Non-Boltzmann stationary distributions and nonequilibrium relations in active baths

Most natural and engineered processes, such as biomolecular reactions, protein folding, and population dynamics, occur far from equilibrium and therefore cannot be treated within the framework of classical equilibrium thermodynamics. Here we experimentally study how some fundamental thermodynamic quantities and relations are affected by the presence of the nonequilibrium fluctuations associated with an active bath. We show in particular that, as the confinement of the particle increases, the stationary probability distribution of a Brownian particle confined within a harmonic potential becomes non-Boltzmann, featuring a transition from a Gaussian distribution to a heavy-tailed distribution. Because of this, nonequilibrium relations (e.g., the Jarzynski equality and Crooks fluctuation theorem) cannot be applied. We show that these relations can be restored by using the effective potential associated with the stationary probability distribution. We corroborate our experimental findings with theoretical arguments.

Figure 1

Emergence of non-Boltzmann position distributions in active baths. Measured position distributions (blue symbols) and corresponding Gaussian distributions (dashed blue lines) are shown for a Brownian particle in a thermal bath confined in a harmonic trapping potential with trapping stiffness (a) $k=0.42\mathrm{fN}{\mathrm{nm}}^{-1}$, (b) $k=3.6\mathrm{fN}{\mathrm{nm}}^{-1}$, and (c) $k=22\mathrm{fN}{\mathrm{nm}}^{-1}$. As $k$ is increased, the particle becomes more confined, but the distribution remains Gaussian. Note the different scales for the position axes. Also shown are measured position distributions (red symbols) and best Gaussian fits (red dashed lines) for a Brownian particle in an active bath in a harmonic trapping potential with trapping stiffness (d) $k=0.42\mathrm{fN}{\mathrm{nm}}^{-1}$, (e) $k=3.6\mathrm{fN}{\mathrm{nm}}^{-1}$, and (f) $k=22\mathrm{fN}{\mathrm{nm}}^{-1}$. As $k$ is increased, the particle gets confined within a length scale comparable to the persistence length in the active bath $[L_{\mathrm{a}}$, red bars in (d)–(f)]; consequently, the distribution deviates from Gaussian and can be fitted with a heavy-tailed $q$-Gaussian distribution (solid black line) with (d) $q=1.013$, (e) $q=1.023$, and (f) $q=1.142$ [(a) $q=0.997$, (b) 1.000, and (c) 1.001, compatibly with a Gaussian profile]. The data are obtained from 200-s-long trajectories acquired at about 400 frames per second (fps); the error bars are obtained by repeating the measurements six times.

Figure 2

Particle in an active bath. (a) Sample trajectories of a microscopic particle (polystyrene, diameter $2R=4.06\pm 0.20\mu \mathrm{m}$) in an active (bacterial) bath (red line) and in a thermal bath (blue line). The trajectories are extracted from a 150-s video sampled at $20\mathrm{fps}$. The inset shows a sample frame from the acquired video in the active bath; the bright spot corresponds to the particle, while the elongated lighter objects are the bacteria. (b) The MSD of the particle in the active bath along one axis (red symbols) features a transition from a superdiffusive regime at short time scales to enhanced diffusion at long time scales, differently from the MSD of the particle in the thermal bath (blue symbols).

Figure 3

Violation of the Crooks fluctuation theorem in an active bath. (a) The driving protocol entails an instantaneous translation of the trapping potential: The red dashed line represents the potential before switching, the red solid line the potential after switching, and the black dashed line the applied work. (b) Applied work as a function of the position of the particle before the potential translation. (c) Experimental probability distribution of work when the protocol shown in (a) is employed on a Brownian particle (silica, diameter $2R=4.23\pm 0.20\mu \mathrm{m}$) held in an optical trap ($k=1.33\mathrm{fN}{\mathrm{nm}}^{-1}$) in an active (bacterial) bath, calculated using Eq. (16). The solid line represents the best Gaussian fit. (d) The red symbols show the ratio of the inverse work probabilities on the particle in active bath; they feature a clear deviation from the Crooks fluctuation theorem [solid line, Eq. (14)]. The data are obtained from 5000 work measurements; the error bars are obtained by repeating the measurements three times.

Figure 4

Recovery of the Crooks fluctuation theorem in an active bath using effective potentials. (a) The driving protocol is the same as in Fig. 3, but effective potentials obtained from the particle's stationary position distribution (black lines) are employed instead of the optical potentials (red lines). The difference between the effective and the optical potentials is particularly evident in the tails. (b) Applied work as a function of the position of the particle before the potential translation. (c) Experimental probability distribution of work when the protocol shown in (a) is employed on a Brownian particle (silica, diameter $2R=4.23\pm 0.20\mu \mathrm{m}$) held in an optical trap ($k=1.33\mathrm{fN}{\mathrm{nm}}^{-1}$) in an active (bacterial) bath, calculated with the effective potential approach [Eq. (19)]. The solid line is the same Gaussian fit as in Fig. 3. (d) The black symbols show the ratio of the inverse work probabilities on the particle in active bath: The use of effective potentials recovers the Crooks fluctuation theorem [solid line, Eq. (14)]. The data are obtained from 5000 work measurements; the error bars are obtained by repeating the measurements three times.

Figure 5

Verification of the Jarzynski equality and integral fluctuation theorem in an active bath using effective potential. (a) Boltzmann weighted average of the applied work as a function of the waiting time $\tau$ between switches of the potential. If the particle is given enough time to reach a stationary state, the Jarzynski equality (15) is satisfied. Dots represent the average over all measured work values for each $\tau$. The rightmost dots ($\tau =360\mathrm{ms}$) correspond to the work distribution in Fig. 4. (b) Verification of the integral fluctuation theorem (11) using effective potentials. The heat $Q_i$ is calculated from each switch using Eq. (20). Note that the particle position probability $p^{\mathrm{s}}\left(x\right)$ is calculated before and after the switch, i.e., at $t_i^{-}$ and $t_i^{+}$, respectively. The data are obtained from 5000 work measurements; the error bars are obtained by repeating the measurements three times.