  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.
