Non-equilibrium by molecular dynamics: a dynamical approach

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Some time ago, Ciccotti and Jacucci [Phys. Rev. Lett. 1975;35:789-792] – on the basis of Onsager regression hypothesis as explained by Kubo – suggested and implemented an original approach to study both stationary and time-dependent situations in non-equilibrium systems. The key idea of their approach was to tackle separately the dynamical evolution from the problem of sampling the initial condition. In this review, we discuss the evolution of this approach dubbed D-NEMD, the dynamical approach to non-equilibrium molecular dynamics, to differentiate it from the stationary NEMD methods. We will go through the D-NEMD theoretical framework and illustrate how it allows to compute time-dependent macroscopic dynamical behaviours by averaging on a large sample of non-equilibrium trajectories starting from an ensemble of initial conditions associated to a suitable distribution (either equilibrium or non-equilibrium) at time zero. We discuss a few case studies where the D-NEMD method is illustrated: first for ‘historical’ applications, like the study of transport properties in the linear and non-linear regimes, including a divertissement on the calculation of time correlation functions in the Gran Canonical ensemble. We finally briefly illustrate D-NEMD applications to the study of problems arising in key hydrodynamic processes such as the formation of convective cells and the relaxation of a non-uniform density profile in a fluid, giving attention to the problem of sampling the conditional probability density ensemble associated with the initial states.