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We describe some general results that constrain the dynamical fluctuationsthat can occur in non-equilibrium steady states, with a focus on moleculardynamics. That is, we consider Hamiltonian systems, coupled to external heatbaths, and driven out of equilibrium by non-conservative forces. We focus onthe probabilities of rare events (large deviations). First, we discuss a PT(parity-time) symmetry that appears in ensembles of trajectories where acurrent is constrained to have a large (non-typical) value. We analyse the heatflow in such ensembles, and compare it with non-equilibrium steady states.Second, we consider pathwise large deviations that are defined by consideringmany copies of a system. We show how the probability currents in such systemscan be decomposed into orthogonal contributions, that are related toconvergence to equilibrium and to dissipation. We discuss the implications ofthese results for modelling non-equilibrium steady states.

Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics