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I guess this question is more about definition than about any physical principle. Can a given physical system (or can there be a physical system) admit microstates that have identical macroscopic variables, but that evolve into states with different macroscopic variables?

I would think it is obvious that something like this can happen, think of a chaotic system, which may be in a state that is macroscopically indistinguishable from one that will remain stable, but that after some time evolves into a system with e.g. large macroscopic local pressure differences, due to arbitrarily small changes in the microstate (and you could set this up in such a way that you get into a new equilibrium state).

The reason why this might be a question of definition is that two ways in which the answer could be "no" would be:

  1. by saying that microstates that evolve into different macrostates were not indistinguishable to begin with
  2. by saying that a system that can get out of equilibrium was not in equilibrium to begin with.

Since this knowledge can not be inferred from any macroscopic information at this moment in time (or possibly even any information about the system), these solutions do not seem to be very satisfactory, but maybe I am mistaken.

In any case, I am interested in the answer to the question "Can indistinguishable microstates evolve into different macroscopic states?"

doetoe
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2 Answers2

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I'll offer one answer, I'm sure there will be others.

Yes, certainly this is true, there can be such systems. For any statistical mechanical system at specified temperature or pressure, there is an implied interaction with the surroundings. The interaction is usually assumed to be weak (and I believe that the old tome "The Principles of Statistical Mechanics" by RC Tolman goes into this in detail). The microscopic evolution of the surroundings is never specified in detail, so effectively this provides a stochastic (or at least unpredictable) element to the time evolution of the system.

Hence, unpredictable (in the sense of not being deterministic) fluctuations occur. A simple example is the Ising ferromagnet, below the critical point. The total magnetization $M$ is a macroscopic parameter. For any given system, assuming finite size, there will be a finite probability per unit time of crossing the free energy barrier between $+M$ and $-M$ states. At low temperatures these crossings will be infrequent, and different systems (indistinguishable, to all intents and purposes) will undergo the transitions at different times. At any moment, some of the systems will be in one macroscopic state, and some in the other.

Similar arguments apply to any phase in which the underlying symmetry has been broken, to give degenerate equilibrium macroscopic states.

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Can indistinguishable microstates evolve into different macroscopic states?

In classical mechanics, yes. A very small change in initial conditions can, after enough time, send the system to completely distant parts of the phase space. E.g. one microstate can be on a trajectory to sustained entropy increase towards long-lasting equilibrium, and a very close but different microstate can, after a long enough time, cause the system to deviate from this typical behaviour, and then start decreasing entropy.

In quantum mechanics, the Schroedinger evolution of quantum state is linear, so while that evolution is obeyed, chaos and its sensitivity to initial condition cannot happen. However, in quantum theory, we also have the other type of evolution, "collapse of quantum state", which is sometimes needed. This is not linear evolution, and is not deterministic, so it can result in very different resulting quantum states, not implied by the state before.

Ján Lalinský
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