One dimensional heat conductivity exponent from random collision model

We study numerically the thermal conductivity coefficient $\kappa$ as a function of system length $L$ for several different quasi one dimensional models: classical gases of hard spheres with both longitudinal and transverse degrees of freedom. We introduce a model that is ergodic and highly chaotic but also conserves energy and momentum, and is very useful because it shows scaling even at small system sizes. We find that $\kappa \sim L^{\alpha }$ over more than two decades, with $\alpha$ very close to the analytical prediction of 1/3.