Probing the limits of effective temperature consistency in actively driven systems

TL;DR

This work tests whether an effective temperature concept can extend to a nonequilibrium active bath by coupling a macroscopic tracer to self-propelled walkers in a harmonic trap. By comparing FDR-based, equipartition-based, and work-fluctuation–relation temperatures—and introducing an effective mass to define a modified kinetic temperature—the study shows that, for a broad range of bath densities $N_b$, these measures converge to a single $T_{\text{eff}}$. The agreement persists under weak perturbations and moderate activity, but breaks down for dilute baths ($N_b$ small) and in regimes with heavy tracers or rapid, entangling collisions, indicating fundamental limits to equilibrium-like thermodynamics in athermal active systems. Overall, the results reveal a surprising robustness of the effective-temperature concept in active matter while clarifying when a single temperature description ceases to apply.

Abstract

We investigate the thermodynamic properties of a single inertial probe driven into a nonequilibrium steady-state by random collisions with self-propelled active walkers. The probe and walkers are confined within a gravitational harmonic potential. We evaluate the robustness of the effective temperature concept in this active system by comparing values of distinct, independently motivated definitions: a generalized fluctuation-dissipation relation, a kinetic temperature, and a work fluctuation relation. Our experiments reveal that, under specific conditions, these independent measurements yield a remarkably consistent effective temperature over a wide range of system configurations. Furthermore, we also identify regimes where this consistency breaks down, which delineates the fundamental limits of extending equilibrium-like thermodynamic concepts to athermal, actively driven systems.

Figures (17)

• Figure 1: The system:A. Experimental setup: A Styrofoam ball (diameter $\sim4$ cm, $1$ g) is trapped in a gravitational harmonic potential, a plastic bowl (diameter 38cm, depth 5cm), and subjected to collisions with $N_b=10$ self-propelled bbots (inset: standard bbot, $4\times1$ cm, $7.1$ g). The ball is repeatedly perturbed with a uniform air stream created by an external fan along the $x$-axis (white arrow) to test a fluctuation-response relation. To enforce an abrupt onset and release of the perturbation, a mechanical shutter is used (denoted by 'S'). B. Exemplary results (with $N_b=10$) for three independent tracer's steady states: (a) an unperturbed state; (b) a weakly perturbed state ($10$ V fan operating voltage, $F_0=\kappa\Delta x_\epsilon\approx 62$$\mu$N); and (c) a strongly perturbed state ($13.5$ V, $F_0\approx 107.8$$\mu$N). These stationary position distributions were obtained from combined time and ensemble averages, using 375 trajectories of 1 minute length. C. Velocity distributions of bbot assemblies with different $N_b=3,4,$ and 10. Instantaneous velocities were extracted from recordings of bbot trajectories tracked over 40 minutes using a frame rate of 30 frames per second (fps). D. Spatial distribution of 10 bbots within the harmonic trap, with a soft boundary (steep curvature) indicated by a dashed circle of radius $10$ cm. E, F. Typical $20$ second trajectories of a single bbot in systems with $N_b=3$ (E) and $N_b=10$ (F) (upper panels), alongside the corresponding tracer trajectories (lower panels).
• Figure 1: Forces and stationary statistics under external air stream.A, B. Average forces acting on the tracer in the absence of bbots, on a flat surface. Measurements were obtained using a force sensor (Mark-10 Force Gauge) with $10$V operating fan voltage: (A) as a function of $x-y$ position; (B) along the main air stream axis $x$ (lower panel) - and as a function of height $z$ (upper panel), at a distance $x=8$cm marked by the vertical dashed line. Error bars are standard deviations of $50$ measurements. C. Stationary probability distributions of the tracer’s position under different air stream intensities in the harmonic trap (unperturbed, $10$V, and $13.5$V fan operating voltage), with $N_b=10$. D. Corresponding position variances. Results were obtained from time and ensemble averages over $M = 375$ realizations of one-minute tracer trajectories. Error bars indicate standard deviations.
• Figure 2: Thermometer consistency:$N_b=\{ 3,4,6,10,12,15,17 \}$ bbots. Effective temperature measurements obtained using three independent methods: the potential temperature $T_{\text{pot}}=T_{\text{eff}}$ (circles, Eq. \ref{['eq:temp']}), the modified kinetic temperature $\tilde{T}_{\text{kin}}$ (squares, Eq. \ref{['eq:temp2']}), and a constant temperature $T_{\text{FR}}$ derived from a work FR (diamonds, Eq. \ref{['eq:FR']}). Notably, these static temperatures validate the FDR of Eq. \ref{['eq:FDR1']} (for $N_b>3$) and define a consistent effective temperature $T_{\text{eff}}$.
• Figure 2: The FDR test under different perturbation amplitudes.A. The mean displacement $\Delta x$ under a perturbation of a tracer in a $N_b=10$ bbot bath, as a function of fan operating voltage $V$ (error bars are standard errors). The applied force $F_0=k\Delta x$ increases above $V=11$V. The tracer's position standard deviation, $\sigma_x$, is used to define the range of small perturbations (solid line). The dashed line is a linear fit of the first 4 data points, with $\Delta x = 0.19\cdot V$, whereas $V=13$V and $13.5$V are out of the linear regime. B. The generalized FDRs with $T_{\text{eff}}\sim\langle \Delta x^2\rangle_0$ are plotted for $V=9, \ 12, \ 13.5$ V, with $N_b=10$. The results present an average over $M=375$ perturbation sequences.
• Figure 3: NESS dynamics and statistics: Data were obtained for active bath configurations with $N_b = \{3,4,6,10,12,15,17\}$ bbots, using a tracer of mass $m = 1 \pm 0.1$ g confined in a potential of stiffness $\kappa = 28.2 \pm 3$ g/s$^2$. Results are obtained from combined time and ensemble averages over an ensemble of 375 one-minute-long tracer trajectories recorded at 30 fps for each $N_b$. A. Rescaled position ($x$) distributions. Dashed line show fits to exponential ($N_b=3$) and Gaussian ($N_b=15$) functions. B. Rescaled velocity ($v_x$) distributions. An exponential fit is shown for $N_b=3$ and a stretched exponential fit for $N_b=15$. C,D. Position (C) and velocity (D) autocorrelation functions (ACFs), showing increasingly damped dynamics with higher $N_b$. E. Position and velocity ACFs for $N_b=10$, fitted with Eq. \ref{['eq:Langevin2']} using $\gamma$ and $\Omega$ as fitting parameters, yielding consistent descriptions for both $C_{xx}$ and $C_{vv}$. F. Extracted relaxation rate ($\gamma$) and trapping frequency ($\Omega$) as functions of $N_b$ (upper panel). The impact of under-sampling on the observed dynamics is pronounced for large $N_b$ ($15$ and $17$). Hollow markers (dashed lines) show measurements at $30$ fps, while colored markers correspond to $60$ fps. The lower frame rate leads to apparent overdamped behavior, whereas the higher frame rate reveals dynamics consistent with a critically-damped regime. The collision frequency ($\tau_c^{-1}$) is plotted versus $N_b$ (lower panel), where $\tau_c$ is the measured mean-free time between tracer-bbot collisions.