Non-equilibrium stationary state of a harmonic crystal with alternating masses

TL;DR

This paper investigates the non-equilibrium steady states of a one-dimensional harmonic chain with alternating masses, revealing oscillating temperature profiles and exact expressions for bulk temperature and current, with effects depending on boundary conditions and system size.

Contribution

It provides exact analytical results for temperature profiles and steady state currents in a harmonic chain with alternating masses, including effects of boundary conditions and system size.

Findings

Temperature profile oscillates with period two in the bulk.

Steady state current and temperature depend on whether system size is odd or even.

Temperature oscillations in noisy systems diminish as system size increases.

Abstract

We analyze the non-equilibrium steady states (NESS) of a one dimensional harmonic chain of $N$ atoms with alternating masses connected to heat reservoirs at unequal temperatures. We find that the temperature profile defined through the local kinetic energy $T(j) \equiv {<p_j^2>}/{m_j}$, oscillates with period two in the bulk of the system. Depending on boundary conditions, either the heavier or the lighter particles in the bulk are hotter. We obtain exact expressions for the bulk temperature profile and steady state current in the limit $N \rightarrow \infty$. These depend on whether $N$ is odd or even. We also study similar temperature oscillations in the NESS of systems with noise in the dynamics. These die out as $N \rightarrow \infty$.