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.

 

[astrokreng]

Hello and welcome to the forum!

The concept of slow variables in thermodynamics is an important one and is related to the idea of equilibrium and time scales. The statement that "diverse combinations of the positions and momenta of particles decay very rapidly" is a result of the fact that in a macroscopic system, there are many microscopic particles and interactions happening at the same time. However, not all of these interactions are relevant to the macroscopic behavior of the system. Instead, only a few macroscopic variables, such as mass, momentum, and energy, are conserved and govern the overall behavior of the system. This is because these variables are conserved and do not change over time, while other microscopic variables may fluctuate and decay rapidly.

As for proving this statement, it is a result of statistical mechanics and can be derived from the laws of thermodynamics. The macroscopic variables are related to conserved quantities, and the decay of other variables can be described using statistical mechanics and the concept of entropy. However, it is important to note that there may be other conserved quantities in a system that are not immediately apparent and may not be related to the macroscopic variables. These can be revealed through more detailed analysis and understanding of the system.

To your second question, it is possible to obtain the relevant macroscopic properties without referring to thermodynamics, but it would require a very detailed knowledge and understanding of the microscopic properties of the system. Thermodynamics provides a simplified and more practical approach to understanding the behavior of macroscopic systems, and it would be difficult to identify and analyze all the relevant microscopic variables without the framework of thermodynamics.

I hope this helps answer your questions. Keep exploring and asking questions, and don't be afraid to emerge from the shadows!