Entropy production in systems with random transition rates

We study the entropy production of a system with a finite number of states connected by random transition rates. The stationary entropy production, driven out of equilibrium both by asymmetric transition rates and by an external probability current, is shown to be composed of two contributions whose exact distributions are calculated in the large system size and close to equilibrium. The first contribution is related to Joule's law for the heat dissipated in a classical electrical circuit whereas the second one has a Gaussian distribution with an extensive average and a finite variance.