Relativistic diffusions and Schwarzschild geometry

Franchi, Jacques; Jan, Yves Le

doi:10.1002/cpa.20140

Cited by 72 publications

(102 citation statements)

References 17 publications

(12 reference statements)

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“…Last but not least, notwithstanding recent progress [395,495,498,[535][536][537][538][538][539][540], the generalization of stochastic concepts and their applications within the framework of general relativity offers many interesting challenges for the future. 80 This can be achieved, for example, by choosing the integral boundaries asã − = (x − x 0 ) 2 /(t − t 0 ) α , α = 1 and a + = ∞, but then the variable a may not be interpreted as a conventional action anymore.…”

Section: Discussionmentioning

confidence: 99%

“…The present section describes the generalization of the Langevin theory of Brownian motions to the framework of special relativity [11][12][13][14][15]17,18,[20][21][22][24][25][26][31][32][33][34]394,395,475,495]. More precisely, we will consider stochastic differential equations (SDEs) that describe Markov processes in relativistic one-particle phase space.…”

Section: Relativistic Brownian Motion Processes In Phase Spacementioning

confidence: 99%

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Relativistic Brownian motion

2009

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a b s t r a c tOver the past one hundred years, Brownian motion theory has contributed substantially to our understanding of various microscopic phenomena. Originally proposed as a phenomenological paradigm for atomistic matter interactions, the theory has since evolved into a broad and vivid research area, with an ever increasing number of applications in biology, chemistry, finance, and physics. The mathematical description of stochastic processes has led to new approaches in other fields, culminating in the path integral formulation of modern quantum theory. Stimulated by experimental progress in high energy physics and astrophysics, the unification of relativistic and stochastic concepts has re-attracted considerable interest during the past decade. Focusing on the framework of special relativity, we review, here, recent progress in the phenomenological description of relativistic diffusion processes. After a brief historical overview, we will summarize basic concepts from the Langevin theory of nonrelativistic Brownian motions and discuss relevant aspects of relativistic equilibrium thermostatistics. The introductory parts are followed by a detailed discussion of relativistic Langevin equations in phase space. We address the choice of time parameters, discretization rules, relativistic fluctuationdissipation theorems, and Lorentz transformations of stochastic differential equations. The general theory is illustrated through analytical and numerical results for the diffusion of free relativistic Brownian particles. Subsequently, we discuss how Langevin-type equations can be obtained as approximations to microscopic models. The final part of the article is dedicated to relativistic diffusion processes in Minkowski spacetime. Since the velocities of relativistic particles are bounded by the speed of light, nontrivial relativistic Markov processes in spacetime do not exist; i.e., relativistic generalizations of the nonrelativistic diffusion equation and its Gaussian solutions must necessarily be non-Markovian. We compare different proposals that were made in the literature and discuss their respective benefits and drawbacks. The review concludes with a summary of open questions, which may serve as a starting point for future investigations and extensions of the theory.

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Section: Discussionmentioning

confidence: 99%

Section: Relativistic Brownian Motion Processes In Phase Spacementioning

confidence: 99%

Relativistic Brownian motion

2009

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“…The implementation of the Brownian motion concept [1][2][3][4][5][6] into special relativity [7,8] represents a longstanding issue in mathematical and statistical physics (classical references are [9][10][11]; more recent contributions include [12][13][14][15][16][17][18][19][20][21][22]; for a kinetic theory approach, see Refs. [23][24][25]).…”

Section: Introductionmentioning

confidence: 99%

One-dimensional non-relativistic and relativistic Brownian motions: a microscopic collision model

Dunkel

Hänggi

2007

Physica A: Statistical Mechanics and its Applications

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We study a simple microscopic model for the one-dimensional stochastic motion of a (non-)relativistic Brownian particle, embedded into a heat bath consisting of (non-)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the non-relativistic case, the integral criteria do correctly identify the Maxwellian momentum distributions as stationary (invariant) solutions. Subsequently, we apply the same criteria to the relativistic case. Surprisingly, we find here that the stationary momentum distributions differ slightly from the standard Ju¨ttner distribution by an additional prefactor proportional to the inverse relativistic kinetic energy. r

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“…As the main result, an analytic expression for the transition PDF is proposed in Eq. (23). The simple relativistic diffusion process defined by this PDF avoids superluminal propagation, and reduces to the Wiener process (2) in the nonrelativistic domain.…”

Section: Discussionmentioning

confidence: 99%

“…This problem has attracted considerable interest during the past 100 years (see, e.g., [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and references therein). Nonetheless, it seems fair to say that a commonly accepted solution is still outstanding.…”

Section: Introductionmentioning

confidence: 99%

Relativistic diffusion processes and random walk models

2007

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The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As well-known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (non-continuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the non-relativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.

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Relativistic diffusions and Schwarzschild geometry

Cited by 72 publications

References 17 publications

Relativistic Brownian motion

Relativistic Brownian motion

One-dimensional non-relativistic and relativistic Brownian motions: a microscopic collision model

Relativistic diffusion processes and random walk models

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