AI Insight
This theoretical study develops a mathematical framework for analyzing non-Markovian jump processes, where future events depend on the entire history of past events rather than just the current state. The researchers show that even when memory effects prevent traditional analysis methods, each transition event can still be described by exact equations using "martingale increments"—unpredictable random components that form a basis for understanding fluctuations. They derive exact relationships connecting spontaneous fluctuations in observables to how the system responds to external perturbations, valid at finite times and frequencies, with a quantifiable gap that measures how strongly memory affects the system's behavior.
Why it matters
This framework provides new tools for analyzing complex biological, chemical, and physical systems where history matters, such as gene expression networks, neural activity, or molecular motors. The ability to extract response properties from spontaneous fluctuations could enable researchers to predict how such systems behave under perturbations without directly intervening, and to test whether proposed memory models adequately capture the system's dynamics.
Understand the Science
⚠️ Preprint – Noch nicht peer-reviewed
Dieser Artikel wurde noch nicht von unabhängigen Experten begutachtet. Die Ergebnisse sind vorläufig und sollten mit Vorsicht interpretiert werden.
Abstract: Modern experiments often record nonequilibrium dynamics as sequences of discrete events whose rates depend on the realized past. We develop a fluctuation-response theory for such non-Markovian jump processes directly on the observed event record. Memory can destroy a closed master equation for the state probabilities. Each transition count nevertheless obeys an exact stochastic equation. After the history-dependent mean event tendency is subtracted, the remaining random increment is a martingale increment–the part of the event that cannot be predicted from the past. Martingale increments associated with different transitions and different times are orthogonal. These increments form a complete orthogonal basis for the random deviation of any observable measured from the record, such as a current, occupation time, or event count. The coefficient of a given increment is the event-consequence kernel. It measures how that event changes the predicted final observable, relative to continuing without the event, for the particular history already realized. Multiplying this kernel by the event intensity gives the history-conditioned response to perturbing the corresponding transition rate. Thus, the intensity-normalized response is exactly the expansion coefficient of that event in the observable fluctuation. This identification yields exact finite-time and finite-frequency fluctuation-response relations. Averaging over histories leaves a nonnegative response-heterogeneity gap. The gap measures how strongly the consequence of the same event varies across histories and tests whether a proposed memory coordinate is response-sufficient. Finite-history versions can be estimated from spontaneous trajectories. Bounding the logarithmic rate sensitivity further gives response-kinetic uncertainty relations controlled by dynamical activity.