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It is practically impossible to model a macroscopic physical system in terms of the microscopic kinematical and dynamical variables of all its particles. Thus one makes a hierarchical reduction in which this complexity is reduced to a small number of collective variables. The theoretical framework for such reductions for systems is statistical mechanics or statistical physics.

One special case of hierarchical reduction is the limit of large volumes #VV, in which the number of particles (of each species) per volume, N/VN/V, stays constant. This is called the thermodynamic limit in statistical physics. Under some standard assumptions like homogeneity (spacial and possibly directional) and stability (no transitory effects), there is a small number of collective variables characterizing the system. Such a description can be (and historically was) postulated as an independent self-consistent phenomenological theory even without going into the details of statistical mechanics; such a description is called equilibrium thermodynamics, which is believed to be deducible from statistical mechanics, as has been partially proved for some classes of systems. Sometimes transitional finite-time phenomena are described either statistically by studying stochastic processes or by a more elaborate hierarchical form of thermodynamics, so-called nonequilibrium thermodynamics.

One of the basic characteristics of a thermodynamical system is its temperature, which has no analogue in fundamental non-statistical physics. Other common thermodynamical variables include pressure, volume, entropy, enthalpy etc.




Mathematically rigorous treatments:

See also

In terms of symplectic geometry (Souriau)

A covariant formalization of thermodynamics in terms of moment maps in symplectic geometry is due to

Review includes

The Souriau model of thermodynamics has been extented for “geometric science of information” (Koszul information geometry) with a general definition of Fisher metric, Euler-Poincaré equation and variational definition of Souriau thermodynamics, as in:

Irreversible thermodynamics

A survey of irreversible thermodynamics is in

For more on this see also rational thermodynamics.

Relativistic thermodynamics

Making sense of thermodynamics when taking into account special relativity and ultimately, possibly, general relativity (gravity) is notoriously subtle (even ignoring the issue of Bekenstein-Hawking entropy).

Shortly after the advent of the relativity theory, Planck, Hassenoerl, Einstein and others advanced separately a formulation of the thermodynamical laws in accordance with the special principle of relativity. This treatment was adopted unchanged including the first edition of this monograph. However it was shown by Ott and indepently by Arzelies, that the old formulation was not quite satisfactory, in particular because generalized forces were used instead of the true mechanical forces in the description of thermodynamical processes.

The papers of Ott and Arzelies gave rise to many controversial discussions in the literature and at the present there is no generally accepted description of relativistic thermodynamics.

(quote from Moller, The theory of relativity, 1952)

A standard textbook has been

but Tolman’s approach has been called into question, see e.g.

See also

Further generalizations

Some formal generalizations of thermodynamical formalism include mixing time and temperature in formalisms with complex time-temperature like Matsubara formalism in QFT.

Mathematical analogies and generalizations include also