  Thermal contact is the archetype of non-equilibrium processes driven by
constant non-equilibrium constraints enforced by reservoirs exchanging
conserved microscopic quantities. In models with a finite number of possible
configurations, if the microscopic dynamics is assumed to be deterministic and
ergodic and to conserve energy according to some specific pattern, and if the
mesoscopic evolution of the global system is approximated by a Markov process
as closely as possible, then the mesoscopic transition rates obey three
constraints. In the limit where macroscopic bodies can be considered as
reservoirs at thermodynamic equilibrium (but with different intensive
parameters) the third constraint becomes modified detailed balance (MDB) ; the
latter is generically expressed in terms of the microscopic exchange entropy
variation. We investigate the generic statistical properties for measurable
quantities that arise from the MDB constraint. For a finite-time evolution
after the system prepared in an equilibrium state has been set in contact with
thermostats at different temperatures, we derive a detailed fluctuation
relation for the excess exchange entropy variation. In the non-equilibrium
stationary state (long-time limit), the proper mathematical definition of a
large deviation function is introduced together with alternative definitions,
and fluctuation relations are rederived. The generalization to systems
exchanging energy, volume and matter with several reservoirs, with a possible
conservative external force acting on the contact system, is given explicitly.
The infinite time limit of any odd cumulant per unit time of exchanged
quantities is expressed in terms of a series involving higher even cumulants
and powers of the thermodynamic forces associated to independent mean currents.
