arXiv subjects

Clement Sire

Publications and source records attributed to Clement Sire.

At least 19 recordsLinked to original sources

arXiv2009

Dipole diffusion in a random electrical potential

We study the Langevin dynamics of a dipole diffusing in a random electrical field E derived from a quenched Gaussian potential. We show that in a suitable adiabatic limit (where the dynamics of the dipole moment is much faster than the dynamics of its position), one can reduce the coupled stochastic equations to an effective Langevin equation for a particle diffusing in an effective potential with a spatially varying and anisotropic local diffusivity $κ_{ij}$. Analytic results, close to the adiabatic limit, for the diffusion constant $κ_e$ are found in one dimension and a finite temperature dynamical transition is found. The system is also studied numerically. In particular, we study the anomalous diffusion exponent in the low temperature regime. Our findings strongly support the conclusion that the location of the dynamical transition and the anomalous diffusion exponents are determined by purely static considerations, i.e. they are independent of the relative values of the diffusion constants of the particle position and its dipole moment.

cond-mat.dis-nn↗

arXiv2009

Critical dynamics of self-gravitating Langevin particles and bacterial populations

We study the critical dynamics of the generalized Smoluchowski-Poisson system (for self-gravitating Langevin particles) or generalized Keller-Segel model (for the chemotaxis of bacterial populations). These models [Chavanis & Sire, PRE, 69, 016116 (2004)] are based on generalized stochastic processes leading to the Tsallis statistics. The equilibrium states correspond to polytropic configurations with index $n$ similar to polytropic stars in astrophysics. At the critical index $n_{3}=d/(d-2)$ (where $d\ge 2$ is the dimension of space), there exists a critical temperature $Θ_{c}$ (for a given mass) or a critical mass $M_{c}$ (for a given temperature). For $Θ>Θ_{c}$ or $M M_{c}$ the system collapses and forms, in a finite time, a Dirac peak containing a finite fraction $M_c$ of the total mass surrounded by a halo. This study extends the critical dynamics of the ordinary Smoluchowski-Poisson system and Keller-Segel model in $d=2$ corresponding to isothermal configurations with $n_{3}\to +\infty$. We also stress the analogy between the limiting mass of white dwarf stars (Chandrasekhar's limit) and the critical mass of bacterial populations in the generalized Keller-Segel model of chemotaxis.

cond-mat.stat-mech↗

arXiv2008

Critical mass of bacterial populations in a generalized Keller-Segel model. Analogy with the Chandrasekhar limiting mass of white dwarf stars

We point out a remarkable analogy between the limiting mass of white dwarf stars (Chandrasekhar's limit) and the critical mass of bacterial populations in a generalized Keller-Segel model of chemotaxis [Chavanis & Sire, PRE, 69, 016116 (2004)]. This model is based on generalized stochastic processes leading to the Tsallis statistics. The equilibrium states correspond to polytropic configurations similar to gaseous polytropes in astrophysics. For the critical index n_3=d/(d-2) (where d is the dimension of space), the theory of polytropes leads to a unique value of the mass M_c that we interpret as a limiting mass. In d=3, we find M_c=202.8956... and in d=2, we recover the well-known result M_c=8 pi (in suitable units). For M M_c, the system collapses and forms a Dirac peak containing a mass M_c surrounded by a halo. This paper exposes the model and shows, by simple considerations, the origin of the critical mass. A detailed description of the critical dynamics of the generalized Keller-Segel model will be given in a forthcoming paper.

cond-mat.stat-mech↗

arXiv2008

Jeans type analysis of chemotactic collapse

We perform a linear dynamical stability analysis of a general hydrodynamic model of chemotactic aggregation [Chavanis & Sire, Physica A, in press (2007)]. Specifically, we study the stability of an infinite and homogeneous distribution of cells against "chemotactic collapse". We discuss the analogy between the chemotactic collapse of biological populations and the gravitational collapse (Jeans instability) of self-gravitating systems. Our hydrodynamic model involves a pressure force which can take into account several effects like anomalous diffusion or the fact that the organisms cannot interpenetrate. We also take into account the degradation of the chemical which leads to a shielding of the interaction like for a Yukawa potential. Finally, our hydrodynamic model involves a friction force which quantifies the importance of inertial effects. In the strong friction limit, we obtain a generalized Keller-Segel model similar to the generalized Smoluchowski-Poisson system describing self-gravitating Langevin particles. For small frictions, we obtain a hydrodynamic model of chemotaxis similar to the Euler-Poisson system describing a self-gravitating barotropic gas. We show that an infinite and homogeneous distribution of cells is unstable against chemotactic collapse when the "velocity of sound" in the medium is smaller than a critical value. We study in detail the linear development of the instability and determine the range of unstable wavelengths, the growth rate of the unstable modes and the damping rate, or the pulsation frequency, of the stable modes as a function of the friction parameter and shielding length. For specific equations of state, we express the stability criterion in terms of the density of cells.

physics.bio-ph↗

arXiv2007

Kinetic and hydrodynamic models of chemotactic aggregation

We derive general kinetic and hydrodynamic models of chemotactic aggregation that describe certain features of the morphogenesis of biological colonies (like bacteria, amoebae, endothelial cells or social insects). Starting from a stochastic model defined in terms of N coupled Langevin equations, we derive a nonlinear mean field Fokker-Planck equation governing the evolution of the distribution function of the system in phase space. By taking the successive moments of this kinetic equation and using a local thermodynamic equilibrium condition, we derive a set of hydrodynamic equations involving a damping term. In the limit of small frictions, we obtain a hyperbolic model describing the formation of network patterns (filaments) and in the limit of strong frictions we obtain a parabolic model which is a generalization of the standard Keller-Segel model describing the formation of clusters (clumps). Our approach connects and generalizes several models introduced in the chemotactic literature. We discuss the analogy between bacterial colonies and self-gravitating systems and between the chemotactic collapse and the gravitational collapse (Jeans instability). We also show that the basic equations of chemotaxis are similar to nonlinear mean field Fokker-Planck equations so that a notion of effective generalized thermodynamics can be developed.

physics.bio-ph↗

arXiv2006

Logotropic distributions

In all spatial dimensions $d$, we study the static and dynamical properties of a generalized Smoluchowski equation which describes the evolution of a gas obeying a logotropic equation of state, $p=A\lnρ$. A logotrope can be viewed as a limiting form of polytrope ($p=Kρ^γ$, $γ=1+1/n$), with index $γ=0$ or $n=-1$. In the language of generalized thermodynamics, it corresponds to a Tsallis distribution with index $q=0$. We solve the dynamical logotropic Smoluchowski equation in the presence of a fixed external force deriving from a quadratic potential, and for a gas of particles subjected to their mutual gravitational force. In the latter case, the collapse dynamics is studied for any negative index $n$, and the density scaling function is found to decay as $r^{-α}$, with $α=\frac{2n}{n-1}$ for $n<-\frac{d}{2}$, and $α=\frac{2d}{d+2}$ for $-\frac{d}{2}\leq n<0$.

cond-mat.stat-mech↗

arXiv2006

Exact solution of a model of time-dependent evolutionary dynamics in a rugged fitness landscape

A simplified form of the time dependent evolutionary dynamics of a quasispecies model with a rugged fitness landscape is solved via a mapping onto a random flux model whose asymptotic behavior can be described in terms of a random walk. The statistics of the number of changes of the dominant genotype from a finite set of genotypes are exactly obtained confirming existing conjectures based on numerics.

cond-mat.stat-mech↗

arXiv2006

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 semi-classical 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 $T_c$, the chemical potential $μ(t)$ vanishes exponentially with a universal rate that we compute exactly. Below $T_c$, we show analytically that $\sqrt{μ(t)}$ 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 $T_c$) and the collapse time $t_{coll}$ (below $T_c$), both behave like $-T_c^{-3}\ln|T-T_c|$, near $T_c$.

cond-mat.stat-mech↗

arXiv2006

Virial theorem and dynamical evolution of self-gravitating Brownian particles and bacterial populations in an unbounded domain

We derive the Virial theorem appropriate to the generalized Smoluchowski-Poisson system describing self-gravitating Brownian particles and bacterial populations (chemotaxis). 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 T_c in dimension d=2 and show how the effective diffusion coefficient and the Einstein relation are modified by self-gravity or chemotactic attraction. We also study the collapse at T=T_c 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 generalized Smoluchowski-Poisson 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. Finally, we propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles and biological populations and recover known models in some particular limits.

cond-mat.stat-mech↗

arXiv2005

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 multi-components 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 non universal 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.

cond-mat.stat-mech↗

arXiv2004

Post-collapse dynamics of self-gravitating Brownian particles in D dimensions

We address the post-collapse dynamics of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. In the canonical ensemble, the post-collapse 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 post-collapse regime is marked by the formation of a ``binary''-like structure surrounded by an almost uniform halo with high temperature. These results are consistent with thermodynamical predictions.

cond-mat.stat-mech↗

arXiv2004

Estimate of blow-up and relaxation time for self-gravitating Brownian particles and bacterial populations

We determine an asymptotic expression of the blow-up time t_coll for self-gravitating Brownian particles or bacterial populations (chemotaxis) close to the critical point. We show that t_coll=t_{*}(eta-eta_c)^{-1/2} with t_{*}=0.91767702..., where eta represents the inverse temperature (for Brownian particles) or the mass (for bacterial colonies), and eta_c is the critical value of eta above which the system blows up. This result is in perfect agreement with the numerical solution of the Smoluchowski-Poisson system. We also determine the asymptotic expression of the relaxation time close but above the critical temperature and derive a large time asymptotic expansion for the density profile exactly at the critical point.

cond-mat.stat-mech↗

arXiv2003

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. The equilibrium states correspond to polytropic distributions. 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 Smoluchowski-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 points arguments and analytical methods. When no equilibrium state exists, we investigate self-similar solutions describing the collapse. These results can be relevant for astrophysical systems, two-dimensional vortices and for the chemotaxis of bacterial populations. Above all, this model constitutes a prototypical dynamical model of systems with long-range interactions which possesses a rich structure and which can be studied in great detail.

cond-mat.stat-mech↗

arXiv1999

Analytical results for random walks in the presence of disorder and traps

In this paper, we study the dynamics of a random walker diffusing on a disordered one-dimensional lattice with random trappings. The distribution of escape probabilities is computed exactly for any strength of the disorder. These probabilities do not display any multifractal properties contrary to previous numerical claims. The explanation for this apparent multifractal behavior is given, and our conclusion are supported by numerical calculations. These exact results are exploited to compute the large time asymptotics of the survival probability (or the density) which is found to decay as $\exp [-Ct^{1/3}\log^{2/3}(t)]$. An exact lower bound for the density is found to decay in a similar way.

cond-mat.stat-mech↗

arXiv1998

Analytical results for random walk persistence

In this paper, we present the detailed calculation of the persistence exponent $θ$ for a nearly-Markovian Gaussian process $X(t)$, a problem initially introduced in [Phys. Rev. Lett. 77, 1420 (1996)], describing the probability that the walker never crosses the origin. New resummed perturbative and non-perturbative expressions for $θ$ are obtained, which suggest a connection with the result of the alternative independent interval approximation (IIA). The perturbation theory is extended to the calculation of $θ$ for non-Gaussian processes, by making a strong connection between the problem of persistence and the calculation of the energy eigenfunctions of a quantum mechanical problem. Finally, we give perturbative and non-perturbative expressions for the persistence exponent $θ(X_0)$, describing the probability that the process remains bigger than $X_0\sqrt{ }$.

cond-mat.stat-mech↗

arXiv1998

Block persistence

We define a block persistence probability $p_l(t)$ as the probability that the order parameter integrated on a block of linear size $l$ has never changed sign since the initial time in a phase ordering process at finite temperature T<T_c. We argue that p_l(t)\sim l^{-zθ_0}f(t/l^z) in the scaling limit of large blocks, where θ_0 is the global (magnetization) persistence exponent and f(x) decays with the local (single spin) exponent θfor large x. This scaling is demonstrated at zero temperature for the diffusion equation and the large n model, and generically it can be used to determine easily θ_0 from simulations of coarsening models. We also argue that θ_0 and the scaling function do not depend on temperature, leading to a definition of θat finite temperature, whereas the local persistence probability decays exponentially due to thermal fluctuations. We also discuss conserved models for which different scaling are shown to arise depending on the value of the autocorrelation exponent λ. We illustrate our discussion by extensive numerical results. We also comment on the relation between this method and an alternative definition of θat finite temperature recently introduced by Derrida [Phys. Rev. E 55, 3705 (1997)].

cond-mat.stat-mech↗

arXiv1997

Spin block persistence at finite temperature

We explore a new definition of the persistence exponent, measuring the probability that a spin never flips after a quench of an Ising-like model at a temperature 0<T<Tc, while the usual definition only makes sense at T=0. This probability is now defined for spin blocks, and a general scaling for it, involving time and block linear size is introduced and illustrated by extensive simulations.

cond-mat.stat-mech↗

arXiv1997

Droplets nucleation, and Smoluchowski's equation with growth and injection of particles

We show that models for homogeneous and heterogeneous nucleation of D-dimensional droplets in a d-dimensional medium are described in mean-field by a modified Smoluchowski equation for the distribution N(s,t) of droplets masses s, with additional terms accounting for exogenous growth from vapor absorption, and injection of small droplets when the model allows renucleation. The corresponding collision kernel is derived in both cases. For a generic collision kernel K, the equation describes a clustering process with clusters of mass s growing between collision with ds/dt=As^β$, and injection of monomers at a rate I(t). General properties of this equation are studied. The gel criterion is determined. Without injection, exact solutions are found with a constant kernel, exhibiting unusual scaling behavior. For a general kernel, under the scaling assumption N(s,t)\sim Y(t)^{-1}f(s/S(t)), we determine the asymptotics of S(t) and Y(t), and derive the scaling equation. Depending on βand K, a great diversity of behaviors is found. For constant injection, there is an asymptotic steady state with N(s,t=\infty)\propto s^{-τ} and τ$is determined. The case of a constant mass injection rate is related to homogeneous nucleation and is studied. Finally, we show how these results shed some new light on heterogeneous nucleation with d=D. For d=D=2 (discs on a plane), numerical simulations are performed, in good agreement with the mean-field results.

cond-mat.stat-mech↗