Nonequilibrium statistical mechanics revealed by Doob <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>h</mml:mi></mml:math> transform and variational autoregressive networks

The nonequilibrium dynamics of stochastic systems are typically characterized by the joint probability distribution and its time evolution within configuration space or through trajectory ensembles. Within the framework of trajectory ensembles, processes occurring on exponentially rare spatio-temporal scales are of particular interest. Recent research has demonstrated that Doob dynamics is a highly effective method for sampling these rare trajectories. Most existing methods directly sample specific transition trajectories within the trajectory space. However, they fundamentally lack the capacity to resolve the time evolution of configuration-space probability distributions associated with these rare trajectories. In this study, we demonstrate how to construct Doob dynamics by approximating the leading eigenstate of the tilted generator using the variational autoregressive networks (VAN) ansatz. This approach enables us to sample the time-evolution information of the probability distribution associated with rare trajectories and to compute the large deviation statistics. We apply our methodology to two typical lattice models: the East and the Fredrickson-Andersen models, to sample large deviation statistics and probability distributions in one-dimensional and two-dimensional scenarios, respectively. Finally, we discuss the limitations of our method and propose potential solutions.