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Transition path theory (TPT) offers a powerful formalism for extracting the rate and mechanism of rare dynamical transitions between metastable states. Most applications of TPT either focus on systems with modestly sized state spaces or use collective variables to try to tame the curse of dimensionality. Increasingly, expressive function approximators like neural networks and tensor networks have shown promise in computing the central object of TPT, the committor function, even in very high dimensional systems. That progress prompts our consideration of how one could use such a high dimensional function to extract mechanistic insight. Here, we present and illustrate a straightforward but powerful way to track how individual dynamical coordinates evolve during a reactive event. The strategy, which involves marginalizing the reactive ensemble, naturally captures the evolution of the dynamical coordinate's distribution, not just its mean reactive behavior.

Thermodynamic uncertainty relations (TURs) relate precision to the dissipation rate, yet the inequalities can be far from saturation. Indeed, in catenane molecular motor simulations, we record precision far below the TUR limit. We further show that this inefficiency can be anticipated by four physical parameters: the thermodynamic driving force, fuel decomposition rate, coupling between fuel decomposition and motor motion, and rate of undriven motor motion. The physical insights might assist in designing molecular motors in the future.

The interplay between stochastic chemical reactions and diffusion can generate rich spatiotemporal patterns. While the timescale for individual reaction or diffusion events may be very fast, the timescales for organization can be much longer. That separation of timescales makes it particularly challenging to anticipate how the rapid microscopic dynamics gives rise to macroscopic rates in the non-equilibrium dynamics of many reacting and diffusing chemical species. Within the regime of stochastic fluctuations, the standard approach is to employ Monte Carlo sampling to simulate realizations of random trajectories. Here, we present an alternative numerically tractable approach to extract macroscopic rates from the full ensemble evolution of many-body reaction diffusion problems. The approach leverages the Doi-Peliti second-quantized representation of reaction-diffusion master equations along with compression and evolution algorithms from tensor networks. By focusing on a Schl\"{o}gl model with one-dimensional diffusion between $L$ otherwise well-mixed sites, we illustrate the potential of the tensor network approach to compute rates from many-body systems, here with approximately $3 \times 10^{15}$ microstates. Specifically, we compute the rate for switching between metastable macrostates, with the expense for computing those rates growing subexponentially in $L$. Because we directly work with ensemble evolutions, we crucially bypass many of the difficulties encountered by rare event sampling techniques$\unicode{x2013}$detailed balance and reaction coordinates are not needed.

Simulations can help unravel the complicated ways in which molecular structure determines function. Here, we use molecular simulations to show how slight alterations of a molecular motor's structure can cause the motor's typical dynamical behavior to reverse directions. Inspired by autonomous synthetic catenane motors, we study the molecular dynamics of a minimal motor model, consisting of a shuttling ring that moves along a track containing interspersed binding sites and catalytic sites. The binding sites attract the shuttling ring while the catalytic sites speed up a reaction between molecular species, which can be thought of as fuel and waste. When that fuel and waste are held in a nonequilibrium steady-state concentration, the free energy from the reaction drives directed motion of the shuttling ring along the track. Using this model and nonequilibrium molecular dynamics, we show that the shuttling ring's direction can be reversed by simply adjusting the spacing between binding and catalytic sites on the track. We present a steric mechanism behind the current reversal, supported by kinetic measurements from the simulations. These results demonstrate how molecular simulation can guide future development of artificial molecular motors.

The thermodynamic uncertainty relation (TUR) quantifies a relationship between current fluctuations and dissipation in out-of-equilibrium overdamped Langevin dynamics, making it a natural counterpart of the fluctuation-dissipation theorem in equilibrium statistical mechanics. For underdamped Langevin dynamics, the situation is known to be more complicated, with dynamical activity also playing a role in limiting the magnitude of current fluctuations. Progress on those underdamped TUR-like bounds has largely come from applications of the information-theoretic Cram\'er-Rao inequality. Here, we present an alternative perspective by employing large deviation theory. The approach offers a general, unified treatment of TUR-like bounds for both overdamped and underdamped Langevin dynamics built upon current fluctuations achieved by scaling time. The bounds we derive following this approach are similar to known results but with differences we discuss and rationalize., Comment: 6 pages, 3 figures

We present an approach based upon binary tree tensor network (BTTN) states for computing steady-state current statistics for a many-particle 1D ratchet subject to volume exclusion interactions. The ratcheted particles, which move on a lattice with periodic boundary conditions subject to a time-periodic drive, can be stochastically evolved in time to sample representative trajectories via a Gillespie method. In lieu of generating realizations of trajectories, a BTTN state can variationally approximate a distribution over the vast number of many-body configurations. We apply the density matrix renormalization group (DMRG) algorithm to initialize BTTN states, which are then propagated in time via the time-dependent variational principle (TDVP) algorithm to yield the steady-state behavior, including the effects of both typical and rare trajectories. The application of the methods to ratchet currents is highlighted in a companion letter, but the approach extends naturally to other interacting lattice models with time-dependent driving. Though trajectory sampling is conceptually and computationally simpler, we discuss situations for which the BTTN TDVP strategy could be more favorable., Comment: 12 pages, 7 figures

The study of Brownian ratchets has taught how time-periodic driving supports a time-periodic steady state that generates nonequilibrium transport. When a single particle is transported in one dimension, it is possible to rationalize the current in terms of the potential, but experimental efforts have ventured beyond that single-body case to systems with many interacting carriers. Working with a lattice model of volume-excluding particles in one dimension, we analyze the impact of interactions on a flashing ratchet's current. To surmount the many-body problem, we employ the time-dependent variational principle with a binary tree tensor network, methods discussed at length in a companion paper. Rather than propagating individual trajectories, the tensor network approach propagates a distribution over many-body configurations via a controllable variational approximation. The calculations, which reproduce Gillespie trajectory sampling, identify and explain a shift in the frequency of maximum current to higher driving frequency as the lattice occupancy increases., Comment: 6 pages, 3 figures