Molecules can undergo reactions and diffusion through space, creating the cornucopia of patterns we observe in nature. Understanding how these patterns emerge is challenging to study due to the immense separation of scales between the fast microscopic dynamics and the macroscopic pattern. A classic example of such a pattern is bistability, where a system will spontaneously switch between two macroscopic states of the system. Quantifying the rate of switching historically has relied on waiting for exponentially rare events in the system to be observed. Ensembles of such events lead to estimates of kinetic rates. In this work we show how to calculate rare macroscopic rates from high-dimensional reaction diffusion systems without resorting to sampling techniques. Instead, we exploit and extract observables such as macroscopic rates by evolving the ensemble of all possible trajectories.

The foundation of this work is based on using the Doi-Peliti formalism to encode the chemical master equation into a second-quantized form. This form allows chemical networks to be readily evolved using efficient tensor network methods. Our results are illustrated using an adapted version of the bistable Schogl model with diffusion. We calculate rates over five-orders of magnitude for large systems (∼ 3 × 1015 microstates) and show strong agreement to kinetic-Monto Carlo simulations and the more advanced forward flux sampling method. Our Doi-Peliti tensor network procedure demonstrates sub-exponential scaling in computational expense, while bypassing complications due to sampling errors or needing intimate knowledge of the reaction network as is the case with more advanced sampling methods.