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Thermodynamics of the space of trajectories governed by a combination of two additive boundary functionals

This paper extends stochastic path-space thermodynamics by analyzing the fluctuation statistics of an Ornstein-Uhlenbeck process with asymmetric drift and a step penalty potential, utilizing the Feynman-Kac formalism and parabolic cylinder theory to derive high-precision eigenvalues and formulate a modified Gallavotti-Cohen fluctuation theorem for a combination of residence time and integral current functionals relevant to molecular motors and ion channels.

Original authors: V. V. Ryazanov

Published 2026-09-30

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CC BY 4.0

Original authors: V. V. Ryazanov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of biology, where molecules jostle and collide in a chaotic thermal bath, the rules of thermodynamics look very different than they do for a steam engine or a cooling cup of coffee. In the macroscopic world, we rely on averages: the average temperature, the average pressure, the average work done. But for a single protein or a tiny molecular motor, these averages often tell us very little. The fluctuations—the random, wild swings in energy and position—are just as large as the average values themselves. To understand how these tiny machines work, scientists have begun to shift their focus from the state of a system at a single moment to the shape of its entire history. Instead of asking where a particle is right now, they ask what path it took to get there. This new perspective, known as path-space thermodynamics, treats the entire trajectory of a moving object as a single, complex object of study, where the time spent in certain regions and the total distance traveled become the fundamental variables, just like temperature or pressure are for a gas.

Building on this idea, a researcher at the Institute for Nuclear Research in Kiev has developed a new mathematical framework to describe how these microscopic paths respond to forces. The study focuses on a specific type of random motion called the Ornstein-Uhlenbeck process, which models a particle tethered by a spring-like force but constantly buffeted by heat. The researcher introduced two new ways to measure the particle's journey: the total time it spends in a specific "positive" region, and the total distance it covers, or its integral current. By combining these two measurements, the study creates a detailed map of how the system behaves when pushed by external forces, such as chemical signals or electrical fields. The goal was to see if these two measurements are linked, and if so, how a change in one affects the other.

To solve this, the researcher translated the problem of random motion into the language of quantum mechanics, using a powerful mathematical tool called the Feynman-Kac formalism. This approach allowed the complex statistics of the particle's path to be treated like the energy levels of a quantum particle moving through a landscape with a sudden step or barrier. The landscape represents the physical constraints of the system: a smooth, spring-like pull, a sudden penalty for crossing a boundary, and a steady push from an external force. By solving the equations that describe this landscape, the researcher calculated the exact energy values that define the system's behavior. Using a high-precision numerical method, the study found specific values for these energy levels, such as 1.255 and 0.399, which represent the free energy of the system under different conditions. These numbers are not just abstract results; they define the likelihood of the particle taking certain paths and reveal how the system transitions between different modes of operation.

The most significant finding of the work is the discovery of a precise relationship between the time a system spends in a particular state and the total current it generates. The researcher derived a formula that acts as a bridge between these two quantities, showing how sensitive the system is to changes in its environment. This relationship is a new version of a famous principle in physics known as the Onsager reciprocity relation, but adapted for the history of a particle's path rather than just its current state. The study shows that this sensitivity is not constant; it depends heavily on the balance of forces within the system. When the system is perfectly balanced, sitting right at the threshold between two states, it is most sensitive to external changes. However, if the system is pushed too far into one state or the other, this sensitivity drops off dramatically, effectively making the system "blind" to coordinated changes in its environment.

The research also highlights the crucial role of thermal noise. In many contexts, noise is seen as a nuisance that obscures signals. Here, the study demonstrates that thermal fluctuations actually act as a lubricant, allowing the system to respond to external forces. Without this noise, the system would get stuck in one state, unable to switch or respond. With it, the system can explore different paths and synchronize its movements with external chemical or electrical cues. The study applies this theory to real biological examples, such as ion channels in cell membranes and molecular motors like kinesin. For an ion channel, the "time spent in the positive region" corresponds to how long the channel stays open, while the "integral current" relates to the total amount of ions that flow through. The model suggests that for these biological machines to work efficiently, they must be tuned to operate right at the edge of stability, where they can switch states quickly and respond to multiple signals at once.

The paper concludes by showing that this new way of looking at trajectories provides a powerful tool for understanding the reliability and efficiency of microscopic machines. By treating the history of a particle's motion as a thermodynamic variable, scientists can now predict how likely a system is to fail or how it will respond to complex, changing environments. The work does not just describe what happens; it provides a way to calculate the exact probability of rare, extreme events, such as a molecular motor stalling or an ion channel opening for an unusually long time. This level of detail is essential for designing better nanomachines or understanding the fundamental limits of biological processes, moving beyond simple averages to a deeper understanding of the chaotic, beautiful paths that life takes at the smallest scales.

Technical Summary: Thermodynamics of the Space of Trajectories Governed by a Combination of Two Additive Boundary Functionals

Problem Statement
Traditional stochastic thermodynamics and large deviation theory have largely relied on integral quantities (e.g., total work, heat, entropy production) or instantaneous phase-space coordinates to describe non-equilibrium systems. However, for mesoscopic and open systems—such as biological macromolecules, nanoscale motors, and quantum devices—where fluctuations are comparable to average values, these standard variables are insufficient. There is a need to incorporate the geometric invariants of random trajectories, specifically "trajectory memory" and history-dependent characteristics. While previous works have extended thermodynamic state spaces to include single boundary functionals (e.g., first-passage times, maxima, or occupation times), the mutual influence and cross-correlations between multiple disparate trajectory invariants remain unexplored. This paper addresses the thermodynamics of a system governed by a combination of two additive boundary functionals: the residence time in a specific half-plane and the integral current (or accumulated position).

Methodology
The study utilizes the Feynman–Kac formalism applied to a one-dimensional Ornstein–Uhlenbeck (OU) process with an asymmetric linear drift and a step penalty potential at the origin. The system is modeled by a stochastic Langevin equation where the thermodynamic potential is derived from the joint moment-generating function of two functionals:

Residence Time ( $\tau_+$ ): The time the process spends in the positive half-plane ( $x > 0$ ).
Integral Current ( $I_T$ ): The integral of the position over time (representing accumulated work or displacement).

The mathematical framework transforms the problem of finding the cumulant generating function (the large deviation rate function) into a stationary Schrödinger equation with a piecewise linear potential. The effective Hamiltonian includes a harmonic term, a linear drift term, and a step-like penalty (Heaviside function).

Analytical Approach: The solution involves matching parabolic cylinder functions (Weber functions) across the discontinuity at $x=0$ . The wave functions in the left ( $x<0$ ) and right ( $x>0$ ) half-planes are derived separately, and a transcendental secular equation is constructed by enforcing continuity of the wave function and its first derivative at the interface.
Numerical Approach: Due to the complexity of the transcendental equation involving non-integer indices, a high-precision numerical scheme is employed. This involves integrating the Schrödinger equation from asymptotic regions toward the matching point using a fourth-order Runge–Kutta method and solving for the principal eigenvalue (ground-state energy) via the secant method.
Perturbation Theory: To derive explicit expressions for cross-sensitivity coefficients, the author applies perturbation theory to the matching equation around the unperturbed state (zero external forces).

Key Contributions and Results

Extension of Thermodynamic Variables: The paper systematically expands the space of thermodynamic variables to include a comprehensive set of boundary functionals. It categorizes these into a hierarchy ranging from integral invariants (Level 1) to global extrema (Level 3) and differential/relative extrema (Level 4), such as "Maximal Deficit" (Maximum Drawdown) and "Time of Return."
Exact Thermodynamic Potential: An exact expression for the dynamic free energy (the principal eigenvalue $E(\lambda_1, \lambda_2)$ ) is obtained for the combined functionals. This function serves as the generating function for the joint distribution of residence time and integral current.
Modified Fluctuation Theorem: A Gallavotti–Cohen-type fluctuation theorem is formulated for the conjugate drift current at a fixed integral occupation time. The theorem relates the probability of a forward trajectory to its reverse, incorporating a symmetry relation for the free energy spectrum that accounts for the step penalty and drift.
Trajectory-Based Onsager Reciprocity: The paper derives an exact expression for the mutual cross-sensitivity coefficient ( $\chi_{12}$ ), which acts as a trajectory-based analogue of Onsager reciprocity relations. This coefficient quantifies the correlation between the residence time in the positive half-plane and the integral current.
- In the symmetric case, $\chi_{12}$ is inversely proportional to $\gamma^{2.5}$ (the restoring force coefficient to the power of 2.5) and proportional to $D^{0.5}$ (the square root of the diffusion coefficient).
- In the asymmetric case (with a baseline energy gap $U_0$ ), the cross-sensitivity is exponentially suppressed by a factor involving $U_0$ and $D$ . This reveals a "sensitivity window" effect: maximal cross-sensitivity occurs only when the system operates near the critical threshold ( $U_0 \approx 0$ ).
Numerical Validation: High-precision numerical calculations confirm the existence of a "kink" (discontinuity in the first derivative) in the ground-state wave function at the potential step, signifying a discrete-event nature of the phase transition. The results show a nonlinear response where increasing the external drift force shifts the probability density maximum into the positive half-plane and lowers the ground-state energy.

Significance and Claims
The paper claims to take a significant step toward expanding the trajectory phase space by moving beyond single integral characteristics or simple extrema to a comprehensive set of differential and relative boundary functionals.

Theoretical Advancement: It establishes a formalism where "residence time" and "integral current" are treated as conjugate thermodynamic variables, allowing for the description of non-equilibrium processes via their history rather than just instantaneous states.
Biophysical Application: The framework is applied to model molecular Brownian motors (e.g., kinesin, F1-ATPase) and ion channels (e.g., mechanosensitive or voltage-gated channels).
- For molecular motors, the step potential models the waiting phase (ATP binding) versus the active transport phase.
- For ion channels, the model links the "open time" (residence time) with the "total conductance" (integral current), demonstrating how thermal noise acts as a "lubricant" to synchronize the channel's geometry and kinetics.
Optimization Insight: The derivation of the cross-sensitivity coefficient provides a criterion for the efficient operation of biomolecular machines: systems must be tuned to operate near a critical point ( $U_0 \approx 0$ ) to maximize their responsiveness to combined chemical and mechanical stimuli.
Robustness: The work demonstrates that additivity is a crucial property for extending thermodynamic spaces, as it preserves the structure of classical thermodynamics (analogous to entropy and energy) and enables the direct application of Large Deviation Theory.

The paper concludes that this approach transforms continuous stochastic thermodynamics into a discrete-event framework, offering new tools for calculating catastrophic drawdown probabilities and optimizing control protocols in fluctuation-dominated regimes.

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