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Precise Thermal Statistics in Discrete-Time Langevin Systems

Mathematical Physics & Probability

Speaker: Niels Gronbech-Jensen, UC Davis
Location: 2112 MSB
Start time: Wed, Feb 6 2019, 4:10PM


Numerical simulations of atomic and molecular ensembles by Molecular Dynamics always involve discretization of time, and as the time step is increased the discrete-time behavior becomes increasingly different from that of the anticipated continuous-time dynamics. This creates a dilemma for any simulation of a dynamical system: use a small time-step, resulting in dynamics that resemble the desired continuous-time behavior at the expense of computational efficiency; or use a large time step that makes the simulation finish sooner at the expense of systematic errors in the simulation behavior and results.
We present a new, simple simulation technique for systems in thermal equilibrium. We briefly review our stochastic Størmer-Verlet algorithm for the evolution of Langevin equations in a manner that preserves proper configurational sampling (diffusion and Boltzmann distribution) in discrete time. The resulting method, which is as simple as conventional Verlet schemes, has been numerically tested on both low-dimensional nonlinear systems as well as more complex molecular ensembles with many degrees of freedom.Additionally, we present a the solution to the “velocity problem”, and we show a simple approach to achieving simultaneous exact measures for both configurational and kinetic sampling in discrete time. We show exact analytic results for linear systems, and demonstrate the applicability and accuracy of the method for both nonlinear and complex systems, which can be accurately simulated at any time step within the stability limit. The method [1] is in the standard form of a Verlet-type algorithm, and is consequently easy to test and apply in existing codes, including Molecular Dynamics.

[1] Molecular Physics (2019): https://doi.org/10.1080/00268976.2019.1570369