Dissipative particle dynamics with energy conservation: equilibrium   properties
  The stochastic differential equations for a model of dissipative particle dynamics, with both total energy and total momentum conservation at every time-step, are presented. The algorithm satisfies detailed balance as well as the fluctuation-dissipation theorems that ensure that the proper thermodynamic equilibrium can be reached. Macroscopic equilibrium probability distributions as well as equations of state for the model are also derived, and an appropriate definition of the free energy of the system is consistently proposed. Several simulations results of equilibrium as well as transport properties, including heat transport and thermal convection in a box, are shown as proof of the internal consistency of the model. 
Allan D. Mackie and Josep Bonet Avalos