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We introduce stochastic models for the transport of heat in systems describedby local collisional dynamics. The dynamics consists of tracer particles movingthrough an array of hot scatterers describing the effect of heat baths at fixedtemperatures. Those models have the structure of Markov renewal processes. Westudy their ergodic properties in details and provide a useful formula for thecumulant generating function of the time integrated energy current. We observethat out of thermal equilibrium, the generating function is not analytic. Whenthe set of temperatures of the scatterers is fixed by the condition that inaverage no energy is exchanged between the scatterers and the system, differentbehaviours may arise. When the tracer particles are allowed to travel freelythrough the whole array of scatterers, the temperature profile is linear. Ifthe particles are locked in between scatterers, the temperature profile becomesnonlinear. In both cases, the thermal conductivity is interpreted as afrequency of collision between tracers and scatterers.

Hot scatterers and tracers for the transfer of heat in collisional dynamics