Transport coefficients from Einstein-Helfand relations using standard and energy-conserving dissipative particle dynamics methods.

In this article we demonstrate that contrary to general belief, the standard Einstein-Helfand (EH) formulas are valid for the evaluation of transport coefficients of systems containing dissipative and random forces provided that for these mesoscopic systems: (i) the corresponding conservation laws are satisfied, and (ii) the transition probabilities satisfy detailed balance. Dissipative particle dynamics (DPD) and energy-conserving DPD methods (DPDE), for instance, are archetypical of such mesoscopic approaches satisfying these properties. To verify this statement, we have derived a mesoscopic heat flux form for the DPDE method, suitable for the calculation of the thermal conductivity from an EH expression. We have compared EH measurements against non-equilibrium simulation values for different scenarios, including many-body potentials, and have found excellent agreement in all cases. The expressions are valid notably for systems with density- and temperature-dependent potentials, such as the recently developed generalised DPDE method (GenDPDE) [Avalos et al. , Phys. Chem. Chem. Phys. , 2019, 21 , 24891]. We thus demonstrate that traditional EH formulas in equilibrium simulations can be widely used to obtain transport coefficients, provided that the appropriate expression for the associated flux is used.