Phase transitions in self-gravitating systems and bacterial populations with a screened attractive potential P.H. Chavanis and L. Delfini 1 $^{1}$Laboratoire de Physique Th´eorique (IRSAMC), CNRS and UPS, Universit´e de Toulouse, F-31062 Toulouse, France We consider a system of particles interacting via a screened Newtonian potential and study phase transitions between homogeneous and inhomogeneous states in the microcanonical and canonical ensembles. Like for other systems with long-range interactions, we obtain a great diversity of microcanonical and canonical phase transitions depending on the dimension of space and on the importance of the screening length. We also consider a system of particles in Newtonian interaction in the presence of a “neutralizing background”. By a proper interpretation of the parameters, our study describes (i) self-gravitating systems in a cosmological setting, and (ii) chemotaxis of bacterial populations in the original Keller-Segel model. PACS numbers: 47.10.A-,47.15.ki Many biological species like bacteria, amoebae, en- dothelial cells, or even ants interact through the phe- nomenon of chemotaxis [1]. These organisms secrete a chemical substance (like a pheromone) that has an at- tractive (or sometimes repulsive) action on the organ- isms themselves. This phenomenon is responsible for the self-organization and morphogenesis of many biological species. It has also been proposed as a leading mechanism for the formation of blood vessels during embriogenesis [2]. On a theoretical point of view, chemotaxis can be de- scribed by the Keller-Segel [3] model or its generalizations [4]. The Keller-Segel model consists in a drift-diffusion equation for the evolution of the density of bacteria ρ ( r , t ) coupled to a reaction-diffusion equation for the evolu- tion of the secreted chemical c ( r , t ). In certain approxi- mations, the reaction-diffusion equation is replaced by a Poisson equation. In that case, the Keller-Segel (KS) [3] model becomes isomorphic to the Smoluchowski-Poisson (SP) system [5] describing self-gravitating Brownian par- ticles (see, e.g., [6] for a description of this analogy). The KS model and SP system have been studied thoroughly in applied mathematics (see Refs. in [7]) and in theoret- ical physics (see Refs. in [5]). is modified so as to allow for the existence of spatially homogeneous distributions at equilibrium. This will be called the modified Newtonian model . In that model, the source of the potential is the deviation between the ac- tual density ρ ( r , t ) and the average density ρ . This is similar to the effect of a “neutralizing background” in plasma physics [10]. This model can be derived from the Keller-Segel model in the limit of vanishing degradation of the chemical [11]. It also appears in cosmology, due to the expansion of the universe, when we work in the comoving frame [12]. It is therefore interesting to con- sider this form of interaction at a general level and study the corresponding phase transitions. We shall also com- pare them with the ones obtained within the ordinary Newtonian model [13–30] (see a review in [31]). The paper is organized as follows. In Sec. II, we discuss several kinetic models taken from astrophysics, plasma physics and biology for which our study applies. We consider either isolated systems described by the mi- crocanonical ensemble (fixed energy E ) or dissipative sys- tems described by the canonical ensemble (fixed temper- ature T ). We characterize their equilibrium states in the mean field approximation: in the microcanonical ensem- ble (MCE), they maximize the entropy at fixed mass and energy and in the canonical ensemble (CE) they min- imize the free energy at fixed mass. In Sec. III, we specifically consider the case of a Newtonian interaction with a neutralizing background. We study phase tran- sitions between homogeneous and inhomogeneous states depending on the dimension of space. In d = 1, the system presents canonical and microcanonical second or- der phase transitions. In d = 2, the system presents an isothermal collapse in CE (zeroth order phase transi- tion) and a first order phase transition in MCE. In d = 3, the system presents an isothermal collapse in CE and a gravothermal catastrophe in MCE (zeroth order phase transitions). In Sec. IV, we perform a similar study for the attractive Yukawa potential with screening length k − 1 0 . In d = 1, there exists a canonical tricritical point ( k$_{0}$ )$_{c}$ R = √$_{2}$$_{π}$ ≃ 4 . 44 and a microcanonical tricritical However, the original KS model [3] also allows for the possibility that the chemical suffers a degradation process which has the effect of reducing the range of the interac- tion. In that case, the Poisson equation is replaced by a screened Poisson equation [8]. In the gravitational anal- ogy, this amounts to replacing the gravitational potential by a screened gravitational potential, i.e. an attractive Yukawa potential. In that case, there exists interest- ing phase transitions between spatially homogeneous and spatially inhomogeneous equilibrium distributions. This is a physical motivation to consider the thermodynamics of N-body systems interacting via an attractive Yukawa potential [9]. This will be called the screened Newto- nian model . We shall also consider a related model where the interaction is not screened but the Poisson equation arXiv:1001.1942v1 [cond-mat.stat-mech] 12 Jan 2010 I. INTRODUCTION