# what measures a system's distance from equilibrium?

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Equilibrium is a difficult concept for me. I know several statistical ensembles or quantum states characterized entirely by a few thermodynamic quantities, eg. the canonical ensemble characterized entirely by the temperature via the Boltzmann weights. A system whose instantaneous macroscopic quantities, eg. pressure, are equal to a corresponding thermal average in some ensemble is said to be in equilibrium.

I'm interested in how systems approach equilibrium. Is there a quantity that measures "distance from equilibrium"?

In some sense entropy defines such a quantity, since it always increases until it is at a maximum at equilibrium. In another sense free energy defines such a quantity, since it is a minimum at equilibrium. The first I suppose is equilibrium defined relative to the microcanonical ensemble, where energy is fixed, while the second is defined relative to the canonical ensemble, which may exchange energy with a reservoir. Similarly the Gibbs free energy defines another kind of distance from equilibrium quantity. I am confused by the dependence on the ensemble here.

I found this wonderful nobel lecture by Ilya Prigogine http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1977/prigogine-lecture.pdf

He explains how AN equilibrium is characterized by A Lyapunov function (free energy functional) which decreases towards its corresponding equilibrium *within some region of the equilibrium*. Far from any equilibrium, these all compete with eachother and the equilibrium that one eventually attains can be very sensitive to these initial conditions (or boundary conditions when we're thinking about a system in contact with a thermodynamic reservoir).

asked Feb 15, 2016
edited Feb 15, 2016

I would say: we don't know! The conceptual foundations of non-equilibrium statistical mechanics are still very murky. One interesting new line of research is looking at concepts beyond entropy, like frenesy.

Thanks for the link, Ruben. I hadn't heard of this (: And yes I very much agree that our understanding is severely lacking. I'd just like to know what all is out there.

## 3 Answers

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Your question has many answers. I can not provide an overview, but instead offer one (maybe naive) answer. It is based on this work: http://arxiv.org/abs/1505.00912.

The generating functional of correlation functions of quantum systems can be expressed as a path integral weighted by a Schwinger-Keldysh action. This is true in and out of equilibrium. In http://arxiv.org/abs/1505.00912, thermal equilibrium is defined through a symmetry of the Keldysh action. A field transformation with the following properties is identified:

• The action that you get for systems that are at thermal equilibrium is invariant under it.
• If this transformation is a symmetry of the Keldysh action, then the corresponding Ward identities are all the fluctuation-dissipation relations.

You can say that being at thermal equilibrium is equivalent to being invariant under such a transformation.

Then, my first guess to evaluate the distance of a generic system from thermal equilibrium would be to apply the field transformation and evaluate how much the action changes. The bigger the change, the bigger the distance from thermal equilibrium.

Note that I have never actually tried this. There surely is a million technical problems to work out before it actually works. Moreover, the paper concentrates on (non-)equilibrium steady states. I do not know if (and how) such a measure can be applied to a system that is time-dependent.

answered Feb 16, 2016 by (350 points)
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Entropy production (not entropy itself) is a good local measure of (not too large) deviation from equilibrium - it vanishes in equilibrium. I recommend the statistical physics book by Linda Reichl. She gives an excellent and thorough introduction into thermodynamics and its microscopic foundations.

The dependence on the ensemble is in more physical terms a dependence on the boundary conditions imposed on the system. Depending on the boundary conditions, equilibrium is approached in different ways. (Intuitively, dissipative systems are parabolic in nonrelativistic physics, and there is an associated initial-boundary value problem like for the heat equation, just more complex. The initial conditions are lost with time and if the boundary conditions are constant in the variables appropriate for the various ensembles, equilibrium is achieved with thermodynamic variables determined by these boundary conditions. The equilibrium state itself is determined by a few thermodynamic variables and independent of which constant boundary conditions are imposed (though not independent of their values).

answered Feb 16, 2016 by (15,787 points)
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It depends on specific variables you are interested in. Otherwise it is meaningless.

The "distance" can be the difference of a variable actual value and its equilibrium value, if an equilibrium is clearly defined in case of non-equilibrium state ;-). In any case this difference-like deviation must appear in some calculations to be meaningful.

EDIT: For example, when a system is out of equilibrium, but it does not experience external influences, it will tend to an equilibrium state. There may be many ways of describing the system "kinetics" in this case, depending on chosen variables. Another example, a system under permanent influence of external conditions, like a heat conduction of a plate. The system may be in a stationary state (a stationary temperature profile from $T_1$ to $T_2$), but it never arrives to the global equilibrium.

answered Jul 30, 2018 by (102 points)
edited Jul 30, 2018

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