Slow variables in thermodynamics

[tun]

After a while reading physicsforums.com from the shadows, I have decided to emerge and ask a question or two.

I was reading a review article on extended irreversible thermodynamics recently, but the EIT isn't relevant - it was something in the intro that caught my eye.

"The variables used in the macroscopic description [thermodynamics] are not arbitrary: they
are directly related to conserved quantities, namely, mass, momentum and energy. All
the other variables - diverse combinations of the positions and momenta of particles - are
not conserved and they decay very rapidly, in such a way that in a very short time
one is left with only the slow conserved variables."

How would one go about proving the statement about the decay of other quantities? Obviously, in the usual microscopic Hamiltonian description of the system we can identify conserved quantities, but does that imply that these are the only conserved quantities of the system (or the only ones that aren't combinations of the basic ones, energy etc)?

A related question which might not make sense: given the microscopic description of the system, is it possible to obtain the relevant macroscopic properties without referring to thermodynamics? e.g. not just calculate the temperature from microscopic properties, but identify it as a variable that characterises the macroscopic system in the first place, to the exclusion of those other "diverse combinations" of variables.

Any thoughts are welcome!

 

[Andy Resnick]

The emergence of macroscopic properties from a microscopic system description has not been demonstrated in any generality, AFAIK. It's a very important problem, especially in material science.

I'm not sure I totally agree with the quoted sentence- mixtures of materials are usually handled by allowing mass (of the constituents) to change, and are not conserved (by adding a chemical potential). I suspect the authors are starting with a Hamiltonian for the system, where H is written in terms of canonical pairs (positions and momenta)-but Hamiltonians can only be written for restricted classes of systems- conservative systems. I don't think it's possible to write a Hamiltonian for a fluid containing a shock wave or singular surface, for example.