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Jarzynski matrix equality: calculating free energy difference by non-equilibrium simulations with an arbitrary initial distribution

The Jarzynski equality (JE), which relates works of non-equilibrium trajectories to the free energy difference of the initial and final states of the non-equilibrium process, provides an efficient way to calculate free energies of systems in simulations and experiments. However, wider applications of the JE are limited by the requirement that the initial distribution of non-equilibrium trajectories must be equilibrium. Here we extend the JE to a matrix form, the Jarzynski matrix equality (JME), which transforms the free energies of metastable conformational regions in the initial system to that of final one. Therefore, we can calculate the free energies from non-equilibrium trajectories which started from an arbitrary initial distribution. We demonstrate the application of the JME in toy models, Lennard-Jones fluids, and polymer chain models, show its good efficiency in calculation of free energy with a satisfactory accuracy. The JME extends applications of the non-equilibrium method in estimate of free energy in complex system where the initial global equilibrium is difficult to reach.