Interlacing relaxation and first passage phenomena in reversible Markovian dynamics
David Hartich (Max-Planck-Institut für BPC Göttingen )
The first passage time quantifies the time a stochastic process reaches a given threshold for the first time, for example, the first instance a diffusing molecule overcomes an energetic barrier. Relaxation processes describe the approach towards equilibrium and, in contrast to first passage processes, are not terminated upon reaching a given threshold. We proof a duality between relaxation and first passage processes in ergodic reversible Markovian dynamics in both discrete and continuous state space. The duality is deduced from a spectral interlacing of relaxation and first passage time scales. For example, the slowest time scale on which a molecule relaxes to equilibrium is always shorter than the slowest first passage time scale. Based on the spectral interlacing we derive the full first passage time statistics from the corresponding relaxation eigenspec- trum in form of a Newtons series of almost triangular matrices [1], which inter alia, allows for the first time to analytically determine the full first passage time statistics for a diffusive exit from a harmonic potential. We further apply our theory to the study of first passage kinetics in a lattice jump process and in a simple discrete folding model. Going beyond single-molecule kinetics we discuss many-particle first passage problems in the few-encounter limit [2], where a response is triggered by a few reactive events, for example, a ‘catastrophic nucleation’ of misfolded proteins. We analyze such few-encounter reaction kinetics in rugged energy landscapes and reveal that even in the presence of time-scale separation these cannot be correctly explained using a single time scale [3].
Literature
[1] D. Hartich and A. Godec, arXiv:1802.10049 (2018).
[2] A. Godec and R. Metzler, Phys. Rev. X 6, 041037 (2016).
[3] D. Hartich and A. Godec, arXiv:1802.10046 (2018).