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The Einstein-Smoluchowski promeasure versus the Boltzmann(-Peierls) equation

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  • Banach, Zbigniew
  • Piekarski, Sławomir

Abstract

Beginning from the specific model of the Boltzmann-Peierls equation for the distribution function f(k, t), the time-dependent theory of fluctuations is developed from Einstein's inversion of Boltzmann's relation. The distribution function f(k, t) that satisfies the Boltzmann-Peierls equation is to be viewed as a random variable characterizing the macro-state of a gas of quasiparticles (phonons, magnons, rotons, etc.). The crucial assumption entering into the present approach is simply that the randomness in the statistical distribution of the initial values of f(k, t) can be characterized by the Einstein-Smoluchowski promeasure μϵ. It is convenient to think of μϵ as being the infinite-dimensional analog of Einstein's distribution law in the Gaussian approximation. The equilibrium promeasure μϵ is used to study the correlations in time of fluctuations in the moments of f(k, t). Some problems associated with a kinetic version of the Green-Kubo approach to transport processes are carefully studied. The exact formula for the thermal conductivity coefficient ϰT, which is derived in this paper, emerges as a portion of the general formalism.

Suggested Citation

  • Banach, Zbigniew & Piekarski, Sławomir, 1992. "The Einstein-Smoluchowski promeasure versus the Boltzmann(-Peierls) equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 180(3), pages 336-358.
    • Handle: RePEc:eee:phsmap:v:180:y:1992:i:3:p:336-358
    DOI: 10.1016/0378-4371(92)90394-6
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