Work distribution and path integrals in general mean-field systems

We consider a mean-field system described by a general collective variable $M$, driven out of equilibrium by the manipulation of a parameter $\mu$. Given a general dynamics compatible with its equilibrium distribution, we derive the evolution equation for the joint probability distribution function of $M$ and the work $W$ done on the system. We solve this equation by path integrals. We show how the Jarzynski equality holds identically at the path integral level and for the classical paths which dominate the expression in the thermodynamic limit. We discuss some implications of our results.