Weakly coupled heat bath models for Gibbs-like invariant states in nonlinear wave equations

Abstract

Thermal bath coupling mechanisms as utilized in molecular dynamics are applied to partial differential equation models. Working from a semi-discrete (Fourier mode) formulation for the Burgers or KdV equation, we introduce auxiliary variables and stochastic perturbations in order to drive the system to sample a target ensemble which may be a Gibbs state or, more generally, any smooth distribution defined on a constraint manifold. We examine the ergodicity of approaches based on coupling of the heat bath to the high wave numbers, with the goal of controlling the ensemble through the fast modes. We also examine different thermostat methods in the extent to which dynamical properties are corrupted in order to accurately compute the average of a desired observable with respect to the invariant distribution. The principal observation of this paper is that convergence to the invariant distribution can be achieved by thermostatting just the highest wave number, while the evolution of the slowest modes is little affected by such a thermostat.

Observation-based correction of dynamical models using thermostats

“…It is easily checked that is stationary under the Fokker–Planck operator associated with (2.2) provided and provided the ‘potential’ A is only a function of y through the first integrals of f . The (extended) target measure is ergodic if the vector fields f and g satisfy a Hörmander condition [ 36 ]. Up to verification of the Hörmander condition, the choice of g is free.…”

Long time behaviour and particle approximation of a generalised Vlasov dynamic

“…(1) at time t. Systems of the form as in (1) and (3) have been studied by many authors. The law of the R 3d -valued process (Q t , P t , Z t ) t≥0 evolving according to the SDE (3) is a solution of Eq.…”

“…The law of the R 3d -valued process (Q t , P t , Z t ) t≥0 evolving according to the SDE (3) is a solution of Eq. It is also a special case of the so-called Nosé-Hoover-Langevin dynamic studied in [16,13,2,11,3]. When B = 0 and A is a gradient of a potential A = ∇V , (3) is called a generalised Langevin dynamic in [12,15].…”

“…When B = 0 and A is a gradient of a potential A = ∇V , (3) is called a generalised Langevin dynamic in [12,15]. In this case, (3) belongs to the class of weakly interacting particle systems and have been used widely in applied sciences, especially in plasma physics, see e.g., [8,9,14,10] and references therein. In these papers (3) is considered as a stochastic perturbation, via the external heat bath variable Z t , of the classical deterministic Hamiltonian system for (Q t , P t ).…”

“…It is also a special case of the so-called Nosé-Hoover-Langevin dynamic studied in [16,13,2,11,3]. Combining both cases, the full dynamic (3) can also be seen as stochastic perturbation (via the augmented variable Z) of Vlasov's dynamic similarly as the generalised Langevin dynamic. The generalised Langevin and the Nosé-Hoover-Langevin dynamics play an important role in molecular dynamics because they maintain canonical sampling of the original Hamiltonian system while being ergodic.…”