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

Delfini

2014

Phys. Rev. E

, 118 route de Narbonne, 31062 Toulouse cedex 4.We study random transitions between two metastable states that appear below a critical temperature in a one dimensional self-gravitating Brownian gas with a modified Poisson equation experiencing a second order phase transition from a homogeneous phase to an inhomogeneous phase [P.H. Chavanis and L. Delfini, Phys. Rev. E 81, 051103 (2010)]. We numerically solve the Nbody Langevin equations and the stochastic Smoluchowski-Poisson system which takes fluctuations (finite N effects) into account. The system switches back and forth between the two metastable states (bistability) and the particles accumulate successively at the center or at the boundary of the domain. We explicitly show that these random transitions exhibit the phenomenology of the ordinary Kramers problem for a Brownian particle in a double-well potential. The distribution of the residence time is Poissonian and the average lifetime of a metastable state is given by the Arrhenius law, i.e. it is proportional to the exponential of the barrier of free energy ∆F divided by the energy of thermal excitation kBT . Since the free energy is proportional to the number of particles N for a system with long-range interactions, the lifetime of metastable states scales as e N and is considerable for N 1. As a result, in many applications, metastable states of systems with longrange interactions can be considered as stable states. However, for moderate values of N , or close to a critical point, the lifetime of the metastable states is reduced since the barrier of free energy decreases. In that case, the fluctuations become important and the mean field approximation is no more valid. This is the situation considered in this paper. By an appropriate change of notations, our results also apply to bacterial populations experiencing chemotaxis in biology. Their dynamics can be described by a stochastic Keller-Segel model that takes fluctuations into account and goes beyond the usual mean field approximation.

Section: Introductionmentioning

confidence: 71%

Random transitions described by the stochastic Smoluchowski-Poisson system and by the stochastic Keller-Segel model

Delfini

2014

Phys. Rev. E

“…where α = (S d Gβmρ 0 ) 1/2 R is the normalized box radius. For d = 2, the thermodynamical parameters can be calculated analytically (see, e.g., [5,37] and Appendix C of [35]). Introducing the pressure at the box P = p(R), the global equation of state of the self-gravitating gas can be written as…”

Section: The Non Degenerate Limit: Classical Isothermal Spheresmentioning

confidence: 99%

“…It turns out that this long-range interaction is similar to the gravitational interaction in astrophysics. Indeed, the Keller-Segel model shows deep analogies with the dynamics of self-gravitating Brownian particles described by the Smoluchowski-Poisson system in the canonical ensemble [37]. In the biological context, the consideration of a "box" in which the bacteria live is completely justified (more than in astrophysics).…”

Section: Introductionmentioning

confidence: 99%

Gravitational phase transitions with an exclusion constraint in position space

2014

Eur. Phys. J. B

Self Cite

Abstract. We discuss the statistical mechanics of a system of self-gravitating particles with an exclusion constraint in position space in a space of dimension d. The exclusion constraint puts an upper bound on the density of the system and can stabilize it against gravitational collapse. We plot the caloric curves giving the temperature as a function of the energy and investigate the nature of phase transitions as a function of the size of the system and of the dimension of space in both microcanonical and canonical ensembles. We consider stable and metastable states and emphasize the importance of the latter for systems with long-range interactions. For d ≤ 2, there is no phase transition. For d > 2, phase transitions can take place between a "gaseous" phase unaffected by the exclusion constraint and a "condensed" phase dominated by this constraint. The condensed configurations have a core-halo structure made of a "rocky core" surrounded by an "atmosphere", similar to a giant gaseous planet. For large systems there exist microcanonical and canonical first order phase transitions. For intermediate systems, only canonical first order phase transitions are present. For small systems there is no phase transition at all. As a result, the phase diagram exhibits two critical points, one in each ensemble. There also exist a region of negative specific heats and a situation of ensemble inequivalence for sufficiently large systems. We show that a statistical equilibrium state exists for any values of energy and temperature in any dimension of space. This differs from the case of the selfgravitating Fermi gas for which there is no statistical equilibrium state at low energies and low temperatures when d ≥ 4. By a proper interpretation of the parameters, our results have application for the chemotaxis of bacterial populations in biology described by a generalized Keller-Segel model including an exclusion constraint in position space. They also describe colloids at a fluid interface driven by attractive capillary interactions when there is an excluded volume around the particles. Connexions with two-dimensional turbulence are also mentioned.

“…(iii) For β < β c , the system forms, in a finite time t coll , a Dirac peak containing (β c /β)N point vortices surrounded by a halo of vortices evolving pseudo self-similarly [52]. A Dirac peak containing all the point vortices is formed in the post-collapse regime t > t coll in a finite time t end [54]. Let us now determine the two-body correlation function of an infinite and homogeneous distribution of point vortices using Eq.…”

Section: Newtonian or Coulombian Potentialmentioning

confidence: 99%

Two-dimensional Brownian vortices

Physica A: Statistical Mechanics and its Applications

2008

Self Cite

We introduce a stochastic model of two-dimensional Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other ("plasma" case) and at negative temperatures, like-sign vortices attract each other ("gravity" case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N -body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N , where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N → +∞. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-Hückel approximation valid at the order O(1/N ). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-Hückel model of electrolytes.