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The microstates of a system are said to be unobservable. I can introduce the entropy as a measure of the number of microstates, which lead to the same macroscopic variables. So in this detailed definition of entropy, there are two theories at work, where one can be viewed as approximation to the other. If I introduce an energy function of the macroscopic system, then there is heat too. Dissipation in the big system can microscopically understood in the small system.


I formulated the above intro rather general, without reference to any particular system.

I guess I can sort a quantum mechanics description underneath the classical one, I can sort quantum field theory description underneath certain quantum mechanical systems, and I can even semi-consider deeper theories in renormalization procedures. However, all the models I know which consider problems involving heat are usually fluids.

Can I, in a similar way as macroscopic fluid dynamics and statistical mechanics is related, introduce dissipative systems in the microscopic system and explain it in terms of even lower level systems? And if it's possible to introduce two levels of dissipation in that way (three theories at work), how are the first and the third related?

To formulate an aspect of the second part more practically: If I have a thermodynamical entropy, then I don't know the absolute value. If I construct a microscopic description I can get such a value. If I now construct another microscopic description, then I'll get another value for the same macroscopic entropy, right?


I never encountered the microscopic-macroscopic entropy and heat viewpoint as such developed as a general meta theory or mathematical theory, independend of a certain physical viewpoint of how the real world is made of.

What is left if I strip off explicit statistical systems (namely these we considered to be of physical relevance so far, especially particles and fields descibed by this and that physical differential equation) of these considerations?

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For a general meta theory or mathematical theory, independend of a certain physical viewpoint of how the real world is made of, see Chapter 10 of my book Classical and Quantum Mechanics via Lie algebras.

What you can get is values for two different definitions of entropy on two different level, not different values for the same macroscopic entropy.

You need to decide on a thermal description level (defined by the set of extensive variables relevant for the chosen means of observation), and each such level has objectively and uniquely associated its own entropy. Different description levels have different entropies.

(Of course, if you calculate the entropy from lower levels in different ways, you may get different values due to different approximations made, but if one were able to do all calculations without approximation, the results would be identical.)

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