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The Boltzmann Equation and Its Place in the Edifice of Statistical Mechanics

Charlotte Werndl¹, Roman Frigg²

¹Department of Philosophy, Faculty of Social Sciences, University of Salzburg, Franziskanergasse 1, 5020 Salzburg, Austria.

Entropy (Basel, Switzerland)

|January 8, 2025

Abstract:

It is customary to classify approaches in statistical mechanics (SM) as belonging either to Boltzmanninan SM (BSM) or Gibbsian SM (GSM). It is, however, unclear how the Boltzmann equation (BE) fits into either of these approaches. To discuss the relation between BE and BSM, we first present a version of BSM that differs from standard presentation in that it uses local field variables to individuate macro-states, and we then show that BE is a special case of BSM thus understood. To discuss the relation between BE and GSM, we focus on the BBGKY hierarchy and note the version of the BE that follows from the hierarchy is "Gibbsian" only in the minimal sense that it operates with an invariant measure on the state space of the full system.

Keywords:

BBGKY hierarchy Boltzmann equation Boltzmannian statistical mechanics Gibbsian statistical mechanics macro-state

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