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

## TL;DR

This paper reviews recent mathematical results on how conserved quantities like energy and angular momentum influence Fourier's law in diffusive mechanical systems, especially when stochastic dynamics are involved.

## Contribution

It highlights new mathematical insights into the diffusive transport of multiple conserved quantities and the role of stochastic perturbations in defining transport coefficients.

## Key findings

- Existence of transport coefficients like thermal conductivity.
- Proofs of local equilibrium and linear response in stochastic models.
- Understanding of temperature and conserved quantity profiles in non-equilibrium states.

## Abstract

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.