Fluctuation-dissipation relations for Markov processes

The fluctuation-dissipation relation is calculated for stochastic models obeying a master equation with continuous time. In the general case of a nonstationary process, there appears to be no simple relation between the response and the correlation. Also, if one considers stationary processes, the linear response cannot be expressed via time-derivatives of the correlation function alone. In this case, an additional function, which has rarely been discussed previously, is required. This so-called asymmetry depends on the two times also relevant for the response and the correlation and it vanishes under equilibrium conditions. The asymmetry can be expressed in terms of the propagators and the transition rates of the master equation but it is not related to any physical observable in an obvious way. It is found that the behavior of the asymmetry strongly depends on the nature of the dynamical variable considered in the calculation of the correlation and the response. If one is concerned with a variable which randomizes with any transition among the states of the system, the asymmetry vanishes in most cases. This is in contrast to the situation for other classes of variables. In particular, for trap models of glassy relaxation, the fluctuation-dissipation ratio strongly depends on the observable and the asymmetry plays a dominant role in the determination of this ratio also if only neutral variables are considered. Some implications of a nonvanishing asymmetry with regard to the definition of an effective temperature are discussed.