Sampling Constraints in Average: The Example of Hugoniot Curves

We present a method for sampling microscopic configurations of a physical system distributed according to a canonical (Boltzmann) measure, with a constraint holding in average. Assuming that the constraint can be controlled by the volume and/or the temperature of the system, and considering an extended ensemble where the control parameter is a dynamical variable, conditional expectations of a nonlinear stochastic process are used to determine the right value of the control variable. A single trajectory discretization is proposed. As an application, we consider the computation of points along the Hugoniot curve, which are equilibrium states obtained after equilibration of a material heated and compressed by a shock wave.