  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.
