Editing Statistical mechanics (section)

=== 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]].
}}