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Entropy 2018, 20(7), 492; https://doi.org/10.3390/e20070492

Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation

Department of Environment, Health and Safety, Imec, 3001 Leuven, Belgium
Current address: Imec, Kapeldreef 75, 3001 Leuven, Belgium
Received: 15 April 2018 / Revised: 8 June 2018 / Accepted: 21 June 2018 / Published: 25 June 2018
(This article belongs to the Special Issue Emergent Quantum Mechanics – David Bohm Centennial Perspectives)
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Abstract

The present work is concerned with the study of a 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 the Langevin equation depends 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 for the drift. 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–Planck equation of the probability density of the differential system, which can be readily estimated from simulations of the random paths. Based on the Fokker–Planck formalism, a new derivation of the transient probability densities is presented. Finally, stochastic simulations are compared to the theoretical results. View Full-Text
Keywords: stochastic differential equations; Monte Carlo simulations; Burgers equation; Langevin equation; fractional velocity stochastic differential equations; Monte Carlo simulations; Burgers equation; Langevin equation; fractional velocity
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Prodanov, D. Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation. Entropy 2018, 20, 492.

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