The time evolution during which macroscopic systems reach thermodynamic equilibrium states proceeds as a continuous sequence of contact structure preserving transformations maximizing the entropy. This viewpoint of mesoscopic thermodynamics and dynamics provides a unified setting for the classical equilibrium and nonequilibrium thermodynamics, kinetic theory, and statistical mechanics. One of the illustrations presented in the paper is a new version of extended nonequilibrium thermodynamics with fluxes as extra state variables.

Reduction of a mesoscopic level to a level with fewer details is made by the time evolution during which the entropy increases. An extension of a mesoscopic level is a construction of a level with more details. In particular, we discuss extensions in which extra state variables are found in the vector fields appearing on the level that we want to extend. Reductions, extensions, and compatibility relations among them are formulated first in an abstract setting and then illustrated in specific mesoscopic theories.

Heat transfer is investigated on three levels of description: Fourier, Cattaneo, and Peierls. The microscopic nature of the heat that becomes important, in particular in nanoscale systems, is characterized by a vector field related to the heat flux on the Cattaneo level and by the phonon distribution function on the Peierls level. All dynamical theories discussed in the paper are fully nonlinear and all are proven to be compatible among themselves, with equilibrium thermodynamics, and with mechanics. An investigation of the first two compatibilities gives rise to potentials having the physical interpretation of nonequilibrium entropies. The compatibility with mechanics is manifested by the Hamiltonian structure of the time-reversible part of the time evolution.