# Minimization of energy for non-equilibrium systems at steady state (NESS)?

Suppose a non-equilibrium system at steady state. Does the steady state corresponds to the state of some minimal "energy-like", like in classical statistical physics?

Example with the Ising model. Suppose a lattice with spins (up or down) on each site. Each spin interact with its nearest neighbours such that each spin tend to have the same configuration than its neighbours. Now suppose one regularly choose one spin randomly, and flip it. This process consumes energy and the system is now at non equilibrium and cannot be described by classical statistical physics any more. How could you describe the steady state?

What books usually call "thermo-dynamics" is really "thermo-statics", the time variable is missing. There are all kinds of steady states and ever since Kirchhoff (~160 years) people have been trying to find some extremum principle to describe these. Kirchhoff showed that stationary electric current is distributed so that the dissipation is minimized and derived Ohm's law as the corresponding Euler-Lagrange equation. The results are correct for isothermal conduction. Later Rayleigh tried to generalize the minimum dissipation concept. Next Prigogine tried to derive steady state thermo- dynamics from his minimum entropy production principle. This has generated an enormous literature, but its practical significance is still being debated. The dream is, of course, to get an extremum principle for thermodynamics that could be used for non-equilibrium steady or nonsteady states just as the Euler-Lagrange equations are useful in mechanics to describe dynamics. Aside from the scholastic argumentation that states can only exist in equilibrium, this subject has been a dream for 150 years, and it is still an active area of research, Prigogine did not (re)solve it.

• Yes, and @David, note that a lot of work has been done specifically for the type of stochastic dynamics that you seem to have in mind; see, e.g., this paper and papers citing the latter. Commented Nov 27, 2015 at 15:39
• Cheers. I spent a little time on non-equilibrium statistical physics and found incredibly hard to a have clear and synthetic view of the current knowledge... Commented Nov 27, 2015 at 17:35

Steady state, as can be seen from the meaning of the word "steady" is "steady" in some way. Usually, there are multiple properties of the system that become steady or constant in time. For a thermodynamic system, these properties can be energy, entropy, temperature, pressure, volume.

Then, it seems impossible to talk about a non-equilibirum system that is in steady state. A non-equilibrium system is by its definition not in a steady state. However, generally, if you let such system be (and external conditions do not change), it will tend to relax to a steady state.

• This is incorrect. There are of course nonequilibrium steady states (and these are studied so much that they have gained their own acronym: NESS). At equilibrium, there are no fluxes, no dissipation. In NESS, the system is submitted to a constant driving force (e.g., it is put in contact with two infinite reservoirs at different temperatures, or different chemical potentials, etc), which will induce fluxes in the system. What characterizes NESS among nonequilibrium states is that their statistical properties are time-invariant. Commented Nov 27, 2015 at 7:31
• It is simply a matter of definitions. There is no clear definition of a "non-equilibrium" system. That is, non-equilibrium based on what variable? In your case, yes, while the system will be experiencing some flux because of potential differences of the baths, it also needs to preserve some quantity to be deemed in a "steady state." In the same way, one can ask, steady state based on what variable? Energy? Temperature? etc. Commented Nov 27, 2015 at 23:18
• No, it's not "a matter of definition", except if you wish for every physicist to use its own set of definitions for everything... There is a very clear definition of equilibrium systems, to which NESS do not belong: equilibrium systems are systems with time-invariant statistical properties and no macroscopic flow of matter or energy. The distinction is not gratuitous: the formalism of equilibrium statistical physics does not apply to NESS (and the same is true of equilibrium thermodynamics). Commented Nov 28, 2015 at 9:04