Realization of nonequilibrium thermodynamic processes using external colored noise

We investigate the dynamics of single microparticles immersed in water that are driven out of equilibrium in the presence of an additional external colored noise. As a case study, we trap a single polystyrene particle in water with optical tweezers and apply an external electric field with flat spectrum but a finite bandwidth of the order of kHz. The intensity of the external noise controls the amplitude of the fluctuations of the position of the particle and therefore of its effective temperature. Here we show, in two different nonequilibrium experiments, that the fluctuations of the work done on the particle obey the Crooks fluctuation theorem at the equilibrium effective temperature, given that the sampling frequency and the noise cutoff frequency are properly chosen.

Figure 1

Experimental setup described in Sec. 3a. A polystyrene sphere of radius $R=500\mathrm{nm}$ is immersed in a microfluidic chamber filled with water and trapped with an infrared laser using a high numerical aperture objective. Random forces are exerted using a Gaussian white noise process applied to two electrodes placed at the two ends of the chamber. The position is detected projecting the forward scattered light of a green laser in a quadrant photodiode.

Figure 2

Position of the trap (thick green dashed curve) as a function of time and time traces of the position of the particle sampled at 1 kHz (blue curve) and 10 kHz (thin red dashed curve) in the dragging experiment. The trap moves at a constant velocity of $\pm 22$ nm/ms.

Figure 3

Position of the particle (blue line, left axis) and trap stiffness (green line, right axis) as functions of time in the isothermal compression-expansion cycle. Sampling rate, $f=1\mathrm{kHz}$.

Figure 4

Work distributions in the forward [$\rho \left(W\right)$, filled symbols] and backward [$\tilde{\rho }(-W)$, open symbols] dragging experiments depicted in Fig. 2. Different symbols and colors correspond to different noise intensities, yielding the following values of the Crooks temperature: $T_c=525\mathrm{K}$ (blue squares) $T_c=775\mathrm{K}$ (red circles), and $T_c=1010\mathrm{K}$ (green triangles). Solid lines are the theoretical values of the work distributions obtained for the same values of kinetic temperatures. Work was calculated from trajectories sampled at $f=10\mathrm{kHz}$.

Figure 5

Effective nonequilibrium kinetic temperature, $T_c$, vs effective equilibrium kinetic temperature, $T_{\mathrm{kin}}$, for different amplitudes of the external noise. Different symbols correspond to results obtained for different sampling rates: 1 kHz (blue squares), 2 kHz (red circles), 5 kHz (green triangles), and 10 kHz (magenta diamonds). Solid black line corresponds to $T_c=T_{\mathrm{kin}}$. Error bars represent statistical errors with a statistical significance of $90\%$.

Figure 6

Values of the quotient $T_c/T_{\mathrm{kin}}$ in the dragging experiment as functions of the sampling frequency for different values of the external field, corresponding to the kinetic temperatures: $T_{\mathrm{kin}}=525\mathrm{K}$ (blue squares), $T_{\mathrm{kin}}=775\mathrm{K}$ (red circles), $T_{\mathrm{kin}}=1010\mathrm{K}$ (green triangles), and $T_{\mathrm{kin}}=1520\mathrm{K}$ (magenta diamonds). We also show the value of $T_c/T_{\mathrm{kin}}$ as a function of the sampling frequency obtained from numerical simulations of the overdamped Langevin equation for an external noise with flat spectrum up to $f_{\mathrm{co}}=3\mathrm{kHz}$ and intensity ${\sigma }^2/2k\gamma =500\mathrm{K}$ (black dashed curve). Inset: $T_{\mathrm{kin}}$ (open blue squares) and $T_c$ (red filled circles) as a function of noise intensity, ${\sigma }^2/2k\gamma$, for the experimental values of the experiment described in Ref. [11], where $f_{\mathrm{co}}=1\mathrm{kHz}$ and the acquisition frequency $f=20\mathrm{kHz}$. Solid lines are included to guide the eye.

Figure 7

Average of the work distributions (blue squares, left axis) and variance (red circles, right axis) for the dragging process at positive velocity described by the protocol shown in Fig. 2 as a function of the sampling frequency. Solid lines are a guide to the eye.

Figure 8

Work distributions in the isothermal compression [$\rho \left(W\right)$, filled symbols] and isothermal expansion [$\tilde{\rho }(-W)$, open symbols] for different values of the noise intensities corresponding to the following nonequilibrium effective temperatures: Without external field, $T_c=300\mathrm{K}$ (blue squares), $T_c=610\mathrm{K}$ (red circles), $T_c=885\mathrm{K}$ (green triangles), $T_c=1920\mathrm{K}$ (magenta diamonds), and $T_c=2950\mathrm{K}$ (orange pentagons). Solid and dashed curves are fits to Eqs. (12) and (13), respectively. Vertical lines of the corresponding colors show the expected value for the free energy change at the given temperatures. Data acquisition rate to calculate the work: $f=2\mathrm{kHz}$.

Figure 9

Experimental values (markers) and theoretical values (solid lines) of the work asymmetry function obtained from the work distributions in Fig. 8. Theoretical curves are computed using the values obtained for $T_{\mathrm{kin}}$. Inset: $T_c$ as a function of $T_{\mathrm{kin}}$ (open magenta circles, error bars are smaller than the symbol size). The solid line has slope 1.