**The challenge of stochastic Stormer-Verlet thermostats generating
correct statistics**

J Finkelstein and CH Cheng and G Fiorin and B Seibold and N Gronbech- Jensen, JOURNAL OF CHEMICAL PHYSICS, 153, 134101 (2020).

DOI: 10.1063/5.0018962

In light of the recently developed complete GJ set of single random
variable stochastic, discrete-time StOrmer-Verlet algorithms for
statistically accurate simulations of Langevin equations **N. GrOnbech-
Jensen, Mol. Phys. 118, e1662506 (2020)**, we investigate two outstanding
questions: (1) Are there any algorithmic or statistical benefits from
including multiple random variables per time step and (2) are there
objective reasons for using one or more methods from the available set
of statistically correct algorithms? To address the first question, we
assume a general form for the discrete-time equations with two random
variables and then follow the systematic, brute-force GJ methodology by
enforcing correct thermodynamics in linear systems. It is concluded that
correct configurational Boltzmann sampling of a particle in a harmonic
potential implies correct configurational free-particle diffusion and
that these requirements only can be accomplished if the two random
variables per time step are identical. We consequently submit that the
GJ set represents all possible stochastic StOrmer-Verlet methods that
can reproduce time step-independent statistics of linear systems. The
second question is thus addressed within the GJ set. Based on numerical
simulations of complex molecular systems, as well as on analytic
considerations, we analyze apparent friction-induced differences in the
stability of the methods. We attribute these differences to an inherent,
friction-dependent discrete-time scaling, which depends on the specific
method. We suggest that the method with the simplest interpretation of
temporal scaling, the GJ-I/GJF-2GJ method, be preferred for statistical
applications.

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