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A question kept bothering me about the Non-Equilibrium Statistical mechanics, can somebody give a simple description of how one approaches this subject. Is there a exact formalism, as we have for Equilibrium Statistical Mechanics, or is it some kind of an approximation. I would also like to know the promising efforts in this field.

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It's just where you allow the thermodynamic potentials to vary slowly in space, and treat them as external sources in the path integral for the partition function. This is an approximation for the true cause of the variation, and it works when there is local equilibrium, meaning there is a relaxation time and a relaxation length smaller than the time and space scales of the variation in the thermodynamic potentials. This is a big area in rigorous work, not so much in physics anymore. –  Ron Maimon Jun 21 '12 at 6:57
@Ron: This case is called local thermodynamic equilibrium. In local non-equilibrium case you cannot establish thermodynamics potentials at all. –  Alexey Bobrick Jun 21 '12 at 8:48
@AlexeyBobrick: In local nonequilibrium, one typically uses insted kinetic theory, where observables dapend on phase space coordinates. But for macroscopic nonequilibrium processes, local equilibrium is fully adequate. –  Arnold Neumaier Jun 21 '12 at 13:08
@Arnold: One might continue, and distinguish between different degrees of local non-equilibria. For example, if there are several particle species, some of them might be in local equilibrium at a given time, some not. It might be that some degrees of freedom are virialized, some not. In a local non-equilibrium case one may or may not apply one-particle approximation for the distribution function. Etc. Full local equlibrium is just an approximation, and there are many cases when is doesn't work, most notably in plasma physics, e.g. in astrophysic context. –  Alexey Bobrick Jun 21 '12 at 14:23
@AlexeyBobrick: I know what it's called. There is no other general theory of statistical mechanics outside of equilibrium, since there is no way to describe the general system's statistics unless you have some equilirium assumptions, or some extra dynamical assumptions. If you don't say "local equilibrium", the problem of statistics of systems includes describing the geometry of all dynamical strange attractors, all turbulent flows, and all biology, and it is obviously impossible to give a general theory. –  Ron Maimon Jun 21 '12 at 14:56

3 Answers 3

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There exist an exact formalism to treat non equilibrium statistical mechanics. You start to write down the Hamiltonian for the N interacting particles. Then you introduce the distribution function in the phase space $f(r_1,r_2...r_n,p_1,p_2,...p_n,t)$.The time evolution of this distribution function is generated by the Hamiltonian and more precisely by the poisson brackets: ${x_i,p_i};{x_i,H};{p_i,H}$. The time evolution equation for f is named Liouvillian. However beautifull this formalism is, it is completly equivalent to solving the motion equation for the N particles, that is to say, it is useless. So on reduces by 2N-1 integrations over $x_i,p_i$ the problem to a 1 particle distribution function. The reduction is exact but one finds that $f_1$ is coupled to $f_{12}$; $f_{12}$ is coupled to $f_{123}$ etc. (BBGKY hierarchy). There are different methods to stop the expansion and the resulting equation for the 1 particle distribution function is named differently depending on the problem: Vlasov's equation, Laundau's equation, Balescu's equation, Fokker-Planck's equation or Boltzmann's equation.

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The macroscopic theory of non-equilibrium physics is called fluid dynamics. This theory is analogous to thermodynamics for the equilibrium case -- there are no assumptions about microscopic degrees of freedom.

The theory that replaces classical statistical mechanics is classical kinetic theory, originally developed by Maxwell and Boltzmann.

The theory that corresponds to quantum statistical mechanics is non-equilibrium quantum (field) theory, also known as the Schwinger-Keldysh formalism.

BBGKY, in both the classical and the quantum case, is an attempt to derive an equation for the single particle distribution function. This is the Boltzmann equation in the classical case, and the Kadanoff-Baym equations in the quantum case.

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In principle, nonequilibrium statistical mechnaics is exact like quantum theory in general. But to do actual computations in realistic systems you need to resort to approximations.

For modern expositions, I'd recommend the books ''The theory of open quantum systems'' by Breuer and Petruccione and ''Beyond equilibrium thermodynamics'' by Öttinger.

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