A $N$-uniform quantitative Tanakas theorem for the conservative Kacs $N$-particle system with Maxwell molecules

This paper considers the space homogenous Boltzmann equation with Maxwell molecules and arbitrary angular distribution. Following Kacs program, emphasis is laid on the the associated conservative Kacs stochastic $N$-particle system, a Markov process with binary collisions conserving energy and total momentum. An explicit Markov coupling (a probabilistic, Markovian coupling of two copies of the process) is constructed, using simultaneous collisions, and parallel coupling of each binary random collision on the sphere of collisional directions. The euclidean distance between the two coupled systems is almost surely decreasing with respect to time, and the associated quadratic coupling creation (the time variation of the averaged squared coupling distance) is computed explicitly. Then, a family (indexed by $delta > 0$) of $N$-uniform weak coupling / coupling creation inequalities are proven, that leads to a $N$-uniform power law trend to equilibrium of order ${sim}_{ t to + infty} t^{-delta} $, with constants depending on moments of the velocity distributions strictly greater than $2(1 + delta)$. The case of order $4$ moment is treated explicitly, achieving Kacs program without any chaos propagation analysis. Finally, two counter-examples are suggested indicating that the method: (i) requires the dependance on $>2$-moments, and (ii) cannot provide contractivity in quadratic Wasserstein distance in any case.

Scalable and Quasi-Contractive Markov Coupling of Maxwell Collision

This paper considers space homogenous Boltzmann kinetic equations in dimension $d$ with Maxwell collisions (and without Grads cut-off). An explicit Markov coupling of the associated conservative (Nanbu) stochastic $N$-particle system is constructed, using plain parallel coupling of isotropic random walks on the sphere of two-body collisional directions. The resulting coupling is almost surely decreasing, and the $L_2$-coupling creation is computed explicitly. Some quasi-contractive and uniform in $N$ coupling / coupling creation inequalities are then proved, relying on $2+alpha$-moments ($alpha >0$) of velocity distributions; upon $N$-uniform propagation of moments of the particle system, it yields a $N$-scalable $alpha$-power law trend to equilibrium. The latter are based on an original sharp inequality, which bounds from above the coupling distance of two centered and normalized random variables $(U,V)$ in $R^d$, with the average square parallelogram area spanned by $(U-U_ast,V-V_ast)$, $(U_ast,V_ast)$ denoting an independent copy. Two counter-examples proving the necessity of the dependance on $>2$-moments and the impossibility of strict contractivity are provided. The paper, (mostly) self-contained, does not require any propagation of chaos property and uses only elementary tools.

On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes

This paper considers Schrodinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of fermion groundstates, defined by the bottom eigenelements of the Schrodinger operator of a fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that the shape derivative of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.

Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction

We discuss variance reduced simulations for an individual-based model of chemotaxis of bacteria with internal dynamics. The variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations. In the companion paper cite{limits}, we have rigorously shown, using a pathwise probabilistic technique, that both processes converge towards the same advection-diffusion process in the diffusive asymptotics. In this work, a direct coupling is achieved between paths of individual bacteria simulated by both models, by using the same sets of random numbers in both simulations. This coupling is used to construct a hybrid scheme with reduced variance. We first compute a deterministic solution of the kinetic density description of the direct gradient sensing model; the deviations due to the presence of internal dynamics are then evaluated via the coupled individual-based simulations. We show that the resulting variance reduction is emph{asymptotic}, in the sense that, in the diffusive asymptotics, the difference between the two processes has a variance which vanishes according to the small parameter.

Individual-based models for bacterial chemotaxis in the diffusion asymptotics

We discuss velocity-jump models for chemotaxis of bacteria with an internal state that allows the velocity jump rate to depend on the memory of the chemoattractant concentration along their path of motion. Using probabilistic techniques, we provide a pathwise result that shows that the considered process converges to an advection-diffusion process in the (long-time) diffusion limit. We also (re-)prove using the same approach that the same limiting equation arises for a related, simpler process with direct sensing of the chemoattractant gradient. Additionally, we propose a time discretization technique that retains these diffusion limits exactly, i.e., without error that depends on the time discretization. In the companion paper cite{variance}, these results are used to construct a coupling technique that allows numerical simulation of the process with internal state with asymptotic variance reduction, in the sense that the variance vanishes in the diffusion limit.