Bernt Wennberg

• Doctor of Engineering
• Chalmers University of Technology

We consider the Lorentz gas in a distribution of scatterers which microscopically converges to a periodic distribution, and prove that the Lorentz gas in the low density limit satisfies a linear Boltzmann equation. This is in contrast with the periodic Lorentz gas, which does not satisfy the Boltzmann equation in the limit.

We consider the Lorentz gas in a distribution of scatterers which microscopically converges to a periodic distribution, and prove that the Lorentz gas in the low density limit satisfies a linear Boltzmann equation. This is in contrast with the periodic Lorentz gas, which does not satisfy the Boltzmann equation in the limit.

We consider a one dimension Kac model with conservation of energy and an exclusion rule: Fix a number of particles $n$, and an energy $E>0$. Let each of the particles have an energy $x_j \geq 0$, with $\sum_{j=1}^n x_j = E$. For some $\epsilon$, the allowed configurations $(x_1,\dots,x_n)$ are those that satisfy $|x_i - x_j| \geq \epsilon$ for all...

The spatially homogeneous BGK equation is obtained as the limit of a model of a many particle system, similar to Mark Kac’s charicature of the spatially homogeneous Boltzmann equation.

The mechanical properties of the extracellular matrix, in particular its stiffness, are known to impact cell migration. In this paper, we develop a mathematical model of a single cell migrating on an elastic matrix, which accounts for the deformation of the matrix induced by forces exerted by the cell, and investigate how the stiffness impacts the...

The spatially homogeneous BGK equation is obtained as the limit if a model of a many particle system, similar to Mark Kac's charicature of the spatially homogeneous Boltzmann equation.

The flocking of animals is often modelled as a dynamical system, in which individuals are represented as particles whose interactions are determined by the current state of the system. Many animals, however, including humans, have predictive capabilities, and presumably base their behavioural decisions---at least partially---upon an anticipated sta...

Collective motion in biology is often modelled as a dynamical system, in which individuals are represented as particles whose interactions are determined by the current state of the system. Many animals, however, including humans, have predictive capabilities, and presumably base their behavioural decisions---at least partially---upon an anticipate...

Context: Mathematical models may help the analysis of biological systems by providing estimates of otherwise un-measurable quantities such as concentrations and fluxes. The variability in such systems makes it difficult to translate individual characteristics to group behavior. Mixed effects models offer a tool to simultaneously assess individual...

We consider a Boltzmann model introduced by Bertin, Droz and Greegoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the equilibria of this Boltzmann model and we rigorous...

We study a modified Kac model where the classical kinetic energy is replaced by an arbitrary energy function $\phi(v)$, $v\in\mathbb{R}$. The aim of this paper is to show that the uniform distribution with respect to the microcanonical measure is $Ce^{-z_0\phi(v)} $-chaotic, $C,z_0\in\mathbb{R}_{+}$. The kinetic energy for relativistic particles is...

The Kac model is a simplified model of an $N$-particle system in which the collisions of a real particle system are modeled by random jumps of pairs of particle velocities. Kac proved propagation of chaos for this model, and hence provided a rigorous validation of the corresponding Boltzmann equation. Starting with the same model we consider an $N$...

The "War of Attrition" is a classical game theoretic model that was first introduced to mathematically describe certain non-violent animal behavior. The original setup considers two participating players in a one-shot game competing for a given prize by waiting. This model has later been extended to several different models allowing more than two p...

We study Boltzmann's collision operator for long-range interactions, i.e., without Grad's angular cut-off assumption. We establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function, in a precise sense. Our estimate is optimal, and gives a unified treatment of both the linear and the nonlin...

The Lorentz gas is a model for a cloud of point particles (electrons) in a distribution of scatterers in space. The scatterers are often assumed to be spherical with a fixed diameter d, and the point particles move with constant velocity between the scatterers, and are specularly reflected when hitting a scatterer. There is no interaction between p...

We consider two models of biological swarm behavior. In these models, pairs of particles interact to adjust their velocities one to each other. In the first process, called 'BDG', they join their average velocity up to some noise. In the second process, called 'CL', one of the two particles tries to join the other one's velocity. This paper establi...

Non-linear mixed effects (NLME) models represent a powerful tool to simultaneously analyse data from several individuals. In this study, a compartmental model of leucine kinetics is examined and extended with a stochastic differential equation to model non-steady-state concentrations of free leucine in the plasma. Data obtained from tracer/tracee e...

We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair-interaction driven master equations. In the spatially homogeneous case, we prove a propagation of c...

The paper discusses a family of Markov processes that represent many particle systems, and their limiting behaviour when the number of particles go to infinity. The first part concerns model of biological systems: a model for sympatric speciation, i.e. the process in which a genetically homogeneous population is split in two or more different speci...

This paper is devoted the the study of the mean field limit for many-particle systems undergoing jump, drift or diffusion processes, as well as combinations of them. The main results are quantitative estimates on the decay of fluctuations around the deterministic limit and of correlations between particles, as the number of particles goes to infini...

To simultaneously model plasma dilution and urinary output in female volunteers. Ten healthy female non-pregnant volunteers, aged 21-39 years (mean 29), with a bodyweight of 58-67 kg (mean 62.5 kg) participated. No oral fluid or food was allowed between midnight and completion of the experiment. The protocol included an infusion of acetated Ringer'...

The observability of nonlinear delay systems has previously been defined in an algebraic setting by a rank condition on modules over noncommutative rings. We introduce an analytic definition of observability to ensure the local uniqueness of state and initial conditions that correspond to a given input–output behaviour. It is shown that an algebrai...

Sympatric speciation, i.e., the evolutionary split of one species into two in the same environment, has been a highly troublesome concept. It has been a questioned if it is actually possible. Even though there have been a number of reported results both in the wild and from controlled experiments in laboratories, those findings are both hard to get...

Recently a new class of Monte Carlo methods, called Time Relaxed Monte Carlo (TRMC), designed for the simulation of the Boltzmann equation close to fluid regimes have been introduced. A generalized Wild sum expansion of the solution is at the basis of the simulation schemes. After a splitting of the equation the time discretization of the collision...

We have analyzed the identifiability of time-lag parameters in nonlinear delay systems using an algebraic framework. The identifiability is determined by the form of the system’s input-output representation. The values of the time lags can be found directly from the input-output equations, if these can be obtained explicitly. Linear-algebraic crite...

The identifiability of the delay parameter for nonlinear systems with a single constant time delay is analyzed. We show the existence of input-output equations and relate the identifiability of the delay parameter to their form. Explicit criteria based on rank calculations are formulated. The identifiability of the delay parameter is shown not to b...

The non-cutoff Boltzmann equation can be simulated using the DSMC method, by a truncation of the collision term. However, even for computing stationary solutions this may be very time consuming, in particular in situations far from equilibrium. By adding an appropriate diffusion, to the DSMC-method, the rate of convergence when the truncation is re...

New experimental techniques in bioscience provide us with high-quality data allowing quantitative mathematical modelling. Parameter estimation is often necessary and, in connection with this, it is important to know whether all parameters can be uniquely estimated from available data, (i.e. whether the model is identifiable). Dealing essentially wi...

We study the stationary states of a Kac equation with a Gaussian thermostat in the case of a noncutoff cross section. We investigate the existence, smoothness and uniqueness of the stationary states. The theoretical results are illustrated by some numerical simulations.

The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties eve...

We derive coupled mass and energy balance laws from a High-Field limit of thermostatted Boltzmann equations. The starting point is a Boltzmann equation for elastic collisions subjected to a large force field. By adding a thermostat correction, it is possible to expand the solutions about a High-Field equilibrium obtained when balancing the thermost...

We consider the Kac equation with a thermostatted force field and prove the existence of a global in time solution that converges weakly to a stationary state. As there is no an obvious candidate for the entropy functional, in this case, the convergence result is obtained via Fourier transform techniques.

The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties eve...

We sought to compare the synthesis and metabolism of VLDL1 and VLDL2 in patients with type 2 diabetes mellitus (DM2) and nondiabetic subjects. We used a novel multicompartmental model to simultaneously determine the kinetics of apolipoprotein (apo) B and triglyceride (TG) in VLDL1 and VLDL2 after a bolus injection of [2H3]leucine and [2H5]glycerol...

The use of stable isotopes in conjunction with compartmental modeling analysis has greatly facilitated studies of the metabolism of the apolipoprotein B (apoB)-containing lipoproteins in humans. The aim of this study was to develop a multicompartment model that allows us to simultaneously determine the kinetics of apoB and triglyceride (TG) in VLDL...

The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties eve...

this paper we consider a scaling which is intermediary between the case considered in [CPR] and in [BGW]: the scatterers still have radius #, but the lattice parameter is # , where 1/2 < # 1. In order to achieve a proper Boltzmann-Grad limit, the probability of finding a scatterer at a lattice site must be # 2#-1 ; we have found it convenient to wr...

We study the stationary Kac equation in the presence of an external force field and a Gaussian thermostat, and investigate the behavior of a solution to this equation for varying field strength. We prove that for a weak field, the stationary density is continuous, but for a strong field the density has a singularity. Monte Carlo simulations illustr...

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space S(R ). The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, wh...

 The paper considers the stability and strong convergence to equilibrium of solutions to the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. Under a cutoff condition on the collision kernel, we prove a strong stability in L 1 topology at any finite time interval, and, for hard and Maxwellian potentials, we prove that the soluti...

We present a recursive Monte Carlo method for the numerical solution of the Boltzmann equation. The method is based on the Wild sum expansion of the solution in the Maxwellian case. The recursive structure is used efficiently to obtain uniform accuracy in time, a very desirable property in many practical applications. Numerical examples of some spa...

Consider the domain $Z_\epsilon=\{x\in\mathbb{R}^n ; {dist}(x,\epsilon\mathbb{Z}^n)> \epsilon^\gamma\}$ and let the free path length be defined as $\tau_\epsilon(x,v)=\inf\{t> 0 ; x-tv\in Z_\epsilon\}.$ In the Boltzmann-Grad scaling corresponding to $\gamma=\frac{n}{n-1}$, it is shown that the limiting distribution $\phi_\epsilon$ of $\tau_\epsilon...

The paper deals with the spatially homogeneous Boltzmann equation for hard potentials. An example is given which shows that, even though it is known that there is only one solution that conserves energy, there may be other solutions for which the energy is increasing; uniqueness holds if and only if energy is assumed to be conserved 1. Introduction...

An abstract is not available.

We consider the question of existence and uniqueness of solutions to the spatially homogeneous Boltzmann equation. The main result is that to any initial data with finite mass and energy, there exists a unique solution for which the same two quantities are conserved. We also prove that any solution which satisfies certain bounds on moments of order...

The paper deals with the spatially homogeneous Boltzmann equation for hard potentials. An example is given which shows that, even though it is known that there is only one solution that conserves energy, there may be other solutions for which the energy is increasing; uniqueness holds if and only if energy is assumed to be conserved.

Consider the domain Z(epsilon) = {x is an element of R-n/dist(x, epsilon Z(n)) > epsilon(gamma)}, and let the free path length be defined as tau(epsilon)(x, omega) = inf{t > 0 \x - t omega is an element of Z(epsilon)}. The distribution of values of tau(epsilon) is studied in the limit as epsilon --> 0 for all gamma greater than or equal to 1. It is...

Elastic collisions are characterized by the conservation of momentum and energy. We consider some geometrical aspects of such collisions, when the energy of one particle can be expressed in terms of the moments as . The geometry of elastic collisions is essential for the regularizing property of the gain term in the Boltzmann equation, which was pr...

LetH(f/M)=flog(f/M)dv be the relative entropy off and the Maxwellian with the same mass, momentum, and energy, and denote the corresponding entropy dissipation term in the Boltzmann equation byD(f)=Q(f,f) logf dv. An example is presented which shows that |D(f)/H(f/M)| can be arbitrarily small. This example is a sequence of isotropic functions, and...

We show, for a particular collision kernel, that the entropy tends to its equilibrium limit for all solutions of the spatially homogeneous Boltzmann equation with finite energy, finite entropy initial data. This verses the natural conjecture following from the Boltzmann H-theorem. All previous results in this direction placed additional restriction...

We prove that the solution of the spatially homogeneous Boltzmann equation is bounded pointwise from below by a Maxwellian, i.e. a function of the formc 1 exp(-c 2v 2). This holds for any initial data with bounded mass, energy and entropy, and for any positive timet≧t 0. The constantsc 1, andc 2, depend on the mass, energy and entropy of the initia...

We prove that the solution of the space homogeneous Boltzmann equation is bounded from below by an exponential C exp{– ―v ―2+λ}, where λ > 0 is arbitrary. The estimate holds uniformly for all t ≥ to, where to is a given positive time.

This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has pre...

We prove existence, uniqueness and stability for solutions of the nonlinear Boltzmann equation in a periodic box in the case when the initial data are sufficiently close to a spatially homogeneous function. The results are given for a range of spaces, including L 1, and extend previous results in L for the non-homogeneous equation, as well as the m...

Recently Desvillettes proved that the solutions to the space homogeneous Boltzmann equation possess all moments, provided that the initial data have 2 + moments in L 1. Here an alternative proof of this statement is presented, together with a proof that the result does not hold for pseudo-Maxwellian molecules. Two implications of the result are dis...

We consider an initial-boundary value problem for the Boltzmann equation with a source term. The spatial domain is assumed to be a rectangular box with specularly reflecting boundaries. It is proved that this problem possesses a unique solution globally in time, provided initial data are sufficiently close to a spatially homogeneous function, and t...

We study the stationary states of a Kac equation with a Gaussian thermostat in the case of a non-cuto cross section. We investigate the existence, smoothness and uniqueness of the stationary states. The theoretical results are illustrated by some numerical simulations.

The topic is the evolution of moments of solutions of the spatially homogeneous Boltzmann equation. The main result is that if the kernel in the collision operator is unbounded (as for molecules interacting with hard potentials, for example) and if the mass, energy and entropy of the initial data are bounded, then the solution of the Boltzmann equa...