## Tuesday, August 31, 2010

### The Statistical Postulate

The Statistical Postulate (Idea 6.4) states that:

"When an isolated system is left alone long enough, it evolves to thermal equilibrium. Equilibrium is not one particular microstate. Rather, it's that probability distribution of microstates having the greatest possible disorder allowed by the physical constraints on the system."

Many of the ideas in thermodynamics leave you with a sense of wondering how anything exciting can happen if a system is at equilibrium. However if you study a subsystem such as a single molecule the fluctuations that occur can represent a distribution of states, of which some will be high energy.

This leaves us with an idea that, at equilibrium, the total energy does not fluctuate because there are no more inputs into the system. However energy can still be transferred within and between subsystems at equilibrium.

1. figure 6.2 describes your quote really well. we see that at the point where the energies of the two systems are equal (that is, they have equilibrated their temperatures) that is when the sum of entropies is at its greatest.
On a note of the fluctuations, we actually find that there is technically flux occuring, just on a minimal scale as energy leaves the system and as energy at the same time enters the system.

2. This is all true, and emphasizes the point that the "states" of statistical mechanics are probability distributions.

A few weeks ago I mentioned in passing that the equilibrium canonical distribution functions are examples of a more general class of probability distributions called exponential families (http://en.wikipedia.org/wiki/Exponential_family).

One very USEFUL aspect of exponential families is that the multiplier in the exponent (i.e. the temperature, for the canonical distribution) is a SUFFICIENT STATISTIC. This means that knowing the statistic tells you everything to know about the probability distribution. In the context of statistical mechanics and thermodynamics, this is the mathematical embodiment of the statement that the thermodynamic parameters (temp., pressure or volume, particle number or chemical potential) are sufficient to uniquely specify the equilibrium state! If the equilibrium state did not have the form it does, then this would not be true, and you would need more information than the thermodynamic parameters to completely specify the distribution.

This is also why one can say that equilibrium has no MEMORY. Again, if the system had memory, you would need to specify more information than just the thermodynamic parameters (e.g. you might have to specify what they were at some previous moment of time...). You don't have to do this for the exponential distributions -- this embodies the idea that the equilibrium state is PATH INDEPENDENT.