Ruelle D. Positivity of entropy production in the presence of a random thermostat. (37K, TeX) ABSTRACT. We study nonequilibrium statistical mechanics in the presence of a thermostat acting by random forces, and propose a formula for the rate of entropy production $e(\mu)$ in a state $\mu$. When $\mu$ is a natural nonequilibrium steady state we show that $e(\mu)\ge0$, and sometimes we can prove $e(\mu)>0$.