Using tensor network states for multi-particle Brownian ratchets

Error-Controlled Coarse-Graining Dynamics with Mean-Field Randomization

In order to comprehend the stochastic behavior of biological systems, it is essential to accurately infer the dynamics of chemical reaction networks. However, computation of the likelihood remains a bottleneck. In this study, we propose the mean-field randomization procedure as a means of efficiently generating error-controlled coarse-graining dynamics. The error is measured by mutual information between the generated trajectories and the coarse-graining procedure. We demonstrate that the exact dynamics can be recovered by resampling, which eliminates the correlation between the dynamics and the procedure. We developed three algorithms to efficiently generate exact or coarse-graining trajectories within a specified error range. By subjecting our algorithms to testing on chemical reaction systems of varying complexities and scales, we observe that they outperform existing state-of-the-art algorithms, and the efficiency of coarse-graining trajectory generation is only weakly dependent on system scales.

Efficient simulations of epidemic models with tensor networks: Application to the one-dimensional susceptible-infected-susceptible model

The contact process is an emblematic model of a nonequilibrium system, containing a phase transition between inactive and active dynamical regimes. In the epidemiological context, the model is known as the susceptible-infected-susceptible model, and it is widely used to describe contagious spreading. In this work, we demonstrate how accurate and efficient representations of the full probability distribution over all configurations of the contact process on a one-dimensional chain can be obtained by means of matrix product states (MPSs). We modify and adapt MPS methods from many-body quantum systems to study the classical distributions of the driven contact process at late times. We give accurate and efficient results for the distribution of large gaps, and illustrate the advantage of our methods over Monte Carlo simulations. Furthermore, we study the large deviation statistics of the dynamical activity, defined as the total number of configuration changes along a trajectory, and investigate quantum-inspired entropic measures, based on the second Rényi entropy.

Emergent information dynamics in many-body interconnected systems

The information implicitly represented in the state of physical systems allows for their analysis using analytical techniques from statistical mechanics and information theory. This approach has been successfully applied to complex networks, including biophysical systems such as virus-host protein-protein interactions and whole-brain models in health and disease, drawing inspiration from quantum statistical physics. Here we propose a general mathematical framework for modeling information dynamics on complex networks, where the internal node states are vector valued, allowing each node to carry multiple types of information. This setup is relevant for various biophysical and sociotechnological models of complex systems, ranging from viral dynamics on networks to models of opinion dynamics and social contagion. Instead of focusing on node-node interactions, we shift our attention to the flow of information between network configurations. We uncover fundamental differences between widely used spin models on networks, such as voter and kinetic dynamics, which cannot be detected through classical node-based analysis. We illustrate the mathematical framework further through an exemplary application to epidemic spreading on a low-dimensional network. Our model provides an opportunity to adapt powerful analytical methods from quantum many-body systems to study the interplay between structure and dynamics in interconnected systems.

Optimal Sampling of Dynamical Large Deviations in Two Dimensions via Tensor Networks

We use projected entangled-pair states (PEPS) to calculate the large deviation statistics of the dynamical activity of the two-dimensional East model, and the two-dimensional symmetric simple exclusion process (SSEP) with open boundaries, in lattices of up to 40×40 sites. We show that at long times both models have phase transitions between active and inactive dynamical phases. For the 2D East model we find that this trajectory transition is of the first order, while for the SSEP we find indications of a second order transition. We then show how the PEPS can be used to implement a trajectory sampling scheme capable of directly accessing rare trajectories. We also discuss how the methods described here can be extended to study rare events at finite times.

Computing time-periodic steady-state currents via the time evolution of tensor network states

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, 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 can be beneficial.