field with a specific color charge (i.e. red, blue, and green) and summation over flavor (i.e. up, down, strange, etc.) is implied.
The gauge covariant derivative of QCD is defined by
$$
D_{\mu} \equiv \partial_\mu - i g_\text{s}\frac{1}{2}\lambda^a G_\mu^a
$$
, where
- are the Dirac matrices,
- is the 8-component (
$$
a = 1, 2, \dots, 8
$$
) SU(3) gauge field,
- are the 3 × 3 Gell-Mann matrices, generators of the SU(3) color group,
- represents the gluon field strength tensor, and
- is the strong coupling constant.
The QCD Lagrangian is invariant under local SU(3) gauge transformations; i.e., transformations of the form
$$
\psi \rightarrow \psi' = U\psi
$$
, where
$$
U = e^{-i g_\text{s}\lambda^a \phi^{a}(x)}
$$
is 3 × 3 unitary matrix with determinant 1, making it a member of the group SU(3), and
$$
\phi^{a}(x)
$$
is an arbitrary function of spacetime.
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#### Electroweak sector
The electroweak sector is a Yang–Mills gauge theory with the symmetry group ,
$$
\mathcal{L}_\text{EW} = \overline{Q}_{\text{L}j} i\gamma^\mu D_{\mu} Q_{\text{L}j} + \overline{u}_{\text{R}j} i\gamma^\mu D_{\mu} u_{\text{R}j} + \overline{d}_{\text{R}j} i\gamma^\mu D_{\mu} d_{\text{R}j} + \overline{\ell}_{\text{L}j} i\gamma^\mu D_{\mu} \ell_{\text{L}j} + \overline{e}_{\text{R}j} i\gamma^\mu D_{\mu} e_{\text{R}j} - \tfrac{1}{4} W_a^{\mu\nu} W_{\mu\nu}^a - \tfrac{1}{4} B^{\mu\nu} B_{\mu\nu},
$$
where the subscript
$$
j
$$
sums over the three generations of fermions;
$$
Q_\text{L}, u_\text{R}
$$
, and
$$
d_\text{R}
$$
are the left-handed doublet, right-handed singlet up type, and right handed singlet down type quark fields; and
$$
\ell_\text{L}
$$
and
$$
e_\text{R}
$$
are the left-handed doublet and right-handed singlet lepton fields.
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The electroweak gauge covariant derivative is defined as
$$
D_\mu \equiv \partial_\mu - ig' \tfrac12 Y_\text{W} B_\mu - ig \tfrac{1}{2} \vec\tau_\text{L} \vec W_\mu
$$
, where
- is the U(1) gauge field,
- is the weak hypercharge – the generator of the U(1) group,
- is the 3-component SU(2) gauge field,
- are the Pauli matrices – infinitesimal generators of the SU(2) group – with subscript L to indicate that they only act on left-chiral fermions,
- and are the U(1) and SU(2) coupling constants respectively,
-
$$
W^{a\mu\nu}
$$
(
$$
a = 1, 2, 3
$$
) and
$$
B^{\mu\nu}
$$
are the field strength tensors for the weak isospin and weak hypercharge fields.
Notice that the addition of fermion mass terms into the electroweak Lagrangian is forbidden, since terms of the form
$$
m\overline\psi\psi
$$
do not respect gauge invariance. Neither is it possible to add explicit mass terms for the U(1) and SU(2) gauge fields.
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Notice that the addition of fermion mass terms into the electroweak Lagrangian is forbidden, since terms of the form
$$
m\overline\psi\psi
$$
do not respect gauge invariance. Neither is it possible to add explicit mass terms for the U(1) and SU(2) gauge fields. The Higgs mechanism is responsible for the generation of the gauge boson masses, and the fermion masses result from Yukawa-type interactions with the Higgs field.
#### Higgs sector
In the Standard Model , the Higgs field is an SU(2) doublet of complex scalar fields with four degrees of freedom:
$$
\varphi = \begin{pmatrix} \varphi^+ \\ \varphi^0 \end{pmatrix}
= \frac{1}{\sqrt{2}} \begin{pmatrix} \varphi_1 + i\varphi_2 \\ \varphi_3 + i\varphi_4 \end{pmatrix},
$$
where the superscripts + and 0 indicate the electric charge
$$
Q
$$
of the components. The weak hypercharge
$$
Y_\text{W}
$$
of both components is 1.
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, the Higgs field is an SU(2) doublet of complex scalar fields with four degrees of freedom:
$$
\varphi = \begin{pmatrix} \varphi^+ \\ \varphi^0 \end{pmatrix}
= \frac{1}{\sqrt{2}} \begin{pmatrix} \varphi_1 + i\varphi_2 \\ \varphi_3 + i\varphi_4 \end{pmatrix},
$$
where the superscripts + and 0 indicate the electric charge
$$
Q
$$
of the components. The weak hypercharge
$$
Y_\text{W}
$$
of both components is 1. Before symmetry breaking, the Higgs Lagrangian is
$$
\mathcal{L}_\text{H} = \left(D_{\mu}\varphi\right)^{\dagger} \left(D^{\mu}\varphi \right) - V(\varphi),
$$
where
$$
D_{\mu}
$$
is the electroweak gauge covariant derivative defined above and
$$
V(\varphi)
$$
is the potential of the Higgs field.
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The weak hypercharge
$$
Y_\text{W}
$$
of both components is 1. Before symmetry breaking, the Higgs Lagrangian is
$$
\mathcal{L}_\text{H} = \left(D_{\mu}\varphi\right)^{\dagger} \left(D^{\mu}\varphi \right) - V(\varphi),
$$
where
$$
D_{\mu}
$$
is the electroweak gauge covariant derivative defined above and
$$
V(\varphi)
$$
is the potential of the Higgs field. The square of the covariant derivative leads to three and four point interactions between the electroweak gauge fields
$$
W^{a}_{\mu}
$$
and
$$
B_{\mu}
$$
and the scalar field
$$
\varphi
$$
.
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Before symmetry breaking, the Higgs Lagrangian is
$$
\mathcal{L}_\text{H} = \left(D_{\mu}\varphi\right)^{\dagger} \left(D^{\mu}\varphi \right) - V(\varphi),
$$
where
$$
D_{\mu}
$$
is the electroweak gauge covariant derivative defined above and
$$
V(\varphi)
$$
is the potential of the Higgs field. The square of the covariant derivative leads to three and four point interactions between the electroweak gauge fields
$$
W^{a}_{\mu}
$$
and
$$
B_{\mu}
$$
and the scalar field
$$
\varphi
$$
. The scalar potential is given by
$$
V(\varphi) = -\mu^2\varphi^{\dagger}\varphi + \lambda \left( \varphi^{\dagger}\varphi \right)^2,
$$
where
$$
\mu^2>0
$$
, so that
$$
\varphi
$$
acquires a non-zero Vacuum expectation value, which generates masses for the Electroweak gauge fields (the Higgs mechanism), and
$$
\lambda>0
$$
, so that the potential is bounded from below.
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The square of the covariant derivative leads to three and four point interactions between the electroweak gauge fields
$$
W^{a}_{\mu}
$$
and
$$
B_{\mu}
$$
and the scalar field
$$
\varphi
$$
. The scalar potential is given by
$$
V(\varphi) = -\mu^2\varphi^{\dagger}\varphi + \lambda \left( \varphi^{\dagger}\varphi \right)^2,
$$
where
$$
\mu^2>0
$$
, so that
$$
\varphi
$$
acquires a non-zero Vacuum expectation value, which generates masses for the Electroweak gauge fields (the Higgs mechanism), and
$$
\lambda>0
$$
, so that the potential is bounded from below. The quartic term describes self-interactions of the scalar field
$$
\varphi
$$
.
The minimum of the potential is degenerate with an infinite number of equivalent ground state solutions, which occurs when
$$
\varphi^{\dagger}\varphi = \tfrac{\mu^2}{2\lambda}
$$
.
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The quartic term describes self-interactions of the scalar field
$$
\varphi
$$
.
The minimum of the potential is degenerate with an infinite number of equivalent ground state solutions, which occurs when
$$
\varphi^{\dagger}\varphi = \tfrac{\mu^2}{2\lambda}
$$
. It is possible to perform a gauge transformation on
$$
\varphi
$$
such that the ground state is transformed to a basis where
$$
\varphi_1 = \varphi_2 = \varphi_4 = 0
$$
and
$$
\varphi_3 = \tfrac{\mu}{\sqrt{\lambda}} \equiv v
$$
. This breaks the symmetry of the ground state. The expectation value of
$$
\varphi
$$
now becomes
$$
\langle \varphi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v \end{pmatrix},
$$
where
$$
v
$$
has units of mass and sets the scale of electroweak physics. This is the only dimensional parameter of the Standard Model and has a measured value of ~.
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The expectation value of
$$
\varphi
$$
now becomes
$$
\langle \varphi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v \end{pmatrix},
$$
where
$$
v
$$
has units of mass and sets the scale of electroweak physics. This is the only dimensional parameter of the Standard Model and has a measured value of ~.
After symmetry breaking, the masses of the W and Z are given by
$$
m_\text{W}=\frac{1}{2}gv
$$
and
$$
m_\text{Z}=\frac{1}{2}\sqrt{g^2+g'^2}v
$$
, which can be viewed as predictions of the theory. The photon remains massless. The mass of the Higgs boson is
$$
m_\text{H}=\sqrt{2\mu^2}=\sqrt{2\lambda}v
$$
. Since
$$
\mu
$$
and
$$
\lambda
$$
are free parameters, the Higgs's mass could not be predicted beforehand and had to be determined experimentally.
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The mass of the Higgs boson is
$$
m_\text{H}=\sqrt{2\mu^2}=\sqrt{2\lambda}v
$$
. Since
$$
\mu
$$
and
$$
\lambda
$$
are free parameters, the Higgs's mass could not be predicted beforehand and had to be determined experimentally.
#### Yukawa sector
The Yukawa interaction terms are:
$$
\mathcal{L}_\text{Yukawa} = (Y_\text{u})_{mn}(\bar{Q}_\text{L})_m \tilde{\varphi}(u_\text{R})_n + (Y_\text{d})_{mn}(\bar{Q}_\text{L})_m \varphi(d_\text{R})_n + (Y_\text{e})_{mn}(\bar{\ell}_\text{L})_m {\varphi}(e_\text{R})_n + \mathrm{h.c.}
$$
where
$$
Y_\text{u}
$$
,
$$
Y_\text{d}
$$
, and
$$
Y_\text{e}
$$
are matrices of Yukawa couplings, with the term giving the coupling of the generations and , and h.c. means Hermitian conjugate of preceding terms.
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Since
$$
\mu
$$
and
$$
\lambda
$$
are free parameters, the Higgs's mass could not be predicted beforehand and had to be determined experimentally.
#### Yukawa sector
The Yukawa interaction terms are:
$$
\mathcal{L}_\text{Yukawa} = (Y_\text{u})_{mn}(\bar{Q}_\text{L})_m \tilde{\varphi}(u_\text{R})_n + (Y_\text{d})_{mn}(\bar{Q}_\text{L})_m \varphi(d_\text{R})_n + (Y_\text{e})_{mn}(\bar{\ell}_\text{L})_m {\varphi}(e_\text{R})_n + \mathrm{h.c.}
$$
where
$$
Y_\text{u}
$$
,
$$
Y_\text{d}
$$
, and
$$
Y_\text{e}
$$
are matrices of Yukawa couplings, with the term giving the coupling of the generations and , and h.c. means Hermitian conjugate of preceding terms. The fields
$$
Q_\text{L}
$$
and
$$
\ell_\text{L}
$$
are left-handed quark and lepton doublets.
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#### Yukawa sector
The Yukawa interaction terms are:
$$
\mathcal{L}_\text{Yukawa} = (Y_\text{u})_{mn}(\bar{Q}_\text{L})_m \tilde{\varphi}(u_\text{R})_n + (Y_\text{d})_{mn}(\bar{Q}_\text{L})_m \varphi(d_\text{R})_n + (Y_\text{e})_{mn}(\bar{\ell}_\text{L})_m {\varphi}(e_\text{R})_n + \mathrm{h.c.}
$$
where
$$
Y_\text{u}
$$
,
$$
Y_\text{d}
$$
, and
$$
Y_\text{e}
$$
are matrices of Yukawa couplings, with the term giving the coupling of the generations and , and h.c. means Hermitian conjugate of preceding terms. The fields
$$
Q_\text{L}
$$
and
$$
\ell_\text{L}
$$
are left-handed quark and lepton doublets. Likewise,
$$
u_\text{R}, d_\text{R}
$$
and
$$
e_\text{R}
$$
are right-handed up-type quark, down-type quark, and lepton singlets.
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The fields
$$
Q_\text{L}
$$
and
$$
\ell_\text{L}
$$
are left-handed quark and lepton doublets. Likewise,
$$
u_\text{R}, d_\text{R}
$$
and
$$
e_\text{R}
$$
are right-handed up-type quark, down-type quark, and lepton singlets. Finally
$$
\varphi
$$
is the Higgs doublet and _ BLOCK9_ is its charge conjugate state.
The Yukawa terms are invariant under the SU(2) × U(1) gauge symmetry of the Standard Model and generate masses for all fermions after spontaneous symmetry breaking.
## Fundamental interactions
The Standard Model describes three of the four fundamental interactions in nature; only gravity remains unexplained. In the Standard Model, such an interaction is described as an exchange of bosons between the objects affected, such as a photon for the electromagnetic force and a gluon for the strong interaction. Those particles are called force carriers or messenger particles.
+
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|
Those particles are called force carriers or messenger particles.
+ The four fundamental interactions of nature Property/InteractionGravitationElectroweakStrongWeakElectromagneticFundamentalResidualMediating particles align="center" observed(Graviton hypothesised)W+, W− and Z0γ (photon)Gluonsπ, ρ and ω mesonsAffected particlesAll particlesW+, W−: Left-handed fermions; Z0: All fermions Electrically chargedQuarks, gluonsHadronsActs on Stress–energy tensorFlavorElectric chargeColor chargeBound states formedPlanets, stars, galaxies, galaxy groups align="center" Atoms, moleculesHadronsAtomic nucleiStrength at the scale of quarks (relative to electromagnetism) (predicted)160 align="center" Strength at the scale of protons/neutrons (relative to electromagnetism) (predicted)1 align="center" 20
Gravity
Despite being perhaps the most familiar fundamental interaction, gravity is not described by the Standard Model, due to contradictions that arise when combining general relativity, the modern theory of gravity, and quantum mechanics. However, gravity is so weak at microscopic scales, that it is essentially unmeasurable.
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The four fundamental interactions of nature Property/InteractionGravitationElectroweakStrongWeakElectromagneticFundamentalResidualMediating particles align="center" observed(Graviton hypothesised)W+, W− and Z0γ (photon)Gluonsπ, ρ and ω mesonsAffected particlesAll particlesW+, W−: Left-handed fermions; Z0: All fermions Electrically chargedQuarks, gluonsHadronsActs on Stress–energy tensorFlavorElectric chargeColor chargeBound states formedPlanets, stars, galaxies, galaxy groups align="center" Atoms, moleculesHadronsAtomic nucleiStrength at the scale of quarks (relative to electromagnetism) (predicted)160 align="center" Strength at the scale of protons/neutrons (relative to electromagnetism) (predicted)1 align="center" 20
Gravity
Despite being perhaps the most familiar fundamental interaction, gravity is not described by the Standard Model, due to contradictions that arise when combining general relativity, the modern theory of gravity, and quantum mechanics. However, gravity is so weak at microscopic scales, that it is essentially unmeasurable. The graviton is postulated to be the mediating particle, but has not yet been proved to exist.
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However, gravity is so weak at microscopic scales, that it is essentially unmeasurable. The graviton is postulated to be the mediating particle, but has not yet been proved to exist.
Electromagnetism
Electromagnetism is the only long-range force in the Standard Model. It is mediated by photons and couples to electric charge. Electromagnetism is responsible for a wide range of phenomena including atomic electron shell structure, chemical bonds, electric circuits and electronics. Electromagnetic interactions in the Standard Model are described by quantum electrodynamics.
### Weak nuclear force
The weak interaction is responsible for various forms of particle decay, such as beta decay. It is weak and short-range, due to the fact that the weak mediating particles, W and Z bosons, have mass. W bosons have electric charge and mediate interactions that change the particle type (referred to as flavor) and charge. Interactions mediated by W bosons are charged current interactions. Z bosons are neutral and mediate neutral current interactions, which do not change particle flavor. Thus Z bosons are similar to the photon, aside from them being massive and interacting with the neutrino. The weak interaction is also the only interaction to violate parity and CP.
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Thus Z bosons are similar to the photon, aside from them being massive and interacting with the neutrino. The weak interaction is also the only interaction to violate parity and CP. Parity violation is maximal for charged current interactions, since the W boson interacts exclusively with left-handed fermions and right-handed antifermions.
In the Standard Model, the weak force is understood in terms of the electroweak theory, which states that the weak and electromagnetic interactions become united into a single electroweak interaction at high energies.
### Strong nuclear force
The strong nuclear force is responsible for hadronic and nuclear binding. It is mediated by gluons, which couple to color charge. Since gluons themselves have color charge, the strong force exhibits confinement and asymptotic freedom. Confinement means that only color-neutral particles can exist in isolation, therefore quarks can only exist in hadrons and never in isolation, at low energies. Asymptotic freedom means that the strong force becomes weaker, as the energy scale increases. The strong force overpowers the electrostatic repulsion of protons and quarks in nuclei and hadrons respectively, at their respective scales.
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Asymptotic freedom means that the strong force becomes weaker, as the energy scale increases. The strong force overpowers the electrostatic repulsion of protons and quarks in nuclei and hadrons respectively, at their respective scales.
While quarks are bound in hadrons by the fundamental strong interaction, which is mediated by gluons, nucleons are bound by an emergent phenomenon termed the residual strong force or nuclear force. This interaction is mediated by mesons, such as the pion. The color charges inside the nucleon cancel out, meaning most of the gluon and quark fields cancel out outside of the nucleon. However, some residue is "leaked", which appears as the exchange of virtual mesons, that causes the attractive force between nucleons. The (fundamental) strong interaction is described by quantum chromodynamics, which is a component of the Standard Model.
## Tests and predictions
The Standard Model predicted the existence of the W and Z bosons, gluon, top quark and charm quark, and predicted many of their properties before these particles were observed. The predictions were experimentally confirmed with good precision.
The Standard Model also predicted the existence of the Higgs boson, which was found in 2012 at the Large Hadron Collider, the final fundamental particle predicted by the Standard Model to be experimentally confirmed.
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The predictions were experimentally confirmed with good precision.
The Standard Model also predicted the existence of the Higgs boson, which was found in 2012 at the Large Hadron Collider, the final fundamental particle predicted by the Standard Model to be experimentally confirmed.
## Challenges
Self-consistency of the Standard Model (currently formulated as a non-abelian gauge theory quantized through path-integrals) has not been mathematically proved. While regularized versions useful for approximate computations (for example lattice gauge theory) exist, it is not known whether they converge (in the sense of S-matrix elements) in the limit that the regulator is removed. A key question related to the consistency is the Yang–Mills existence and mass gap problem.
Experiments indicate that neutrinos have mass, which the classic Standard Model did not allow. To accommodate this finding, the classic Standard Model can be modified to include neutrino mass, although it is not obvious exactly how this should be done.
If one insists on using only Standard Model particles, this can be achieved by adding a non-renormalizable interaction of leptons with the Higgs boson. On a fundamental level, such an interaction emerges in the seesaw mechanism where heavy right-handed neutrinos are added to the theory.
This is natural in the left-right symmetric extension of the Standard Model and in certain grand unified theories.
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On a fundamental level, such an interaction emerges in the seesaw mechanism where heavy right-handed neutrinos are added to the theory.
This is natural in the left-right symmetric extension of the Standard Model and in certain grand unified theories. As long as new physics appears below or around 1014 GeV, the neutrino masses can be of the right order of magnitude.
Theoretical and experimental research has attempted to extend the Standard Model into a unified field theory or a theory of everything, a complete theory explaining all physical phenomena including constants. Inadequacies of the Standard Model that motivate such research include:
- The model does not explain gravitation, although physical confirmation of a theoretical particle known as a graviton would account for it to a degree. Though it addresses strong and electroweak interactions, the Standard Model does not consistently explain the canonical theory of gravitation, general relativity, in terms of quantum field theory. The reason for this is, among other things, that quantum field theories of gravity generally break down before reaching the Planck scale. As a consequence, we have no reliable theory for the very early universe.
- Some physicists consider it to be ad hoc and inelegant, requiring 19 numerical constants whose values are unrelated and arbitrary. Although the Standard Model, as it now stands, can explain why neutrinos have masses, the specifics of neutrino mass are still unclear.
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Some physicists consider it to be ad hoc and inelegant, requiring 19 numerical constants whose values are unrelated and arbitrary. Although the Standard Model, as it now stands, can explain why neutrinos have masses, the specifics of neutrino mass are still unclear. It is believed that explaining neutrino mass will require an additional 7 or 8 constants, which are also arbitrary parameters.
- The Higgs mechanism gives rise to the hierarchy problem if some new physics (coupled to the Higgs) is present at high energy scales. In these cases, in order for the weak scale to be much smaller than the Planck scale, severe fine tuning of the parameters is required; there are, however, other scenarios that include quantum gravity in which such fine tuning can be avoided. There are also issues of quantum triviality, which suggests that it may not be possible to create a consistent quantum field theory involving elementary scalar particles.
- The model is inconsistent with the emerging Lambda-CDM model of cosmology. Contentions include the absence of an explanation in the Standard Model of particle physics for the observed amount of cold dark matter (CDM) and its contributions to dark energy, which are many orders of magnitude too large.
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The model is inconsistent with the emerging Lambda-CDM model of cosmology. Contentions include the absence of an explanation in the Standard Model of particle physics for the observed amount of cold dark matter (CDM) and its contributions to dark energy, which are many orders of magnitude too large. It is also difficult to accommodate the observed predominance of matter over antimatter (matter/antimatter asymmetry). The isotropy and homogeneity of the visible universe over large distances seems to require a mechanism like cosmic inflation, which would also constitute an extension of the Standard Model.
Currently, no proposed theory of everything has been widely accepted or verified.
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Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanical force fields. The method is applied mostly in chemical physics, materials science, and biophysics.
Because molecular systems typically consist of a vast number of particles, it is impossible to determine the properties of such complex systems analytically; MD simulation circumvents this problem by using numerical methods. However, long MD simulations are mathematically ill-conditioned, generating cumulative errors in numerical integration that can be minimized with proper selection of algorithms and parameters, but not eliminated.
For systems that obey the ergodic hypothesis, the evolution of one molecular dynamics simulation may be used to determine the macroscopic thermodynamic properties of the system: the time averages of an ergodic system correspond to microcanonical ensemble averages.
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However, long MD simulations are mathematically ill-conditioned, generating cumulative errors in numerical integration that can be minimized with proper selection of algorithms and parameters, but not eliminated.
For systems that obey the ergodic hypothesis, the evolution of one molecular dynamics simulation may be used to determine the macroscopic thermodynamic properties of the system: the time averages of an ergodic system correspond to microcanonical ensemble averages. MD has also been termed "statistical mechanics by numbers" and "Laplace's vision of Newtonian mechanics" of predicting the future by animating nature's forces and allowing insight into molecular motion on an atomic scale.
## History
MD was originally developed in the early 1950s, following earlier successes with Monte Carlo simulationswhich themselves date back to the eighteenth century, in the Buffon's needle problem for examplebut was popularized for statistical mechanics at Los Alamos National Laboratory by Marshall Rosenbluth and Nicholas Metropolis in what is known today as the Metropolis–Hastings algorithm. Interest in the time evolution of N-body systems dates much earlier to the seventeenth century, beginning with Isaac Newton, and continued into the following century largely with a focus on celestial mechanics and issues such as the stability of the Solar System.
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## History
MD was originally developed in the early 1950s, following earlier successes with Monte Carlo simulationswhich themselves date back to the eighteenth century, in the Buffon's needle problem for examplebut was popularized for statistical mechanics at Los Alamos National Laboratory by Marshall Rosenbluth and Nicholas Metropolis in what is known today as the Metropolis–Hastings algorithm. Interest in the time evolution of N-body systems dates much earlier to the seventeenth century, beginning with Isaac Newton, and continued into the following century largely with a focus on celestial mechanics and issues such as the stability of the Solar System. Many of the numerical methods used today were developed during this time period, which predates the use of computers; for example, the most common integration algorithm used today, the Verlet integration algorithm, was used as early as 1791 by Jean Baptiste Joseph Delambre. Numerical calculations with these algorithms can be considered to be MD done "by hand".
As early as 1941, integration of the many-body equations of motion was carried out with analog computers. Some undertook the labor-intensive work of modeling atomic motion by constructing physical models, e.g., using macroscopic spheres. The aim was to arrange them in such a way as to replicate the structure of a liquid and use this to examine its behavior. J.D. Bernal describes this process in 1962, writing:...
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The aim was to arrange them in such a way as to replicate the structure of a liquid and use this to examine its behavior. J.D. Bernal describes this process in 1962, writing:... I took a number of rubber balls and stuck them together with rods of a selection of different lengths ranging from 2.75 to 4 inches. I tried to do this in the first place as casually as possible, working in my own office, being interrupted every five minutes or so and not remembering what I had done before the interruption. Following the discovery of microscopic particles and the development of computers, interest expanded beyond the proving ground of gravitational systems to the statistical properties of matter. In an attempt to understand the origin of irreversibility, Enrico Fermi proposed in 1953, and published in 1955, the use of the early computer MANIAC I, also at Los Alamos National Laboratory, to solve the time evolution of the equations of motion for a many-body system subject to several choices of force laws. Today, this seminal work is known as the Fermi–Pasta–Ulam–Tsingou problem. The time evolution of the energy from the original work is shown in the figure to the right.
In 1957, Berni Alder and Thomas Wainwright used an IBM 704 computer to simulate perfectly elastic collisions between hard spheres.
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The time evolution of the energy from the original work is shown in the figure to the right.
In 1957, Berni Alder and Thomas Wainwright used an IBM 704 computer to simulate perfectly elastic collisions between hard spheres. In 1960, in perhaps the first realistic simulation of matter, J.B. Gibson et al. simulated radiation damage of solid copper by using a Born–Mayer type of repulsive interaction along with a cohesive surface force. In 1964, Aneesur Rahman published simulations of liquid argon that used a Lennard-Jones potential; calculations of system properties, such as the coefficient of self-diffusion, compared well with experimental data. Today, the Lennard-Jones potential is still one of the most frequently used intermolecular potentials. It is used for describing simple substances (a.k.a. Lennard-Jonesium) for conceptual and model studies and as a building block in many force fields of real substances.
## Areas of application and limits
First used in theoretical physics, the molecular dynamics method gained popularity in materials science soon afterward, and since the 1970s it has also been commonly used in biochemistry and biophysics. MD is frequently used to refine 3-dimensional structures of proteins and other macromolecules based on experimental constraints from X-ray crystallography or NMR spectroscopy.
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## Areas of application and limits
First used in theoretical physics, the molecular dynamics method gained popularity in materials science soon afterward, and since the 1970s it has also been commonly used in biochemistry and biophysics. MD is frequently used to refine 3-dimensional structures of proteins and other macromolecules based on experimental constraints from X-ray crystallography or NMR spectroscopy. In physics, MD is used to examine the dynamics of atomic-level phenomena that cannot be observed directly, such as thin film growth and ion subplantation, and to examine the physical properties of nanotechnological devices that have not or cannot yet be created. In biophysics and structural biology, the method is frequently applied to study the motions of macromolecules such as proteins and nucleic acids, which can be useful for interpreting the results of certain biophysical experiments and for modeling interactions with other molecules, as in ligand docking. In principle, MD can be used for ab initio prediction of protein structure by simulating folding of the polypeptide chain from a random coil.
The results of MD simulations can be tested through comparison to experiments that measure molecular dynamics, of which a popular method is NMR spectroscopy.
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In principle, MD can be used for ab initio prediction of protein structure by simulating folding of the polypeptide chain from a random coil.
The results of MD simulations can be tested through comparison to experiments that measure molecular dynamics, of which a popular method is NMR spectroscopy. MD-derived structure predictions can be tested through community-wide experiments in Critical Assessment of Protein Structure Prediction (CASP), although the method has historically had limited success in this area. Michael Levitt, who shared the Nobel Prize partly for the application of MD to proteins, wrote in 1999 that CASP participants usually did not use the method due to "... a central embarrassment of molecular mechanics, namely that energy minimization or molecular dynamics generally leads to a model that is less like the experimental structure". Improvements in computational resources permitting more and longer MD trajectories, combined with modern improvements in the quality of force field parameters, have yielded some improvements in both structure prediction and homology model refinement, without reaching the point of practical utility in these areas; many identify force field parameters as a key area for further development.
MD simulation has been reported for pharmacophore development and drug design. For example, Pinto et al. implemented MD simulations of Bcl-xL complexes to calculate average positions of critical amino acids involved in ligand binding.
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MD simulation has been reported for pharmacophore development and drug design. For example, Pinto et al. implemented MD simulations of Bcl-xL complexes to calculate average positions of critical amino acids involved in ligand binding. Carlson et al. implemented molecular dynamics simulations to identify compounds that complement a receptor while causing minimal disruption to the conformation and flexibility of the active site. Snapshots of the protein at constant time intervals during the simulation were overlaid to identify conserved binding regions (conserved in at least three out of eleven frames) for pharmacophore development. Spyrakis et al. relied on a workflow of MD simulations, fingerprints for ligands and proteins (FLAP) and linear discriminant analysis (LDA) to identify the best ligand-protein conformations to act as pharmacophore templates based on retrospective ROC analysis of the resulting pharmacophores. In an attempt to ameliorate structure-based drug discovery modeling, vis-à-vis the need for many modeled compounds, Hatmal et al. proposed a combination of MD simulation and ligand-receptor intermolecular contacts analysis to discern critical intermolecular contacts (binding interactions) from redundant ones in a single ligand–protein complex. Critical contacts can then be converted into pharmacophore models that can be used for virtual screening.
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In an attempt to ameliorate structure-based drug discovery modeling, vis-à-vis the need for many modeled compounds, Hatmal et al. proposed a combination of MD simulation and ligand-receptor intermolecular contacts analysis to discern critical intermolecular contacts (binding interactions) from redundant ones in a single ligand–protein complex. Critical contacts can then be converted into pharmacophore models that can be used for virtual screening.
An important factor is intramolecular hydrogen bonds, which are not explicitly included in modern force fields, but described as Coulomb interactions of atomic point charges. This is a crude approximation because hydrogen bonds have a partially quantum mechanical and chemical nature. Furthermore, electrostatic interactions are usually calculated using the dielectric constant of a vacuum, even though the surrounding aqueous solution has a much higher dielectric constant. Thus, using the macroscopic dielectric constant at short interatomic distances is questionable. Finally, van der Waals interactions in MD are usually described by Lennard-Jones potentials based on the Fritz London theory that is only applicable in a vacuum. However, all types of van der Waals forces are ultimately of electrostatic origin and therefore depend on dielectric properties of the environment.
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Finally, van der Waals interactions in MD are usually described by Lennard-Jones potentials based on the Fritz London theory that is only applicable in a vacuum. However, all types of van der Waals forces are ultimately of electrostatic origin and therefore depend on dielectric properties of the environment. The direct measurement of attraction forces between different materials (as Hamaker constant) shows that "the interaction between hydrocarbons across water is about 10% of that across vacuum". The environment-dependence of van der Waals forces is neglected in standard simulations, but can be included by developing polarizable force fields.
## Design constraints
The design of a molecular dynamics simulation should account for the available computational power. Simulation size (n = number of particles), timestep, and total time duration must be selected so that the calculation can finish within a reasonable time period. However, the simulations should be long enough to be relevant to the time scales of the natural processes being studied. To make statistically valid conclusions from the simulations, the time span simulated should match the kinetics of the natural process. Otherwise, it is analogous to making conclusions about how a human walks when only looking at less than one footstep. Most scientific publications about the dynamics of proteins and DNA use data from simulations spanning nanoseconds (10−9 s) to microseconds (10−6 s).
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Otherwise, it is analogous to making conclusions about how a human walks when only looking at less than one footstep. Most scientific publications about the dynamics of proteins and DNA use data from simulations spanning nanoseconds (10−9 s) to microseconds (10−6 s). To obtain these simulations, several CPU-days to CPU-years are needed. Parallel algorithms allow the load to be distributed among CPUs; an example is the spatial or force decomposition algorithm.
During a classical MD simulation, the most CPU intensive task is the evaluation of the potential as a function of the particles' internal coordinates. Within that energy evaluation, the most expensive one is the non-bonded or non-covalent part. In big O notation, common molecular dynamics simulations scale by
$$
O(n^2)
$$
if all pair-wise electrostatic and van der Waals interactions must be accounted for explicitly. This computational cost can be reduced by employing electrostatics methods such as particle mesh Ewald summation (
$$
O(n \log(n))
$$
), particle-particle-particle mesh (P3M), or good spherical cutoff methods (
$$
O(n)
$$
).
Another factor that impacts total CPU time needed by a simulation is the size of the integration timestep. This is the time length between evaluations of the potential.
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Another factor that impacts total CPU time needed by a simulation is the size of the integration timestep. This is the time length between evaluations of the potential. The timestep must be chosen small enough to avoid discretization errors (i.e., smaller than the period related to fastest vibrational frequency in the system). Typical timesteps for classical MD are on the order of 1 femtosecond (10−15 s). This value may be extended by using algorithms such as the SHAKE constraint algorithm, which fix the vibrations of the fastest atoms (e.g., hydrogens) into place. Multiple time scale methods have also been developed, which allow extended times between updates of slower long-range forces.
For simulating molecules in a solvent, a choice should be made between an explicit and implicit solvent. Explicit solvent particles (such as the TIP3P, SPC/E and SPC-f water models) must be calculated expensively by the force field, while implicit solvents use a mean-field approach. Using an explicit solvent is computationally expensive, requiring inclusion of roughly ten times more particles in the simulation. But the granularity and viscosity of explicit solvent is essential to reproduce certain properties of the solute molecules. This is especially important to reproduce chemical kinetics.
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But the granularity and viscosity of explicit solvent is essential to reproduce certain properties of the solute molecules. This is especially important to reproduce chemical kinetics.
In all kinds of molecular dynamics simulations, the simulation box size must be large enough to avoid boundary condition artifacts. Boundary conditions are often treated by choosing fixed values at the edges (which may cause artifacts), or by employing periodic boundary conditions in which one side of the simulation loops back to the opposite side, mimicking a bulk phase (which may cause artifacts too).
### Microcanonical ensemble (NVE)
In the microcanonical ensemble, the system is isolated from changes in moles (N), volume (V), and energy (E). It corresponds to an adiabatic process with no heat exchange. A microcanonical molecular dynamics trajectory may be seen as an exchange of potential and kinetic energy, with total energy being conserved.
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It corresponds to an adiabatic process with no heat exchange. A microcanonical molecular dynamics trajectory may be seen as an exchange of potential and kinetic energy, with total energy being conserved. For a system of N particles with coordinates
$$
X
$$
and velocities
$$
V
$$
, the following pair of first order differential equations may be written in Newton's notation as
$$
F(X) = -\nabla U(X)=M\dot{V}(t)
$$
$$
V(t) = \dot{X} (t).
$$
The potential energy function
$$
U(X)
$$
of the system is a function of the particle coordinates
$$
X
$$
. It is referred to simply as the potential in physics, or the force field in chemistry. The first equation comes from Newton's laws of motion; the force
$$
F
$$
acting on each particle in the system can be calculated as the negative gradient of
$$
U(X)
$$
.
For every time step, each particle's position
$$
X
$$
and velocity
$$
V
$$
may be integrated with a symplectic integrator method such as Verlet integration. The time evolution of
$$
X
$$
and
$$
V
$$
is called a trajectory.
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For every time step, each particle's position
$$
X
$$
and velocity
$$
V
$$
may be integrated with a symplectic integrator method such as Verlet integration. The time evolution of
$$
X
$$
and
$$
V
$$
is called a trajectory. Given the initial positions (e.g., from theoretical knowledge) and velocities (e.g., randomized Gaussian), we can calculate all future (or past) positions and velocities.
One frequent source of confusion is the meaning of temperature in MD. Commonly we have experience with macroscopic temperatures, which involve a huge number of particles, but temperature is a statistical quantity. If there is a large enough number of atoms, statistical temperature can be estimated from the instantaneous temperature, which is found by equating the kinetic energy of the system to nkBT/2, where n is the number of degrees of freedom of the system.
A temperature-related phenomenon arises due to the small number of atoms that are used in MD simulations. For example, consider simulating the growth of a copper film starting with a substrate containing 500 atoms and a deposition energy of 100 eV. In the real world, the 100 eV from the deposited atom would rapidly be transported through and shared among a large number of atoms (
$$
10^{10}
$$
or more) with no big change in temperature.
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A temperature-related phenomenon arises due to the small number of atoms that are used in MD simulations. For example, consider simulating the growth of a copper film starting with a substrate containing 500 atoms and a deposition energy of 100 eV. In the real world, the 100 eV from the deposited atom would rapidly be transported through and shared among a large number of atoms (
$$
10^{10}
$$
or more) with no big change in temperature. When there are only 500 atoms, however, the substrate is almost immediately vaporized by the deposition. Something similar happens in biophysical simulations. The temperature of the system in NVE is naturally raised when macromolecules such as proteins undergo exothermic conformational changes and binding.
### Canonical ensemble (NVT)
In the canonical ensemble, amount of substance (N), volume (V) and temperature (T) are conserved. It is also sometimes called constant temperature molecular dynamics (CTMD). In NVT, the energy of endothermic and exothermic processes is exchanged with a thermostat.
A variety of thermostat algorithms are available to add and remove energy from the boundaries of an MD simulation in a more or less realistic way, approximating the canonical ensemble.
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In NVT, the energy of endothermic and exothermic processes is exchanged with a thermostat.
A variety of thermostat algorithms are available to add and remove energy from the boundaries of an MD simulation in a more or less realistic way, approximating the canonical ensemble. Popular methods to control temperature include velocity rescaling, the Nosé–Hoover thermostat, Nosé–Hoover chains, the Berendsen thermostat, the Andersen thermostat and Langevin dynamics. The Berendsen thermostat might introduce the flying ice cube effect, which leads to unphysical translations and rotations of the simulated system.
It is not trivial to obtain a canonical ensemble distribution of conformations and velocities using these algorithms. How this depends on system size, thermostat choice, thermostat parameters, time step and integrator is the subject of many articles in the field.
### Isothermal–isobaric (NPT) ensemble
In the isothermal–isobaric ensemble, amount of substance (N), pressure (P) and temperature (T) are conserved. In addition to a thermostat, a barostat is needed. It corresponds most closely to laboratory conditions with a flask open to ambient temperature and pressure.
In the simulation of biological membranes, isotropic pressure control is not appropriate.
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It corresponds most closely to laboratory conditions with a flask open to ambient temperature and pressure.
In the simulation of biological membranes, isotropic pressure control is not appropriate. For lipid bilayers, pressure control occurs under constant membrane area (NPAT) or constant surface tension "gamma" (NPγT).
### Generalized ensembles
The replica exchange method is a generalized ensemble. It was originally created to deal with the slow dynamics of disordered spin systems. It is also called parallel tempering. The replica exchange MD (REMD) formulation tries to overcome the multiple-minima problem by exchanging the temperature of non-interacting replicas of the system running at several temperatures.
## Potentials in MD simulations
A molecular dynamics simulation requires the definition of a potential function, or a description of the terms by which the particles in the simulation will interact. In chemistry and biology this is usually referred to as a force field and in materials physics as an interatomic potential. Potentials may be defined at many levels of physical accuracy; those most commonly used in chemistry are based on molecular mechanics and embody a classical mechanics treatment of particle-particle interactions that can reproduce structural and conformational changes but usually cannot reproduce chemical reactions.
The reduction from a fully quantum description to a classical potential entails two main approximations.
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Potentials may be defined at many levels of physical accuracy; those most commonly used in chemistry are based on molecular mechanics and embody a classical mechanics treatment of particle-particle interactions that can reproduce structural and conformational changes but usually cannot reproduce chemical reactions.
The reduction from a fully quantum description to a classical potential entails two main approximations. The first one is the Born–Oppenheimer approximation, which states that the dynamics of electrons are so fast that they can be considered to react instantaneously to the motion of their nuclei. As a consequence, they may be treated separately. The second one treats the nuclei, which are much heavier than electrons, as point particles that follow classical Newtonian dynamics. In classical molecular dynamics, the effect of the electrons is approximated as one potential energy surface, usually representing the ground state.
When finer levels of detail are needed, potentials based on quantum mechanics are used; some methods attempt to create hybrid classical/quantum potentials where the bulk of the system is treated classically but a small region is treated as a quantum system, usually undergoing a chemical transformation.
### Empirical potentials
Empirical potentials used in chemistry are frequently called force fields, while those used in materials physics are called interatomic potentials.
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When finer levels of detail are needed, potentials based on quantum mechanics are used; some methods attempt to create hybrid classical/quantum potentials where the bulk of the system is treated classically but a small region is treated as a quantum system, usually undergoing a chemical transformation.
### Empirical potentials
Empirical potentials used in chemistry are frequently called force fields, while those used in materials physics are called interatomic potentials.
Most force fields in chemistry are empirical and consist of a summation of bonded forces associated with chemical bonds, bond angles, and bond dihedrals, and non-bonded forces associated with van der Waals forces and electrostatic charge. Empirical potentials represent quantum-mechanical effects in a limited way through ad hoc functional approximations. These potentials contain free parameters such as atomic charge, van der Waals parameters reflecting estimates of atomic radius, and equilibrium bond length, angle, and dihedral; these are obtained by fitting against detailed electronic calculations (quantum chemical simulations) or experimental physical properties such as elastic constants, lattice parameters and spectroscopic measurements.
Because of the non-local nature of non-bonded interactions, they involve at least weak interactions between all particles in the system. Its calculation is normally the bottleneck in the speed of MD simulations.
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Because of the non-local nature of non-bonded interactions, they involve at least weak interactions between all particles in the system. Its calculation is normally the bottleneck in the speed of MD simulations. To lower the computational cost, force fields employ numerical approximations such as shifted cutoff radii, reaction field algorithms, particle mesh Ewald summation, or the newer particle–particle-particle–mesh (P3M).
Chemistry force fields commonly employ preset bonding arrangements (an exception being ab initio dynamics), and thus are unable to model the process of chemical bond breaking and reactions explicitly. On the other hand, many of the potentials used in physics, such as those based on the bond order formalism can describe several different coordinations of a system and bond breaking. Examples of such potentials include the Brenner potential for hydrocarbons and its
further developments for the C-Si-H and C-O-H systems. The ReaxFF potential can be considered a fully reactive hybrid between bond order potentials and chemistry force fields.
### Pair potentials versus many-body potentials
The potential functions representing the non-bonded energy are formulated as a sum over interactions between the particles of the system.
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The ReaxFF potential can be considered a fully reactive hybrid between bond order potentials and chemistry force fields.
### Pair potentials versus many-body potentials
The potential functions representing the non-bonded energy are formulated as a sum over interactions between the particles of the system. The simplest choice, employed in many popular force fields, is the "pair potential", in which the total potential energy can be calculated from the sum of energy contributions between pairs of atoms. Therefore, these force fields are also called "additive force fields". An example of such a pair potential is the non-bonded Lennard-Jones potential (also termed the 6–12 potential), used for calculating van der Waals forces.
$$
U(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6} \right]
$$
Another example is the Born (ionic) model of the ionic lattice. The first term in the next equation is Coulomb's law for a pair of ions, the second term is the short-range repulsion explained by Pauli's exclusion principle and the final term is the dispersion interaction term. Usually, a simulation only includes the dipolar term, although sometimes the quadrupolar term is also included.
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The first term in the next equation is Coulomb's law for a pair of ions, the second term is the short-range repulsion explained by Pauli's exclusion principle and the final term is the dispersion interaction term. Usually, a simulation only includes the dipolar term, although sometimes the quadrupolar term is also included. When nl = 6, this potential is also called the Coulomb–Buckingham potential.
$$
U_{ij}(r_{ij}) = \frac {z_i z_j}{4 \pi \epsilon_0} \frac {1}{r_{ij}} + A_l \exp \frac {-r_{ij}}{p_l} + C_l r_{ij}^{-n_l} + \cdots
$$
In many-body potentials, the potential energy includes the effects of three or more particles interacting with each other. In simulations with pairwise potentials, global interactions in the system also exist, but they occur only through pairwise terms. In many-body potentials, the potential energy cannot be found by a sum over pairs of atoms, as these interactions are calculated explicitly as a combination of higher-order terms. In the statistical view, the dependency between the variables cannot in general be expressed using only pairwise products of the degrees of freedom.
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In many-body potentials, the potential energy cannot be found by a sum over pairs of atoms, as these interactions are calculated explicitly as a combination of higher-order terms. In the statistical view, the dependency between the variables cannot in general be expressed using only pairwise products of the degrees of freedom. For example, the Tersoff potential, which was originally used to simulate carbon, silicon, and germanium, and has since been used for a wide range of other materials, involves a sum over groups of three atoms, with the angles between the atoms being an important factor in the potential. Other examples are the embedded-atom method (EAM), the EDIP, and the Tight-Binding Second Moment Approximation (TBSMA) potentials, where the electron density of states in the region of an atom is calculated from a sum of contributions from surrounding atoms, and the potential energy contribution is then a function of this sum.
### Semi-empirical potentials
Semi-empirical potentials make use of the matrix representation from quantum mechanics. However, the values of the matrix elements are found through empirical formulae that estimate the degree of overlap of specific atomic orbitals. The matrix is then diagonalized to determine the occupancy of the different atomic orbitals, and empirical formulae are used once again to determine the energy contributions of the orbitals.
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However, the values of the matrix elements are found through empirical formulae that estimate the degree of overlap of specific atomic orbitals. The matrix is then diagonalized to determine the occupancy of the different atomic orbitals, and empirical formulae are used once again to determine the energy contributions of the orbitals.
There are a wide variety of semi-empirical potentials, termed tight-binding potentials, which vary according to the atoms being modeled.
### Polarizable potentials
Most classical force fields implicitly include the effect of polarizability, e.g., by scaling up the partial charges obtained from quantum chemical calculations. These partial charges are stationary with respect to the mass of the atom. But molecular dynamics simulations can explicitly model polarizability with the introduction of induced dipoles through different methods, such as Drude particles or fluctuating charges. This allows for a dynamic redistribution of charge between atoms which responds to the local chemical environment.
For many years, polarizable MD simulations have been touted as the next generation. For homogenous liquids such as water, increased accuracy has been achieved through the inclusion of polarizability. Some promising results have also been achieved for proteins. However, it is still uncertain how to best approximate polarizability in a simulation.
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Some promising results have also been achieved for proteins. However, it is still uncertain how to best approximate polarizability in a simulation. The point becomes more important when a particle experiences different environments during its simulation trajectory, e.g. translocation of a drug through a cell membrane.
### Potentials in ab initio methods
In classical molecular dynamics, one potential energy surface (usually the ground state) is represented in the force field. This is a consequence of the Born–Oppenheimer approximation. In excited states, chemical reactions or when a more accurate representation is needed, electronic behavior can be obtained from first principles using a quantum mechanical method, such as density functional theory. This is named Ab Initio Molecular Dynamics (AIMD). Due to the cost of treating the electronic degrees of freedom, the computational burden of these simulations is far higher than classical molecular dynamics. For this reason, AIMD is typically limited to smaller systems and shorter times.
Ab initio quantum mechanical and chemical methods may be used to calculate the potential energy of a system on the fly, as needed for conformations in a trajectory. This calculation is usually made in the close neighborhood of the reaction coordinate. Although various approximations may be used, these are based on theoretical considerations, not on empirical fitting.
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This calculation is usually made in the close neighborhood of the reaction coordinate. Although various approximations may be used, these are based on theoretical considerations, not on empirical fitting. Ab initio calculations produce a vast amount of information that is not available from empirical methods, such as density of electronic states or other electronic properties. A significant advantage of using ab initio methods is the ability to study reactions that involve breaking or formation of covalent bonds, which correspond to multiple electronic states. Moreover, ab initio methods also allow recovering effects beyond the Born–Oppenheimer approximation using approaches like mixed quantum-classical dynamics.
### Hybrid QM/MM
QM (quantum-mechanical) methods are very powerful. However, they are computationally expensive, while the MM (classical or molecular mechanics) methods are fast but suffer from several limits (require extensive parameterization; energy estimates obtained are not very accurate; cannot be used to simulate reactions where covalent bonds are broken/formed; and are limited in their abilities for providing accurate details regarding the chemical environment). A new class of method has emerged that combines the good points of QM (accuracy) and MM (speed) calculations. These methods are termed mixed or hybrid quantum-mechanical and molecular mechanics methods (hybrid QM/MM).
The most important advantage of hybrid QM/MM method is the speed.
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These methods are termed mixed or hybrid quantum-mechanical and molecular mechanics methods (hybrid QM/MM).
The most important advantage of hybrid QM/MM method is the speed. The cost of doing classical molecular dynamics (MM) in the most straightforward case scales O(n2), where n is the number of atoms in the system. This is mainly due to electrostatic interactions term (every particle interacts with every other particle). However, use of cutoff radius, periodic pair-list updates and more recently the variations of the particle-mesh Ewald's (PME) method has reduced this to between O(n) to O(n2). In other words, if a system with twice as many atoms is simulated then it would take between two and four times as much computing power. On the other hand, the simplest ab initio calculations typically scale O(n3) or worse (restricted Hartree–Fock calculations have been suggested to scale ~O(n2.7)). To overcome the limit, a small part of the system is treated quantum-mechanically (typically active-site of an enzyme) and the remaining system is treated classically.
In more sophisticated implementations, QM/MM methods exist to treat both light nuclei susceptible to quantum effects (such as hydrogens) and electronic states.
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To overcome the limit, a small part of the system is treated quantum-mechanically (typically active-site of an enzyme) and the remaining system is treated classically.
In more sophisticated implementations, QM/MM methods exist to treat both light nuclei susceptible to quantum effects (such as hydrogens) and electronic states. This allows generating hydrogen wave-functions (similar to electronic wave-functions). This methodology has been useful in investigating phenomena such as hydrogen tunneling. One example where QM/MM methods have provided new discoveries is the calculation of hydride transfer in the enzyme liver alcohol dehydrogenase. In this case, quantum tunneling is important for the hydrogen, as it determines the reaction rate.
### Coarse-graining and reduced representations
At the other end of the detail scale are coarse-grained and lattice models. Instead of explicitly representing every atom of the system, one uses "pseudo-atoms" to represent groups of atoms. MD simulations on very large systems may require such large computer resources that they cannot easily be studied by traditional all-atom methods. 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.
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