Nose-Hoover-Langevin NHL method

The Nose-Hoover-Langevin (NHL) method is based on a simple idea replace the chain in the Nos6-Hoover Chain, whose sole purpose is to maintain a Gaussian distribution in the auxiliary variable, by a stochastic Langevin-type thermostat. The method was first proposed in [323]. The proof of ergodicity (more precisely the confirmation of the Hormander condition), for a problem with harmonic internal interactions, was given in [226] and we roughly follow the treatment from this paper. 

Let us first demonstrate that the Nos6-Hoover-Langevin method, is, in fact, a thermostat. We observe that we have already shown that the Nos6-Hoover method preserves the extended canonical distribution pp /2). Therefore, by the 

In order to show that this thermostat is ergodic, we need to demonstrate that the assumptions presented in Sect. 6.4.4 are valid. Assumption l(ii) requires us to verify that a Hormander condition (see Definition 6.1) holds for the Nosd-Hoover-Langevin system. Because of the complexity of the high order commutators, we will work here with the assumption of a quadratic potential in the physical model. In some sense this is the most difficult case for a thermostat, but paradoxically, the assumption facilitates the mathematical analysis. We will assume throughout the following that Nd = ISt, i.e. that there are no constraints in the system. Our potential will therefore be assumed to be 

We will further assume M = I. If this is not the case, then one may introduce the rescaling q = M / q,p = in order to rewrite the system in a form that does not involve M. Thus the system under study is 

Theorem 8.1 If A is symmetric positive definite and has distinct eigenvalues, then (8.30)-(8.32) satisfies the Hdrmander condition on D x R. 

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