132 research outputs found

### Role of conserved quantities in Fourier's law for diffusive mechanical systems

Energy transport can be influenced by the presence of other conserved
quantities. We consider here diffusive systems where energy and the other
conserved quantities evolve macroscopically on the same diffusive space-time
scale. In these situations the Fourier law depends also from the gradient of
the other conserved quantities. The rotor chain is a classical example of such
systems, where energy and angular momentum are conserved. We review here some
recent mathematical results about diffusive transport of energy and other
conserved quantities, in particular for systems where the bulk Hamiltonian
dynamics is perturbed by conservative stochastic terms. The presence of the
stochastic dynamics allows to define the transport coefficients (thermal
conductivity) and in some cases to prove the local equilibrium and the linear
response argument necessary to obtain the diffusive equations governing the
macroscopic evolution of the conserved quantities. Temperature profiles and
other conserved quantities profiles in the non-equilibrium stationary states
can be then understood from the non-stationary diffusive behaviour. We also
review some results and open problems on the two step approach (by weak
coupling or kinetic limits) to the heat equation, starting from mechanical
models with only energy conserved.Comment: Review Article for the CRAS-Physique, final versio

### Microscopic derivation of an adiabatic thermodynamic transformation

We obtain macroscopic adiabatic thermodynamic transformations by space-time
scalings of a microscopic Hamiltonian dynamics subject to random collisions
with the environment. The microscopic dynamics is given by a chain of
oscillators subject to a varying tension (external force) and to collisions
with external independent particles of "infinite mass". The effect of each
collision is to change the sign of the velocity without changing the modulus.
This way the energy is conserved by the resulting dynamics. After a diffusive
space-time scaling and cross-graining, the profiles of volume and energy
converge to the solution of a deterministic diffusive system of equations with
boundary conditions given by the applied tension. This defines an irreversible
thermodynamic transformation from an initial equilibrium to a new equilibrium
given by the final tension applied. Quasi-static reversible adiabatic
transformations are then obtained by a further time scaling. Then we prove that
the relations between the limit work, internal energy and thermodynamic entropy
agree with the first and second principle of thermodynamics.Comment: New version accepted for the publication in Brazilian Journal of
Probability and Statistic

### Toward the Fourier law for a weakly interacting anharmonic crystal

For a system of weakly interacting anharmonic oscillators, perturbed by an
energy preserving stochastic dynamics, we prove an autonomous (stochastic)
evolution for the energies at large time scale (with respect to the coupling
parameter). It turn out that this macroscopic evolution is given by the so
called conservative (non-gradient) Ginzburg-Landau system of stochastic
differential equations. The proof exploits hypocoercivity and hypoellipticity
properties of the uncoupled dynamics

### Equilibrium fluctuation for an anharmonic chain with boundary conditions in the Euler scaling limit

We study the evolution in equilibrium of the fluctuations for the conserved
quantities of a chain of anharmonic oscillators in the hyperbolic space-time
scaling. Boundary conditions are determined by applying a constant tension at
one side, while the position of the other side is kept fixed. The Hamiltonian
dynamics is perturbed by random terms conservative of such quantities. We prove
that these fluctuations evolve macroscopically following the linearized Euler
equations with the corresponding boundary conditions, even in some time scales
larger than the hyperbolic one

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