Estimating Relaxation Times from a Single Trajectory

Complex systems such as protein conformational fluctuations and supercooled liquids exhibit a long relaxation time and are considered to posses multiple relaxation times. We analytically obtain the exact correlation function for stochastic processes with multiple relaxation times. We show that the time-averaged correlation function calculated by a trajectory whose length is shorter than the longest relaxation time exhibits an apparent aging behavior. We propose a method to extract relaxation times from a single trajectory. This method successfully extracts relaxation times of a stochastic process with multiple states when a state can be characterized by the values of the trajectory. As an application of this method, we estimate several relaxation times smaller than the longest relaxation time in conformational fluctuations of a small protein.

💡 Research Summary

The paper addresses a fundamental challenge in the analysis of complex stochastic systems—how to extract multiple relaxation time scales when only a single, possibly short, trajectory is available. Complex systems such as protein conformational fluctuations, super‑cooled liquids, and polymer dynamics often display a long‑time power‑law decay in their correlation functions, which can be interpreted as a superposition of several exponential relaxation modes. Traditional approaches (e.g., time‑structure based independent component analysis, Markov state models, relaxation‑mode analysis) require many independent trajectories or high‑dimensional data, which are frequently unavailable in experiments or costly molecular simulations.

Theoretical framework
The authors first construct a general stochastic model in which N independent dichotomous processes (I_k(t)) (each taking two values (I_k^{\pm})) are summed to form the observable signal (I(t)=\sum_{k=1}^{N} I_k(t)). Each dichotomous process is characterized by a waiting‑time probability density function (PDF) (\psi_k(t)). By counting the number of stochastic switches (N_t) up to time t and using the independence of the processes, they derive the un‑normalized correlation function (\hat C_k(t)). When (\psi_k(t)) is exponential, the normalized correlation reduces to a simple exponential decay (C_k(t)=\exp(-t/\tau_k)) with relaxation time (\tau_k). If (\psi_k(t)) follows a power‑law (\psi_k(t)\propto t^{-\alpha-1}) ((\alpha>1) for stationarity), the correlation exhibits a long‑time algebraic tail (C_k(t)\propto t^{-\alpha}). The authors also discuss the equilibrium renewal case, where the first‑waiting‑time distribution differs from (\psi_k), leading to modified exponents.

The analysis is then generalized to a multi‑state Markov chain with transition probabilities (p_{ij}=p_j) and state values (I_k). Assuming exponential waiting times for each state, the overall correlation function again becomes a sum of exponentials, each weighted by the variance of the corresponding state. This shows that the dichotomous superposition is a special case of a broader class of renewal processes.

The paper further treats alternating renewal processes, where the + and – states have distinct PDFs (\psi_{+}(t)) and (\psi_{-}(t)). By Laplace transforming the transition probabilities (W_{hh’}(t)) and using equilibrium probabilities (p_{eq}^{\pm}), they obtain a single exponential decay with an effective relaxation time (\tau_{\text{eff}} = \tau_{+}\tau_{-}/(\tau_{+}+\tau_{-})). This illustrates how two different time scales can collapse into one observable decay when the process is observed only through a binary signal.

Finally, the authors consider a superposition of Ornstein‑Uhlenbeck (OU) processes. Each OU component obeys (dI_i = -\nu_i I_i dt + \sqrt{2\nu_i},\eta_i \xi_i(t)) and has an autocorrelation (\hat C_i(t)=\eta_i e^{-\nu_i t}). Summing N such processes yields a correlation function that is simply the sum of exponentials with amplitudes (\eta_i). This provides a continuous‑Gaussian analogue of the dichotomous superposition and is directly relevant to fluctuating‑diffusivity models.

Apparent aging and time‑averaged correlations
In many experiments, only a single long trajectory of length (T) is available, and ensemble averages cannot be computed. The authors therefore define a time‑averaged correlation function
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