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.     

Brownian particles with long- and short-range interactions

We develop the kinetic theory of Brownian particles with long- and short-range interactions. Since the particles are in contact with a thermal bath fixing the temperature T, they are described by the canonical ensemble. We consider both overdamped and inertial models. In the overdamped limit, the evolution of the spatial density is governed by the generalized mean field Smoluchowski equation including a mean field potential due to long-range interactions and a generically nonlinear barotropic pressure due to short-range interactions. This equation describes various physical systems such as self-gravitating Brownian particles (Smoluchowski-Poisson system), bacterial populations experiencing chemotaxis (Keller-Segel model) and colloidal particles with capillary interactions. We also take into account the inertia of the particles and derive corresponding kinetic and hydrodynamic equations generalizing the usual Kramers, Jeans, Euler and Cattaneo equations. For each model, we provide the corresponding form of free energy and establish the H-theorem and the virial theorem. Finally, we show that the same hydrodynamic equations are obtained in the context of nonlinear mean field Fokker-Planck equations associated with generalized thermodynamics. However, in that case, the nonlinear pressure is due to the bias in the transition probabilities from one state to the other leading to non-Boltzmannian distributions while in the former case the distribution is Boltzmannian but the nonlinear pressure arises from the two-body correlation function induced by the short-range potential of interaction. As a whole, our paper develops connections between the topics of long-range interactions, short-range interactions, nonlinear mean field Fokker-Planck equations and generalized thermodynamics. It also justifies from a kinetic theory based on microscopic processes, the basic equations that were introduced phenomenologically to describe self-gravitating Brownian particles, chemotaxis and colloidal suspensions with attractive interactions.     

Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.     

Hydrodynamics of Brownian particles

When considering the hydrodynamics of Brownian particles, one is confronted to a difficult closure problem. One possibility to close the hierarchy of hydrodynamic equations is to consider a strong friction limit. This leads to the Smoluchowski equation that reduces to the ordinary diffusion equation in the absence of external forces. Unfortunately, this equation has infinite propagation speed leading to some difficulties. Another possibility is to make a Local Thermodynamic Equilibrium (L.T.E) assumption. This leads to the damped Euler equation with an isothermal equation of state. However, this approach is purely phenomenological. In this paper, we provide a preliminary discussion of the validity of the L.T.E assumption. To that purpose, we consider the case of free Brownian particles and harmonically bound Brownian particles for which exact analytical results can be obtained [S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943)]. For these systems, we find that the L.T.E. assumption is not unreasonable and that it can be improved by introducing a time dependent kinetic temperature T_kin(t)=γ(t)T instead of the bath temperature T. We also compare hydrodynamic equations and generalized diffusion equations with time dependent diffusion coefficients.     

A Non-Maxwellian Steady Distribution for One-Dimensional Granular Media

We consider a nonlinear Fokker–Planck equation for a one-dimensional granular medium. This is a kinetic approximation of a system of nearly elastic particles in a thermal bath. We prove that homogeneous solutions tend asymptotically in time toward a unique non-Maxwellian stationary distribution.     

Multiscaling for classical nanosystems: Derivation of Smoluchowski & Fokker–Planck equations

Using multiscale analysis and methods of statistical physics, we show that a solution to the N-atom Liouville equation can be decomposed via an expansion in terms of a smallness parameter , wherein the long scale time behavior depends upon a reduced probability density that is a function of slow-evolving order parameters. This reduced probability density is shown to satisfy the Smoluchowski equation up to O(²) for a given range of initial conditions. Furthermore, under the additional assumption that the nanoparticle momentum evolves on a slow time scale, we show that this reduced probability density satisfies a Fokker–Planck equation up to O(²). This approach has applications to a broad range of problems in the nanosciences.     

Jeans type instability for a chemotactic model of cellular aggregation

We consider an inertial model of chemotactic aggregation generalizing the Keller-Segel model and we study the linear dynamical stability of an infinite and homogeneous distribution of cells (bacteria, amoebae, endothelial cells, ...) when inertial effects are accounted for. These inertial terms model cells directional persistance. We determine the condition of instability and the growth rate of the perturbation as a function of the cell density and the wavelength of the perturbation. We discuss the differences between overdamped (Keller-Segel) and inertial models. Finally, we show the analogy between the instability criterion for biological populations and the Jeans instability criterion in astrophysics.     

Kernel-based regression of drift and diffusion coefficients of stochastic processes

To improve the estimation of drift and diffusion coefficients of stochastic processes in case of a limited amount of usable data due to e.g. non-stationarity of natural systems we suggest to use kernel-based instead of histogram-based regression. We propose a method for bandwidth selection and compare it to a widely used cross-validation method. Kernel-based regression reveals an enhanced ability to estimate drift and diffusion especially for a small amount of data. This allows one to improve resolvability of changes in complex dynamical systems as evidenced by an exemplary analysis of electroencephalographic data recorded from a human epileptic brain.     

Statistical Properties of the Polarized Radiation of a Dye–Solution Laser

Within the framework of the method of polarization components we obtained the Fokker–Planck equation for the intensity–distribution function of an individual component. Solutions for the Ornstein–Uhlenbeck process show that the experimentally observed special features in the behavior of the distribution function of the intensity and degree of polarization of laser radiation in the vicinity of the threshold are well described in the approximation of statistical independence of polarization components. However, since the Ornstein–Uhlenbeck process includes states not realizable physically for the given case, an exact solution of the Fokker–Planck equation is constructed by the method of expansion in eigenstates. It is shown that this solution is totally correct physically and yields virtually the same values for the distribution functions of the intensity and degree of polarization of radiation as the dependences obtained earlier for the Ornstein–Uhlenbeck process.     

Radiation Polarization Distribution Function of a Dye Laser

The Fokker–Planck equation for the distribution function of the intensity of an individual component has been solved in the approximation of the Ornstein–Uhlenbeck process within the framework of the formalism of the method of polarization components. Based on this solution, we have constructed the distribution functions of the degree of lasing radiation polarization, analyzed experimental data for a certain geometry of laser pumping, and determined the values of the distribution parameters, including the loss coefficients for the polarization component.     

Elimination of Fast Chaotic Degrees of Freedom: On the Accuracy of the Born Approximation

We apply standard projection operator techniques known from nonequilibrium statistical mechanics to eliminate fast chaotic degrees of freedom in a low-dimensional dynamical system. Through the usual perturbative approach we end up in second order with a stochastic system where the fast chaotic degrees of freedom are modelled by Gaussian white noise. The accuracy of the perturbation expansion is analysed in detail by the discussion of an exactly solvable model.     

The Becker–Döring Equations and the Lifshitz–Slyozov Theory of Coarsening

In this paper the relation between the kinetic set of Becker–Döring (BD) equations and the classical Lifshitz–Slyozov (LS) theory of coarsening is studied. A model that resembles the LS theory but keeps some of the nucleation effects is derived. For this model a solution is described that shows how the kinetic effects explain the particular solution selected in the LS theory. By means of a renormalization procedure, a discrete group of transformations is shown to play an important role in describing the structure of the solution near the critical size of the LS theory.     

On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds

We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails: in particular, Boltzmann-type equations with (smoothed) soft potentials. We compensate the lack of uniform-in-time estimates by the use of precise logarithmic Sobolev-type inequalities, and the assumption that the initial datum decays rapidly at large velocities. Our method not only gives explicit results on the times of convergence, but is also able to cover situations in which compactness arguments apparently do not apply (even mere convergence to equilibrium was an open problem for soft potentials).     

Nonlinear analysis of the gradient drift instability

An analytical study of the gradient drift instability in the equatorial electrojet of wavelengths in the order of one kilometer is presented. Different mechanisms, linear, non-local and turbulent, are found in the literature to explain the predominance of the 1 km wavelength in the electrojet. In the present work a simplified model is proposed in which the nonlinear evolution of three coupled modes is followed. By considering that one of the modes attains the stationary state, the evolution of the other two is obtained, and it is found that they follow equations of the Lotka–Volterra type. A stable stationary nonlinear solution for these equations is also found, and the conditions under which periodic solutions are possible are analyzed.     

Distribution of wealth in a network model of the economy

We show, analytically and numerically, that wealth distribution in the Bouchaud–Mézard network model of the economy is described by a three-parameter generalized inverse gamma distribution. In the mean-field limit of a network with any two agents linked, it reduces to the inverse gamma distribution.     

Topologically Nontrivial Sectors of Maxwell Field Theory on Riemann Surfaces

In this Letter, the Maxwell field theory is considered on a closed and orientable Riemann surface of genus h 1. The solutions of the Maxwell equations corresponding to nontrivial values of the first Chern class are explicitly constructed for any metric in terms of the prime form.     

A nonextensive maximum entropy approach to a family of nonlinear reaction–diffusion equations

We consider a monoparametric family of reaction–diffusion equations endowed with both a nonlinear diffusion term and a nonlinear reaction one that possess exact time-dependent particular solutions of the Tsallis’ maximum entropy (MaxEnt) form. The evolution of these solutions is governed by a system of three coupled nonlinear ordinary differential equations that are integrated numerically. A simple population dynamics interpretation provides a qualitative understanding of the behaviour of the q-MaxEnt solutions. When the reaction term vanishes the time-dependent distributions studied here reduce to the previously known Tsallis’ MaxEnt solutions for the nonlinear diffusion equation.     

A new type of sample tube for reducing convection effects in PGSE-NMR measurements of self-diffusion coefficients of liquid samples

Pulsed gradient spin-echo (PGSE) NMR measurements of the self-diffusion coefficients of low viscosity liquids are greatly hampered by the effects of convection especially away from ambient temperature. Here we report on a new NMR tube designed to minimize the deleterious effects of convection. In this tube, which derives from a Shigemi symmetrical NMR tube, the sample is contained in an annulus formed from a concentric cylinder of susceptibility matched glass. The performance of this tube was demonstrated by conducting measurements on the electrochemically important LiN(SO3CF3)2 (LiTFSI)-diglyme (DG) system. Calibrations were first made using DG at column heights of 2, 3, and 4-mm in the temperature range between -40 and 100 degrees C. Measurements of the diffusion coefficients of the lithium, anion, and DG were then performed to probe the solvent-ion and ion-ion interactions in the DG doped with LiTFSI. Changes in the 1H, 7Li, and 19F PGSE-NMR attenuation curves at -40 degrees C provided clear evidence of interactions between the DG and lithium ion.     

Interaction of Three Electromagnetic Fields with a Resonance Medium in EPR. The Case of a Small Local Field in Comparison with the Saturating One

A system of Bloch equations modified with allowance for the presence of a dipole–dipole reservoir for the case where the local magnetic field is small in comparison with the saturating one is suggested. The system is used for solving the problem of interaction of three electromagnetic fields: a saturating field, a probe one, and the third - a combination field resulting from the interaction of the first two in a resonance medium. The imaginary and real parts of the system susceptibility at the probe-field frequency have been investigated in detail at both different frequencies of interacting waves and coinciding ones (degenerate case). For the degenerate case, the dependence of the coefficient of the parametric connection of waves on the frequency is considered. The results of the present work are compared with those obtained by us earlier for the case where the local magnetic field is much in excess of the saturating one (Provotorov's case). It is shown that in the problem considered the amplification of weak waves when they pass through the layer of an absorbing resonance medium is inaccessible.