Biomedical subjects

Pierre-Henri Chavanis

Publications and source records attributed to Pierre-Henri Chavanis.

13 recordsLinked to original sources

PubMed2006

Dynamics of the Bose-Einstein condensation: analogy with the collapse dynamics of a classical self-gravitating Brownian gas.

We consider the dynamics of a gas of free bosons within a semiclassical Fokker-Planck equation for which we give a physical justification. In this context, we find a striking similarity between the Bose-Einstein condensation in the canonical ensemble, and the gravitational collapse of a gas of classical self-gravitating Brownian particles. The paper is mainly devoted to the complete study of the Bose-Einstein "collapse" within this model. We find that at the Bose-Einstein condensation temperature Tc, the chemical potential mu(t) vanishes exponentially with a universal rate that we compute exactly. Below Tc, we show analytically that square root mu(t) vanishes linearly in a finite time t coll. After t coll, the mass of the condensate grows linearly with time and saturates exponentially to its equilibrium value for large time. We also give analytical results for the density scaling functions, for the corrections to scaling, and for the exponential relaxation time. Finally, we find that the equilibration time (above Tc) and the collapse time T coll(below Tc) both behave like -T -3 c ln|T-Tc|, near Tc.

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PubMed2006

Virial theorem and dynamical evolution of self-gravitating Brownian particles in an unbounded domain. I. Overdamped models.

We derive the virial theorem appropriate to the generalized Smoluchowski-Poisson (GSP) system describing self-gravitating Brownian particles in an overdamped limit. We extend previous works by considering the case of an unbounded domain and an arbitrary equation of state. We use the virial theorem to study the diffusion (evaporation) of an isothermal Brownian gas above the critical temperature Tc in dimension d = 2 and show how the effective diffusion coefficient and the Einstein relation are modified by self-gravity. We also study the collapse at T = Tc and show that the central density increases logarithmically with time instead of exponentially in a bounded domain. Finally, for d > 2, we show that the evaporation of the system is essentially a pure diffusion slightly slowed down by self-gravity. We also study the linear dynamical stability of stationary solutions of the GSP system representing isolated clusters of particles and investigate the influence of the equation of state and of the dimension of space on the dynamical stability of the system.

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PubMed2006

Virial theorem and dynamical evolution of self-gravitating Brownian particles in an unbounded domain. II. Inertial models.

We propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles in d dimensions. We go beyond the usual approximations by considering inertial effects and finite-N effects while previous works use a mean-field approximation valid in a proper thermodynamic limit (N --> +infinity) and consider an overdamped regime (xi --> +infinity). We recover known models in some particular cases of our general description. We derive the expression of the virial theorem for self-gravitating Brownian particles and study the linear dynamical stability of isolated clusters of particles and uniform systems by using techniques introduced in astrophysics. We investigate the influence of the equation of state, of the dimension of space, and of the friction coefficient on the dynamical stability of the system. We obtain the exact expression of the critical temperature Tc for a multicomponents self-gravitating Brownian gas in d = 2. We also consider the limit of weak frictions, xi --> 0, and derive the orbit-averaged Kramers equation.

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PubMed2005

Relaxation of the distribution function tails for systems described by Fokker-Planck equations.

We study the formation and the evolution of velocity distribution tails for systems with weak long-range interactions. In the thermal bath approximation, the evolution of the distribution function of a test particle is governed by a Fokker-Planck equation where the diffusion coefficient depends on the velocity. We extend the theory of Potapenko et al [Phys. Rev. E 56, 7159 (1997)] developed for power-law diffusion coefficients to the case of an arbitrary form of diffusion coefficient and friction force. We study how the structure and the progression of the front depend on the behavior of the diffusion coefficient and friction force for large velocities. Particular emphasis is given to the case where the velocity dependence of the diffusion coefficient is Gaussian. This situation arises in Fokker-Planck equations associated with one dimensional systems with long-range interactions such as the Hamiltonian mean field (HMF) model and in the kinetic theory of two-dimensional point vortices in hydrodynamics. We show that the progression of the front is extremely slow (logarithmic) in that case so that the convergence towards the equilibrium state is peculiar. Our general formalism can have applications for other physical systems such as optical lattices.

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PubMed2005

Self-gravitating Brownian systems and bacterial populations with two or more types of particles.

We study the thermodynamical properties of a self-gravitating gas with two or more types of particles. Using the method of linear series of equilibria, we determine the structure and stability of statistical equilibrium states in both microcanonical and canonical ensembles. We show how the critical temperature (Jeans instability) and the critical energy (Antonov instability) depend on the relative mass of the particles and on the dimension of space. We then study the dynamical evolution of a multicomponent gas of self-gravitating Brownian particles in the canonical ensemble. Self-similar solutions describing the collapse below the critical temperature are obtained analytically. We find particle segregation, with the scaling profile of the slowest collapsing particles decaying with a nonuniversal exponent that we compute perturbatively in different limits. These results are compared with numerical simulations of the two-species Smoluchowski-Poisson system. Our model of self-attracting Brownian particles also describes the chemotactic aggregation of a multi-species system of bacteria in biology.

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PubMed2004

Statistical mechanics and thermodynamic limit of self-gravitating fermions in D dimensions.

We discuss the statistical mechanics of a system of self-gravitating fermions in a space of dimension D. We plot the caloric curves of the self-gravitating Fermi gas giving the temperature as a function of energy and investigate the nature of phase transitions as a function of the dimension of space. We consider stable states (global entropy maxima) as well as metastable states (local entropy maxima). We show that for D> or =4, there exists a critical temperature (for sufficiently large systems) and a critical energy below which the system cannot be found in statistical equilibrium. Therefore, for D> or =4, quantum mechanics cannot stabilize matter against gravitational collapse. This is similar to a result found by Ehrenfest (1917) at the atomic level for Coulomb forces. This makes the dimension D=3 of our Universe very particular with possible implications regarding the anthropic principle. Our study joins a long tradition of scientific and philosophical papers that examined how the dimension of space affects the laws of physics.

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PubMed2004

Postcollapse dynamics of self-gravitating Brownian particles and bacterial populations.

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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PubMed2004

Anomalous diffusion and collapse of self-gravitating Langevin particles in D dimensions.

We address the generalized thermodynamics and the collapse of a system of self-gravitating Langevin particles exhibiting anomalous diffusion in a space of dimension D. This is a basic model of stochastic particles in interaction. The equilibrium states correspond to polytropic configurations similar to stellar polytropes and polytropic stars. The index n of the polytrope is related to the exponent of anomalous diffusion. We consider a high-friction limit and reduce the problem to the study of the nonlinear Smoluchowacute;ski-Poisson system. We show that the associated Lyapunov functional is the Tsallis free energy. We discuss in detail the equilibrium phase diagram of self-gravitating polytropes as a function of D and n, and determine their stability by using turning point arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions of the nonlinear Smoluchowski-Poisson system describing the collapse. Our stability analysis of polytropic spheres can be used to settle the generalized thermodynamical stability of self-gravitating Langevin particles as well as the nonlinear dynamical stability of stellar polytropes, polytropic stars and polytropic vortices. Our study also has applications concerning the chemotactic aggregation of bacterial populations.

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PubMed2003

Generalized thermodynamics and Fokker-Planck equations: applications to stellar dynamics and two-dimensional turbulence.

We introduce a class of generalized Fokker-Planck equations that conserve energy and mass and increase a generalized entropy functional until a maximum entropy state is reached. Nonlinear Fokker-Planck equations associated with Tsallis entropies are a special case of these equations. Applications of these results to stellar dynamics and vortex dynamics are proposed. Our prime result is a relaxation equation that should offer an easily implementable parametrization of two-dimensional turbulence. Usual parametrizations (including a single turbulent viscosity) correspond to the infinite temperature limit of our model. They forget a fundamental systematic drift that acts against diffusion as in Brownian theory. Our generalized Fokker-Planck equations can have applications in other fields of physics such as chemotaxis for bacterial populations. We propose the idea of a classification of generalized entropies in "classes of equivalence" and provide an aesthetic connection between topics (vortices, stars, bacteria, em leader ) which were previously disconnected.

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PubMed2002

Thermodynamics and collapse of self-gravitating Brownian particles in D dimensions.

We address the thermodynamics and the collapse of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. We study the equilibrium density profile and phase diagram of isothermal spheres and, for 2<D<10, determine the onset of instability in the series of equilibria. We also study the dynamics of self-gravitating Brownian particles in a high friction limit leading to the Smoluchowski-Poisson system. Self-similar solutions describing the collapse are investigated analytically and numerically. In the canonical ensemble (fixed temperature), we derive the analytic form of the density scaling profile which decays as f(x) approximately x(-alpha), with alpha=2. In the microcanonical ensemble (fixed energy), we show that f decays as f(x) approximately x(-alpha(max)), where alpha(max) is a nontrivial exponent. We derive exact expansions for alpha(max) and f in the limit of large D. Finally, we solve the problem in D=2, which displays rather rich and peculiar features with, in particular, the formation of a Dirac peak in the density profile.

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PubMed2002

Thermodynamics of self-gravitating systems.

We study the thermodynamics and the collapse of a self-gravitating gas of Brownian particles. We consider a high-friction limit in order to simplify the problem. This results in the Smoluchowski-Poisson system. Below a critical energy or below a critical temperature, there is no equilibrium state and the system develops a self-similar collapse leading to a finite time singularity. In the microcanonical ensemble, this corresponds to a "gravothermal catastrophe" and in the canonical ensemble to an "isothermal collapse." Self-similar solutions are investigated analytically and numerically.

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PubMed2002

Phase transitions in self-gravitating systems: self-gravitating fermions and hard-sphere models.

We discuss the nature of phase transitions in self-gravitating systems both in the microcanonical and in the canonical ensemble. We avoid the divergence of the gravitational potential at short distances by considering the case of self-gravitating fermions and hard-sphere models. Depending on the values of the parameters, three kinds of phase transitions (of zero, first, and second order) are evidenced. They separate a "gaseous" phase with a smoothly varying distribution of matter from a "condensed" phase with a core-halo structure. We propose a simple analytical model to describe these phase transitions. We determine the value of energy (in the microcanonical ensemble) and temperature (in the canonical ensemble) at the transition point and we study their dependence on the degeneracy parameter (for fermions) or on the size of the particles (for a hard-sphere gas). Scaling laws are obtained analytically in the asymptotic limit of a small short distance cutoff. Our analytical model captures the essential physics of the problem and compares remarkably well with the full numerical solutions. We also stress some analogies with the liquid-gas transition and with the Blume-Emery-Griffiths model with infinite range interactions. In particular, our system presents two tricritical points at which the transition passes from first order to second order.

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PubMed2002

Effective velocity created by a point vortex in two-dimensional hydrodynamics.

We complete previous investigations on the statistics of velocity fluctuations arising from a random distribution of point vortices in two-dimensional hydrodynamics. We show that, in a statistical sense, the velocity created by a point vortex is shielded by cooperative effects on a distance Lambda approximately n(-1/2), the intervortex separation. For R>>Lambda, the "effective" velocity decays as R-2 instead of the ordinary law R(-1) recovered for R<<Lambda. These results are similar to those obtained by Agekyan [Sov. Astron. 5, 809 (1962)] in his investigations on the fluctuations of the gravitational field. They give further support to our previous observation that the statistics of velocity fluctuations are (marginally) dominated by the contribution of the nearest neighbor.

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