The Fundamental Assumption of Statistical Mechanics

[cdot]

Fundamental assumtption:
"a closed system is equally likely to be in any of its g accessible micro- states, and all accessible micro- states are assumed to be equally probable."

There's just a few things I don't understand about this,

1. Isn't saying that a closed system is equally likely to be in any of it's g accesible micro-states the exact same thing as saying that all accesible micro-states are equally probable? Why say both? Isn't that redundant?

2. Am I to assume that this definition applies only to closed systems in equlibrium throughout the system itself? For example, at a given total energy, volume, and number of particles, common sense says that a microstate corresponding to most of the energy of the molecules in this room being concentrated in some corner is NOT just as likely as the energy being distributed evenely. But according to the fundamental assumption, as long as the properties of the system are compatible with the system parameters then the micro-state is accesible and equally probable. So even if all the energy of the molecules in a room were concentrated in one corner, as long as the TOTAL energy, volume, and number of particles were compatible with system parameters, then this state is accesible (but it should not be equally probable?).

 

[matteo137]

1. No. If a state is accessible by the system it does not mean that the probability to find the system in that state is the same as finding it in an other one.

2. A gas of ideal particles, what are the micro states? they are the positions and the velocities of every single particle.

Am I to assume that this definition applies only to closed systems in equlibrium throughout the system itself?

If you start talking about thermal equilibrium than the micro states are probably not equally probable: you have to follow the Maxwell/Boltzmann distribution.

most of the energy of the molecules in this room being concentrated in some corner

is not a micro state.

 

[tom26]

You probably understand this by now, but to answer your second question, the reason it's very unlikely to be found in a state in which all of the atoms are in one corner is that there are (relatively) few of those states, compared to the number of homogeneous ones.

The simple analogy is tossing a million coins. Every sequence HHTTHTHTTHHT... is as likely as any other, but there are far more sequences which have, say 500,000 heads and 500,000 tails, than there are sequences which have 1,000,000 heads and no tails (there's only one of those). And so it's true that HHHHHHHH... is just as likely as any other sequence, but it's also true that you're far more likely to get 500,000 heads and 500,000 tails (in some order). There's no contradiction here.

On the other hand, the assumption in statistical mechanics, that all states are equally probable is a bit harder to justify (that's why we assume it :) ). It's related to the ergodic hypothesis (specifically, the ergodicity of the microcanonical ensemble - actually to some people this just is the ergodic hypothesis) which is fundamentally a very, very hard mathematical problem about measure-preserving maps. But some partial results from that field (e.g. Von Neumann's work) can give us confidence that it holds for most systems we'd expect it to (gases, liquids, etc.).