Time-dependence of the effective temperatures of a two-dimensional Brownian gyrator with cold and hot components

We consider a model of a two-dimensional molecular machine—called Brownian gyrator—that consists of two coordinates coupled to each other and to separate heat baths at temperatures respectively T _x and T _y. We consider the limit in which one component is passive, because its bath is 'cold', T _x → 0, while the second is in contact with a 'hot' bath, T _y > 0, hence it entrains the passive component in a stochastic motion. We derive an asymmetry relation as a function of time, from which time dependent effective temperatures can be obtained for both components. We find that the effective temperature of the passive element tends to a constant value, which is a fraction of T _y, while the effective temperature of the driving component grows without bounds, in fact exponentially in time, as the steady-state is approached. (paper)