Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dynamics of stochastic oscillator chains with harmonic and FPUT potentials

Emilio N. M. Cirillo, Matteo Colangeli, Claudio Giberti, Lamberto Rondoni

TL;DR

The paper studies stochastic oscillator chains on $\mathbb{Z}$ with a three-region inverse-temperature profile $\beta(x)$ to induce nonequilibrium stationary states. It analyzes two Markov dynamics: a reversible Metropolis rule with energy $\mathcal{H}(\boldsymbol{x})$ and invariant measure $\pi(\boldsymbol{x})=e^{-\beta_0 \mathcal{H}(\boldsymbol{x})}/Z$, and an irreversible variant that uses $\beta(x_j)$ and breaks detailed balance at interfaces. Interactions are either harmonic with $V(r)=\frac{g_2}{2}(r-a)^2$ or $V(r)=\frac{g_2}{2}(r-a)^2+\frac{g_4}{4}(r-a)^4$, leading to different stationary profiles and mobility. Numerical simulations reveal that harmonic chains tend to accumulate near the hot bath under irreversible dynamics, while $\beta$-FPUT chains remain closer to mechanical equilibrium; the results illustrate how nonuniform driving and the form of the interaction govern transport in low-dimensional stochastic systems.

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 $N$ 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.

Dynamics of stochastic oscillator chains with harmonic and FPUT potentials

TL;DR

The paper studies stochastic oscillator chains on with a three-region inverse-temperature profile to induce nonequilibrium stationary states. It analyzes two Markov dynamics: a reversible Metropolis rule with energy and invariant measure , and an irreversible variant that uses and breaks detailed balance at interfaces. Interactions are either harmonic with or , leading to different stationary profiles and mobility. Numerical simulations reveal that harmonic chains tend to accumulate near the hot bath under irreversible dynamics, while -FPUT chains remain closer to mechanical equilibrium; the results illustrate how nonuniform driving and the form of the interaction govern transport in low-dimensional stochastic systems.

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 oscillators performing continuous-time random walks on the integer lattice with exponentially distributed waiting times. The oscillators are bound by confining forces to two particles that do not move, placed at positions and , respectively, and they feel the presence of baths with given inverse temperatures: to the left, in the middle, and 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.

Paper Structure

This paper contains 6 sections, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The model
  3. The reversible dynamics
  4. The irreversible dynamics
  5. Numerical simulations
  6. Conclusions

Figures (5)

  • Figure 1: Oscillator chain considered in Sec. \ref{['sec:sec2']} with $N=5$. Particles interact via either a harmonic or a $\beta$-FPUT potential and are in contact with three external thermal reservoirs, which fix the inverse temperature at $\beta_L$ on the left, $\beta_B$ in the center and $\beta_R$ on the right.
  • Figure 2: Top row: normalized average position profiles obtained with the harmonic potential in Eq. \ref{['harm']}, using the Metropolis dynamics defined in Eq. \ref{['eq:intens_1']} (left panel) and its variant given in Eq. \ref{['eq:intens_2']} (right panel), in the inhomogeneous case with $\beta_L\in\{0.001,0.005,0.01\}$, $\beta_B = 0.03$ and $\beta_R = 0.1$, with $N = 21$, $a = 10$, $\delta=1.5\ a$ and $g_2=1$. The errorbars represent one standard deviation. Bottom row: average intensity profiles corresponding to the Metropolis dynamics (left panel) and to Eq. \ref{['eq:intens_2']} (right panel). Simulations are performed over $5\times10^7$ time steps. Various forms of contact resistance arise in all cases, analogously to common deterministic models.
  • Figure 3: Normalized average position (top row) and average intensity profiles (bottom row) profiles for the dynamics in Eq. \ref{['eq:intens_1']} (left panel) and in Eq. \ref{['eq:intens_2']} (right panel) with the $\beta$-FPUT potential, with $g_4=1$. Values of the other parameters are the same as in Fig. \ref{['fig:fig2']}.
  • Figure 4: Normalized average position (top row) and average intensity profiles (bottom row) profiles for the dynamics in Eq. \ref{['eq:intens_1']} (left panel) and in Eq. \ref{['eq:intens_2']} (right panel) with the harmonic potential, with $\delta=11\ a$. Values of the other parameters are the same as in Fig. \ref{['fig:fig2']}. Note: the discontinuity in the bottom left panel is real.
  • Figure 5: Normalized average position (top row) and average intensity profiles (bottom row) for the dynamics in Eq. \ref{['eq:intens_1']} (left panel) and in Eq. \ref{['eq:intens_2']} (right panel) with the $\beta$-FPUT potential, with $\delta=11\ a$ and $g_4=1$. Values of the other parameters are the same as in Fig. \ref{['fig:fig2']}.