🔬 condensed matter

Free-Energy Differences from Nonequilibrium Fluctuations in High Dissipation

The paper introduces EquiNET, a method that leverages repeated trajectory fluctuations to accurately estimate free-energy differences in high-dissipation regimes where conventional fluctuation theorems fail due to the statistical rarity of negative entropy production events.

Original authors: Sreekanth K Manikandan, Giovanni Volpe

Published 2026-08-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Sreekanth K Manikandan, Giovanni Volpe

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 atoms and molecules, energy is never still. Even when a system appears calm, its particles are constantly jostling, driven by the heat of their surroundings. Scientists have long sought a way to map the hidden energy landscapes that dictate how these tiny systems behave, such as how a protein folds into its functional shape or how a chemical reaction proceeds. The height of the hills and the depth of the valleys in these landscapes determine stability and speed, but measuring them directly is notoriously difficult. For decades, researchers have relied on a clever workaround: they pull a molecule out of its resting state and measure the effort required to do so. By repeating this process thousands of times and averaging the results, they can theoretically deduce the energy difference between the starting and ending points. However, this method hits a wall when the process is messy or fast. In these high-dissipation scenarios, the math requires the researcher to find rare, almost impossible events where the system seems to give back energy rather than consume it. These events are so infrequent that gathering enough of them to get a reliable answer would take longer than the age of the universe.

A team of researchers at the University of Gothenburg has found a way around this bottleneck. They introduced a new technique, which they call EquiNET, that bypasses the need to wait for these rare, lucky events. Instead of hunting for the needle in the haystack, their method looks at the shape of the entire haystack. By analyzing the subtle fluctuations in how a molecule moves during a standard pulling experiment, the team can infer the total amount of energy lost to friction and heat. Once they know how much energy was wasted, they can subtract that loss from the total work done to reveal the true energy difference between the two states. This approach works even when the process is highly irreversible and the usual methods fail completely.

The researchers tested their idea first on a simple, one-dimensional model of a particle moving through a changing energy landscape. In this simulation, they drove a particle from one stable position to another, creating a scenario where the forward and backward paths were so different that they barely overlapped. Standard methods would have been unable to find the connection between them. Yet, EquiNET successfully reconstructed the energy difference by learning from the trajectory data alone. It did not need the rare backward-moving events; it simply learned the rules of the motion from the common forward ones. The results matched the known theoretical values perfectly, proving that the method could extract the hidden energy landscape from the noise of ordinary movement.

To see if this would hold up in a more complex, realistic setting, the team applied EquiNET to a model of a polymer hairpin, a structure that mimics how a strand of DNA or a protein folds. They simulated the process of mechanically pulling this hairpin apart, breaking the bonds that hold it together, and then letting it snap back. They varied the strength of the bonds to create scenarios ranging from mild to extremely difficult, where the energy loss was massive. In the hardest cases, the forward and backward experiments produced completely different sets of results with no overlap, rendering traditional statistical tools useless. EquiNET, however, thrived in this chaos. By training a specialized computer program on the paths the hairpin took, the system learned to predict the energy loss at every moment of the process. When they combined this predicted loss with the measured work, the calculated energy difference was accurate, even though the data came from a regime where conventional approaches would have given up.

One of the most practical aspects of this discovery is that it works even when scientists cannot see the whole picture. In real-world experiments, it is often impossible to track every single atom in a molecule; researchers usually only see a few key points, such as the ends of a strand or specific labeled spots. The team showed that EquiNET can still function with this limited view. When they fed the system only the coordinates of a few beads on the hairpin, it could not determine the exact energy difference, but it could provide a strict mathematical boundary. It told them that the true energy difference must be below a certain value. This bound was surprisingly tight, staying very close to the actual value, which suggests that even with partial information, scientists can gain reliable insights into the stability of complex biological structures.

The researchers also investigated how much data is needed to make these predictions accurate. They found that the method converges quickly; in their simulations, they only needed about one thousand trajectories to recover a large fraction of the true energy loss. This is a significant improvement over older methods, which might require millions or billions of attempts to find the few rare events they depend on. The team also discovered a way to check if their experiment was long enough. By watching the rate of energy loss as the simulation ended, they could tell if the molecule had truly settled back into a calm state. If the energy loss rate did not drop to zero, it meant the molecule was still relaxing, and the observation window needed to be extended. This provides a built-in diagnostic tool for experimentalists to know when they have gathered enough information.

Ultimately, this work offers a new path for understanding the energy of life's machinery. By shifting the focus from rare, improbable events to the common, everyday fluctuations of a system, EquiNET allows researchers to measure energy landscapes in conditions that were previously too messy to analyze. Whether studying how proteins fold, how drugs bind to targets, or how synthetic materials respond to stress, this technique provides a robust way to see the invisible energy that drives them. It turns the problem of high dissipation from a dead end into a solvable puzzle, using the very noise that once made the measurement impossible.

Technical Summary: EquiNET for Free-Energy Estimation from Nonequilibrium Fluctuations

Problem Statement
Equilibrium free-energy differences ( $\Delta F$ ) are fundamental to understanding the stability and function of physical and biological systems. While fluctuation theorems, such as the Jarzynski equality and the Crooks fluctuation theorem, theoretically allow the reconstruction of $\Delta F$ from nonequilibrium work fluctuations, they face a critical practical limitation in high-dissipation regimes. The convergence of these methods relies on the sampling of rare trajectories characterized by negative entropy production ( $\Delta S_{tot} < 0$ ). In strongly dissipative systems (where average entropy production exceeds a few tens of $k_B T$ ), these rare events occur exponentially infrequently, rendering conventional estimators statistically inefficient and requiring impractically large numbers of trajectories to converge. Furthermore, total entropy production is generally not directly measurable in experiments, as it requires knowledge of both heat dissipation and system entropy changes, the latter of which depends on inaccessible microscopic state probabilities.

Methodology: EquiNET
The authors introduce EquiNET, a data-driven technique that infers equilibrium free-energy differences by directly estimating total entropy production from trajectory fluctuations, bypassing the need for rare-event sampling. The method is grounded in the short-time Thermodynamic Uncertainty Relation (TUR), which expresses the instantaneous entropy production rate, $\sigma(t)$ , as the solution to a variational optimization problem:
$\sigma(t) = \frac{1}{dt} \max_{\mathbf{d}} \left[ \frac{2k_B \langle J_{\mathbf{d}} \rangle^2}{\text{Var}(J_{\mathbf{d}})} \right]$
where $J_{\mathbf{d}}$ is a weighted stochastic current defined via the Stratonovich product. This formulation relies only on the first two cumulants (mean and variance) of the current, avoiding the higher-order statistics and exponential averages required by fluctuation theorems.

To solve this optimization for complex, high-dimensional systems, the authors employ a time-conditioned, $E(3)$ -equivariant graph neural network (EGNN).

Architecture: The system (e.g., a polymer) is represented as a graph where nodes correspond to beads and edges encode interactions (backbone, native contacts). The network takes bead coordinates and time as input, outputting a vector field $\mathbf{d}(\mathbf{x}, t)$ that approximates the thermodynamic force field.
Training: The network parameters are optimized end-to-end by maximizing the variational objective in the TUR equation using trajectory data.
Inference: Once trained, the network yields the instantaneous entropy production rate $\sigma(t)$ . Integrating $\sigma(t)$ over the protocol duration provides the total entropy production $\langle \Delta S_{tot} \rangle$ .
Free-Energy Calculation: The equilibrium free-energy difference is then recovered using the relation $\Delta F = \langle W \rangle - T\langle \Delta S_{tot} \rangle$ , where $\langle W \rangle$ is the measured average work.

The framework naturally extends to partial observations (where only a subset of degrees of freedom is measured). In such cases, the inferred entropy production provides a rigorous lower bound on the true entropy production, yielding a corresponding rigorous bound on $\Delta F$ .

Key Results and Validation
The authors validate EquiNET through numerical simulations in two distinct regimes:

Analytically Tractable 1D System: A particle driven between two equilibrium states by a time-dependent potential.
- In this case, the variational optimization admits an exact analytical solution using a polynomial basis.
- Results show that EquiNET accurately recovers the equilibrium free-energy difference even when the forward and reverse work distributions are completely separated (no overlap), a regime where Crooks' theorem fails with the available data.
Polymer Hairpin Model: A high-dimensional biomolecular system undergoing force-induced unfolding.
- Simulations were performed with varying native-contact strengths ( $\epsilon$ ), creating regimes of moderate to strong dissipation where forward and reverse work distributions showed no overlap even with $10^5$ trajectories.
- Full Observation: Using the coordinates of all beads, EquiNET recovered $\Delta F$ values in excellent agreement with a ground-truth reference obtained from slow-pulling simulations (where distributions do overlap).
- Partial Observation: When trained only on the $x$ -coordinates of a subset of beads (mimicking experimental limitations like optical tweezers or FRET), EquiNET provided rigorous upper bounds on $\Delta F$ that remained very close to the true values.
- Convergence: The method converges rapidly; approximately $10^3$ trajectories were sufficient to recover a large fraction of the true entropy production in the forward process, significantly outperforming the data requirements of fluctuation-theorem-based approaches.

Significance and Claims
The paper claims that EquiNET offers a scalable and efficient alternative to fluctuation theorems for estimating free-energy differences in strongly dissipative regimes. By shifting the focus from rare negative-entropy events to the inference of average entropy production via lower-order statistics, the method avoids the exponential data inefficiency that plagues current techniques.

The authors emphasize that while the method requires knowledge of the interaction network to construct the graph neural network, it does not require direct access to heat or full microscopic state probabilities. The framework is presented as a general tool for analyzing high-dimensional systems, such as complex biomolecules, and is particularly valuable when experimental access is limited to partial observables, providing rigorous bounds where exact estimates are otherwise unattainable. The work establishes a bridge between stochastic thermodynamics and physics-informed machine learning, demonstrating that trajectory fluctuations contain sufficient information to reconstruct equilibrium properties without relying on the "rare event" assumption.

Technical Summary: EquiNET for Free-Energy Estimation from Nonequilibrium Fluctuations

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