Partial Entropy production of active particles with hidden states in potentials

This paper extends a perturbative framework to calculate the partial entropy production of active particles with hidden self-propulsion in generic confining potentials, successfully reproducing exact results for active Ornstein-Uhlenbeck particles and deriving new rates for run-and-tumble particles in harmonic traps.

The Gist

Imagine you are watching a busy city street through a foggy window. You can see people walking, but you can't see their faces, their intentions, or whether they are walking because they want to go somewhere or because they are being pushed by a crowd.

This paper is about trying to figure out if the people on the street are just wandering randomly (which would be "equilibrium," like a calm day) or if they are actually being driven by some hidden force (which is "non-equilibrium," like a parade or a panic).

Here is the breakdown of the research in simple terms:

The Problem: The Foggy Window

In physics, if you watch a system closely enough, you can usually tell if it's out of balance. If you see a ball rolling uphill, you know something is pushing it. This "push" creates entropy production, which is essentially a measure of how much the system is "wasting" energy to keep moving.

However, in the real world (especially in biology), we often can't see everything. We might see a bacteria moving, but we can't see the tiny internal engine (the "hidden state") that is driving it.

• The Trick: If you hide the engine, the bacteria might look like it's just jittering randomly. It might look like it's obeying the laws of a calm, balanced system, even though it's actually working hard.
• The Goal: The authors wanted to create a mathematical tool to detect that hidden "work" even when the engine is invisible, specifically when the particle is trapped in a potential (like a valley or a bowl).

The Analogy: The Run-and-Tumble Hiker

The authors use a specific example called a "Run-and-Tumble" particle. Imagine a hiker in a foggy forest:

1. The Run: The hiker walks in a straight line for a while.
2. The Tumble: The hiker stops, spins around randomly, and picks a new direction.

Scenario A: The Free Forest (No Hills)
If the forest is perfectly flat, and you only see the hiker's path (but not which way they are facing), the path looks perfectly symmetrical. If you played the video backward, it would look exactly the same. The hiker looks like they are just wandering randomly.

• Result: The "Partial Entropy" (the measure of hidden work) is zero. You can't tell they are active.

Scenario B: The Hilly Forest (The Potential)
Now, imagine the forest is a bowl (a harmonic potential). The hiker is at the bottom.

• Going Down: When the hiker is pushed by their internal engine down the hill, they move fast.
• Going Up: When they are pushed up the hill, they have to fight gravity, so they move slow.
• The Clue: If you watch the video backward, the hiker would be seen moving slowly down the hill and quickly up the hill. That looks weird! It breaks the symmetry.
• Result: Even though you can't see the engine, the shape of the path (the "kinks" in the trajectory) gives it away. The "Partial Entropy" is positive.

What They Did

The authors developed a new mathematical recipe (a "perturbative framework") to calculate exactly how much "hidden work" is being done just by looking at the path of the particle.

1. The Formula: They created a complex equation that sums up all the little details of the path. It looks at how the particle moves and how the "hidden engine" (the self-propulsion) correlates with the shape of the valley it's in.
2. The Surprise: They found that for certain types of particles (like the "Active Ornstein-Uhlenbeck" particle, which is like a hiker with a very smooth, jittery engine), if they are in a perfect bowl, the hidden work might still look like zero. But for other types (like the "Run-and-Tumble" hiker), the hidden work is very clearly visible in the path, even without seeing the engine.

The Key Takeaway

The paper proves that hiding the engine doesn't always hide the evidence.

• If a particle is in a flat area, hiding its engine makes it look perfectly normal (equilibrium).
• But if the particle is in a "valley" (a potential), the way it moves up and down the sides creates a unique signature. The particle rushes down and creeps up. This asymmetry reveals that the system is not in equilibrium, even if you can't see the internal motor.

They calculated exactly how strong this signal is for two common types of active particles. They found that for the "Run-and-Tumble" particle in a bowl, the signal is very weak (it requires looking at very high-order details of the path), but it is definitely there.

In short: You can't always tell if a system is "alive" or "active" just by looking at it. But if you know the shape of the environment it's in, you can often deduce that it's doing work, even if you can't see the engine driving it.

Technical Summary

Technical Summary: Partial Entropy Production of Active Particles with Hidden States in Potentials

Problem Statement
Stochastic systems operating far from equilibrium often exhibit broken time-reversal symmetry (TRS). However, when internal degrees of freedom (such as self-propulsion states) are hidden from observation, the visible dynamics of an active particle may appear time-reversible, yielding a zero estimate for entropy production despite the system being out of equilibrium. This "partial" observability poses a challenge for experimentalists attempting to quantify non-equilibrium behavior. While previous work established that hidden states can restore TRS in free particles (e.g., a free run-and-tumble particle), the behavior of such systems under the influence of a confining potential remained an open question. Specifically, it was unclear how a potential $V(x)$ affects the partial entropy production rate (partial EPR) when self-propulsion is unobserved.

Methodology
The authors extend a perturbative framework previously introduced for free particles to the case of active particles in a generic confining potential. The system is modeled by an overdamped Langevin equation:
$$\dot{x}(t) = \nu w(t) - \partial_x V(x(t)) + \xi(t)$$
where $x(t)$ is the observable position, $w(t)$ is a hidden dimensionless stochastic self-propulsion process, $\nu$ is the propulsion speed, and $\xi(t)$ is Gaussian white noise.

The derivation proceeds by:

1. Path Probability Formulation: Defining the full entropy production rate (EPR) via the Kullback-Leibler divergence between forward and time-reversed path probabilities of the joint system $\{x(t), w(t)\}$.
2. Marginalization: Constructing the partial EPR ($\dot{S}_x$) by marginalizing over the hidden variable $w(t)$, effectively considering only the path probability of the observable position $x(t)$.
3. Perturbative Expansion: Utilizing the Onsager-Machlup functional to express the ratio of forward to reversed path probabilities. The authors expand the logarithm of this ratio in powers of the coupling parameter $\nu/D$. This results in a series representation of the partial EPR involving cumulants of the self-propulsion process $w(t)$ and correlation functions of the position $x(t)$ and its derivatives.
4. Symmetry Analysis: Systematically analyzing how properties of $w(t)$ (zero mean, time-translation invariance, parity, and time-reversal symmetry) and the potential $V(x)$ lead to cancellations of terms in the expansion.
5. Specific Models: Applying the general framework to two canonical models: the Active Ornstein-Uhlenbeck Particle (AOUP) and the Run-and-Tumble (RnT) particle, specifically within a harmonic potential $V(x) = \frac{1}{2}kx^2$. For the RnT case, the authors employ Doi-Peliti field theory to calculate the necessary high-order correlation functions.

Key Contributions and Results

• General Perturbative Framework: The paper derives a general series expansion (Eq. 15) for the partial EPR of an active particle in a generic potential. This expression relates the partial EPR to the cumulants of the hidden self-propulsion and the correlators of the position trajectory.
• Symmetry-Induced Cancellations: The authors demonstrate that for active particles with zero-mean self-propulsion ($\langle w \rangle = 0$) and specific symmetries (parity and time-reversal), the leading-order terms in the expansion vanish. In a free system ($V=0$), these symmetries can result in a zero partial EPR. However, the presence of a potential introduces terms that do not vanish under parity reversal of velocity, generally restoring a non-zero partial EPR.
• Active Ornstein-Uhlenbeck Particle (AOUP): For an AOUP in a harmonic potential, the authors reproduce an exact result showing that the partial EPR vanishes ($\dot{S}_x = 0$). This is attributed to the Gaussian nature of the AOUP noise, which limits the expansion to the second cumulant, and the specific cancellation of terms in the harmonic potential.
• Run-and-Tumble (RnT) Particle: For an RnT particle in a harmonic potential, the partial EPR is non-zero. Due to the discrete nature of the RnT process and the symmetries of the potential, the leading-order contribution appears at order $n=4$ in the expansion (proportional to $(\nu/D)^4$).
  • The explicit result for the leading-order partial EPR is derived as:
    $$\dot{S}_x = \left(\frac{\nu^2}{D}\right)^4 \frac{2\alpha k}{3(\alpha + k)(\alpha + 3k)(2\alpha + k)(2\alpha + 3k)(3\alpha + 2k)} + O\left(\frac{\nu^{10}}{D^5}\right)$$
  • This result is significantly higher order than the full EPR (which scales as $\nu^2/D$), indicating that the time-reversal asymmetry of the spatial trajectory is a subtle effect in the presence of hidden states and a confining potential.
• Vanishing of Lower Orders: The paper rigorously demonstrates that for the RnT particle in a harmonic potential, terms of order $n=2$ and $n=3$ vanish due to the interplay between the time-translation invariance of the steady state and the specific structure of the potential derivatives.

Significance and Claims
The paper claims to provide a systematic method for calculating the partial entropy production of active particles with hidden states in arbitrary potentials. The primary significance lies in quantifying how much time-reversal symmetry is broken in the observable dynamics of a non-equilibrium system when internal states are hidden.

The authors highlight that:

1. Potentials Break Hidden Symmetry: While hidden states can render free active particles indistinguishable from equilibrium (zero partial EPR), the introduction of a confining potential generally breaks this symmetry, making the system detectable as non-equilibrium through spatial trajectories alone.
2. Order of Magnitude: The partial EPR is often of a much higher order in the propulsion parameter $\nu$ than the full EPR. For the RnT particle, the partial EPR scales as $\nu^8/D^4$ (due to the $(\nu/D)^4$ prefactor and the $\nu^4$ scaling of the position correlators), whereas the full EPR scales as $\nu^2/D$. This suggests that detecting non-equilibrium behavior in partially observed systems requires high precision or specific conditions.
3. Uniqueness of AOUP: The work identifies the AOUP in a harmonic potential as a unique case where the partial EPR remains zero in the steady state, distinguishing it from non-Gaussian active processes like RnT.

The authors remain modest regarding future applications, noting that calculating the required correlation functions is challenging and that the thermodynamic interpretation of partial EPR remains a subject for future research. They suggest the framework could be extended to time-varying potentials or potentials dependent on hidden variables, but do not propose specific experimental protocols.