We study Landauer's Principle for Repeated Interaction Systems (RIS) consisting of a reference quantum system S in contact with a structured environment E made of a chain of independent quantum probes; S interacts with each probe, for a fixed duration, in sequence. We first adapt Landauer's lower bound, which relates the energy variation of the environment E to a decrease of entropy of the system S during the evolution, to the peculiar discrete time dynamics of RIS. Then we consider RIS with a structured environment E displaying small variations of order T −1 between the successive probes encountered by S, after n ≃ T interactions, in keeping with adiabatic scaling. We establish a discrete time non-unitary adiabatic theorem to approximate the reduced dynamics of S in this regime, in order to tackle the adiabatic limit of Landauer's bound. We find that saturation of Landauer's bound is related to a detailed balance condition on the repeated interaction system, reflecting the non-equilibrium nature of the repeated interaction system dynamics. This is to be contrasted with the generic saturation of Landauer's bound known to hold for continuous time evolution of an open quantum system interacting with a single thermal reservoir in the adiabatic regime.

We show that elements of control theory, together with an application of Harris' ergodic theorem, provide an alternate method for showing exponential convergence to a unique stationary measure for certain classes of networks of quasi-harmonic classical oscillators coupled to heat baths. With the system of oscillators expressed in the formA encodes the harmonic part of the force and −F corresponds to the gradient of the anharmonic part of the potential, the hypotheses under which we obtain exponential mixing are the following: A is dissipative, the pair (A, B) satisfies the Kalman condition, F grows sufficiently slowly at infinity (depending on the dimension d), and the vector fields in the equation of motion satisfy the weak Hörmander condition in at least one point of the phase space.Different sufficient conditions for the hypotheses of the main theorem to hold are given in more concrete terms throughout Sections 4 and 5. In the former, we give criteria for the dissipativity, Kalman and growth conditions in terms of more physical quantities for networks of oscillators based on [JPS17]. In the latter, we give a perturbative condition for the weak Hörmander condition to hold.

We study heat fluctuations in the two-time measurement framework. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0.On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat variation distribution exhibits heavy tails. The tails can be as heavy as preventing the existence of a fourth moment of the heat variation.