Editing Statistical mechanics (section)

== Non-equilibrium statistical mechanics ==
{{see also|Non-equilibrium thermodynamics}}

Many physical phenomena involve quasi-thermodynamic processes out of equilibrium, for example:
* [[Thermal conduction|heat transport by the internal motions in a material]], driven by a temperature imbalance,
* [[Electrical conduction|electric currents carried by the motion of charges in a conductor]], driven by a voltage imbalance,
* spontaneous [[chemical reaction]]s driven by a decrease in free energy,
* [[friction]], [[dissipation]], [[quantum decoherence]],
* systems being pumped by external forces ([[optical pumping]], etc.),
* and irreversible processes in general.
All of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)

In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as [[Liouville's theorem (Hamiltonian)|Liouville's equation]] or its quantum equivalent, the [[von Neumann equation]]. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. These ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's [[Gibbs entropy]] is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.

Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.

=== 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.

{{unordered list
|1 = ''[[Boltzmann transport equation]]'': An early form of stochastic mechanics appeared even before the term "statistical mechanics" had been coined, in studies of [[kinetic theory of gases|kinetic theory]]. [[James Clerk Maxwell]] had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. [[Ludwig Boltzmann]] subsequently showed that, by taking this [[molecular chaos]] for granted as a complete randomization, the motions of particles in a gas would follow a simple [[Boltzmann transport equation]] that would rapidly restore a gas to an equilibrium state (see [[H-theorem]]).

The Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the "interesting" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped [[semiconductor]]s (in [[transistor]]s), where the electrons are indeed analogous to a rarefied gas.

A quantum technique related in theme is the [[random phase approximation]].

|2 = ''[[BBGKY hierarchy]]'':
In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The [[BBGKY hierarchy]] (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.

|3 = ''[[Keldysh formalism]]'' (a.k.a. NEGF—non-equilibrium Green functions):
A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach is often used in electronic [[quantum transport]] calculations.

|4 = Stochastic [[Liouville's theorem (Hamiltonian)|Liouville equation]].
}}

=== 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}}

This provides an indirect avenue for obtaining numbers such as [[Ohm's law|ohmic conductivity]] and [[thermal conductivity]] by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation–dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.

A few of the theoretical tools used to make this connection include:
* [[Fluctuation–dissipation theorem]]
* [[Onsager reciprocal relations]]
* [[Green–Kubo relations]]
* [[Ballistic conduction#Landauer-Büttiker formalism|Landauer–Büttiker formalism]]
* [[Mori–Zwanzig formalism]]
* [[GENERIC formalism]]

=== 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 |title=Effects of electron-electron collisions with small energy transfers on quantum localisation |journal=Journal of Physics C: Solid State Physics |date=30 December 1982 |volume=15 |issue=36 |pages=7367–7386 |doi=10.1088/0022-3719/15/36/018 |bibcode=1982JPhC...15.7367A }}</ref><ref>{{cite journal |last1=Aleiner |first1=I. L. |last2=Blanter |first2=Ya. M. |title=Inelastic scattering time for conductance fluctuations |journal=Physical Review B |date=28 February 2002 |volume=65 |issue=11 |pages=115317 |doi=10.1103/PhysRevB.65.115317 |url=http://resolver.tudelft.nl/uuid:e7736134-6c36-47f4-803f-0fdee5074b5a |arxiv=cond-mat/0105436 |bibcode=2002PhRvB..65k5317A }}</ref>