Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension

We investigate the motion of a run-and-tumble particle (RTP) in one dimension. We find the exact probability distribution of the particle with and without diffusion on the infinite line, as well as in a finite interval. In the infinite domain, this probability distribution approaches a Gaussian form in the long-time limit, as in the case of a regular Brownian particle. At intermediate times, this distribution exhibits unexpected multi-modal forms. In a finite domain, the probability distribution reaches a steady state form with peaks at the boundaries, in contrast to a Brownian particle. We also study the relaxation to the steady state analytically. Finally we compute the survival probability of the RTP in a semi-infinite domain. In the finite interval, we compute the exit probability and the associated exit times. We provide numerical verifications of our analytical results

Self-propelled particles with selective attraction-repulsion interaction - From microscopic dynamics to coarse-grained theories

In this work we derive and analyze coarse-grained descriptions of self-propelled particles with selective attraction-repulsion interaction, where individuals may respond differently to their neighbours depending on their relative state of motion (approach versus movement away). Based on the formulation of a nonlinear Fokker-Planck equation, we derive a kinetic description of the system dynamics in terms of equations for the Fourier modes of a one-particle density function. This approach allows effective numerical investigation of the stability of possible solutions of the system. The detailed analysis of the interaction integrals entering the equations demonstrates that divergences at small wavelengths can appear at arbitrary expansion orders. Further on, we also derive a hydrodynamic theory by performing a closure at the level of the second Fourier mode of the one-particle density function. We show that the general form of equations is in agreement with the theory formulated by Toner and Tu. Finally, we compare our analytical predictions on the stability of the disordered homogeneous solution with results of individual-based simulations. They show good agreement for sufficiently large densities and non-negligible short-ranged repulsion. Disagreements of numerical results and the hydrodynamic theory for weak short-ranged repulsion reveal the existence of a previously unknown phase of the model consisting of dense, nematically aligned filaments, which cannot be accounted for by the present Toner and Tu type theory of polar active matter.Comment: revised version, 37pages, 11 figure

The Brownian Mean Field model

We discuss the dynamics and thermodynamics of the Brownian Mean Field (BMF) model which is a system of N Brownian particles moving on a circle and interacting via a cosine potential. It can be viewed as the canonical version of the Hamiltonian Mean Field (HMF) model. We first complete the description of this system in the mean field approximation. Then, we take fluctuations into account and study the stochastic evolution of the magnetization both in the homogeneous phase and in the inhomogeneous phase. We discuss its behavior close to the critical point

Levy flights in quenched random force fields

Levy flights, characterized by the microscopic step index f, are for f<2 (the case of rare events) considered in short range and long range quenched random force fields with arbitrary vector character to first loop order in an expansion about the critical dimension 2f-2 in the short range case and the critical fall-off exponent 2f-2 in the long range case. By means of a dynamic renormalization group analysis based on the momentum shell integration method, we determine flows, fixed point, and the associated scaling properties for the probability distribution and the frequency and wave number dependent diffusion coefficient. Unlike the case of ordinary Brownian motion in a quenched force field characterized by a single critical dimension or fall-off exponent d=2, two critical dimensions appear in the Levy case. A critical dimension (or fall-off exponent) d=f below which the diffusion coefficient exhibits anomalous scaling behavior, i.e, algebraic spatial behavior and long time tails, and a critical dimension (or fall-off exponent) d=2f-2 below which the force correlations characterized by a non trivial fixed point become relevant. As a general result we find in all cases that the dynamic exponent z, characterizing the mean square displacement, locks onto the Levy index f, independent of dimension and independent of the presence of weak quenched disorder.Comment: 27 pages, Revtex file, 17 figures in ps format attached, submitted to Phys. Rev.

Evolution of galaxies due to self-excitation

These lectures will cover methods for studying the evolution of galaxies since their formation. Because the properties of a galaxy depend on its history, an understanding of galaxy evolution requires that we understand the dynamical interplay between all components. The first part will emphasize n-body simulation methods which minimize sampling noise. These techniques are based on harmonic expansions and scale linearly with the number of bodies, similar to Fourier transform solutions used in cosmological simulations. Although fast, until recently they were only efficiently used for small number of geometries and background profiles. These same techniques may be used to study the modes and response of a galaxy to an arbitrary perturbation. In particular, I will describe the modal spectra of stellar systems and role of damped modes which are generic to stellar systems in interactions and appear to play a significant role in determining the common structures that we see. The general development leads indirectly to guidelines for the number of particles necessary to adequately represent the gravitational field such that the modal spectrum is resolvable. I will then apply these same excitation to understanding the importance of noise to galaxy evolution.Comment: 24 pages, 7 figures, using Sussp.sty (included). Lectures presented at the NATO Advanced Study Institute, "The Restless Universe: Applications of Gravitational N-Body Dynamics to Planetary, Stellar and Galactic Systems," Blair Atholl, July 200

Symplectic tomography as classical approach to quantum systems

By using a generalization of the optical tomography technique we describe the dynamics of a quantum system in terms of equations for a purely classical probability distribution which contains complete information about the system.Comment: 12 pages, LATEX,preprint of Camerino University, to appear in Phys.Lett.A (1996