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## Section: New Results

### Joule effect in chains of oscillators

In physics, the rotor chain has been investigated as an example of a system with two conserved quantities (angular momentum and energy), for which the thermal conductivity is finite (and therefore energy diffuses). Numerics shows an unexpected behaviour of the chain when the latter is connected at the boundaries to two thermostats, and a mechanical force imposes an average angular momentum at one boundary: the stationary temperature profile coincides with the values of the thermostats, but in the middle of the chain it raises to a much higher value. This behaviour is related to the presence of two conserved quantities and is sometimes referred to as Joule effect. Since the rotor model is too difficult to be treated analytically, T. Komorowski, S. Olla and M. Simon investigate in [25] the harmonic chain of oscillators, perturbed with a stochastic noise, which makes the heat transport diffusive, namely: the noise destroys the conservation law of the total momentum, but keeps the other two conservation laws (energy and stretch) intact. The boundaries of the chain are connected to two Langevin thermostats and an external force acting on one boundary puts the system in a non-equilibrium stationary state. The authors rigorously derive the Joule effect for a particular value of the noise intensity.