# Fluctuation relations

**Introduction**

Recently discovered and formulated fluctuation theorems (FTs) reveal deep relationship between nonequilibrium fluctuations and equilibrium properties of small systems, both classical and quantum. These theorems can be employed, e.g., in physics, chemistry, and biology. However, FTs are not only of fundamental interest. Indeed, they also provide a tool for experimental monitoring of small systems. Moreover, the FTs are closely connected to the concept of work in classical and quantum mechanics occurring under an external driving protocol.

**Classical fluctuation relations**

Classical fluctuation relations have been a hot topic of research during the last two decades. It has been shown that the Helmholtz free-energy difference between two equilibrium configurations of a classical system is related by an equality to the ensemble average of finite-time nonequilibrium measurements of the work performed over the system controlled by an external parameter [Phys. Rev. E 56 (5), 5018 (1997)].

Thus after the averaging over multiple stochastic trajectories of the system dynamics, one can obtain valuable information about the system and its environment. This can be used in both the fundamental research and device engineering. We have experimentally studied the Jarzynski and Crooks relations for charge transfer in a single-electron box [Phys. Rev. Lett. 109, 180601 (2012)] as well as the distribution of entropy production relating to the so-called integral fluctuation theorem [Nat. Phys. 9, 644-648 (2013)]. Further work in collaboration with the PICO group and MSP group at the Aalto University will be carried out in future.

**Fig.** (a) Sketch of the single-electron box used to study the entropy production together with a scanning electron micrograph of a typical sample. (b) Trace of the measured detector signal under a sinusoidal protocol used for the gate voltage drive.

**Quantum fluctuation relations**

In the quantum world, the act of measurement has a special role. The strong projective measurement alters the state of the measured system and, as a result, alters the achievable time-evolution. This is significant for fluctuation relations as it turns out that to easily retrieve the quantum analogues of the classical ones, one is again required to measure the system twice determining the stochastic trajectories in the process (see for example [Rev. Mod. Phys. 81, 1665 (2009); Rev. Mod. Phys. 83, 771 (2011)]). This two-measurement approach can also be extended to open quantum system relating the measurement results to open system partition functions.

The theoretical analysis of these systems can use both the master equation and stochastic wavefunction formalisms [J. Opt.Soc. Am. B 10, 524-538 (1993)] to treat the complex quantum evolution. Such treatment additionally allows possible control over the information flow and, hence, is connected to quantum information theory. Other approaches to determine the quantum fluctuation relations such as weak measurements or studying the quantum mechanical expectations exist and we are actively pursuing them. Both the selection of the dynamical basis for driven systems and the composite approach to quantum driving contribute to modifying the fluctuation relations presenting an interesting field of study.