Trends to equilibrium for a class of relativistic diffusions

Angst, Jürgen

doi:10.1063/1.3659685

Cited by 6 publications

(4 citation statements)

References 27 publications

(44 reference statements)

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“…Many types of relativistic Ornstein–Uhlenbeck (OU) processes directly driven by a standard Wiener process have been proposed, and shown to converge to either Jüttner or modified-Jüttner stationary distributions, see Dunkel and Hänggi (2009) and Angst (2011) for references to this extensive literature.…”

Section: Relativistic Maxwell–boltzmann Distributionsmentioning

confidence: 99%

Central limit theorems under special relativity

McKeague

2015

Statistics & Probability Letters

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Several relativistic extensions of the Maxwell–Boltzmann distribution have been proposed, but they do not explain observed lognormal tail-behavior in the flux distribution of various astrophysical sources. Motivated by this question, extensions of classical central limit theorems are developed under the conditions of special relativity. The results are related to CLTs on locally compact Lie groups developed by Wehn, Stroock and Varadhan, but in this special case the asymptotic distribution has an explicit form that is readily seen to exhibit lognormal tail behavior.

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Section: Relativistic Maxwell–boltzmann Distributionsmentioning

confidence: 99%

Central limit theorems under special relativity

McKeague

2015

Statistics & Probability Letters

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“…We do not know whether the small temperature assumption is only a technical condition or a necessary physical one. However in [2], see also [4], it is shown that spatially homogeneous solutions converge exponential fast to equilibrium for all temperatures, although the result is proved not in…”

Section: −1mentioning

confidence: 99%

Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations

Calogero¹

2012

Communications in Partial Differential Equations

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A class of linear kinetic Fokker-Planck equations with a non-trivial diffusion matrix and with periodic boundary conditions in the spatial variable is considered. After formulating the problem in a geometric setting, the question of the rate of convergence to equilibrium is studied within the formalism of differential calculus on Riemannian manifolds. Under explicit geometric assumptions on the velocity field, the energy function and the diffusion matrix, it is shown that global regular solutions converge in time to equilibrium with exponential rate. The result is proved by estimating the time derivative of a modified entropy functional, as recently proposed by Villani. For spatially homogeneous solutions the assumptions of the main theorem reduce to the curvature bound condition for the validity of logarithmic Sobolev inequalities discovered by Bakry and Emery. The result applies to the relativistic Fokker-Planck equation in the low temperature regime, for which exponential trend to equilibrium was previously unknown.

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“…(3.18). The method may also be useful in the mathematical study of the spectral properties of relativistic diffusion [13].…”

Section: Discussionmentioning

confidence: 99%

Momentum relaxation of a relativistic Brownian particle

2012

Phys. Rev. E

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The momentum relaxation of a relativistic Brownian particle immersed in a fluid is studied on the basis of the Fokker-Planck equation for the relativistic Ornstein-Uhlenbeck process. An analytical expression is derived for the short-time relaxation rate. The relaxation spectrum has both discrete and continuum components. It is shown that the Fokker-Planck equation under consideration is closely related to the Schrödinger equation for the hydrogen atom. Hence it follows that there is an infinite number of discrete states. The momentum autocorrelation function is calculated numerically for a strongly relativistic particle.

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Trends to equilibrium for a class of relativistic diffusions

Cited by 6 publications

References 27 publications

Central limit theorems under special relativity

Central limit theorems under special relativity

Exponential Convergence to Equilibrium for Kinetic Fokker-Planck Equations

Momentum relaxation of a relativistic Brownian particle

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