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Mathematical Physics Seminar

Stefano Olla - Diffusive behavior of conserved quantities in systems with thermal and mechanical boundary forcing: hydrodynamic limits and non-equilibrium stationary states

Stefano Olla

Date & time: Thursday, 23 July 2020 at 11:00AM - 12:00PM

Stefano Olla - CEREMADE, Université Paris-Dauphine

Thursday, July 23, 11:00AM

“Diffusive behavior of conserved quantities in systems with thermal and mechanical boundary forcing: hydrodynamic limits and non-equilibrium stationary states.”

We consider some models of interacting particles with at least two locally conserved quantities (including energy) with boundaries in contact with heat bath at different temperatures and mechanical forces.
One of the most interesting dynamics is a chain of rotors (also known as XY-model), with constant torques applied at the boundaries. Energy and angular momentum are the locally conserved quantities, and by the symmetries of the system the corresponding Onsager matrix depends only on the thermal conductivity and momentum diffusivity. There is no ballistic propagation in this system and the macroscopic behavior of the conserved quantities is diffusive. The corresponding macroscopic diffusive system of PDE show explicitly an increase of local temperature proportional to the square of the gradient of the profile of momentum. In the (non-equilibrium) stationary state (NESS), the equations predict interesting profiles where temperatures inside the system are much higher than at the boundaries, confirmed by previous numerical simulations. Furthermore “up-hill” diffusion and negative linear response for the energy transport can appear in these NESS.
Rigorous mathematical results can be proven in more simple stochastic dynamics: a chain of harmonic oscillators with random exchange of momentum between n.n. particles, in contact with Langevin heat bath and a constant exterior pulling tension. The locally conserved quantities are here the energy and the local volume stretch. Similar behavior can be proven in the NESS of this dynamics.
Joint works with Alessandra Iacobucci and Gabriel Stoltz (rotors), and with Tomasz Komorowski and Marielle Simon (harmonic chain with noise).
— S. Olla, Role of conserved quantities in Fourier's law for diffusive mechanical systems, Comptes Rendus Physique, Volume 20, Issue 5, July–August 2019, Pages 429-441
— A. Iacobucci, S. Olla, G. Stoltz, Thermo-mechanical transport in rotor chains, 2020,
— T. Komorowski, S. Olla, M. Simon, An open microscopic model of heat conduction: evolution and non-equilibrium stationary states, Communications in Mathematical Sciences, Vol. 18, No. 3 (2020), pp. 751-780. 

— T. Komorowski, S. Olla, M. Simon : Hydrodynamic limit for a chain with thermal and mechanical boundary forces,

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