Brownian Duet: A Novel Tale of Thermodynamic Efficiency

We calculate analytically the stochastic thermodynamic properties of an isothermal Brownian engine driven by a duo of time-periodic forces, including its Onsager coefficients, the stochastic work of each force, and the corresponding stochastic entropy production. We verify the relations between different operational regimes, maximum power, maximum efficiency, and minimum dissipation, and reproduce the signature features of the stochastic efficiency. All of these results are experimentally tested without adjustable parameters on a colloidal system.

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The steam engine was the basis of the industrial revolution, and it also prompted the discovery and development of thermodynamics. The second law of thermodynamics, which limits the efficiency of transforming a heat flux into work, similarly limits the transformation between different forms of work while operating at the same temperature. Such “isothermal engines” are, in fact, ubiquitous in nature. Indeed, on the smallest scales of cellular biology, temperature variations are rare and temperature fluctuations dominate. In such a noisy, fluctuating world, cells must accomplish tasks such as movement or signal detection. Stochastic thermodynamics generalizes thermodynamics to include the effects of such fluctuations and places basic limits on all systems, molecular (e.g., ATP) or otherwise. Here, using this theory, we calculate analytically the stochastic thermodynamic properties of an isothermal Brownian engine consisting of a single Brownian particle confined in a harmonic well and driven by a duo of time-periodic forces.

Our experimental work focuses on a micron-sized colloidal particle (i.e., a silica bead) held in a one-dimensional feedback trap subject to specific electric forces applied every 5 ms. We evaluate the response properties of this Brownian particle to a quadratic potential, and we establish a relation between these quantities and the fluctuating properties. We verify the relations among different operational regimes, maximum power, maximum efficiency, and minimum dissipation and reproduce the signature features of the stochastic efficiency.

We expect that our findings will pave the way for modern-day applications of the tenets of thermodynamics developed nearly 300 years ago.

Figure 1

Experimentally determined $\dot{\overline{W_i}}/F_{1,0}^{\prime }$, $i=1$, 2, for the forward (a) and time-reversed (b) processes. Typical errors are around $\pm 0.1k_{B}T/t_r$. Error bars are not shown, as they are smaller than the markers indicating the mean values of the data. Solid lines are least-squares linear fits.

Figure 2

Asymmetry of Onsager coefficients. (a) $(L_{12}-L_{21})/(L_{12}+L_{21})$ versus temporal asymmetry. The red markers denote the experimental results, the solid black line the theoretical curve. ${\mathcal{T}}^{\prime }=2.5$ and $F_{2,0}^{\prime }=5$. $F_{1,0}^{\prime }$ varies from $-10$ to $+10$ for each $ϵ$. Work rates ${\dot{\overline{W}}}_i$ are calculated from 150 cycles. (b) Illustration of the driving function $g_2(t^{\prime })$, for $ϵ=0.5$ (red line, asymmetric), compared to the symmetric case, $ϵ=0$ (black line, symmetric).

Figure 3

Power (a), efficiency (b), and entropy production (c) as a function of $F_1^{\prime }$. The solid lines are least-squares fits.

Figure 4

Efficiency distributions for symmetric driving. The solid lines show the theoretical probability densities, calculated—not fit—from Eqs. (20) and (21). Error bars are calculated as the square root of counts in each histogram bin. Figure (a) shows the probability density for 1 cycle, (b) for 2 cycles and (c) for 4 cycles.

Figure 5

Large deviation function for the efficiency $\mathcal{J}(\eta )$, defined in Eq. (23), for symmetric driving. Solid line is calculated using Eq. (25).

Figure 6

Median of the efficiency distribution as a function of the number of cycles $n$. Dotted line indicates macroscopic efficiency, $\overline{\eta }\approx -0.93$. Error bars estimated via the bootstrap method [58].