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thermodynamics

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

One special case of hierarchial reduction is the limit of large volumes V, in which the number of particles (of each species) N/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 deducable 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 hierarchial 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.

entropy, relative entropy, second law of thermodynamics, generalized second law, KMS state

References

General

A formalization in terms of symplectic geometry is in chapter IV “Statistical mechanics” of

  • Jean-Marie Souriau, Structure of dynamical systems. Asymplectic view of physics . Translated from the French by C. H. Cushman-de Vries. Translation edited and with a preface by R. H. Cushman and G. M. Tuynman. Progress in Mathematics, 149. Birkhäuser Boston, Inc., Boston, MA, 1997

as well as in

See also

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

  • John Baez, Mike Stay, Algorithmic thermodynamics, pdf, cafe
  • M. Marcolli, R. Thomgren, Thermodynamical semirings, arXiv/1108.2874
  • M. Zinsmeister, Thermodynamic formalism and holomorphic dynam- ical systems, Amer. Math. Soc. 2000.
  • I. Itenberg, G. Mikhalkin, Geometry in the tropical limit, arXiv/1108.3111

Revised on July 21, 2012 13:21:13 by Urs Schreiber (89.204.139.182)
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