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New hierarchy for the Liouville equation, irreversibility and Fokker‐Planck‐like structures
Author(s) -
AlvarezEstrada R.F.
Publication year - 2002
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 - 10.1002/1521-3889(200205)11:5<357::aid-andp357>3.0.co;2-v
Subject(s) - bbgky hierarchy , ansatz , fokker–planck equation , physics , statistical physics , distribution (mathematics) , hierarchy , mathematical physics , distribution function , phase space , hermite polynomials , detailed balance , brownian motion , classical mechanics , mathematical analysis , mathematics , quantum mechanics , differential equation , economics , market economy
The issue of irreversibility is revisited for a closed system formed by N classical non‐relativistic particles inside a volume Ω , interacting through two‐body potentials, for large N and Ω . The classical phase‐space distribution function f , multiplied by suitable Hermite polynomials and integrated over all momenta, yields new moments. The Liouville equation and the initial distribution f in imply a new non‐equilibrium linear infinite hierarchy for the moments. That hierarchy differs from the BBGKY one for distribution functions and displays some suggestive Fokker‐Planck‐like structures. A physically motivated ansatz for f in (which introduces statistical assumptions), used by previous authors, is chosen. All moments of order n ≥ n 0 are expressed in terms of those of order n 0 — 1 and of f in . The properties of the Fokker‐Planck‐like structures (hermiticity, non‐negative eigenvalues) allow for implementing a natural long‐time approximation in the hierarchy, so as to introduce relaxation to equilibrium and irreversibility, consistently with the hydrodynamical balance equations. Further (more restrictive) assumptions and approximations lead to new irreversible models, generalizing non‐trivially the Fokker‐Planck equation. They are described through a truncated hierarchy of linear equations for moments of order n ≤ n 0 — 1 ( n 0 being finite). The connections with Brownian particle dynamics and Fluid Dynamics are analyzed, for consistency.