Transport in the Stochastic Lorenz System

We study transport in the stochastic Lorenz system mathematically, computationally and using a circuit model. The circuit model provides a very efficient method for computing long time averages of polynomials in the variables $X,Y,$ and $Z$ with real-time updates. In particular, we use this approach to the quantity $\langle XY \rangle$, which is the heat transport corresponding with Rayleigh-B\'enard convection. We interpret our results in the framework of analytical stochastic upper bounds [1] for $\langle XY \rangle$ versus $\rho$ (the reduced Rayleigh number), as well as against numerical solutions. For a given $\rho$ we find a rich dependence of the transport on both noise color and amplitude due to the detailed coupling of noise with Unstable Periodic Orbits.\\ {\bf [1]} {S. Agarwal and J. S. Wettlaufer, Maximal stochastic transport in the Lorenz equations, {\em Phys. Lett. A} {\bf 380}, 142 (2016).}