Entropy production in thermostats II

We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field $E$ has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfies $$K_{w}(\sigma)+{1/4}|E_{\sigma}|^2<0$$ for every 2-plane $\sigma$, where $K_{w}$ is the sectional curvature of the associated Weyl connection and $E_{\sigma}$ is the orthogonal projection of $E$ onto $\sigma$. Then the entropy production of any SRB measure is positive unless $E$ has a global potential. A related non-perturbative result is also obtained for certain generalized thermostats on surfaces.