# Markov process and equilibrium statistical mechanics

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Are all systems studied under equilibrium statistical mechanics going through nothing but reversible Markov process?

Let me elaborate, When we study the basic of Statistical Mechanics... All of the systems are considered heurestically ,however, the theory of Markov chain doesn't consider the physical nature of the system but, considers the basic essence of any system and tries to explain its temporal behaviour...when it does this it encounters things like Reducible Markov chain which I feel is closely connected to the idea of ergodicity...stationarity which is closely connected to equilibrium. I asked the question to clarify if all the processes studied by "equilibrium" stat mech can also be explained by Markov chains?

recategorized Nov 21, 2017

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No.

Classical statistical mechanics usually approximates the classical N-body dynamics by dissipative stochastic equations for macroscopic variables, which are either of the form of master equations (generalizing Markov chains to an infinite number of states) or of the form of stochastic differential equations. In both cases we end up with Markov processes. Their stationary states lead to equilibrium thermodynamics.

Quantum statistical mechanics usually approximates the quantum N-body or quantum field dynamics by dissipative stochastic equations for measurable variables which are only sometimes classical Markov processes. In general they have rather the form of quantum stochastic processes, which are generalizations of classical Markov processes to the case of noncommuting quantities.

However, more refined models lead in both cases to non-Markovian classical or quantum stochastic dynamics with memory. Usually the memory last for a very short time only and hence can be neglected. But for some materials the memory is long-living enough to be of industrial value.

Useful references are books by Accardi et al. (Quantum theory and its stochastic limit, 2013), Breuer & Petruccione (The theory of open quantum systems, 2002). or Gardiner and Zoller (Quantum noise,  2004).

answered Nov 21, 2017 by (14,537 points)

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