Hormander condition

The fact that the Hormander condition holds is, as we know from Chap. 6, only part of the story. However based on known results for Langevin dynamics, we conjecture that ergodicity will hold if (i) the Hormander condition holds, (ii) U is sufficiently smooth, and (iii) the configurational phase space is compact, e.g. periodic boundary conditions are employed. However we stress that each possible method will ultimately need to be carefully and systematically checked to verify the ergodic properties. 

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. 

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... 

Demonstrating the Hormander condition is sufficient for Assumption l(ii) in Sect. 6.4.4. Assumption l(i) can be verified by proving that solutions are able to access an open set around any pointy e D x R, We omit the proof of Assumption l(i) for Nos6-Hoover-Langevin dynamics as it is follows the same structure as the proof of Lemma 6.1 for Langevin dynamics. The details of the proof can be found in [283], with examples for more general SDEs given in [256, 257]. 

For this system, it is possible to show that the Hormander condition holds. Setting all parameters to 1 and taking M = I (without loss of generality in this demonstration) we write our SDE in the form d

[Pg.357]

If p = 0 and (7 = 0 then we are on Mq. If p 0 then U2 and kq are linearly independent. If q 0 then us and o are linearly independent. From these we establish that the Hormander condition holds everywhere except on a space of codimension 2 (that is except on the one-dimensional set Mo). 

The statement can be proved by solving (5.67) for the function g( ), and taking into consideration the condition (5.68). The theoretical basis of the procedure can be found in Hormander (1964), and the statement may be interpreted in four different ways. 

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