Statistical mechanics of gravitating systems

Padmanabhan, Τ.

doi:10.1016/0370-1573(90)90051-3

Cited by 632 publications

(1,029 citation statements)

References 51 publications

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“…This is similar to the Boltzmann-Poisson equation appearing in astrophysics (in the statistical mechanics of stellar systems [27,38] and in isothermal models of stars [39]) and in vortex dynamics (in the statistical mechanics of 2D point vortices [35]). If we restrict ourselves to axisymmetric solutions, this equation can be solved analytically in d = 2.…”

Section: Chemotaxis Of Bacterial Populationsmentioning

confidence: 69%

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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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Section: Chemotaxis Of Bacterial Populationsmentioning

confidence: 69%

“…This equation of state (and its extension to finite N systems) has been obtained by various authors using different methods [25,26,27,13,14,28].…”

Section: The Smoluchowski-poisson Systemmentioning

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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“…This corresponds to the Kirkwood approximation in plasma physics. Another possibility is to use the Klimontovich approach and develop a quasilinear theory (see, e.g., [3] and Appendix B). A third possibility is to use a projection operator formalism, e.g.…”

Section: F the Landau Equationmentioning

confidence: 99%

“…Its derivation is classical but we shall need some intermediate steps in order to justify the expression (98) of the temporal correlation function. We follow the approach of Padmanabhan [3] but we consider an arbitrary potential of interaction and we take into account collective effects. We start from the Klimontovich equation …”

Section: Appendix B: the Lenard-balescu Equation And The Temporal Cormentioning

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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“…Therefore, non-ideal gas equations of state (as often assumed and used in the literature [8,9,14,16]) imply the presence of additional non-gravitational forces. Such non-ideal equations of state appear for quantum gases [2].…”

Section: Statistical Mechanics Of the Self-gravitating Gasmentioning

confidence: 99%

Statistical mechanics of the self-gravitating gas: II. Local physical magnitudes and fractal structures

2002

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We complete our study of the self-gravitating gas by computing the fluctuations around the saddle point solution for the three statistical ensembles (grand canonical, canonical and microcanonical). Although the saddle point is the same for the three ensembles, the fluctuations change from one ensemble to the other. The zeroes of the small fluctuations determinant determine the position of the critical points for each ensemble. This yields the domains of validity of the mean field approach. Only the S wave determinant exhibits critical points. Closed formulae for the S and P wave determinants of fluctuations are derived. The local properties of the self-gravitating gas in thermodynamic equilibrium are studied in detail. The pressure, energy density, particle density and speed of sound are computed and analyzed as functions of the position. The equation of state turns out to be locally p( r) = T ρ V ( r) as for the ideal gas. Starting from the partition function of the self-gravitating gas, we prove in this microscopic calculation that the hydrostatic description yielding locally the ideal gas equation of state is exact in the N = ∞ limit. The dilute nature of the thermodynamic limit (N ∼ L → ∞ with N/L fixed) together with the long range nature of the gravitational forces play a crucial role in obtaining such ideal gas equation. The self-gravitating gas being inhomogeneous, we have P V /[N T ] = f (η) ≤ 1 for any finite volume V . The inhomogeneous particle distribution in the ground state suggests a fractal distribution with Haussdorf dimension D, D is slowly decreasing with increasing density, 1 < D < 3. The average distance between particles is computed in Monte Carlo simulations and analytically 1 in the mean field approach. A dramatic drop at the phase transition is exhibited, clearly illustrating the properties of the collapse.

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Statistical mechanics of gravitating systems

Cited by 632 publications

References 51 publications

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

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

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

Statistical mechanics of the self-gravitating gas: II. Local physical magnitudes and fractal structures

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