The entropy concept for non-equilibrium states

Elliott H. Lieb, Jakob Yngvason

Research output: Contribution to journalReview articlepeer-review

62 Scopus citations


In earlier work, we presented a foundation for the second law of classical thermodynamics in terms of the entropy principle. More precisely, we provided an empirically accessible axiomatic derivation of an entropy function defined on all equilibrium states of all systems that has the appropriate additivity and scaling properties, and whose increase is a necessary and sufficient condition for an adiabatic process between two states to be possible. Here, after a brief review of this approach, we address the question of defining entropy for non-equilibrium states. Our conclusion is that it is generally not possible to find a unique entropy that has all relevant physical properties. We do show, however, that one can define two entropy functions, called S- and S+, which, taken together, delimit the range of adiabatic processes that can occur between non-equilibrium states. The concept of comparability of states with respect to adiabatic changes plays an important role in our reasoning.

Original languageEnglish (US)
Article number20130408
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2158
StatePublished - Oct 8 2013

All Science Journal Classification (ASJC) codes

  • General Engineering
  • General Physics and Astronomy
  • General Mathematics


  • Entropy
  • Non-equilibrium thermodynamics
  • Second law of thermodynamics


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