Thermal conductivity of the Toda lattice with conservative noise

We study the thermal conductivity of the one dimensional Toda lattice perturbed by a stochastic dynamics preserving energy and momentum. The strength of the stochastic noise is controlled by a parameter $\gamma$. We show that heat transport is anomalous, and that the thermal conductivity diverges with the length $n$ of the chain according to $\kappa(n) \sim n^\alpha$, with $0 < \alpha \leq 1/2$. In particular, the ballistic heat conduction of the unperturbed Toda chain is destroyed. Besides, the exponent $\alpha$ of the divergence depends on $\gamma$.