# In Monte Carlo: does nonequilibrium imply stationary state?

I'm currently responding to referees about a manuscript on the dynamics of the Ising model. I have been using the term nonequilibrium to refer to any state that is not characteristic of the equilibrium distribution of that system. For example, the simulation gets stuck (due to the dynamics) in a state far from the ground state. Basically nonequilibrium = not equilibrium.

A referee complained that nonequilibrium states are stationary states stabilized by some external driving force.

Who is right? And is there an authoritative definition that I could cite somewhere?

This is actually a tricky question. First your use of "non equilibrium" is incorrect. Without more information on your MC simulations, especially on the applied biases and simulation process, one cannot state if you are in an out-of-equilibrium (OOE) regime (not state) or not. The way you define it as a state far from the ground state can be a "local" equilibrium in which the system is trapped. This can be, for example, a glassy state obtained from quenching (and yes I know the definition of glass as OOE states is a blurred field). In such a case, the system does not need an external force to be kept far from the ground state (because of the energy barrier it needs a really long time to pass) and it is actually not "out-of-equilibrium" in a thermodynamic sense. It is just a very unlikely state of the equilibrium distribution that would eventually relax using standard techniques (in an infinitely long time, depending on your MC moves).

The referee's definition you give is related to the regime of your whole system. If your system is in a steady-state with a bias or a force which maintains it in such a state, then you are in a truly OOE regime. Removing the bias is supposed to let the system free to go back in an equilibrium state, even a local one (not especially the ground state).

I think you should refer clearly to the source of your unlikely configurations and clarify this point in your manuscript.

Sidenote: As an example, some MC simulations use a so-called "local equilibrium" technique to simulate OOE regimes at the atomic/spin level: https://www.sciencedirect.com/science/article/pii/037843719390463E

• Thanks! Can you help me find the right word for what I need to say? Basically, I'm discussing states that the MC becomes stuck in after a quench. I was using the word nonequilibrium because I wanted to always make it clear that I was referring to behavior that is the result of the quench process (i.e. not the equilibrium behavior). These states are local energy minima that perhaps could be described as a "local equilibrium." Is there a more general term I could use? – taciteloquence May 18 at 10:52
• Since your systems are trapped far from equilibrium after quenching I guess you could simply call them "quenched systems" or "glassy states". The idea of quenching is that you will have disordered systems in conditions where you should have more ordered states so maybe it is superfluous to precise that they are disordered. As I said, I don't know you paper and work so in the end it's up to you. – G.Clavier May 18 at 14:06

No, nonequilibrium doesn't imply stationarity, but, at the sime time, using "nonequilibrium state" to denote those local energy minima can indeed be confusing, as it might seem to refer to nonequilibrium steady states, which is what your referee is probably getting to.

One possible name for these local minima (besides "local minima") could be "non-equilibrated states" or even "non-asymptotic states", but I don't think that's standard, so if used it should be defined.