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Perturbation treatment of non‐linear transport via the Robertson statistical formalism
Author(s) -
Nettleton R.E.
Publication year - 1999
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/(sici)1521-3889(199905)8:5<425::aid-andp425>3.0.co;2-9
Subject(s) - dissipative system , physics , phase space , perturbation (astronomy) , liouville equation , mathematical physics , heat equation , convection–diffusion equation , statistical physics , mathematical analysis , classical mechanics , quantum mechanics , mathematics , thermodynamics , quantum
A perturbation solution is found for the differential equation defining an operator Tˆ used by Robertson to relate the information‐theoretic phase‐space distribution σ to the solution ρ of the classical Liouville equation. This relation provides a closure, used in obtaining an exact equation for σ . Multiplying the latter equation by F , a phase‐space function odd under momentum reversal, of which heat and diffusion fluxes are among the examples, one gets an exact equation for ∂ 〈 F 〉/ ∂ t . 〈 F 〉 is the phase space integral of ρF . The dissipative terms in ∂ 〈 F 〉/ ∂ t can be expanded, like Tˆ, in successive orders O (〈 F 〉 n ). For a model in which equilibrium ensemble fluctuations relax exponentially, terms linear and O (〈 F 〉 3 ) are calculated. The non‐linear terms exhibit an explicit time‐dependence for short times. In a steady state induced by external driving forces, the explicit time‐dependence disappears, in agreement with existing phenomenology. For simplicity, spatial uniformity is assumed. A generalization is required for large temperature or velocity gradients.