Thermodynamics of self-gravitating systems

Chavanis, Pierre‐Henri; Rosier, Carole; Sire, Clément

doi:10.1103/physreve.66.036105

Cited by 160 publications

(425 citation statements)

References 32 publications

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“…(24) we see that the instability criteria (24) and (29) coincide as announced. We now turn to the nonlinear dynamical stability problem.…”

Section: E Nonlinear Dynamical Stabilitysupporting

confidence: 50%

“…Finally, we note that the criteria (34) and (35) are the same as (29) and (24). This implies that, for homogeneous systems, the conditions of linear and nonlinear dynamical stability coincide.…”

Section: E Nonlinear Dynamical Stabilitymentioning

confidence: 76%

“…The evolution of the distribution function of these Brownian particles is described by non-local Fokker-Planck equations which are the canonical counterpart of the kinetic equations governing the evolution of Hamiltonian systems. For example, a gas of self-gravitating Brownian particles [24] is governed by the Kramers-Poisson and Smoluchowski-Poisson systems. These equations decrease the Boltzmann free energy at fixed mass and temperature.…”

Section: Introductionmentioning

confidence: 99%

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

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

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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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“…(24) we see that the instability criteria (24) and (29) coincide as announced. We now turn to the nonlinear dynamical stability problem.…”

Section: E Nonlinear Dynamical Stabilitysupporting

confidence: 50%

Section: E Nonlinear Dynamical Stabilitymentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

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

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

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204

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“…Finally, in Sec. 6, we provide a summary of the results obtained by Chavanis & Sire [6,7,8,9,10,11,12,13,14] for selfgravitating Brownian particles and adapt them to the context of chemotaxis to clearly show the link between these two problems.…”

Section: Introductionmentioning

confidence: 99%

“…In a series of papers, Chavanis & Sire [6,7,8,9,10,11,12,13,14] have studied a model of self-gravitating Brownian particles in various dimensions of space. In statistical mechanics, this model is associated with the canonical ensemble in which the temperature is fixed.…”

Section: Introductionmentioning

confidence: 99%

Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions

Chavanis¹

2007

Physica A: Statistical Mechanics and its Applications

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We show that the critical mass M c = 8π of bacterial populations in two dimensions in the chemotactic problem is the counterpart of the critical temperature T c = GM m/4k B of self-gravitating Brownian particles in two-dimensional gravity. We obtain these critical values by using the Virial theorem or by considering stationary solutions of the Keller-Segel model and Smoluchowski-Poisson system. We also consider the case of one dimensional systems and develop the connection with the Burgers equation. Finally, we discuss the evolution of the system as a function of M or T in bounded and unbounded domains in dimensions d = 1, 2 and 3 and show the specificities of each dimension. This paper aims to point out the numerous analogies between bacterial populations, self-gravitating Brownian particles and, occasionally, two-dimensional vortices.

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Self‐attracting Fermi–Dirac particles in canonical and microcanonical setting

Stańczy

2005

Math Methods in App Sciences

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Thermodynamics of self-gravitating systems

Cited by 160 publications

References 32 publications

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

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

Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions

Self‐attracting Fermi–Dirac particles in canonical and microcanonical setting

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