  This work presents a formalism to derive field quantities and conservation
laws from the atomistic using the theory of distributions as the mathematical
tool. By defining temperature as a derived quantity as that in molecular
kinetic theory and atomistic simulations, a field representation of the
conservation law of linear momentum is derived and expressed in terms of
temperature field, leading to a unified atomistic and continuum description of
temperature and a new conservation equation of linear momentum that,
supplemented by an interatomic potential, completely governs thermal and
mechanical processes across scales from the atomic to the continuum. The
conservation equation can be used to solve atomistic trajectories for systems
at finite temperatures, as well as the evolution of field quantities in space
and time, with atomic or multiscale resolution. Four sets of numerical examples
are presented to demonstrate the efficacy of the formulation in capturing the
effect of temperature or thermal fluctuations, including phonon density of
states, thermally activated dislocation motion, dislocation formation during
epitaxial processes, and attenuation of longitudinal acoustic waves as a result
of their interaction with thermal phonons.
