Thermodynamics of a Colloidal Particle in a Time-Dependent Nonharmonic Potential

We study the motion of an overdamped colloidal particle in a time-dependent nonharmonic potential. We demonstrate the first lawlike balance between applied work, exchanged heat, and internal energy on the level of a single trajectory. The observed distribution of applied work is distinctly non-Gaussian in good agreement with numerical calculations. Both the Jarzynski relation and a detailed fluctuation theorem are verified with good accuracy.

Figure 1

Particle-wall interaction potentials for three different intensities of the lower optical tweezers [decreasing power from (1) to (3)]. The solid line shows the fit according to Eq. (1) with ${\kappa }^{-1}\approx 25\mathrm{nm}$. Inset: light pressure vs tweezers intensity. The light pressure is a linear function of the laser intensity.

Figure 2

Measured tweezers intensity and particle trajectory. During the first pulse the particle is pressed towards the surface. During the second pulse thermal fluctuations support the particle and it is able to move away from the wall. Hence the applied work is positive for the first pulse and negative for the second.

Figure 3

(a) The quantities $-Q$, $W$, and $\Delta V$ for about 100 periods of the protocol $I(\tau )$. (b) Distribution histogram of $\delta =W-Q-\Delta V$, the experimentally observed “deviation” from the first law of thermodynamics.

Figure 4

Non-Gaussian work distribution. The data were taken from about 16 000 trajectories, where the average work done on the particle was about $2.4k_{\mathrm{B}}T$. The solid line shows the Pearson type III distribution [26] corresponding to the theoretically obtained moments. Inset: logarithm of the ratio of the probability to find trajectories with work $-W$ to those with work $+W$. The solid line shows the expected curve (9). The deviation is due to the poor statistics of large negative work values $W\lesssim -4k_{\mathrm{B}}T$.