Thermodynamics of Micro- and Nano-Systems Driven by Periodic Temperature Variations

We introduce a general framework for analyzing the thermodynamics of small systems that are driven by both a periodic temperature variation and some external parameter modulating their energy. This setup covers, in particular, periodic micro- and nano-heat engines. In a first step, we show how to express total entropy production by properly identified time-independent affinities and currents without making a linear response assumption. In linear response, kinetic coefficients akin to Onsager coefficients can be identified. Specializing to a Fokker-Planck-type dynamics, we show that these coefficients can be expressed as a sum of an adiabatic contribution and one reminiscent of a Green-Kubo expression that contains deviations from adiabaticity. Furthermore, we show that the generalized kinetic coefficients fulfill an Onsager-Casimir-type symmetry tracing back to microscopic reversibility. This symmetry allows for nonidentical off-diagonal coefficients if the driving protocols are not symmetric under time reversal. We then derive a novel constraint on the kinetic coefficients that is sharper than the second law and provides an efficiency-dependent bound on power. As one consequence, we can prove that the power vanishes at least linearly when approaching Carnot efficiency. We illustrate our general framework by explicitly working out the paradigmatic case of a Brownian heat engine realized by a colloidal particle in a time-dependent harmonic trap subject to a periodic temperature profile. This case study reveals inter alia that our new general bound on power is asymptotically tight.

Popular Summary

Reciprocating heat engines like James Watt’s famous steam engine convert thermal energy into useful mechanical work by cyclically compressing and expanding a working fluid whose temperature is altered periodically. In recent decades, much work has focused on the miniaturization of such devices. A comprehensive understanding of the fundamental principles governing the performance of these miniaturized devices is accordingly of paramount importance. Here, we develop a general framework for the thermodynamic description of small periodically driven systems. We also derive a universal constraint on the power output of mesoscopic heat engines.

Our approach is inspired by concepts of irreversible thermodynamics, a powerful theory originally designed for the characterization of nonequilibrium steady states. We show that the central notions of irreversible thermodynamics can be applied consistently to periodically driven systems on the level of cycle averages. In particular, we recover the generic bilinear expression for the total rate of entropy production in terms of fluxes and affinities. Assuming linear response conditions, we obtain a generalization of Onsager’s celebrated reciprocal relations and, using a novel method, prove a general bound on the power of stochastic micro-heat-engines. We show that the power and efficiency of such systems are quadratically related and that the power vanishes as the efficiency approaches the Carnot value.

Since our analysis is entirely classical, our results might have to be reassessed on even smaller scales where genuine quantum effects such as coherence and entanglement become important. Exploring this issue constitutes a challenging and exciting subject for future research.

Figure 1

Nonequilibrium periodic state of a system with 1 degree of freedom $x$ in a sinusoidally shifted harmonic potential, represented by gray parabolas, which is embedded in an environment with periodically changing temperature as indicated by the periodic color gradient. The solid line in the $x\text{−}t$ plane shows the motion of the center of the Gaussian phase-space distribution, which lags behind the position of the potential minimum shown as a dashed line. At any time $t$, the width of the colored region equals the width of the phase-space distribution, which varies according to the temperature.

Figure 2

Plots of the maximum power (57) in units of $P_0\equiv T_{\mathrm{c}}{\mathcal{F}}_q^2{L}_{qq}$ as a function of the normalized efficiency $\overline{\eta }\equiv \eta /{\eta }_{\mathrm{C}}$ for selected values of the asymmetry parameter $x\ge 1$ in the upper and $x<-1$ in the lower panel. For $x\to \pm \infty$, convergence to $P/P_0=\overline{\eta }$ is observed. The dotted black lines correspond to the maxima of Eq. (57) for $0\le x<\infty$ in the first and $-\infty <x<-1$ in the second plot, thus indicating the relation between maximum power and efficiency at maximum power. The apparent divergence for negative $x$ occurs in the limit $x\to -1$ [48].

Figure 3

Operation cycle of a Brownian heat engine. The vertical axis corresponds to the normalized time-dependent strength of the harmonic trap in units of $\wp \equiv \kappa (t)/{\kappa }_0$, the horizontal axis to the normalized width $\mathcal{V}\equiv ⟨x^2⟩/(2x_0^2)$ of the distribution function. This plot is analogous to the pressure volume diagram of a macroscopic heat engine such that the area encircled by the colored lines quantifies the work extracted per operation cycle. Specifically, the plots were obtained using the protocols (73) and (80) for different values of the shape parameter $d$, $2\mu {\kappa }_0\mathcal{T}=1$, ${\eta }_{\mathrm{C}}=T_{\mathrm{c}}{\mathcal{F}}_q=1/10$ and $\overline{\eta }=1/2$. The small graphics show sketches of the potential (gray line) and the phase-space distribution, whose color reflects the temperature of the heat bath, at the respective edges of the cycle. For further explanations, see Sec. 4.

Figure 4

Plots of the temperature protocol ${\gamma }_q(t,d)$ defined in Eq. (80) (dashed lines) and the corresponding optimal protocol ${\gamma }_w^{*}(t,\eta ,d)$ for the trap strength (solid lines) obtained from Eq. (74) for four different values of the parameter $d$ as functions of $t/\mathcal{T}$ [53]. For all plots, we have set $2\mu {\kappa }_0\mathcal{T}=1$ and $\overline{\eta }=1/2$. Additionally, the optimal protocol ${\gamma }_w^{*}(t)$ has been rescaled by the dimensionless factor ${\kappa }_0{\eta }_{\mathrm{C}}/\kappa$.