Stochastic systems with Bose-Hubbard interactions: Effects of bias on particles on a 1D lattice

TL;DR

This paper introduces a driven 1D lattice model with Bose-Hubbard-type on-site repulsion $U$ and a directional bias $g$, examining its non-equilibrium steady states under periodic and open boundary conditions. By combining exact results in limiting regimes (zero and infinite $\beta U$) with Monte Carlo simulations for intermediate couplings, it shows that periodic systems exhibit non-monotonic correlations in $\beta U$, an emergent ASEP-like current at large $\beta U$, and a density-periodic current due to stacked particle backgrounds. Open boundaries yield step-like or plateaued density profiles controlled by the interplay between drive and repulsion, including macroscopic depletion in certain regimes. The work highlights how drive, interaction, and boundary conditions jointly shape current and density structures, offering a minimal framework bridging ASEP, ZRP, and Bose-Hubbard physics with potential relevance to tilted optical lattices and related non-equilibrium systems.

Abstract

Driven non-equilibrium lattice models have wide-ranging applications in contexts such as mass transport, traffic flow, and transport in biological systems. In this work, we investigate the steady-state properties of a one-dimensional lattice system that allows multiple particle occupancy on each site. The particles undergo stochastic nearest-neighbor jumps influenced by both a directional bias and on-site repulsive interactions of the Bose-Hubbard type. With periodic boundary conditions, we observe a non-monotonic dependence of inter-site correlation functions on the interaction strength. At large interaction strengths, the state consists of quiescent stacks of stationary particles along with an emergent asymmetric simple exclusion process(ASEP), and the particle current exhibits a periodic dependence on density. In contrast, with open boundary conditions, the system displays step-like density profiles reminiscent of those in tilted Bose-Hubbard systems, and a regime with a macroscopic number of empty sites followed by a steep parameter-dependent increase in density. Our results highlight how the interplay between drive, interaction, and boundary conditions leads to distinctive signatures on the current and density profiles in the steady state in different regimes.

Figures (12)

• Figure 1: Qualitative phase diagram showing different regimes in a one-dimensional periodic lattice. On the horizontal axis, the system reduces to a ZRP, while on the vertical axis, it has an equilibrium product measure steady state. The diagram illustrates regions of low and high correlation, the emergent ASEP regime, and the domain where linear response theory is valid. Boundaries between these regimes are smooth crossovers rather than sharp transitions. The emergent ASEP limit is formally reached when $\beta U \to \infty$, but its characteristic behavior appears already for $\beta U \gg 1$.
• Figure 2: Illustration of stochastic particle dynamics in a one-dimensional lattice with six sites. Each circle represents a particle at a site, and arrows denote possible hopping transitions of a single particle between nearest-neighbor sites. The particle which attempts a move is shaded for reference. The transition rates $W_1(n_i, n_j)$ and $W_2(n_i, n_j)$ depend on the occupation numbers $n_i$ and $n_j$ of the departure and arrival sites, respectively. For instance, $W_2(4,0)$ represents the hopping rate of a particle moving from site 2 (with 4 particles) to site 1 (empty).
• Figure 3: Steady-state current $j$ as a function of the biasing parameter $g$ for different values of $\rho$ at $\beta U = 0.5$. Other simulation parameters: System size $L=150$, $10^5$ Monte Carlo steps, $5L^2$ relaxation time. The dashed black line correspond to the emergent ASEP limit for fractional part of $\rho$ fixed at 0.5. The ZRP and the ASEP limits serve as upper and lower bounds of the current respectively.
• Figure 4: Steady-state current $j$ as a function of density $\rho$ for varying biasing parameter $g$ and interaction strength $\beta U$. Other simulation parameters, as for Fig. \ref{['fig:current_vs_g']}. Dashed lines indicate the limiting case(s).
• Figure 5: Steady-state current $j$ as a function of the interaction strength $\beta U$ for different values of $g$ at fixed $\rho = 1.5$. Other simulation parameters, as for Fig. \ref{['fig:current_vs_g']}. Dashed lines indicate the limiting value of $j$ at large $\beta U$.