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Active Jarzynski and Crooks Relations for Driven Jump-Diffusions

This paper establishes exact finite-time Active Jarzynski and Crooks relations for driven Markov jump-diffusions by deriving a trajectory-level entropy decomposition relative to an instantaneous steady state, thereby enabling the recovery of free-energy differences and the characterization of active irreversibility in non-Gaussian, non-conservative systems.

Original author: Shuyuan Fan, Bing Miao, Jinqiao Duan, Qi Zhang

Published 2026-11-03

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CC BY 4.0

Original author: Shuyuan Fan, Bing Miao, Jinqiao Duan, Qi Zhang

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of microscopic physics, scientists have long relied on a set of rules to understand how energy moves and how systems settle down. Imagine a tiny particle floating in water, jostled by invisible molecules. If you push that particle with a steady hand, you can predict how much work you did and how much heat was lost, provided the particle eventually finds a calm, resting state. For decades, this "calm state" was the anchor for a famous rule called the Jarzynski equality. This rule allows researchers to calculate the hidden energy difference between two states of a system just by watching how much work is done during a messy, fast process. It is a powerful tool, but it assumes the system behaves like a simple, passive object that eventually stops moving once the pushing stops.

However, the universe is full of things that do not sit still. Inside living cells, or in swarms of synthetic micro-robots, tiny particles are often powered by their own internal engines. They consume fuel to move, jump, and change direction on their own, creating a constant, churning energy that never settles into a simple rest. These are called active systems. When scientists try to apply the old rules to these energetic, self-driving particles, the math breaks down. The constant internal motion creates a kind of hidden friction and heat that the traditional formulas cannot see, making it impossible to tell how much work was done by the outside hand versus how much energy was burned by the particle's own engine. This gap has left researchers unable to accurately measure the energy landscapes of these complex, living-like systems.

A team of researchers has now built a new mathematical framework to bridge this gap, specifically for systems that move by both drifting smoothly and making sudden, random jumps. They focused on a class of active systems where the internal driving force breaks the usual balance of nature, creating a state that is always in motion. The team's central achievement is a new way to separate the work done by an external observer from the energy dissipated by the system's own internal activity. They discovered that even in these chaotic, jumping systems, there is a hidden order that can be revealed if you compare the actual path of a particle with a hypothetical "mirror" path. In this mirror world, the particle follows the same rules but moves in a way that cancels out its internal currents, effectively turning off its engine while keeping the external forces the same.

By comparing the real, active journey with this quiet, mirror journey, the researchers found they could isolate a specific type of energy loss they call "active housekeeping heat." This is the energy the system burns just to keep its internal currents flowing, distinct from the work done by the external protocol. They proved that if you add this hidden housekeeping cost to the external work, a new, exact equality emerges. This new rule, which they call the Active Jarzynski equality, works just as reliably as the old one, but it accounts for the system's internal engine. It allows scientists to calculate the true free-energy difference between two states, even when the system is constantly churning with its own non-stop activity. The researchers demonstrated that this new relation holds true even when the system is driven by a changing protocol, such as a trap that squeezes and expands over time.

The team also developed a companion rule, the Active Crooks relation, which looks at the probability of seeing a certain amount of work in a forward experiment versus a reversed one. In standard physics, the ratio of these probabilities depends only on the work and the energy difference. In these active systems, however, the ratio is skewed by the internal activity. The researchers showed that this skew is governed by the average of the hidden housekeeping heat. This means that two particles might experience the exact same amount of external work, but because one happened to burn more internal fuel during its journey, it is statistically less likely to appear in the forward experiment than its time-reversed twin. This subtle asymmetry, which was previously invisible, is now quantifiable.

To test these ideas, the researchers simulated a two-dimensional particle trapped in a potential well that could change its stiffness. In one scenario, they kept the trap fixed and let the particle's internal engine run. The particle moved and jumped, accumulating a significant amount of internal heat, yet the external work and the energy difference remained zero. The new formulas correctly identified that the system was generating entropy purely through its own activity. In a second scenario, they changed the stiffness of the trap over time, compressing and expanding it. Here, both external work and internal heat were present. The simulations confirmed that the new Active Jarzynski equality held perfectly, recovering the correct energy difference by combining the external work with the measured internal heat. They also found that in their specific model, the internal activity did not distort the work distribution in a way that broke the symmetry, allowing the classical and active versions of the rules to coexist.

This work provides a rigorous method for untangling the complex energy flows in driven, non-equilibrium systems. It offers a way to recover the fundamental energy differences of a system even when it is being pushed by both external forces and its own internal, non-conservative drives. The researchers suggest that these findings could be directly applied to experiments with optically trapped active colloids, engineered micro-swimmers that run and tumble, or single molecules being manipulated in active, gel-like environments. By using these new relations, scientists can simultaneously reconstruct the effective energy landscapes of these systems and quantify the hidden dissipation that was previously impossible to measure, turning the chaotic noise of active matter into a clear signal of thermodynamic truth.

Technical Summary: Active Jarzynski and Crooks Relations for Driven Jump-Diffusions

Problem Statement
Modern statistical physics seeks to quantify how external driving and internal fluctuations govern thermodynamic irreversibility. While classical stochastic thermodynamics successfully describes systems coupled to thermal baths obeying microscopic reversibility and detailed balance, this paradigm fails for active systems undergoing discontinuous, non-Gaussian transitions (e.g., self-propelled colloids, molecular motors, transcriptional bursting).

In driven active jump-diffusions, three interlocking challenges prevent the direct application of standard fluctuation theorems:

Lack of a Common Steady State: Time-dependent protocols often preclude a single time-independent steady state to serve as a thermodynamic baseline.
Entropy Decomposition: Mechanical protocol work ( $W$ ) must be distinguished from continuous and jump-induced active dissipation (housekeeping entropy, $\Delta s_{act}$ ).
Path Measure Singularity: Trajectory comparisons must account for jump-rate asymmetry. Forbidden reverse jumps can render forward and reversed path measures singular, disrupting standard work fluctuation symmetries.

Methodology
The authors establish exact finite-time fluctuation relations for driven Markov jump-diffusions (encompassing infinite-activity Lévy processes) by introducing a dual-gauge framework. The core methodological innovations include:

Instantaneous Steady State (ISS): The system is analyzed relative to a moving thermodynamic reference, the ISS $\pi_t$ , defined at each frozen protocol value. This separates transient state relaxation from the dissipation sustained by steady-state currents.
Conjugate Dual Dynamics: The authors construct a dual process ( $P_\pi$ ) that preserves the frozen ISS density $\pi_t$ but reverses its stationary currents (both diffusive and jump fluxes). For a common initial distribution, the logarithmic ratio of the original path measure ( $P$ ) and the dual path measure ( $P_\pi$ ) yields the active entropy production $\Delta s_{act}$ .
Exponential Gauge Tilting: To incorporate mechanical work, the authors apply an exponential gauge transformation to the path measures. By weighting trajectories with $e^{-W}$ and comparing them against the dual dynamics, they derive exact relations between path probabilities.
Path Reversal vs. Protocol Reversal: The framework carefully distinguishes between direct path reversal (time-reversal operator $\Theta$ ) and physical protocol reversal (running the experiment backward with time-reversed parameters). This distinction is crucial for recovering entropy production in the presence of jump asymmetries.

Key Contributions and Results

Trajectory-Level Decomposition:
The total entropy production $\Delta s_{tot}$ over a time interval $[0, T]$ is decomposed into:
$\Delta s_{tot} = W + \Delta s_{act} - \Delta F + \Delta s_{rel}$
where $W$ is mechanical work, $\Delta s_{act}$ is active housekeeping heat, $\Delta F$ is the equilibrium free-energy difference, and $\Delta s_{rel}$ is a relaxation term related to the Kullback–Leibler divergence between the evolving density and the ISS.
Active Jarzynski Equality:
When the ISS admits a Gibbsian form $\pi_t(x) = Z_t^{-1} e^{-H(t,x)}$ , the authors derive an exact finite-time relation:
$\langle e^{-(W + \Delta s_{act})} \rangle_{\pi_0} = e^{-\Delta F}$
This equality incorporates the active housekeeping entropy $\Delta s_{act}$ alongside mechanical work $W$ . It coexists with the conventional Jarzynski equality ( $\langle e^{-W} \rangle_{\pi_0} = e^{-\Delta F}$ ), which holds when the active contribution vanishes or is ignored. The derivation relies on the stationarity of the Gibbs density under the frozen dual dynamics.
Active Crooks Relation:
By comparing forward paths with the reversed records of a protocol-reversed experiment, the authors establish:
$\frac{p_F(w)}{p_R(-w)} = \frac{e^{w - \Delta F}}{\langle e^{-\Delta s_{act}} \mid W = w \rangle_{\pi_0}}$
Here, the asymmetry between forward ( $p_F$ ) and reverse ( $p_R$ ) work distributions is governed by the conditional exponential average of active dissipation. This factor quantifies how autonomous active fluctuations skew work statistics away from the classical Crooks relation.
Validation via Examples:
The theory is illustrated using a two-dimensional overdamped particle in a harmonic potential subject to chiral active drift and Poisson jumps.
- In a fixed landscape, active entropy accumulates ( $\langle \Delta s_{act} \rangle > 0$ ) without free-energy change, yet the exponential average $\langle e^{-\Delta s_{act}} \rangle = 1$ .
- Under a periodic stiffness protocol, both mechanical and Active Jarzynski relations are shown to hold simultaneously.
- In a specific symmetric case, the conditional active factor equals unity, recovering the classical Crooks relation despite non-zero mean active entropy.

Significance
The paper claims to provide a rigorous framework for:

Recovering Free-Energy Differences: It enables the reconstruction of free-energy differences in driven jump-diffusions with Gibbsian instantaneous steady states, even in the presence of non-conservative active driving.
Characterizing Active Irreversibility: It offers a method to quantify hidden housekeeping dissipation through trajectory statistics, separating it from mechanical work.
Unifying Thermodynamics: The dual-gauge framework unifies non-equilibrium thermodynamics at both the ensemble and trajectory levels for systems lacking detailed balance.

The authors note that these results are directly applicable to optically trapped active colloids, engineered microswimmers (with run-and-tumble or Lévy-flight kinematics), and single-molecule manipulation in active viscoelastic media. The work does not propose new experimental setups but rather provides the theoretical tools to interpret finite-time trajectory measurements in existing active matter systems.

Technical Summary: Active Jarzynski and Crooks Relations for Driven Jump-Diffusions

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