Preprint Article Version 1 This version is not peer-reviewed

Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation

Version 1 : Received: 9 May 2018 / Approved: 10 May 2018 / Online: 10 May 2018 (06:11:01 CEST)

A peer-reviewed article of this Preprint also exists.

Prodanov, D. Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation. Entropy 2018, 20, 492. Prodanov, D. Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation. Entropy 2018, 20, 492.

Journal reference: Entropy 2018, 20, 492
DOI: 10.3390/e20070492

Abstract

The present work is concerned with the study of the generalized Langevin equation and its link to the physical theories of statistical mechanics and scale relativity. It is demonstrated that the form of the coefficients of Langevin equation depend critically on the assumption of continuity of the reconstructed trajectory. This in turn demands for the fluctuations of the diffusion term to be discontinuous in time. This paper further investigates the connection between the scale-relativistic and stochastic mechanics approaches, respectively, with the study of the Burgers equation, which in this case appears as a stochastic geodesic equation. By further demanding time reversibility of the drift the Langevin equation can also describe equivalent quantum-mechanical systems in a path-wise manner. The resulting statistical description obeys the Fokker-Plank equation of the probability density of the differential system, which can be readily estimated from Monte Carlo simulations of the random paths. Based on the Fokker-Plank formalism a new derivation of the transient probability densities is presented. Finally, stochastic simulations are compared to the theoretical results.

Subject Areas

monte-carlo simulations; burgers equation; langevin equation; fractional velocity

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