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

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 ...

• Is the nonequilibirum system here means open system, or a close system not yet reach the equilibrium? Aug 14 '13 at 2:23
• 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. Aug 14 '13 at 13:12
• @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. Aug 14 '13 at 13:21
• 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. Sep 27 '15 at 21:45
• 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 tome point during the time evolution, all three of them switch time for RG scale. Sep 27 '15 at 21:47