# Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

## TL;DR

This paper models heat transfer in a 2D harmonic lattice in a viscous environment with a point heat source, deriving analytical solutions for the steady-state temperature distribution and validating them against numerical simulations.

## Contribution

It introduces a new analytical formula for the steady-state kinetic temperature distribution in a 2D lattice with a point heat source, derived from stochastic differential equations.

## Key findings

- Analytical solution matches numerical results well except near the source.
- Derived differential-difference equation describes non-stationary heat propagation.
- Singularity occurs in the analytical solution at the source point.

## Abstract

We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite system of linear partial differential equations for covariance variables. The most important results of the paper are the deterministic differential-difference equation describing non-stationary heat propagation in the lattice and the analytical formula in the integral form for its steady-state solution describing kinetic temperature distribution caused by a point heat source of a constant intensity. The comparison between numerical solution of stochastic equations and obtained analytical solution demonstrates a very good agreement everywhere except for the main diagonals of the lattice (with respect to the point source position), where the analytical solution is singular.