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

In many fields of physics, astrophysics and biology, one is confronted with the description of the evolution of a system of particles which self-consistently attract each other over large distances. One difficulty and richness of the problem arises from the long-range nature of the potential of interaction [1]. This is the case, for example, in biology in relation with the process of chemotaxis [2]. The chemotactic aggregation of bacterial populations (like Escherichia coli) or amoebae (like Dictyostelium discoideum) is usually studied in the framework of the Keller-Segel model [3] which describes the collective motion of organisms that are attracted by a chemical substance (pheromone) that they produce themselves. The Keller-Segel (KS) model involves a drift-diffusion equation describing the evolution of the concentration of the bacteria in the gradient of concentration of the secreted chemical. In the simplest formulation, the concentration of the chemical is related to the concentration of bacteria by a Poisson equation (this is valid in a limit of large diffusivity of the chemical and for sufficiently large concentrations) [4]. These equations have been studied by applied mathematicians who obtained rigorous results for the existence and unicity of the solutions and for the conditions of blow-up, modeling chemotactic collapse, in different dimensions of space [5]. In particular, in d = 2, there exists a critical mass M c (independent on the size of the domain) above which the system collapses and forms a Dirac peak. Alternatively, for M < M c , the system spreads to infinity in an unbounded domain or tends to a stationary state in a bounded domain.

Gravity is another example of long-range attractive potential of interaction. 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. A lot of analytical results have been obtained and an almost complete description of the system, for all the phases of the dynamics (pre-collapse and post-collapse), has been given in the overdamped limit of the model. In that limit, the evolution of the density of the self-gravitating Brownian gas is governed by the Smoluchowski-Poisson (SP) system. The Smoluchowski equation is a driftdiffusion equation of a Fokker-Planck type. For self-gravitating particles, the gravitational potential inducing the drift is produced by the density of particles through the Newton-Poisson equation. It turns out that the Smoluchowski-Poisson system is isomorphic to the simplified version of the Keller-Segel model of chemotaxis provided that the parameters are suitably re-interpreted, as discussed in [11]. In particular, for the 2D self-gravitating Brownian gas, there exists a critical temperature T c (independent on the size of the domain) below which the system collapses and forms a Dirac peak. This is the counterpart of the critical mass of bacterial populations. For T > T c , the system evaporates in an unbounded domain or tends to a statistical equilibrium state in a bounded domain.

The object of this paper is to emphasize the parallel between these two systems. In Secs. 2 and 3, we use the Virial theorem to derive the critical mass of bacterial populations and the critical temperature of self-gravitating Brownian particles in two dimensions. In Sec. 4, we show that these critical values can also be obtained by considering stationary solutions of the Keller-Segel model and Smoluchowski-Poisson system. In Sec. 5, we consider the one dimensional problem and point out the connection between the Smoluchowski-Poisson system (or the Keller-Segel model) and the Burgers equation. We use this analogy to provide the general solution of these equations in d = 1 in bounded and unbounded domains. 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.

The dynamical evolution of biological populations like bacteria, amoebae, cells… that are attracted by a substance that they emit themselves, is often described by the Keller-Segel (KS) model [3]. In its simplest form, it can be written as (1) ∂ρ ∂t = D∆ρ -χ∇ • (ρ∇c),

(2) ∆c = -λρ.

Equation ( 1) is a drift-diffusion equation for the cell density ρ(r, t). The diffusion term takes into account the erratic motion of the cells (like in Brownian theory) and the drift term with χ > 0 takes into account the chemotactic attraction (one could also consider the case χ < 0 where the secreted substance is a noxious substance, like a poison, so that chemotaxis is repulsive). It is directed along the gradient of concentration c(r, t) of the secreted chemical.

In the simplest formulation [4], the production of the chem

…(Full text truncated)…

This content is AI-processed based on ArXiv data.