A microscopic heat engine with many hidden variables

In the microscopic world where life's machines operate, the rules of motion are different from the heavy, predictable world of cars and cranes. Here, at the scale of single molecules, thermal fluctuations—the constant, jittery jostling of heat—compete with mechanical forces and chemical energy. Because these tiny engines are so small, their behavior cannot be described by simple, deterministic laws where a push always produces a predictable movement. Instead, scientists must describe them using probabilities and the flow of energy, tracking how often a molecule moves forward versus backward and how much disorder, or entropy, is created in the process. A central question in this field is how to understand these machines when we can only see part of them. Most experimental tools can track just one visible movement, like the step of a motor protein along a track, while the internal gears, chemical cycles, and hidden rotations that power the movement remain invisible. This creates a blind spot: when a motor stops moving, does it mean it has reached a state of perfect balance and rest, or is it still burning energy in the dark?

A new study by Mesfin Asfaw Taye addresses this blind spot by examining a microscopic motor that is driven by a single visible coordinate coupled to several hidden internal phases. The researchers investigated a specific type of motor where the visible part and the hidden parts are linked by a shared, periodic interaction, much like gears that turn together. They asked what happens when the motor is "stalled," meaning the external load is so strong that the visible part stops moving on average. In a simple, fully visible system, a stop in movement would mean the entire system has reached equilibrium and stopped generating heat. However, this new work shows that for a motor with hidden variables, a silent visible coordinate does not guarantee that the machine has stopped working. The motor can remain in a state of constant, hidden activity, dissipating energy even while the visible part appears perfectly still.

The researchers developed a precise mathematical framework to separate the energy dissipation of such a stalled motor into two distinct parts. The first part is "reciprocal," meaning it is directly tied to the visible movement. Even though the average speed is zero, the motor still experiences local, rapid fluctuations in speed as it jitters back and forth. These fluctuations create a specific amount of heat that can be calculated and measured. The second part is "orthogonal," representing hidden currents that circulate in directions completely independent of the visible coordinate. These hidden loops can continue to burn energy and generate heat without changing the visible motion in any way. The study proves that a single observed coordinate can only determine the reciprocal part of the energy loss. The orthogonal part remains a "no-go" sector; it is thermodynamically active but mechanically invisible to the observer. Two motors could look identical in every way, with the same visible statistics and the same stall load, yet one could be wasting significantly more energy than the other due to these hidden, orthogonal circulations.

To make these invisible processes detectable, the team explored how to infer the total energy loss using only the data from the visible coordinate. They found that while a simple snapshot of the motor's position tells us nothing about the energy being wasted at a stall, a more detailed look at the motor's movement over time reveals the truth. By analyzing how the motor's velocity fluctuates and how it responds to a tiny, controlled push, researchers can reconstruct the exact amount of heat dissipated through the visible channel. This method, based on a known relationship between fluctuations and response, allows scientists to calculate the "reciprocal" energy loss without ever seeing the hidden gears. However, this method still cannot detect the orthogonal hidden currents. To find those, one would need to measure additional internal markers or have prior knowledge of the motor's internal structure.

The author tested these ideas by simulating a motor with two hidden phases and a sinusoidal interaction potential. In these simulations, they kept the visible behavior exactly the same while varying the hidden currents. They confirmed that the total energy dissipation increased quadratically with the strength of the hidden orthogonal circulation, even though the visible motor remained indistinguishable from a version with no hidden circulation. The simulations verified that the stall load—the force required to stop the motor—remained constant, and that the local currents within the visible coordinate continued to fluctuate and generate heat even when the average speed was zero. This numerical experiment demonstrated that the hidden dissipation is a real, measurable quantity that is completely decoupled from the visible motion.

The study also compared this reciprocal motor to a nonreciprocal model, where the hidden part drives the visible part without any back-reaction, like a clock that turns a hand but is not affected by the hand's movement. In that scenario, the visible current provides no exact constraint on the hidden energy loss at all. Furthermore, the researchers applied their findings to a heat engine operating between two reservoirs of different temperatures. They showed that even at mechanical stall, where no useful work is produced, the engine can still be leaking heat through hidden channels. This hidden leak ensures that the engine's efficiency remains strictly below the theoretical maximum, known as the Carnot limit, because the total entropy production is positive.

Ultimately, this work clarifies the limits of what we can know about microscopic machines from partial observation. It establishes that mechanical silence is not the same as thermodynamic silence. A stalled motor is not necessarily at rest; it may be churning with hidden activity. The amount of energy it wastes can be partially reconstructed if the coupling between the visible and hidden parts is reciprocal and if we can measure the motor's response to small forces. However, there is a fundamental limit: any hidden circulation that is orthogonal to the coupling direction leaves no trace on the visible coordinate. To fully understand the energy budget of such a machine, one must either resolve the hidden variables directly or accept that a portion of the dissipation will always remain a mystery, hidden in the null space of the coupling.