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Recently there has been growing interest in extending the thermodynamicmethod from static configurations to dynamical trajectories. In this approach,ensembles of trajectories are treated in an analogous manner to ensembles ofconfigurations in equilibrium statistical mechanics: generating functions ofdynamical observables are interpreted as partition sums, and the statisticalproperties of trajectory ensembles are encoded in free-energy functions thatcan be obtained through large-deviation methods in a suitable large time limit.This establishes what one can call a 'thermodynamics of trajectories'. In thispaper we go a step further, and make a first connection to fluctuation theoremsby generalising them to this dynamical context. We show that an effective'meta-dynamics' in the space of trajectories gives rise to the celebratedJarzynski relation connecting an appropriately defined 'meta-work' with changesin dynamical generating functions. We demonstrate the potential applicabilityof this result to computer simulations for two open quantum systems, atwo-level system and the micromaser. We finally discuss the behavior of theJarzynski relation across a first-order trajectory phase transition.

Meta-work and the analogous Jarzynski relation in ensembles of dynamical trajectories