Relativistic Brownian motion: From a microscopic binary collision model to the Langevin equation

Dunkel, Jörn; Hänggi, Peter

doi:10.1103/physreve.74.051106

Cited by 26 publications

(53 citation statements)

References 56 publications

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“…From a mathematical perspective, SDEs [74] determine well-defined models of stochastic processes; from the physicist's point of view, their usefulness for the description of a real system is a priori an open issue. Therefore, the derivation of nonrelativistic Langevin equations from microscopic models has attracted considerable interest over the past sixty years [21,[90][91][92][93][94][95][96]. Efforts in this direction not only helped to clarify the applicability of SDEs to physical problems but led, among others, also to the concept of quantum Brownian motion [92,[97][98][99][100][101][102][103][104][105][106][107][108][109][110].…”

Section: Historical Backgroundmentioning

confidence: 99%

“…As an alternative to non-Markovian diffusion models in spacetime, one can consider relativistic Markov processes in phase space [11][12][13][14][15][16][17][18][19][20][21][22]24,26,[31][32][33][34][220][221][222][332][333][334][335][336][337][338]. Typical examples are processes described by Fokker-Planck equations (FPEs) or Langevin equations [11][12][13][14][15]17,18,20,21,[24][25][26][31][32][33][34][391][392][393][394][395].…”

Section: Relativistic Markov Processes In Phase Spacementioning

confidence: 99%

“…A complementary approach towards relativistic stochastic processes in phase space starts from Langevin equations [11][12][13][14][15]17,18,20,21,[24][25][26]31,32,[391][392][393][394][395]. Stochastic differential equations of the Langevin type yield explicit sample trajectories for the stochastic motion of a relativistic Brownian particle.…”

Section: Relativistic Markov Processes In Phase Spacementioning

confidence: 99%

“…• One constructs relativistically acceptable Markov processes in phase space by considering not only the position coordinate X (t) of the diffusing particle, but also its momentum coordinate P(t) [11][12][13][14][15][16][17][18][19][20][21][22]24,26,31,32,[220][221][222][332][333][334][335][336][337][338].…”

Section: Relativistic Diffusion Processes: Problems and General Stratmentioning

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: Historical Backgroundmentioning

confidence: 99%

Section: Relativistic Markov Processes In Phase Spacementioning

confidence: 99%

Section: Relativistic Markov Processes In Phase Spacementioning

confidence: 99%

Section: Relativistic Diffusion Processes: Problems and General Stratmentioning

confidence: 99%

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Relativistic Brownian motion

2009

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“…[23][24][25]). In two recent papers [26,27] we have discussed in detail how one can construct Langevin equations for relativistic Brownian motions (see Debbasch et al [28,29] and Zygadlo [30] for similar approaches, and also Dunkel and Ha¨nggi [31]). Thereby, it was demonstrated that, in general, the relativistic Langevin equation per se cannot uniquely determine the corresponding Fokker-Planck equation (FPE).…”

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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A new look at Bell's inequalities and Nelson's theorem

Schulz¹

2009

Ann. Phys.

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In 1985, Edward Nelson, who formulated the theory of stochastic mechanics, made an interesting remark on Bell's theorem. Nelson analysed the latter in the light of classical fields that behave randomly. He found that if a stochastic hidden variable theory fulfils certain conditions, the inequality of Bell can be violated. Moreover, Nelson was able to prove that this may happen without any instantaneous communication between the two spatially separated measurement stations. Since Nelson's article got almost overlooked by physicists, we try to review his comments on Bell's theorem. We argue that a modification of stochastic mechanics published recently by Fritsche and Haugk can be extended to a theory which fulfils the requirements of Nelson's analysis. The article proceeds to derive the quantum mechanical formalism of spinning particles and the Pauli equation from this version of stochastic mechanics. Then, we investigate Bohm's version of the EPR experiment. Additionally, other setups, like entanglement swapping or time and position correlations, are shortly explained from the viewpoint of our local hidden-variable model. Finally, we mention that this theory could perhaps be relativistically extended.Comment: The stochastic model presented in this article was defect. The parts of this work that are correct are now in a separate article on arxiv.or

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Relativistic Brownian motion: From a microscopic binary collision model to the Langevin equation

Cited by 26 publications

References 56 publications

Relativistic Brownian motion

Relativistic Brownian motion

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

A new look at Bell's inequalities and Nelson's theorem

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