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On the covariance of the Fokker-Planck equation

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  • Garrido, L.

Abstract

We study the covariance of the Fokker-Planck equation under general gross variables transformations by means of Riemann differential geometry using an affine connection Γϱμν unsymmetric in their lower indices and without assuming that the covariant derivative of the diffusion tensor Dμν;ϱ be zero. We come to the conclusion that to achieve our aim we only need the value of the contraction Γϱϱν, all other components of the affine connection remaining completely arbitrary. We argue, therefore, that the most economic way of presenting the covariance of the Fokker-Planck equation is by means of exterior differential calculus. As an application we study physical systems under detailed balance showing that for them the irreversible part of the contravariant drift vector, that is then uniquely determined, is related only to a symmetric tensor while its reversible component is exclusively related to an antisymmetric tensor. A criticism of a compact Fokker-Planck equation is also included.

Suggested Citation

  • Garrido, L., 1980. "On the covariance of the Fokker-Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 100(1), pages 140-152.
    • Handle: RePEc:eee:phsmap:v:100:y:1980:i:1:p:140-152
    DOI: 10.1016/0378-4371(80)90155-7
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