Dynamics of stochastic chains with harmonic and FPUT potentials

Abstract 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 particles performing continuous-time random walks on the integer lattice <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">Z</m:mi> </m:math> $\mathbb{Z}$ with exponentially distributed waiting times. The particles 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. In both cases, the dynamics drive the system toward the minimization of an appropriate energy functional, even under non-uniform temperature profiles.