THE RANDOM BATCH METHOD FOR N-BODY QUANTUM DYNAMICS

This paper discusses a numerical method for computing the evolution of large inter-acting system of quantum particles.The idea of the random batch method is to replace the total interaction of each particle with the N-1 other particles by the interaction with p ＜＜ N particles chosen at random at each time step,multiplied by (N-1)/p.This re-duces the computational cost of computing the interaction potential per time step from O(N2) to O(N).For simplicity,we consider only in this work the case p =1 — in other words,we assume that N is even,and that at each time step,the N particles are orga-nized in N/2 pairs,with a random reshuffling of the pairs at the beginning of each time step.We obtain a convergence estimate for the Wigner transform of the single-particle reduced density matrix of the particle system at time t that is both uniform in N ＞ 1 and independent of the Planck constant h.The key idea is to use a new type of distance on the set of quantum states that is reminiscent of the Wasserstein distance of exponent 1 (or Monge-Kantorovich-Rubinstein distance) on the set of Borel probability measures on Rd used in the context of optimal transport.     

Reflected mean-field backward stochastic differential equations. Approximation and associated nonlinear PDEs

Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections. On the other hand, we will prove that this type of reflected mean-field BSDEs can also be obtained as the limit equation of the mean-field BSDEs by penalization method. Finally, we give the probabilistic interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean-field BSDEs.     

A maximum principle for fully coupled stochastic control systems of mean-field type

The present paper considers an optimal control problem for fully coupled forward–backward stochastic differential equations (FBSDEs) of mean-field type in the case of controlled diffusion coefficient. Moreover, the control domain is not assumed to be convex. By virtue of a reduction method, we establish the necessary optimality conditions of Pontryagin's type. As an application, a linear–quadratic stochastic control problem is studied.     

Open-source tools for dynamical analysis of Liley's mean-field cortex model

Mean-field models of the mammalian cortex treat this part of the brain as a two-dimensional excitable medium. The electrical potentials, generated by the excitatory and inhibitory neuron populations, are described by nonlinear, coupled, partial differential equations that are known to generate complicated spatio-temporal behaviour. We focus on the model by Liley et al. (Network: Computation in Neural Systems 13 (2002) 67–113). Several reductions of this model have been studied in detail, but a direct analysis of its spatio-temporal dynamics has, to the best of our knowledge, never been attempted before. Here, we describe the implementation of implicit time-stepping of the model and the tangent linear model, and solving for equilibria and time-periodic solutions, using the open-source library PETSc. By using domain decomposition for parallelization, and iterative solving of linear problems, the code is capable of parsing some dynamics of a macroscopic slice of cortical tissue with a sub-millimetre resolution.     

Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions

We consider a system of real-valued spins interacting with each other through a mean-field Hamiltonian that depends on the empirical magnetisation of the spins. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. Using large deviation theory we show that there exists an explicitly computable crossover time $t_c\in [0,\infty ]$ from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness ($t_c=0$), short-time conservation and large-time loss of Gibbsianness ($t_c\in (0,\infty )$), and preservation of Gibbsianness ($t_c=\infty$). Depending on the potential, the system can be Gibbs or non-Gibbs at the crossover time $t=t_c$.     

Propagation of chaos for the Keller–Segel equation over bounded domains

In this paper we rigorously justify the propagation of chaos for the parabolic–elliptic Keller–Segel equation over bounded convex domains. The boundary condition under consideration is the no-flux condition. As intermediate steps, we establish the well-posedness of the associated stochastic equation as well as the well-posedness of the Keller–Segel equation for bounded weak solutions.     

Collective behavior models with vision geometrical constraints: Truncated noises and propagation of chaos

We consider large systems of stochastic interacting particles through discontinuous kernels which has vision geometrical constrains. We rigorously derive a Vlasov–Fokker–Planck type of kinetic mean-field equation from the corresponding stochastic integral inclusion system. More specifically, we construct a global-in-time weak solution to the stochastic integral inclusion system and derive the kinetic equation with the discontinuous kernels and the inhomogeneous noise strength by employing the 1-Wasserstein distance.     

带合作行为的平均场模型的依全变差稳定性

该文考虑一个带合作行为的平均场模型的稳定性问题. 应用耦合方法, 建立了相应于这个平均场模型的扩散过程的依全变差稳定性.     

Stochastic Schroedinger equation from optimal observable evolution

In this article, we consider a set of trial wave-functions denoted by |Q〉 and an associated set of operators A_α which generate transformations connecting those trial states. Using variational principles, we show that we can always obtain a quantum Monte-Carlo method where the quantum evolution of a system is replaced by jumps between density matrices of the form D = |Q_a〉〈Q_b|, and where the average evolutions of the moments of A_α up to a given order k, i.e., 〈A_α1〉,〈A_α1A_α2〉,…,〈A_α1?A_αk〉, are constrained to follow the exact Ehrenfest evolution at each time step along each stochastic trajectory. Then, a set of more and more elaborated stochastic approximations of a quantum problem is obtained which approach the exact solution when more and more constraints are imposed, i.e., when k increases. The Monte-Carlo process might even become exact if the Hamiltonian H applied on the trial state can be written as a polynomial of A_α. The formalism makes a natural connection between quantum jumps in Hilbert space and phase-space dynamics. We show that the derivation of stochastic Schroedinger equations can be greatly simplified by taking advantage of the existence of this hierarchy of approximations and its connection to the Ehrenfest theorem. Several examples are illustrated: the free wave-packet expansion, the Kerr oscillator, a generalized version of the Kerr oscillator, as well as interacting bosons or fermions.