  Thermal conductivity in dielectric crystals is the result of the relaxation
of lattice vibrations described by the phonon Boltzmann transport equation.
Remarkably, an exact microscopic definition of the heat carriers and their
relaxation times is still missing: phonons, typically regarded as the relevant
excitations for thermal transport, cannot be identified as the heat carriers
when most scattering events conserve momentum and do not dissipate heat flux.
This is the case for two-dimensional or layered materials at room temperature,
or three-dimensional crystals at cryogenic temperatures. In this work we show
that the eigenvectors of the scattering matrix in the Boltzmann equation define
collective phonon excitations, termed here relaxons. These excitations have
well defined relaxation times, directly related to heat flux dissipation, and
provide an exact description of thermal transport as a kinetic theory of the
relaxon gas. We show why Matthiessen's rule is violated, and construct a
procedure for obtaining the mean free paths and relaxation times of the
relaxons. These considerations are general, and would apply also to other
semiclassical transport models, such as the electronic Boltzmann equation. For
heat transport, they remain relevant even in conventional crystals like
silicon, but are of the utmost importance in the case of two-dimensional
materials, where they can revise by several orders of magnitude the relevant
time- and length-scales for thermal transport in the hydrodynamic regime.
