Mean-field treatment of the many-body Fokker–Planck equation

Martzel, Nicolas; Aslangul, Claude

doi:10.1088/0305-4470/34/50/305

Cited by 31 publications

(36 citation statements)

References 13 publications

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“…By applying a mean field approximation scheme to a many component Brownian gas [153,154], as worked out in several contexts in Refs. [155][156][157][158][159][160][161], Kramers system of equations can readily be generalized to…”

Section: Applicationsmentioning

confidence: 99%

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

Lagos

Simões

2011

Physica A: Statistical Mechanics and its Applications

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a b s t r a c tWe consider a charged Brownian gas under the influence of external and non-uniform electric, magnetic and mechanical fields, immersed in a non-uniform bath temperature. With the collision time as an expansion parameter, we study the solution to the associated Kramers equation, including a linear reactive term. To the first order we obtain the asymptotic (overdamped) regime, governed by transport equations, namely: for the particle density, a Smoluchowski-reactive like equation; for the particle's momentum density, a generalized Ohm's-like equation; and for the particle's energy density, a Maxwell-Cattaneo-like equation. Defining a nonequilibrium temperature as the mean kinetic energy density, and introducing Boltzmann's entropy density via the one particle distribution function, we present a complete thermohydrodynamical picture for a charged Brownian gas. We probe the validity of the local equilibrium approximation, Onsager relations, variational principles associated to the entropy production, and apply our results to: carrier transport in semiconductors, hot carriers and Brownian motors. Finally, we outline a method to incorporate non-linear reactive kinetics and a mean field approach to interacting Brownian particles.

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Section: Applicationsmentioning

confidence: 99%

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

Lagos

Simões

2011

Physica A: Statistical Mechanics and its Applications

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“…We shall use and generalize the method of Martzel & Aslangul [39,40]. We start from the general Markov process…”

Section: A the Non-local Kramers Equationmentioning

confidence: 99%

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

204

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We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the FokkerPlanck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N → +∞, a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of the Vlasov equation. To order 1/N , the collision term of a homogeneous system has the form of the Lenard-Balescu operator. It reduces to the Landau operator when collective effects are neglected. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard-Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N → +∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ → +∞, or for large times t ≫ ξ −1 , it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, two-dimensional vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker-Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.

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“…In this context, the friction is due to the presence of an inert gas and the stochastic force is due to classical Brownian motion, turbulence, or any other stochastic effect. Starting from the N-body Fokker-Planck equation and using a mean-field approximation [25,26], we can derive the nonlocal Kramers equation…”

Section: Analogy Between Self-gravitating Brownian Particles and mentioning

confidence: 99%

Postcollapse dynamics of self-gravitating Brownian particles and bacterial populations

Sire¹,

Chavanis²

2004

Phys. Rev. E

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We address the postcollapse dynamics of a self-gravitating gas of Brownian particles in D dimensions in both canonical and microcanonical ensembles. In the canonical ensemble, the postcollapse evolution is marked by the formation of a Dirac peak with increasing mass. The density profile outside the peak evolves self-similarly with decreasing central density and increasing core radius. In the microcanonical ensemble, the postcollapse regime is marked by the formation of a "binarylike" structure surrounded by an almost uniform halo with high temperature. These results are consistent with thermodynamical predictions in astrophysics. We also show that the Smoluchowski-Poisson system describing the collapse of self-gravitating Brownian particles in a strong-friction limit is isomorphic to a simplified version of the Keller-Segel equations describing the chemotactic aggregation of bacterial populations. Therefore, our study has direct applications in this biological context.

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Mean-field treatment of the many-body Fokker–Planck equation

Cited by 31 publications

References 13 publications

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Postcollapse dynamics of self-gravitating Brownian particles and bacterial populations

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