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We consider heat transfer in an infinite two-dimensional square harmonicscalar 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 ofa system of stochastic ordinary differential equations. We perform acontinualization procedure and derive an infinite system of linear partialdifferential equations for covariance variables. The most important results ofthe paper are the deterministic differential-difference equation describingnon-stationary heat propagation in the lattice and the analytical formula inthe integral form for its steady-state solution describing kinetic temperaturedistribution caused by a point heat source of a constant intensity. Thecomparison between numerical solution of stochastic equations and obtainedanalytical solution demonstrates a very good agreement everywhere except forthe main diagonals of the lattice (with respect to the point source position),where the analytical solution is singular.

continuum mechanics and thermodynamics

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