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{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}

{{main|Mechanics|Statistical ensemble (mathematical physics)|l2=Statistical ensemble}}

 

In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:

In physics, two types of mechanics are usually examined: [[classical mechanics]] and [[quantum mechanics]]. For both types of mechanics, the standard mathematical approach is to consider two concepts:

== Statistical thermodynamics ==

== Statistical thermodynamics ==

 

The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the [[classical thermodynamics]] of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in [[thermodynamic equilibrium]], and the microscopic behaviours and motions occurring inside the material.

The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the [[classical thermodynamics]] of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in [[thermodynamic equilibrium]], and the microscopic behaviours and motions occurring inside the material.

=== Fundamental postulate ===

=== Fundamental postulate ===

 

A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).<ref name="gibbs" />

A [[sufficient condition|sufficient]] (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.).<ref name="gibbs" />

{{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}

{{main|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}

 

There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.<ref name="gibbs"/> These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.

There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.<ref name="gibbs"/> These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.

=== Calculation methods ===

=== Calculation methods ===

 

Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.

Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.

====Exact====

====Exact====

 

There are some cases which allow exact solutions.

There are some cases which allow exact solutions.

{{main|Monte Carlo method}}

{{main|Monte Carlo method}}

 

One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.

One approximate approach that is particularly well suited to computers is the [[Monte Carlo method]], which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.

==== Other ====

==== Other ====

 

* For rarefied non-ideal gases, approaches such as the [[cluster expansion]] use [[perturbation theory]] to include the effect of weak interactions, leading to a [[virial expansion]].<ref name="balescu" />

* For rarefied non-ideal gases, approaches such as the [[cluster expansion]] use [[perturbation theory]] to include the effect of weak interactions, leading to a [[virial expansion]].<ref name="balescu" />

{{see also|Non-equilibrium thermodynamics}}

{{see also|Non-equilibrium thermodynamics}}

 

There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:

There are many physical phenomena of interest that involve quasi-thermodynamic processes out of equilibrium, for example:

=== Stochastic methods ===

=== Stochastic methods ===

 

One approach to non-equilibrium statistical mechanics is to incorporate [[stochastic]] (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from [[Black hole information paradox|hypothetical situations involving black holes]], a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as [[Chaos theory|chaotic]] or [[pseudorandom]] influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

One approach to non-equilibrium statistical mechanics is to incorporate [[stochastic]] (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from [[Black hole information paradox|hypothetical situations involving black holes]], a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as [[Chaos theory|chaotic]] or [[pseudorandom]] influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

=== Near-equilibrium methods ===

=== Near-equilibrium methods ===

 

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in [[linear response theory]]. A remarkable result, as formalized by the [[fluctuation-dissipation theorem]], is that the response of a system when near equilibrium is precisely related to the [[Statistical fluctuations|fluctuations]] that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.<ref name="balescu"/>{{rp|664}}

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in [[linear response theory]]. A remarkable result, as formalized by the [[fluctuation-dissipation theorem]], is that the response of a system when near equilibrium is precisely related to the [[Statistical fluctuations|fluctuations]] that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium.<ref name="balescu"/>{{rp|664}}

=== Hybrid methods ===

=== Hybrid methods ===

 

An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ([[weak localization]], [[conductance fluctuations]]) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic [[dephasing]] by interactions between various electrons by use of the Keldysh method.<ref>{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid =  | pmc = |bibcode = 1982JPhC...15.7367A }}</ref><ref>{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid =  | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}</ref>

An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ([[weak localization]], [[conductance fluctuations]]) in the conductance of an electronic system is the use of the Green-Kubo relations, with the inclusion of stochastic [[dephasing]] by interactions between various electrons by use of the Keldysh method.<ref>{{Cite journal | last1 = Altshuler | first1 = B. L. | last2 = Aronov | first2 = A. G. | last3 = Khmelnitsky | first3 = D. E. | doi = 10.1088/0022-3719/15/36/018 | title = Effects of electron-electron collisions with small energy transfers on quantum localisation | journal = Journal of Physics C: Solid State Physics | volume = 15 | issue = 36 | pages = 7367 | year = 1982 | pmid =  | pmc = |bibcode = 1982JPhC...15.7367A }}</ref><ref>{{Cite journal | last1 = Aleiner | first1 = I. | last2 = Blanter | first2 = Y. | doi = 10.1103/PhysRevB.65.115317 | title = Inelastic scattering time for conductance fluctuations | journal = Physical Review B | volume = 65 | issue = 11 | pages = 115317 | year = 2002 | pmid =  | pmc = |arxiv = cond-mat/0105436 |bibcode = 2002PhRvB..65k5317A | url = http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a }}</ref>

== History ==

== History ==

 

In 1738, Swiss physicist and mathematician [[Daniel Bernoulli]] published ''Hydrodynamica'' which laid the basis for the [[kinetic theory of gases]]. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as [[heat]] is simply the kinetic energy of their motion.<ref name="uffink"/>

In 1738, Swiss physicist and mathematician [[Daniel Bernoulli]] published ''Hydrodynamica'' which laid the basis for the [[kinetic theory of gases]]. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as [[heat]] is simply the kinetic energy of their motion.<ref name="uffink"/>