Is there a systematic way to determine the relevant variables needed to describe a nonequilibrium system?

Question

In strong nonequilirium, the statistical operator describing the system depends on an infinite number of variables (BBGKY-hierarchy), contains information about all the previous states starting from an initial condition $\rho(t_0) = \rho_{rel}(t_0)$

$$ \rho(t) = \frac{1}{1-t_0}\int\limits_{t_0}^t \exp^{i(t_1-t)L}\rho_{rel}(t_1)dt_1 $$

and satisfies the inhomogenous Neumann equation

$$ \frac{\partial\rho(t)}{\partial t} + iL\rho(t) = -\epsilon(\rho(t)-)\rho_{rel}(t) $$

However, to describe the macroscopic state of a system at each time by appropriate observables

$$ \langle B_n(t) \rangle = Tr\{\rho_{rel}(t)B_n\} $$

it is often enough to use only the relevant (known) information contained in the relevant statistical operator, which can be obtained by maximizing the entropy and using in addition to the conserved quantities the mean values of additional variables as constraints

$$ \rho_{rel}(t) = \exp^{- \Phi(t)-\sum F_n(t)B_n} $$

where

$$ \Phi(t) = \ln Tr \left( \exp^{-\sum F_n(t)B_n} \right) $$

is the Messieux-Planck function.

After reading about some different applications of this MaxEnt-formalism, determining what are the appropriate relevant observables to determine the state of a nonequilibrium system looked often unsatisfactorally heuristic and handwaving to me.

So my question is:

Is there a general systematic method, at best motivated by some "first principles", to obtain the relevant variables needed to describe the relevant variables needed to describe the evolution of a nonequilibrium system?

A probably very stupid aside: the evolution of a system far away from equilibrium with many degrees of freedom needed to describe it towards its equilibrium state characterized by the conserved quantities (or their conjugate variables) only, remainds me of the coarse graining needed to describe a system at an effective scale and therefore renormalization comes to mind, not sure if there is a relationship between these two things or not ...

Comments

I think that relevance is completely arbitrary. You start by looking at something and you want to know how it evolves through time. In your context the meaning of "relevant " is very different from the one in renormalization I think. Look at people in protein folding , they don't look always at the same macrovariables and yet they are interested in only the folding of a protein.

This post imported from StackExchange Physics at 2015-09-29 08:06 (UTC), posted by SE-user gatsu

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I think that relevance is completely arbitrary. You start by looking at something and you want to know how it evolves through time. In your context the meaning of "relevant " is very different from the one in renormalization I think. Look at people in protein folding , they don't look always at the same macrovariables and yet they are interested in only the folding of a protein.

This post imported from StackExchange Physics at 2014-03-09 16:11 (UCT), posted by SE-user gatsu

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I don't think that your question has a simple answer. The greatest difficulty with out-of-equilibrium dynamics is that we don't really know what we are looking for. Observables can be relevant or not depending on the questions that you are asking.

This post imported from StackExchange Physics at 2015-09-29 08:06 (UTC), posted by SE-user Steven Mathey

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About your Renormalisation Group insight: You are not alone. Check out these papers: http://arxiv.org/abs/0710.4627, http://arxiv.org/abs/1001.0098, http://arxiv.org/abs/1212.2117v1. At some point during the time evolution, all three of them switch time for RG scale.

This post imported from StackExchange Physics at 2015-09-29 08:06 (UTC), posted by SE-user Steven Mathey

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Thanks @StevenMathey for these useful pointers!

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I am not sure I get it. The information on the initial conditions is not related to the relevant variables? Otherwise, although I am not sure it answers your question, some people are trying to define in a systematic way reaction coordinates. Peter Bohluis is one of them.

This post imported from StackExchange Physics at 2014-03-09 16:11 (UCT), posted by SE-user gatsu

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@gatsu as far as I understand it, the system is completely described by the known relevant variables at the initial time, but in the course of time they can become irrelevand and/or new ones can become relevant.

This post imported from StackExchange Physics at 2014-03-09 16:11 (UCT), posted by SE-user Dilaton

Answer 1

The correct set of variables doesn't depend on the initial conditions but on the time scale.

At a given time scale one can take as constant all observables that don't change noticeably during the time span of interest, and can average over all degrees of freedom that oscillate much faster. This leaves the degrees of freedom that change noticeably but oscillate only little. These need to be taken into account in the maximum entropy principle.

An exception to this rule is the case where there is an external periodic driving force (e.g., monochromatic laser light), which needs to be taken into account even if it has a much higher frequency than the time scale of interest.

As for the identification of these modes: Typically there are three levels of modeling: equilibrium, hydromechanic, and kinetic. Equilibrium considers only the long term behavior with constant boundary conditions. The hydromechanic regime treats spatial distributions as variable, and is the one relevant for most everyday applications. The kinetic regime treats phase space distributions as variable, which is relevant if the momenta do not equilibrate fast enough (e.g., in a plasma and in semiconductors). In quantum field theoretic terms, all these apart from the energy density are 1-particle operators.