I got a question I\'m not sure how to state precisely or is it even valid. Any h

Question

I got a question I'm not sure how to state precisely or is it even valid. Any help is most welcomed.

I stripped the question of all details because I wanted to emphasize my problem, but should someone think they would bring any clarity (it is a solid state problem) I'll present them.

Ok, let say I have two interacting systems. One of them is a system (S1) in a thermodynamical equilibrium and the other is a well defined classical system (S2). I know how to derive S2 from microcanonical state of S1 and how surrounding of a S1 depends on S2. It is very unclear how to combine these two mathematically but here is a kick - I THINK that S2 is changing much more slowly than S1. So, I was thinking of a iterative approach: to run a Monte Carlo to solve S1, then derive S2, then adjust conditions of S1 based on the new state of S2 and rerun MC etc. So my questions would be: is this approach valid if I assume that S1 is changing adiabatically? Is there a practical way to verify adiabatic change? Is there any circumstance where calculation like this is valid? It feels that if the S2 can't kick S1 out of equilibrium, then I got a powerful edge to clear this problem up - but is this true?

Explanation / Answer

I'm not exactly an expert on this, but here goes.

When you say "One of them is a system (S1) in a thermodynamical equilibrium", you are already assuming that the equilibration time of S1 is much faster than the time required for S2 to change significantly. I don't know what this assumption is based on, but the equilibration time of S1 (by itself) can certainly be found out. For example, if S1 is the gas in a room and S2 is a door opening, then how long does it take for the gas to respond to the motion of the door? Well, at the very least, it takes as long as the propagation time of a pressure wave travelig across the room. That's just a lower bound of course. In a simulation, you can try, for example, moving all the gas particles out of some area in the simulation, and see how long it takes for the system to bring the local density back up to normal.

If it is true that S1 is in thermodynamic equilibrium with the instantaneous configuration of S2, then it is possible (as you propose) to do a time-evolution of S2, where in each time-step, S1 is in a random configuration drawn from its equilibrium distribution. To get the right answer, you would need to make sure that in the time required for S2 to do anything significant, you have gone through so many time-steps that S1 has thoroughly sampled its ensemble of configurations. I do not believe this is a wise approach. Instead, I think it's better to calculate the free energy of S1 as a function of the configuration of S2. Take the derivative of that function in order to model the effects of S1 as a deterministic classical force acting on S2.

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