On the Application of Non-Gaussian Noise in Stochastic Langevin Simulations

In light of recent advances in time-step independent stochastic integrators for Langevin equations, we revisit the considerations for using non-Gaussian distributions for the thermal noise term in discrete-time thermostats. We find that the desirable time-step invariance of the modern methods is rooted in the Gaussian noise, and that deviations from this distribution will distort the Boltzmann statistics arising from the fluctuation-dissipation balance of the integrators. We use the GJ stochastic Verlet methods as the focus of our investigation since these methods are the ones that contain the most accurate thermodynamic measures of existing methods. Within this set of methods we find that any distribution of applied noise, which satisfies the two first moments given by the fluctuation-dissipation theorem, will result in correct, time-step independent results that are generated by the first two moments of the system coordinates. However, if non-Gaussian noise is applied, undesired deviations in higher moments of the system coordinates will appear to the detriment of several important thermodynamic measures that depend especially on the fourth moments. The deviations, induced by non-Gaussian noise, become significant with the one-time-step velocity attenuation, thereby inhibiting the benefits of the methods. Thus, we conclude that the application of Gaussian noise is necessary for reliable thermodynamic results when using modern stochastic thermostats with large time steps.