# Fast and slow thermal processes in harmonic scalar lattices

## TL;DR

This paper presents an analytical approach to describe both fast and slow thermal processes in harmonic scalar lattices, capturing short-term oscillations and long-term ballistic heat transfer with high accuracy.

## Contribution

The paper introduces an exact continuum equation for thermal evolution in scalar lattices and analytically solves it, distinguishing between fast oscillatory and slow ballistic heat transfer processes.

## Key findings

- Fast process involves high-frequency temperature oscillations within ten atomic vibration periods.
- Slow process is characterized by ballistic heat transfer with wave-like temperature propagation.
- The theory accurately predicts temperature evolution at both short and long time scales.

## Abstract

An approach for analytical description of thermal processes in harmonic lattices is presented. We cover longitudinal and transverse vibrations of chains and out-of-plane vibrations of two-dimensional lattices with interactions of an arbitrary number of neighbors. Motion of each particle is governed by a single scalar equation and therefore the notion "scalar lattice" is used. Evolution of initial temperature field in an infinite lattice is investigated. An exact equation describing the evolution is derived. Continualization of this equation with respect to spatial coordinates is carried out. The resulting continuum equation is solved analytically. The solution shows that the kinetic temperature is represented as the sum of two terms, one describing short time behavior, the other large time behavior. At short times, the temperature performs high-frequency oscillations caused by redistribution of energy among kinetic and potential forms (fast process). Characteristic time of this process is of order of ten periods of atomic vibrations. At large times, changes of the temperature are caused by ballistic heat transfer (slow process). The temperature field is represented as a superposition of waves having the shape of initial temperature distribution and propagating with group velocities dependent on the wave vector. Expressions describing fast and slow processes are invariant with respect to substitution $t$ by $-t$. However examples considered in the paper demonstrate that these processes are irreversible. Numerical simulations show that presented theory describes the evolution of temperature field at short and large time scales with high accuracy.