Canonical Nonequilibrium Statistics and Applications to Fermi-Bose Systems

Ebeling,

doi:10.5488/cmp.3.2.285

Cited by 18 publications

(25 citation statements)

References 16 publications

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“…With regard to our subsequent discussion of relativistic Brownian motions, it will be important to keep in mind that the nonlinear Langevin equations like Eq. (5b) provide a tool for constructing Brownian motion processes with arbitrary stationary velocity and momentum distributions [19,425].…”

Section: Einstein Fluctuation-dissipation Relationmentioning

confidence: 99%

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: Einstein Fluctuation-dissipation Relationmentioning

confidence: 99%

Relativistic Brownian motion

2009

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“…In general, these systems can be defined by the non-holonomic constraint, which is suggested in [11]. (2) The Fermi-Bose canonical-dissipative systems [13] which are defined by the distribution functions in the form…”

Section: Definitionmentioning

confidence: 99%

“…then we have the canonical non-Hamiltonian systems, which are considered in [12,13]. These systems are called canonical dissipative systems.…”

Section: Non-canonical Distributionsmentioning

confidence: 99%

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Stationary solutions of Liouville equations for non-Hamiltonian systems

Tarasov

2005

Annals of Physics

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We consider the class of non-Hamiltonian and dissipative statistical systems with distributions that are determined by the Hamiltonian. The distributions are derived analytically as stationary solutions of the Liouville equation for non-Hamiltonian systems. The class of non-Hamiltonian systems can be described by a non-holonomic (non-integrable) constraint: the velocity of the elementary phase volume change is directly proportional to the power of non-potential forces. The coefficient of this proportionality is determined by Hamiltonian. The constant temperature systems, canonical-dissipative systems, and Fermi-Bose classical systems are the special cases of this class of non-Hamiltonian systems.

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“…In this context, the concept of canonical-dissipative systems has been developed to address self-propagating systems, self-excited systems, and in particular limitcycle oscillators that are supplied by energy sources [1][2][3][4][5]. Several studies have demonstrated that the canonical-dissipative approach can yield helpful insights into this kind of open systems [6][7][8][9][10][11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space

Mongkolsakulvong¹

2010

Condens. Matter Phys.

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We consider limit cycle oscillators in terms of canonical-dissipative systems that exhibit a short-range interaction in velocity and position space as described by the Dirac delta function. We derive analytical expressions for stationary distribution functions in phase space and energy space and propose a numerical simulation scheme for the simulation of the many body system as well. We show that the short-range interaction squeezes or stretches energy distribution functions depending on whether the interaction can be regarded as attractive or repulsive. In addition to the interaction effect, we show that energy distribution functions become narrower when limit cycle attractors become stronger. Finally, energy distributions become broader when the pumping energy is increased. The latter effect however disappears in the high energy domain.

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Canonical Nonequilibrium Statistics and Applications to Fermi-Bose Systems

Cited by 18 publications

References 16 publications

Relativistic Brownian motion

Relativistic Brownian motion

Stationary solutions of Liouville equations for non-Hamiltonian systems

Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space

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