Distribution of entropy production in a single-electron box

Abstract

Recently, the fundamental laws of thermodynamics have been reconsidered for small systems. The discovery of the fluctuation relations [1][2][3][4][5] has spurred theoretical [1,6,7,[9][10][11][12][13] and experimental [14-23] studies on thermodynamics of systems with few degrees of freedom. The concept of entropy production has been extended to the microscopic level by considering stochastic trajectories of a system coupled to a heat bath. However, the experimental observation of the microscopic entropy production remains elusive. We measure distributions of the microscopic entropy production in a single-electron box consisting of two islands with a tunnel junction. The islands are coupled to separate heat baths at different temperatures, maintaining a steady thermal nonequilibrium. As Jarzynski equality between work and free energy is not applicable in this case, the entropy production becomes the relevant parameter. We verify experimentally that the integral and detailed fluctuation relations are satisfied. Furthermore, the coarse-grained entropy production [10-12, 23, 24] from trajectories of electronic transitions is related to the bare entropy production by a universal formula. Our results reveal the fundamental roles of irreversible entropy production in non-equilibrium small systems.Entropy production is a hallmark of irreversible thermodynamic processes. The concept of a stochastic microscopic trajectory allows one to define entropy for small systems [1]. However, such trajectories depend on the scale of observation. If one only accesses mesoscopic degrees of freedom, one observes coarse-grained trajectories of mesoscopic states. The corresponding entropy production then differs from the bare entropy production without coarse-graining. In fact, it has been recently shown that coarse-graining of the slow background degrees of freedom for stochastic dynamics may actually lead to a modification of the fluctuation relations for entropy [23]. To clarify the concept of microscopic entropy production in non-equilibrium, accurate measurements are needed for systems, where the concepts of stochastic dynamics and time scale separation between the system and the heat bath are well-defined.A single-electron box (SEB) device at low temperatures is an excellent test bench for thermodynamics in small systems [22,25,26].The SEB employed here is shown in Fig. 1(a). The electrons in the normal-metal copper island (N) can tunnel to the superconducting Al island (S) through the aluminum oxide insulator (I). The sample fabrication [27] methods are similar to those in Ref. [22], but the design is different in that the S side of the junction does not overlap with the normal conductor in order to intentionally weaken the relaxation of energy in S [28]. Moreover, the main results in Ref. [22] were extracted from measurements at the temperature of 220 mK, whereas these measurements are conducted at 140 mK. Lower temperature further weakens the relaxation significantly [28], leading to and elevated temperature in S. We denote by n ...

Heat fluctuations and fluctuation theorems in the case of multiple reservoirs

“…Using the identities 13) and setting λ n → β nk − λ n it is easy to show that 14) and the symmetry (6.9) follows.…”

“…These techniques permit direct experimental access to the probability distribution functions for the work or for the heat exchanged with the environment [1][2][3][4][5][6][7][8][9][10][11][12][13]. These methods have also yielded access to the experimental verification of the recent fluctuation theorems which relate the probability of observing entropy-generated trajectories with that of observing entropy-consuming trajectories [6,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Towards quantum thermodynamics in electronic circuits

“…Here, and in refs [17,18] it is worth noting that a large number of experiments is possible -up to millions of repetitions -meaning that sufficient statistics can be collected, unlike in typical experiments on molecules [2]. In another experiment, more general fluctuation relations under time-dependent driving conditions were tested with the two islands in the SEB at unequal temperatures [29]. In the case with two baths, the Jarzynski equality fails naturally, but fluctuation relations based on trajectory entropy [30] and thermodynamic entropy recover eqs.…”

Moments of work in the two-point measurement protocol for a driven open quantum system

“…[1][2][3][4], provide a powerful tool to analyze the thermodynamic nature of nonequilibrium processes beyond the linear response regime. When the microscopic system can be described in terms of classical mechanics, the fluctuation theorems have been examined for several systems [5][6][7][8][9][10][11]. However, when described in terms of quantum mechanics, the situation is more problematic.…”