Searcharxiv⌕ Search

arXiv subjects

Thomas Ihle

Publications and source records attributed to Thomas Ihle.

At least 37 records · Page 2Linked to original sources
arXiv2016

Chapman-Enskog expansion for the Vicsek model of self-propelled particles

Using the standard Vicsek model, I show how the macroscopic transport equations can be systematically derived from microscopic collision rules. The approach starts with the exact evolution equation for the N-particle probability distribution, and after making the mean-field assumption of Molecular Chaos leads to a multi-particle Enskog-type equation. This equation is treated by a non-standard Chapman-Enskog expansion to extract the macroscopic behavior. The expansion includes terms up to third order in a formal expansion parameter $ε$, and involves a fast time scale. A self-consistent closure of the moment equations is presented that leads to a continuity equation for the particle density and a Navier-Stokes-like equation for the momentum density. Expressions for all transport coefficients in these macroscopic equations are given explicitly in terms of microscopic parameters of the model. The transport coefficients depend on specific angular integrals which are evaluated asymptotically in the limit of infinitely many collision partners, using an analogy to a random walk. The consistency of the Chapman-Enskog approach is checked by an independent calculation of the shear viscosity using a Green-Kubo relation.

cond-mat.stat-mech↗
arXiv2015

Active matter beyond mean-field: Ring-kinetic theory for self-propelled particles

A ring-kinetic theory for Vicsek-style models of self-propelled agents is derived from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle pre-collisional correlations and cluster formation which both seem important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small density expansion of the collision operator and derive the first two equations of the BBGKY-hierarchy. An algorithm is presented that numerically solves the evolution equation for the two-particle correlations on a lattice. Agent-based simulations are performed and informative quantities such as orientational and density correlation functions are compared with those obtained by ring-kinetic theory. Excellent quantitative agreement between simulations and theory is found at not too small noises and mean free paths. This shows that there is parameter ranges in Vicsek-like models where the correlated closure of the BBGKY-hierarchy gives correct and nontrivial results. We calculate the dependence of the orientational correlations on distance in the disordered phase and find that it seems to be consistent with a power law with exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed.

cond-mat.stat-mech↗
arXiv2015

Large density expansion of a hydrodynamic theory for self-propelled particles

Recently, an Enskog-type kinetic theory for Vicsek-type models for self-propelled particles has been proposed [T. Ihle, Phys. Rev. E 83, 030901 (2011)]. This theory is based on an exact equation for a Markov chain in phase space and is not limited to small density. Previously, the hydrodynamic equations were derived from this theory and its transport coefficients were given in terms of infinite series. Here, I show that the transport coefficients take a simple form in the large density limit. This allows me to analytically evaluate the well-known density instability of the polarly ordered phase near the flocking threshold at moderate and large densities. The growth rate of a longitudinal perturbation is calculated and several scaling regimes, including three different power laws, are identified. It is shown that at large densities, the restabilization of the ordered phase at smaller noise is analytically accessible within the range of validity of the hydrodynamic theory. Analytical predictions for the width of the unstable band, the maximum growth rate and for the wave number below which the instability occurs are given. In particular, the system size below which spatial perturbations of the homogeneous ordered state are stable is predicted to scale with $\sqrt{M}$ where $M$ is the average number of collision partners. The typical time scale until the instability becomes visible is calculated and is proportional to M.

cond-mat.stat-mech↗
arXiv2014

Tricritical points in a Vicsek model of self-propelled particles with bounded confidence

We study the orientational ordering in systems of self-propelled particles with selective interactions. To introduce the selectivity we augment the standard Vicsek model with a bounded-confidence collision rule: a given particle only aligns to neighbors who have directions quite similar to its own. Neighbors whose directions deviate more than a fixed restriction angle $α$ are ignored. The collective dynamics of this systems is studied by agent-based simulations and kinetic mean field theory. We demonstrate that the reduction of the restriction angle leads to a critical noise amplitude decreasing monotonically with that angle, turning into a power law with exponent 3/2 for small angles. Moreover, for small system sizes we show that upon decreasing the restriction angle, the kind of the transition to polar collective motion changes from continuous to discontinuous. Thus, an apparent tricritical point is identified and calculated analytically. We also find that at very small interaction angles the polar ordered phase becomes unstable with respect to the apolar phase. We show that the mean-field kinetic theory permits stationary nematic states below a restriction angle of $0.681 π$. We calculate the critical noise, at which the disordered state bifurcates to a nematic state, and find that it is always smaller than the threshold noise for the transition from disorder to polar order. The disordered-nematic transition features two tricritical points: At low and high restriction angle the transition is discontinuous but continuous at intermediate $α$. We generalize our results to systems that show fragmentation into more than two groups and obtain scaling laws for the transition lines and the corresponding tricritical points. A novel numerical method to evaluate the nonlinear Fredholm integral equation for the stationary distribution function is also presented.

cond-mat.stat-mech↗
arXiv2014

Towards a quantitative kinetic theory of polar active matter

A recent kinetic approach for Vicsek-like models of active particles is reviewed. The theory is based on an exact Chapman-Kolmogorov equation in phase space. It can handle discrete time dynamics and "exotic" multi-particle interactions. A nonlocal mean-field theory for the one-particle distribution function is obtained by assuming molecular chaos. The Boltzmann approach of Bertin et al., Phys. Rev. E 74, 022101 (2006) and J. Phys. A: Math. Theor. 42, 445001 (2009), is critically assessed and compared to the current approach. In Boltzmann theory, a collision starts when two particles enter each others action spheres and is finished when their distance exceeds the interaction radius. The average duration of such a collision, $τ_0$, is measured for the Vicsek model with continuous time-evolution. If the noise is chosen to be close to the flocking threshold, the average time between collisions is found to be roughly equal to $τ_0$ at low densities. Thus, the continuous-time Vicsek-model near the flocking threshold cannot be accurately described by a Boltzmann equation, even at very small density because collisions take so long that typically other particles join in, rendering Boltzmann's binary collision assumption invalid. Hydrodynamic equations for the phase space approach are derived by means of a Chapman-Enskog expansion. The equations are compared to the Toner-Tu theory of polar active matter. New terms, absent in the Toner-Tu theory, are highlighted. Convergence problems of Chapman-Enskog and similar gradient expansions are discussed.

cond-mat.stat-mech↗
arXiv2013

Invasion-wave induced first-order phase transition in systems of active particles

An Enskog-like kinetic equation for self-propelled particles is solved numerically. I study a density instability near the transition to collective motion and find that while hydrodynamics breaks down, the kinetic approach leads to soliton-like supersonic waves with steep leading kinks and Knudsen numbers of order one. These waves show hysteresis, modify the transition threshold and lead to an abrupt jump of the global order parameter if the noise level is changed. Thus they provide a mechanism to change the second-order character of the phase transition to first order.

physics.bio-ph↗
arXiv2012

Kinetic theory for systems of self-propelled particles with metric-free interactions

A model of self-driven particles similar to the Vicsek model [Phys. Rev. Lett. 75 (1995) 1226] but with metric-free interactions is studied by means of a novel Enskog-type kinetic theory. In this model, N particles of constant speed v0 try to align their travel directions with the average direction of a fixed number of closest neighbors. At strong alignment a global flocking state forms. The alignment is defined by a stochastic rule, not by a Hamiltonian. The corresponding interactions are of genuine multi-body nature. The theory is based on a Master equation in 3N-dimensional phase space, which is made tractable by means of the molecular chaos approximation. The phase diagram for the transition to collective motion is calculated and compared to direct numerical simulations. A linear stability analysis of a homogeneous ordered state is performed using the kinetic but not the hydrodynamic equations in order to achieve high accuracy. In contrast to the regular metric Vicsek-model no instabilities occur. This confirms previous direct simulations that for Vicsek-like models with metric-free interactions, there is no formation of density bands and that the flocking transition is continuous.

cond-mat.stat-mech↗
arXiv2010

Kinetic Theory of Flocking: Derivation of Hydrodynamic Equations

It is shown how to explicitly coarse-grain the microscopic dynamics of the Vicsek model for self-propelled agents. The macroscopic transport equations are derived by means of an Enskog-type kinetic theory. Expressions for all transport coefficients at large particle speed are given. The phase transition from a disordered to a flocking state is studied numerically and analytically.

physics.bio-ph↗
arXiv2008

Transport coefficients of multi-particle collision algorithms with velocity-dependent collision rules

Detailed calculations of the transport coefficients of a recently introduced particle-based model for fluid dynamics with a non-ideal equation of state are presented. Excluded volume interactions are modeled by means of biased stochastic multiparticle collisions which depend on the local velocities and densities. Momentum and energy are exactly conserved locally. A general scheme to derive transport coefficients for such biased, velocity dependent collision rules is developed. Analytic expressions for the self-diffusion coefficient and the shear viscosity are obtained, and very good agreement is found with numerical results at small and large mean free paths. The viscosity turns out to be proportional to the square root of temperature, as in a real gas. In addition, the theoretical framework is applied to a two-component version of the model, and expressions for the viscosity and the difference in diffusion of the two species are given.

cond-mat.soft↗
arXiv2007

Mesoscopic model for the fluctuating hydrodynamics of binary and ternary mixtures

A recently introduced particle-based model for fluid dynamics with continuous velocities is generalized to model immiscible binary mixtures. Excluded volume interactions between the two components are modeled by stochastic multiparticle collisions which depend on the local velocities and densities. Momentum and energy are conserved locally, and entropically driven phase separation occurs for high collision rates. An explicit expression for the equation of state is derived, and the concentration dependence of the bulk free energy is shown to be the same as that of the Widom-Rowlinson model. Analytic results for the phase diagram are in excellent agreement with simulation data. Results for the line tension obtained from the analysis of the capillary wave spectrum of a droplet agree with measurements based on the Laplace's equation. The introduction of "amphiphilic" dimers makes it possible to model the phase behavior and dynamics of ternary surfactant mixtures.

cond-mat.soft↗
arXiv2006

Static and dynamic properties of a particle-based algorithm for non-ideal fluids and binary mixtures

A recently introduced particle-based model for fluid dynamics with effective excluded volume interactions is analyzed in detail. The interactions are modeled by means of stochastic multiparticle collisions which are biased and depend on local velocities and densities. Momentum and energy are exactly conserved locally. The isotropy and relaxation to equilibrium are analyzed and measured. It is shown how a discrete-time projection operator technique can be used to obtain Green-Kubo relations for the transport coefficients. Because of a large viscosity no long-time tails in the velocity auto-correlation and stress correlation functions were seen. Strongly reduced self-diffusion due to caging and an order/disorder transition is found at high collision frequency, where clouds consisting of at least four particles form a cubic phase. These structures were analyzed by measuring the pair-correlation function above and below the transition. Finally, the algorithm is extended to binary mixtures which phase-separate above a critical collision rate.

cond-mat.soft↗
arXiv2006

Dynamic correlations in stochastic rotation dynamics

The dynamic structure factor, vorticity and entropy density dynamic correlation functions are measured for Stochastic Rotation Dynamics (SRD), a particle based algorithm for fluctuating fluids. This allows us to obtain unbiased values for the longitudinal transport coefficients such as thermal diffusivity and bulk viscosity. The results are in good agreement with earlier numerical and theoretical results, and it is shown for the first time that the bulk viscosity is indeed zero for this algorithm. In addition, corrections to the self-diffusion coefficient and shear viscosity arising from the breakdown of the molecular chaos approximation at small mean free paths are analyzed. In addition to deriving the form of the leading correlation corrections to these transport coefficients, the probabilities that two and three particles remain collision partners for consecutive time steps are derived analytically in the limit of small mean free path. The results of this paper verify that we have an excellent understanding of the SRD algorithm at the kinetic level and that analytic expressions for the transport coefficients derived elsewhere do indeed provide a very accurate description of the SRD fluid.

cond-mat.soft↗
arXiv2005

Constructing thermodynamically consistent models with a non-ideal equation of state

A recently introduced particle-based model for fluid dynamics with continuous velocities is generalized to model fluids with excluded volume effects. This is achieved through the use of biased stochastic multi-particle collisions which depend on local velocities and densities and conserve momentum and kinetic energy. The equation of state is derived and criteria for the correct choice of collision probabilities are discussed. In particular, it is shown how a naive implementation can lead to inconsistent density fluctuations.

cond-mat.soft↗
arXiv2005

Equilibrium calculation of transport coefficients for a fluid-particle model

A recently introduced particle-based model for fluid flow, called Stochastic Rotation Dynamics, can be made Galilean invariant by introducing a random shift of the computational grid before collisions. In this paper, it is shown how the Green-Kubo relations derived previously can be resummed to obtain exact expressions for the collisional contributions to the transport coefficients. It is also shown that the collisional contribution to the microscopic stress tensor is not symmetric, and that this leads to an additional viscosity. The resulting identification of the transport coefficients for the hydrodynamic modes is discussed in detail, and it is shown that this does not impose restrictions on the applicability of the model. The collisional contribution to the thermal conductivity, which becomes important for small mean free path and small average particle number per cell, is also derived.

cond-mat.soft↗
arXiv2005

Consistent particle-based algorithm with a non-ideal equation of state

A thermodynamically consistent particle-based model for fluid dynamics with continuous velocities and a non-ideal equation of state is presented. Excluded volume interactions are modeled by means of biased stochastic multiparticle collisions which depend on the local velocities and densities. Momentum and energy are exactly conserved locally. The equation of state is derived and compared to independent measurements of the pressure. Results for the kinematic shear viscosity and self-diffusion constants are presented. A caging and order/disorder transition is observed at high densities and large collision frequency.

cond-mat.soft↗