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  • Writer's pictureUshnish Ray

Tensor Networks approach to Large Deviation Theory

Updated: Apr 19, 2019

In this article (arxiv:1904.07336) we explore the properties of certain lattice systems that model the simple exclusion process (SEP) in 1D and quasi-2D (in the form of multiple lanes). SEPs are very general models that capture the behavior of many out-of-equilibrium systems of interest; for example, the behavior of ribosomes on mRNA, traffic flow etc. Therefore, understanding their behavior has broad implications.

Tackling such nonequilibrium systems has received renewed interest because of a powerful new approach in statistical mechanics called Large deviation theory (LDT). In essence, LDT allows us to approach dynamical properties in ways that are analogous to traditional equilibrium statistical mechanics employing concepts such as the free-energy, entropy, order parameter etc.

Much of recent research activities in the nonequilibrium statistical mechanics community have involved calculating the cumulant generating function (the analog of free energy) and rate function (the analog of entropy) in different archetypical nonequilibrium systems. For all but the simplest systems, we must resort to numerical techniques to compute them and many approaches are being developed.

In previous posts I have pointed out two such techniques that I have worked on: Transition Path Sampling (TPS) and the Cloning Algorithm (CA). These are exact techniques but suffer from sampling issues under important circumstances. My prior work has involved mitigating such issues to make realistic calculations feasible.

An alternate route to Monte-Carlo sampling involves exploiting a fixed point solution motivated by a generalized variational principle (GVP) that allows obtaining approximate solutions that can be systematically improved. The idea is to posit an ansatz with free parameters that can be optimized according to the GVP. In the past we have used cluster ansatz following mean-field theories to tackle such problems. However, it turns out that the Matrix Product State (MPS) ansatz is a much more powerful and flexible ansatz for the SEP systems we consider.

Fig. 1: Density profiles associated with different dynamical phases. Shown are typical configurations associated with different phases and the average over all possible configuration. (Image courtesy Phillip Helms)

Aside from validation against (exact) Funtional Bethe ansatz in 1D, we explore a part of the asymmetric SEP (ASEP) in quasi-2D involving multiple lanes with different boundary conditions. We see important distinctions in the way current is transported across such systems with rich phenomenology in the spectrum of rare trajectories. The system exhibits various dynamical phases such as the maximal current phase (MC) with growing particle correlations in the system, the shock phase (S) where the system spontaneously forms a wavefront to minimize current flow across the system and the minimal current phases (HD+LD = high density + low density) where the system quickly assembles into configurations where no current flow is permitted. These are most easily visualized via their density profiles (shown in Fig. 1).

Using MPS ansatz we are also able to compute site dependent observables that yield information on the microscopic current flow and activity (the total hops) across the system. Shown in Fig. 2 is such behavior for a lattice 2x20 system with different boundary conditions. The key point here is that although the systems are exhibiting similar bulk phases their microscopics are quite nuanced. Read our paper to find out more!

Fig. 2: Columns: closed and open system Rows: MC and LD+HD phases. The arrows show the total hops in the different directions. (Image courtesy Phillip Helms)

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