Experimental Demonstration of Violations of the Second Law of Thermodynamics for Small Systems and Short Time Scales

We experimentally demonstrate the fluctuation theorem, which predicts appreciable and measurable violations of the second law of thermodynamics for small systems over short time scales, by following the trajectory of a colloidal particle captured in an optical trap that is translated relative to surrounding water molecules. From each particle trajectory, we calculate the entropy production/consumption over the duration of the trajectory and determine the fraction of second law–defying trajectories. Our results show entropy consumption can occur over colloidal length and time scales.

Figure 1

Histogram of the dimensionless, time-averaged entropy production, ${\Sigma }_t/t$, from 540 experimental trajectories of a colloidal particle in an optical trap at times $t={10}^{-2}$ (black bars) and $t=2.0$ (gray bars) seconds after initiation of stage translation. The bin size is $\Delta {\Sigma }_t/t=1.0$.

Figure 2

The number ratio of entropy-consuming (${\Sigma }_t<0$) to entropy-producing (${\Sigma }_t>0$) trajectories (data points) and the entropy production averaged over entropy-producing trajectories, $⟨\exp (-{\Sigma }_t){⟩}_{{\Sigma }_t>0}$ (line) versus duration, $t$, of 540 experimental trajectories. In accord with the IFT, both experimentally determined measures agree over time.

Figure 3

The number ratio of entropy-consuming (${\Sigma }_t<0$) to entropy-producing (${\Sigma }_t>0$) trajectories (data points) and the entropy production averaged over entropy-producing trajectories, $⟨\exp (-{\Sigma }_t){⟩}_{{\Sigma }_t>0}$ (line) versus duration, $t$, calculated from 343 500 simulated trajectories. The inset depicts the 2D MD simulation cell (to scale) containing $N_s=32$ particles at a number density of 0.30, $N_w=24$ wall particles, and an optical trap with constant $k=1$ that translates at ${\mathrm{v}}_{\mathrm{o}\mathrm{p}\mathrm{t}}=0.5$. Only the first particle, depicted by the filled circle, “feels” the optical trap, represented by the shaded region. The initial temperature of both wall and solution particles was $T=1.0$. All MD quantities are reported in reduced units.

Figure 4

Logarithm of the number ratio of trajectories with entropy production $-{\Sigma }_t$ to those with entropy production ${\Sigma }_t$ versus ${\Sigma }_t$ found from over 310 000 MD trajectories.