Experimental Test of the Differential Fluctuation Theorem and a Generalized Jarzynski Equality for Arbitrary Initial States

Nonequilibrium processes of small systems such as molecular machines are ubiquitous in biology, chemistry, and physics but are often challenging to comprehend. In the past two decades, several exact thermodynamic relations of nonequilibrium processes, collectively known as fluctuation theorems, have been discovered and provided critical insights. These fluctuation theorems are generalizations of the second law and can be unified by a differential fluctuation theorem. Here we perform the first experimental test of the differential fluctuation theorem using an optically levitated nanosphere in both underdamped and overdamped regimes and in both spatial and velocity spaces. We also test several theorems that can be obtained from it directly, including a generalized Jarzynski equality that is valid for arbitrary initial states, and the Hummer-Szabo relation. Our study experimentally verifies these fundamental theorems and initiates the experimental study of stochastic energetics with the instantaneous velocity measurement.

Figure 1

(a) Experimental scheme. A silica nanosphere (blue sphere) is trapped in an optical tweezer formed by a focused 1550-nm laser beam (magenta). A series of 532-nm laser pulses (green) exerts an optical force on the nanosphere to drive nonequilibrium processes. Within each pulse, an optical force is rapidly ramped from $f_{\mathrm{off}}$ at time $t_1$ to $f_{\mathrm{on}}$ at time $t_2$ during the forward process (green pulse). The reverse process from time $t_3$ to $t_4$ is the time-reversed correspondence of the forward process. (b) An example of experimental data. Vertical slides represent the measured time snapshots of the probability distributions at times $t_1$, $t_2$, $t_3$, and $t_4$ as illustrated in (a). Black curves represent experimental phase-space trajectories during forward processes initialized at $(x_1,v_1)$ and finalized at $(x_2,v_2)$, and during reverse processes initialized at the $(x_2,-v_2)$ and finalized at $(x_1,-v_1)$. Here, $x_1=-19\mathrm{nm}$, $x_2=55\mathrm{nm}$, $v_1=-7\mathrm{mm}/\mathrm{s}$, and $v_2=7\mathrm{mm}/\mathrm{s}$. The nanosphere is levitated in air at 50 torr, and $f_{\mathrm{off}}=0$, $f_{\mathrm{on}}=340\mathrm{fN}$.

Figure 2

Examples of arbitrary nonequilibrium initial states of trajectory ensembles prepared by an information-based method. (a) Nonequilibrium initial states with narrow distributions in position or velocity. (b) An exotic nonequilibirum state with a $P$-shaped distribution in the phase space. (c) Uniform distribution within a rectangle ($-15\mathrm{nm}<x<0$, $-10\mathrm{mm}/\mathrm{s}<v<0$) in the phase space. (d) A microcanonical ensemble with the energy shell $1.3k_{B}T<E<1.35k_{B}T$. The number of experimental trajectories started from each nonequilibrium initial state is labeled next to its distribution.

Figure 3

Testing the differential fluctuation theorem in the underdamped regime. (a) Optical force. (b),(c) Measured position and velocity trajectories. A single trajectory is shown in blue, and the averaged trajectory of over one million trajectories is shown in red. The gray shaded regions in (a)–(c) denote the forward and reverse intervals, respectively. It takes roughly $4.6\mu \mathrm{s}$ for the optical force strength to switch from the 10% to 90% level. (d) An example of probabilities $P_F(W|x_1\to x_2)$ and $P_R(-W|x_2\to x_1)$ in position coordinate. (e) An example of probabilities $P_F(W|v_1\to v_2)$ and $P_R(-W|-v_2\to -v_1)$ in velocity coordinate. The label of the horizontal axis in (d) and (e) is $W(k_{B}T)$. (f),(g) Testing the DFT in position and velocity spaces. The small markers with different colors represent measurements of ${\log }_e[(P_R(-W,x_2\to x_1))/(P_F(W,x_1\to x_2))]$ and ${\log }_e[(P_R(-W,-v_2\to -v_1))/(P_F(W,v_1\to v_2))]$ for 121 different $\{x_1,x_2\}$ and $\{v_1,v_2\}$ combinations, respectively. The big magenta markers are results for parameters shown in (d) and (e), respectively. The black lines represent $-\beta (W-\mathrm{\Delta }F)$.

Figure 4

Testing fluctuation theorems in the underdamped regime. (a),(b) Testing GJE in position and velocity spaces for a fast ramp (red, $4.6\mu \mathrm{s}$ from 10% to 90% levels) and a slow ramp (blue, $40\mu \mathrm{s}$ from 10% to 90% levels). Markers represent the measured $P_R(x_f=x)/P_F(x_i=x)$ and $P_R(v_f=-v)/P_F(v_i=v)$ in position and velocity spaces, respectively. (c),(d) Testing HSR in position and velocity spaces for a fast ramp (red) and a slow ramp (blue). Markers represent the measured $P_R(x_i=x)/P_F(x_f=x)$ and $P_R(v_i=-v)/P_F(v_f=v)$ in position and velocity spaces, respectively. The error bars of $P_R(x)/P_F(x)$ and $P_R(-v)/P_F(v)$ represent the standard deviation of the measurements for 20 equal divisions in each subset $x$ and $v$, respectively. The markers represent their mean values. In (a)–(d), the shaded line represents $⟨e^{-\beta (W-\mathrm{\Delta }F)}⟩$, where its thickness represents the uncertainty of 600 work (Joule) calibrations.