Generalization of the Einstein relation for single trajectories in deterministic subdiffusion.

Diffusion coefficients are intrinsically random in subdiffusion attributable to power-law trapping. Using deterministic biased and unbiased diffusion models, we investigate the Einstein relation for single trajectories in subdiffusion. The difference in the generalized Lyapunov exponent between biased and unbiased deterministic diffusions is related to the velocity under a bias. By Hopf's ergodic theorem, the ratios between the velocities and the Lyapunov exponents for single trajectories converge to a universal constant, which is proportional to the strength of the bias. Based on a certain transport coefficient obtained from a single trajectory, we provide a relation for the transport coefficients divided by the Lyapunov exponent and generalize the Einstein relation for single trajectories.