Ring Kinetic Theory Research Articles

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

Recently, Hanke et al. [Phys. Rev. E 88, 052309 (2013)] showed that mean-field kinetic theory fails to describe collective motion in soft active colloids and that correlations must not be neglected. Correlation effects are also expected to be essential in systems of biofilaments driven by molecular motors and in swarms of midges. To obtain correlations in an active matter system from first principles, we derive a ring-kinetic theory for Vicsek-style models of self-propelled agents 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 precollisional correlations and cluster formation, which are both 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 Bogoliubov-Born-Green-Kirkwood-Yvon (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 are 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 an exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed.

Spatial correlations in dilute granular flows: A kinetic model study

The development of spatial correlations in the hydrodynamic fields of a freely evolving granular fluid is studied. The analysis is based on a model kinetic theory for dilute granular gases, which includes an equation for the pair correlation function. For small inelasticity, the results agree with those derived previously from fluctuating hydrodynamics, and also with preliminary calculations using ring kinetic theory for inelastic hard spheres. A critical discussion of the domain of validity of a perturbative calculation of the hydrodynamic modes is given.

Ring kinetic theory for an idealized granular gas

The dynamics of inelastic hard spheres is described in terms of the binary collision expansion, yielding the corresponding pseudo-Liouville equation and BBGKY hierarchy for the reduced distribution functions. Based on cluster expansion techniques we derive the Boltzmann and ring kinetic equations for inelastic hard spheres. In the simple ring approximation, we calculate the structure factor S ⊥( k, t) of vorticity fluctuations in a freely evolving, dilute granular gas. The kinetic theory result agrees with the result derived previously from fluctuating hydrodynamics. If the fluctuations in the flow field can be considered incompressible, S ⊥( k, t) determines the spatial correlations in the flow velocities, which are of dynamic origin and exhibit long range r − d -behavior. The analytic results are compared with molecular dynamics simulations.

Chaotic properties of a one-dimensional Lorentz gas

We present some chaotic properties of a one-dimensional Lorentz lattice gas (a particle moving on a lattice among fixed probabilistic scatterers) in the frame of the thermodynamical formalism. Mean field theory has allowed us to predict the escape rate and the Lyapunov exponent, but it fails for the topological entropy. A reasonable improvement is obtained by taking correlations into account (ring kinetic theory), the theoretical results are compared to numerical simulations.

Velocity autocorrelation function in lattice gases from the ring kinetic theory. Comparison with numerical simulations

We obtain the complete time dependence of the velocity autocorrelation function (VACF) for lattice gas cellular automata, usingring kinetic theory. This theory accounts for the simplest correlated collisions that improve on the molecular chaos approach, and yields a closed equation for the VACF that we evaluate for both infinite and finite systems. We compare our analytical results with numerical simulations at all times, as well as with long-time results of the mode coupling theories, finding a very good agreement for all times at all densities.

Stress-stress correlation functions in lattice gases beyond Boltzmann's approximation

The complete time dependence of the stress-stress correlation functions in lattice gas cellular automata is calculated from the ring kinetic theory using numerical and analytical methods. This provides corrections, typically of 10-20%, to the usual molecular chaos calculations, where correlation functions decay exponentially. The resulting correlation function crosses over from an initial exponential decay to the long-time behavior calculated from mode coupling theory. The present theory, applied to the viscosity, accounts for a substantial part of the observed difference between the Boltzmann theory and simulations.

Ring kinetic theory for tagged-particle problems in lattice gases.

The kinetic theory for tagged-particle problems in lattice-gas cellular automata is extended beyond Boltzmann's mean-field approximation by including correlated ring-type collisions. This theory provides explicit expressions for the velocity autocorrelation function (VACF) for all times in terms of the ring-collision integral, as well as corrections to the Boltzmann values of the transport coefficients. For times long compared to the mean free time, the ring integral equation yields the phenomenological mode-coupling theory and the long-time tails. For intermediate times it describes a slow transition from initial exponential decay to the long-time tails. At short times the ring kinetic theory is exact. In particular, deviations from the Boltzmann result in the VACF of three-dimensional systems after two time steps are calculated explicitly and compared with computer simulations.

Microscopic theory of condensed phase chemical reactions. I. Pair phase space kinetic equation

A repeated ring kinetic theory describing the reversible chemical reaction A+B?C+D is obtained. We derive a kinetic equation for the equilibrium time correlation function of the solute pair phase space density. The solute–solvent and solvent–solvent interactions are described by a hard sphere potential. The solute–solute interactions consists of both hard and soft potential terms. This choice permits the use of a pseudo-Liouville operator formalism to obtain the kinetic theory. This greatly simplifies the derivation by comparison with a Liouville operator formalism for continuous potentials. The solute is dilute such that the kinetic theory describes isolated solute pairs interacting with the solvent. The static memory function of the kinetic equation is treated exactly. It describes solute propagation induced by direct interaction, Enskog collisions with the solvent and by the potential of mean force operating between the solute pair. The dynamic memory function is obtained at the repeated ring level of approximation. It describes solute propagation by dynamically correlated collisions (collective effects). Specialization of the reactive theory to exclusively hard sphere interactions is given. Also, nonreactive kinetic equations with elastic hard and/or soft solute–solute interactions are presented. In a companion paper we begin an analysis of the configuration space projection of these phase space kinetic equations and the associated transport coefficients of diffusion and reaction.

Kinetic energy relaxation of a test particle in a dense fluid

The rate constant kε for the relaxation of the kinetic energy of a solute test hard sphere in a dense hard sphere solvent is studied. Microscopic boundary layer and ring kinetic theory methods are used to construct a simple expression for kε. It is found that kε−1 has three additive contributions arising from (a) uncorrelated binary collisions, (b) coupling to the solvent energy density, and (c) coupling to the solvent shear modes. Thus kε−1 is determined by the Enskog collisional relaxation rate and the rates of the hydrodynamic flows in the solvent of heat and momentum. The competition between these effects is studied as a function of particle size and mass ratios and solvent density. The regimes examined at higher density vary from the identical solute–solvent case, where kε is mainly collisional with a small hydrodynamic heat flow correction, to the large, massive solute case, where kε is governed by solvent momentum flow according to a Stokes law type relation.

Kinetic theory derivation of a pair configuration space diffusion equation

A repeated ring kinetic theory is derived to describe the thermal fluctuations of two test particles in a fluid. The kinetic equation incorporates the interaction of the two particles via the potential of the mean force, their uncorrelated Enskog propagation, and multiply correlated collisions among the test particles and the bath particles. By projecting on the configuration space of the test particles, an equation of the form of a diffusion equation is obtained, but with a microscopically defined diffusion tensor. The configuration and time dependent diffusion tensor is given by a time correlation function and can be evaluated for a given molecular model.

A stochastic theory of translational correlation functions in gases and liquids

This paper presents a theory of translational relaxation in hard sphere systems. We construct self-equilibrium time correlation functions for the self-density and for the velocity of a molecule by techniques related to the theory of stochastic point processes. Our stochastic theory yields, at low density, the Boltzmann–Enskog kinetic theory. At intermediate densities we obtain enhancement (t−3/2 decay) and at high density reduction of the velocity autocorrelation function relative to the Boltzmann–Enskog exponential decay in accord with molecular dynamics simulations. The self-diffusion coefficient is correspondingly enhanced and reduced. We also compare our formulation with kinetic theories based on the BBGKY hierarchy. In particular, the relationship of our theory to ring kinetic theory is explored.