Abstract
Langevin dynamics is a versatile stochastic model used in biology, chemistry, engineering, physics and computer science. Traditionally, in thermal equilibrium, one assumes (i) the forces are given as the gradient of a potential and (ii) a fluctuation-dissipation relation holds between stochastic and dissipative forces; these assumptions ensure that the system samples a prescribed invariant Gibbs-Boltzmann distribution for a specified target temperature. In this article, we relax these assumptions, incorporating variable friction and temperature parameters and allowing nonconservative force fields, for which the form of the stationary state is typically not known a priori. We examine theoretical issues such as stability of the steady state and ergodic properties, as well as practical aspects such as the design of numerical methods for stochastic particle models. Applications to nonequilibrium systems with thermal gradients and active particles are discussed.
Highlights
Langevin dynamics is a system of stochastic differential equations of the form dq dt dp dt
While the above described regularized stochastic Cucker-Smale dynamics is a valid extension of the original model which ensures that the conditions of Theorem 3 are satisfied and geometric ergodicity for (34) and (35) holds, the form of the corresponding invariant measure does depend in a non-trivial way on Γ, σ and e and unless Γ is constant in q, one cannot find a closed form of the invariant measure
In this article we have provided a general treatment of the convergence of Langevin dynamics to a stationary state, including for systems with configuration-dependent friction and noise, as well as nonconservative forces
Summary
Langevin dynamics is a system of stochastic differential equations of the form dq dt dp dt. We do not assume a conservative force field Such generalized forms of Langevin dynamics can be used to model diffusion (or thermal) gradients by particle simulation [1,2,3], as well as a variety of models for flocking [4,5,6], protein folding [7] and bacterial suspensions [8]. In the case of the more general systems considered in this article, the calculation of the Ornstein-Uhlenbeck solution, which is required at each step of our splitting-based numerical methods, becomes potentially demanding from a computational standpoint. We use our theory to infer attractive steady states for the system, and characterize flocking tendencies by the use of two order parameters: one modelling the formation of consensus and the other characterizing the peculiarity which can be viewed as the average internal energy of isolated clumps of matter
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