Re: Max-entropy (cont)

["Rauber, Joel Phys" <RAUBERJ@mg.sdstate.edu>]

David, you have probably convinced me that I don't want to consider a 
canonical ensemble, as that might be viewed as implying an infinite heat 
reservoir,
My point is that statistical mechanics is intended to provide a basis for 
thermodynamics and is statistical in nature!

E.g. Consider two finite systems systems A and B in thermal contact.  In 
principle one can write the entropy S as S(U_A) , where U_A is the internal 
energy of A.  Statistical mechanics predicts that there are fluctuations in 
the energy of system A around the mean <U_A>.  These fluctuations are 
inherent in the macrostate.  The equilibrium state of the system does not 
just consist of the system staying at the maximum value of S(U_A); it 
includes the fluctuations of the system around the mean energy.

For finite systems, there are finite probabilities of large fluctuations; 
although for "large" systems the probabilities of large fluctuations become 
exceedingly small; to the point of being negligible in practice; the reason 
I don't worry about all the air molecules in my office congregating over in 
the corner. But one can apply statistical mechanics to "small" systems and 
"medium" systems.

________________________________________

> The entropy of a system in equilibrium remains constant -- as do some other
> macroscopic thermodynamic parameters.  When a system is in equilibrium its
> macrostate is time invariant.

Less important point:

Hard sphere ideal gas in a constant volume container which has been isolated 
from its environment for a long time.  Presumably it is in an equilibrium 
macrostate given by the thermodynamic coordinates P,V and T  (only two of 
which need to be specified).

What are the statistical mechanics underpinnings of this state.  Take 
Pressure P.

presumably the pressure is related to the impulse delivered to the walls of 
the container when the spheres collide with the walls.  I would maintain 
that it is only the average value of this impulse that is constant in time; 
and only if averaged over some suitably long period of time.  Otherwise it 
is a varying quantity when I look carefully at the microscopic details.

________________________________

Entropy of mixing:  Gas A and Gas B originally separated by a removable 
partition.  Remove the partition and wait along time T_o.

Continue to observe the system.  The 2nd law at the thermodynamic level says 
that the gases will never unmix, or even partially unmix.

Looked at in terms of the statistical situation, there is a finite 
probability that they will unmix.

Reducto ad absurdum example:
Gas A is two molecules in 1/2 liter, Gas B is two molecules in 1/2 Liter, 
 apply the procedure above; the probability is rather large that the system 
may find itself in the unmixed state after T_o.


Joel