Ergodic Stochastic-Dynamic Thermostats

In this section, we consider the combination of stochastic perturbation with a deterministic thermostat. Methods constructed in this way can be ergodic for the canonical distribution while also providing flexibility in way equilibrium is achieved. We distinguish in (6.16) between multiplicative noise, where B = B(z) varies with z and additive noise, where B is constant. In our treatment of this topic we will only consider additive noise. The presence of multiplicative noise may complicate discretization. As we shall see, the reliance on additive noise improves the performance of discretization schemes. 

To give this some specificity for molecular dynamics, we write the stochastic dynamical model as 

Here r,t are symmetric, square matrices of appropriate dimension, W is a vector of Wiener processes, and W is a vector of m Wiener processes, all independent of each other. We assume that the deterministic thermostat represented by the and g terms preserves the extended distribution p( )pp, where p( ) = 

The additivity property of thermostats mentioned above holds also for stochastic thermostats, since the Fokker-Planck equation is constructed from linear operations on the vector fields involved, thus it is enough to require that the newly introduced terms satisfy appropriate fluctuation-dissipation relationships so that they are compatible with the extended Gibbs distribution of the deterministic part. That is, we require 

Thus we know that the Gibbs distribution will be invariant. It remains to prove that the SDE (8.22)-(8.24) is ergodic. Certainly this would be expected in the case where a full Langevin dynamics contribution is present (F positive definite), but as we shall see next there are cases where noise only contacts the auxiliary degrees of freedom but the system is nonetheless ergodic. 

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