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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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  • $\begingroup$ 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. $\endgroup$
    – Ron Maimon
    Jun 21, 2012 at 6:57
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    $\begingroup$ @Ron: This case is called local thermodynamic equilibrium. In local non-equilibrium case you cannot establish thermodynamics potentials at all. $\endgroup$ Jun 21, 2012 at 8:48
  • $\begingroup$ @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. $\endgroup$ Jun 21, 2012 at 13:08
  • $\begingroup$ @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. $\endgroup$ Jun 21, 2012 at 14:23
  • $\begingroup$ @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. $\endgroup$
    – Ron Maimon
    Jun 21, 2012 at 14:56

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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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  • $\begingroup$ The problem with that approach is that it is incompatible with irreversibility and it fails to give thermodynamically-valid expressions for non-conserved quantities. $\endgroup$
    – juanrga
    Jun 1, 2018 at 12:03
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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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There is no consistent and general formulation of non-equilibrium statistical mechanics because of three problems: (i) the lack of equilibrium constraints, (ii) irreversibility --the time-arrow problem--, and (iii) the lack of a companion thermodynamics theory.

(i) Equilibrium is unique and the constraints allow one to define an universally valid ensemble for any isolated system and use the Gibbs formalism. Outside equilibrium there is no single ensemble valid for the general case, so the traditional Gibbs approach fails and it has to be generalized in many ways.

(ii) Non-equilibrium phenomena is inherently irreversible, but Hamilton or Schrödinger equations are time-reversible. So there is a inconsistency. Contrary to a myth spread in part of the literature, it is not possible to derive irreversible equations of motion from reversible equations. What some non-rigorous people does is to give us pseudo-derivations. A pseudo-derivation implies starting from a time-reversible equation and then at some state to introduce an extra-dynamical assumption that breaks the time-reversibility and produces an irreversible equation. Besides the mathematical inconsistency of the whole approach (it is as assuming $x>5$ at the start of a derivation then using $x<5$ latter), the assumptions introduced are only valid for specific cases, and usually they are derived from phenomenology.

Some pseudoderivations are found in the monograph from Zwanzig: Nonequilibrium Statistical Mechanics. He gives pseudoderivations for some elementary equations as the Langevin and Pauli master equations. This monograph contains a really comical chapter titled The paradoxes of irreversibility where he tries to convince us that Universe is time-reversible and irreversibility we observe in Nature or in experiments is false because "What we know about irreversibility is obtained by experiments on a human time scale." Besides the invalid mathematics in that chapter and rest of the monograph, he doesn't seem to grasp that the time-reversible equations he uses to make bold claims such as "The ice cube will eventually reappear", have been also derived on the same time scale.

(iii) Many derivations in equilibrium statistical theory are guided by thermodynamic reasoning and physical values for statistical mechanics parameters are obtained by direct comparison with thermodynamic macroscopic expressions. E.g. when we associated Lagrange $\beta$ with thermodynamic temperature $T$. Everyone agrees on equilibrium thermodynamics of macroscopic systems. There is no agreement outside equilibrium. We have TIP (Thermodynamics of Irreversible Processes), which is only valid for states near equilibrium and not too fast processes. Then we have rational thermodynamics, which was created in response to TIP. Extended Irreversible Thermodynamics (EIT) was developed to extend TIP, without the flaws of rational thermodynamics. Recently some authors have tried to combine both rational and EIT on a new approach named Rational Extended Thermodynamics (RET), but not everyone agrees, and there are other approaches I do not mention here.


There are many approaches to try to build a consistent and general nonequilibrium statistical mechanics and its associated non-equilibrium thermodynamics. Three of them are more popular (in no particular order):

  1. The Russian school approach of Zubarev and coworkers. This approach starts from an irreversible generalization of the Liouville equation: The Zubarev equation

$$\frac{\partial \rho}{\partial t} = L \rho -i\epsilon \{\rho - \rho_\mathrm{rel}\} $$

and then consistently derives kinetic and transport equations and associated thermodynamics expressions. The $i\epsilon$ term is an infinitesimal source that breaks time-symmetry and produces the correct boundaries for the solutions.

This approach is described in many papers including the original two-volume book "Statistical Mechanics of Nonequilibrium Processes". A brief review of this approach with relevant literature is on this article.

  1. Eu approach. This approach also starts with an irreversible equation of motion: The Eu equation

$$\frac{\partial \rho}{\partial t} = L \rho + R[\rho]$$

where $R[\rho]$ is a multi-body collision term postulated by Eu to generalize both Schrödinger and Newtonian mechanics. There is no explicit form for this collision term in the general case. However, Eu gives a set of three conditions that $R[\rho]$ must satisfy in order to build a consistent nonequilibrium statistical mechanics and thermodynamics.

The approach is described in many papers as well. A comprehensive monograph is available: Nonequilibrium Statistical mechanics, which also describes associated non-equilibrium thermodynamics .

  1. The Brussels-Austin school approach of Prigogine and coworkers. This approach also starts from an irreversible generalization of dynamics, but instead extending the Liouville equation with irreversible terms (as in the above approaches), this school extends the functional space of solutions of the equation to include solutions $\tilde{\rho}=\tilde{\rho}(t)$ with broken time-symmetry

$$\frac{\partial \tilde{\rho}}{\partial t} = L \tilde{\rho}$$

This approach has evolved radically during decades, and their goal is not so 'modest' as the other two approaches (whose only goal is to formulate non-equilibrium statistical mechanics and consistent thermodynamics). The Austin-Brussels school also want to identify the origin of irreversibility and solve foundational problems such as the measurement problem in quantum mechanics. In their more recent work, the source of irreversibility are resonances in LPS (Large Poincaré Systems). An analysis of the evolution of this school in found in this paper and its continuation. There is no modern monograph I know, and recent ideas are spread over many publications. Basic resumes of the modern approach of this school are found in the review papers for classical and quantum systems, which only contain a minimal discussion of the application of the new formalism to nonequilibrium statistical mechanics topics. The early formalism and its application to statistical mechanics is found in the classic Non-equilibrium Statistical mechanics monograph by Prigogine.


All the three approaches I have mentioned are the state of the art in this topic and still open to objections and further generalizations. For instance the form of the relaxation kernel associated to the source term in the Zubarev equation and the choice of the relevant reference state $\rho_\mathrm{rel}$ is not well-justified yet, although the whole approach works in many applications. The collision term in Eu equation is derived from a direct generalization of the Boltzmann equation, but the Boltzmann equation lacks non-Markovian effects and fluctuations, so it is far from proved that the form of the Eu equation is general enough to describe any system and empirical setup. Finally, there are both mathematical and physical objections to the generalized space chosen by the Brussels-Austin school to solve the Liouville equation, and so on.

We are still far from formulating a general non-equilibrium statistical mechanics or even a generally valid and accepted non-equilibrium thermodynamics.

If you are only interested in basic applications relying on simple equations (Boltzmann, Pauli, Langevin, Fokker-Planck,...) and you do not care about rigor and consistency select the Zwanzig monograph. If you are really interested in foundational issues, including the origin of irreversibility, check the recent work of Prigogine and coworkers (although Prigogine's classic monograph discusses a good number of applications, the whole formalism was designed for foundational issues and it is, in my opinion, too difficult --too detailed-- for most practical applications). The monographs from Eu and Zubarev are a good intermediate point. They emphasize some foundational issues (e.g. why the Boltzmann equation cannot be really derived from time-reversible Newtonian equations) whereas provide a good number of applications of the formalisms to non-equilibrium systems.

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  • $\begingroup$ I think people are a little too negative when describing the status of non-equilibrium physics. Obviously, no physical theory that actually describes nature is mathematically rigorous. The fact of the matter is that there are first principles, parameter free (starting from the interaction of elementary particles) calculations of the conductivity of metals, the viscosity of fluids, and many other transport and non-equilibrium phenomena. $\endgroup$
    – Thomas
    Jun 2, 2018 at 17:40
  • $\begingroup$ .. and that these calculations agree with experiment. $\endgroup$
    – Thomas
    Jun 2, 2018 at 17:47
  • $\begingroup$ @Thomas No one is talking about mathematical rigor, but about correct physical models. Of course, there are models whose results agree with experiment in very specific (and usually simple settings). The problem is when those models don't give answers or give wrong results. The history of nonequilibrium physics is full of such problems, and some of them are mentioned, in the literature I cited, as motivation to why new formalisms and new models are needed. $\endgroup$
    – juanrga
    Jun 3, 2018 at 13:49
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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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