  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.

Statistical Mechanics;Computational Physics