Cells and organisms follow aligned structures in their environment, a process that can generate persistent migration paths. Kinetic transport equations are a popular modelling tool for describing biological movements at the mesoscopic level, yet their formulations usually assume a constant turning rate. Here we relax this simplification, extending to include a turning rate that varies according to the anisotropy of a heterogeneous environment. We extend known methods of parabolic and hyperbolic scaling and apply the results to cell movement on micropatterned domains. We show that inclusion of orientation dependence in the turning rate can lead to persistence of motion in an otherwise fully symmetric environment and generate enhanced diffusion in structured domains.

Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision model.

We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel.

The fine description of complex fluids can be carried out by describing the evolution of each individual constituent (e.g. each particle, each macromolecule, etc.). This procedure, despite its conceptual simplicity, involves many numerical issues, the most challenging one being that related to the computing time required to update the system configuration by describing all the interactions between the different individuals. Coarse grained approaches allow alleviating the just referred issue: the system is described by a distribution function providing the fraction of entities that at certain time and position have a particular conformation. Thus, mesoscale models involve many different coordinates, standard space and time, and different conformational coordinates whose number and nature depend on the particular system considered. Balance equation describing the evolution of such distribution function consists of an advection-diffusion partial differential equation defined in a high dimensional space. Standard mesh-based discretization techniques fail at solving high-dimensional models because of the curse of dimensionality. Recently the authors proposed an alternative route based on the use of separated representations. However, until now these approaches were unable to address the case of advection dominated models due to stabilization issues. In this paper this issue is revisited and efficient procedures for stabilizing the advection operators involved in the Boltzmann and Fokker-Planck equation within the PGD framework are proposed.

This paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one- and two-dimensional flow simulations, which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution. The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the “initial” cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface. It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory. The generalized moment limiter is employed there to suppress the possible numerical oscillation. In the second part, the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level, i.e. free transport and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order flux evolution is based on the solution (i.e. the particle velocity distribution function) of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme. By comparing with the exact solutions or the numerical solutions obtained the second-order or third-order accurate gas-kinetic scheme, the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows, and the accuracy of the high-order GKS depends on the choice of the (numerical) collision time.

Discrete-velocity approximations represent a popular way for computing the Boltzmanncollision operator. The direct numerical evaluation of such methods involve a prohibitivecost, typically O(N2d + 1)where d is the dimension of the velocity space. In this paper, followingthe ideas introduced in [C. Mouhot and L. Pareschi, C. R. Acad. Sci. Paris Sér. IMath. 339 (2004) 71–76, C. Mouhot and L. Pareschi, Math.Comput. 75 (2006) 1833–1852], we derive fast summation techniquesfor the evaluation of discrete-velocity schemes which permits to reduce the computationalcost from O(N2d + 1) to O(N̅dNd log2N), N̅ ≪ N, with almost no loss of accuracy.

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and L∞ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

The study of the fluctuations in the steady state of a heated granular system isreviewed. A Boltzmann-Langevin description can be built requiring consistency with theequations for the one- and two-particle correlation functions. From the Boltzmann-Langevinequation, Langevin equations for the total energy and the transverse velocity field arederived. The existence of a fluctuation-dissipation relation for the transverse velocityfield is also studied.

An overview of recent results pertaining to the hydrodynamic description (both Newtonianand non-Newtonian) of granular gases described by the Boltzmann equation for inelasticMaxwell models is presented. The use of this mathematical model allows us to get exactresults for different problems. First, the Navier–Stokes constitutive equations withexplicit expressions for the corresponding transport coefficients are derived by applyingthe Chapman–Enskog method to inelastic gases. Second, the non-Newtonian rheologicalproperties in the uniform shear flow (USF) are obtained in the steady state as well as inthe transient unsteady regime. Next, an exact solution for a special class of Couetteflows characterized by a uniform heat flux is worked out. This solution shares the samerheological properties as the USF and, additionally, two generalized transportcoefficients associated with the heat flux vector can be identified. Finally, the problemof small spatial perturbations of the USF is analyzed with a Chapman–Enskog-like methodand generalized (tensorial) transport coefficients are obtained.

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure on R. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets of R). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variables x̃1, x̃2, …, where these random variables have an infinite second moment and zero mean. Then, with Tn := ∑j=1ηnλj,nx̃j, with max1 ≤ j ≤ ηnλj,n → 0 (as n → +∞), and ∑j=1ηnλj,n2 = 1, n = 1, 2, …, the classical central limit theorem suggests that T should in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

We consider a body immersed in a perfect gas and moving under the action of a constant force.Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity $V_\infty$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $$|V(t)-V_\infty|\approx \frac{C}{t^{d+1}},$$ where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions.

In this paper, we present some interesting connections between anumber of Riemann-solver free approaches to the numerical solutionof multi-dimensional systems of conservation laws. As a main part,we present a new and elementary derivation of Fey's Method ofTransport (MoT) (respectively the second author's ICE version ofthe scheme) and the state decompositions which form the basis of it.The only tools that we use are quadrature rules applied to themoment integral used in the gas kinetic derivation of the Eulerequations from the Boltzmann equation, to the integration in timealong characteristics and to space integrals occurring in the finitevolume formulation. Thus, we establish a connection between theMoT approach and the kinetic approach. Furthermore,Ostkamp's equivalence result between her evolution Galerkin schemeand the method of transport is lifted up from the level ofdiscretizations to the level of exact evolution operators,introducing a new connection between the MoT and theevolution Galerkin approach. At the same time, we clarifysome important differences between these two approaches.

We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.

In this paper we introduce numerical schemes for aone-dimensional kinetic model of the Boltzmann equation withdissipative collisions and variable coefficient of restitution. Inparticular, we study the numerical passage of the Boltzmannequation with singular kernel to nonlinear friction equations inthe so-called quasi elastic limit. To this aim we introduce aFourier spectral method for the Boltzmann equation [CITE]and show that the kernel modes that define the spectral methodhave the correct quasi elastic limit providing a consistentspectral method for the limiting nonlinear friction equation.

We have performed a kinetic study of the electron dynamic relaxation inside a Au film subjected to a subpicosecond laser pulse. For this purpose, we have developed a time-dependent numerical solution of the Boltzmann equation for the electrons inside the film considering the collision integrals due to electron–electron and electron–phonon collisions and a perturbation term due to the laser pulse. Our results show that, after the pulse excitation, electron distributions are very far from equilibrium. Therefore it is not possible, especially in the first part of the temporal evolution, to describe the relaxation of the electron distribution through a two-temperature model.

We state and prove a Korn-like inequality for a vector field in abounded open set of $\mathbb{R}^N$ , satisfying a tangency boundary condition.This inequality, which is crucial in our study of the trend towardsequilibrium for dilute gases, holds true if and only if the domain is notaxisymmetric. We give quantitative, explicit estimates on how thedeparture from axisymmetry affects the constants; a Monge–Kantorovichminimization problem naturally arises in this process. Variants in the axisymmetric case are briefly discussed.

Previous models on low-dimensional thermoelectric investigation deal with the quasi twodimensional electron transport due to quantum confinement effect. The formation of sub-bands in quantum well requires that electron wave reflections or transmissions at the interface are strictly in the specular direction and the superimposed wave function keeps phase coherence. However, due to the interface non-ideality or roughness, electrons can lose coherence such that their transport will deviate from that described by two-dimensional quantum well limit theories. In this paper, we report a theoretical approach to investigate the classical size effect on in-plane thermoelectric transport at low dimensions. A theoretical model based on Boltzmann equation is established with interface scattering treated as partial specular and partial diffuse scattering boundary condition. With the infinite quantum well assumption, the classical size effect in the quantum-classical mixed regime is quantitatively demonstrated. Factors that affecting classical size effect, such as quantum well width and relaxation length, are discussed.