Dynamics of stochastic oscillator chains with harmonic and FPUT potentials

Inspired by recent studies on deterministic oscillator models, we introduce a stochastic one-dimensional model for a chain of interacting particles. The model consists of $N$ oscillators performing continuous-time random walks on the integer lattice $\mathbb{Z}$ with exponentially distributed waiting times. The oscillators are bound by confining forces to two particles that do not move, placed at positions $x_0$ and $x_{N+1}$, respectively, and they feel the presence of baths with given inverse temperatures: $β_L$ to the left, $β_B$ in the middle, and $β_R$ to the right. Each particle has an index and interacts with its nearest neighbors in index space through either a quadratic potential or a Fermi-Pasta-Ulam-Tsingou type coupling. This local interaction in index space can give rise to effective long-range interactions on the spatial lattice, depending on the instantaneous configuration. Particle hopping rates are governed either by the Metropolis rule or by a modified version that breaks detailed balance at the interfaces between regions with different baths.

💡 Research Summary

The paper introduces a stochastic one‑dimensional chain of N interacting particles moving on the integer lattice ℤ via continuous‑time random walks with exponential waiting times. Two immobile boundary particles are fixed at positions x₀ = −a(N+1)/2 and x_{N+1}= a(N+1)/2, providing fixed ends. The lattice is divided into three spatial regions—left, bulk, and right—each coupled to a thermal reservoir characterized by inverse temperatures β_L, β_B, and β_R, respectively. This creates a step‑function temperature profile β(x) that induces a non‑uniform environment for the particles.

Interactions are defined in “index space”: particle k interacts only with its immediate index neighbors k−1 and k+1 through a pair potential V(r). Two potentials are considered: (i) a harmonic (quadratic) potential V(r)= (g₂/2)(r−a)², and (ii) a β‑FPUT potential V(r)= (g₂/2)(r−a)² + (g₄/4)(r−a)⁴, which adds a quartic non‑linearity. The total Hamiltonian is H(x)=∑{k=0}^{N} V(x{k+1}−x_k). The mechanical equilibrium configuration x^e is linear in the index (x_k = x₀ + a k).

Two stochastic dynamics are studied. The reversible dynamics uses the standard Metropolis rates c(x→x′)=min{1, exp