## Final Report Summary - NTFSS (Nonequilibrium Thermodynamics and Fluctuations in Small Systems)

Thermodynamics has been particularly successful in describing macroscopic systems at equilibrium. In recent years, there has been an enormous amount of interest in the thermodynamic description of small systems, inspired and developed in response to advances in bioengineering and nanotechnology that were not accessible in the past. This has led to spectacular theoretical advances, driven by the overarching recognition that the familiar rules governing large systems, especially large systems in equilibrium, do not apply in the small nonequilibrium world. These theoretical breakthroughs offer a new vision on the relation between the micro and macro worlds, in particular the role of dynamical randomness, the formulation of fluctuation and work theorems and the microscopic foundations of the second law. Parallel to the theoretical advance, these developments suggest to introduce new concepts and a novel ways to measure, operate or function in systems of small scale. Rather than miniaturising the construction and principles of macroscopic machines, one can build on the specificities of the small scale, such as the presence of a strong stochastic component, to propose new designs and new modes of operation. This project contributed to these developments on the following aspects:

i) Event- versus time-based current statistics

A system out of equilibrium is crossed by nonzero net currents. These currents provide sensitive information about the internal system dynamics. With the development of more sensitive scientific instruments, fluctuations of currents crossing very small devices can be measured. A current is a number of events (energy or particle transfers) in a given time interval. A lot of attention had been devoted recently to the study of current fluctuations since they are closely related to fluctuation theorems. These approaches were based on the probability that in a given time interval a certain number of events have occurred. I proposed to investigate current fluctuations in a different way than this event-based approach by considering the probability distribution of the time needed for a transfer of a given number of events to occur. I have shown in [1] that this time-based approach can lead to different statistics than the event-based approach. For normal statistics (fast decaying waiting time distribution) the two approaches lead to similar results, but in the case of anomalous statistics (slowly decaying waiting time distribution) differences arise. This suggests that experimental measurements of currents using the time-based approach could be a useful tool to identify anomalous statistics.

ii) Efficiency at maximum power

A fundamental result of thermodynamics is that best efficiencies of an engine (mechanical, thermal, chemical) can be reached when working reversibly, i.e. very close to equilibrium. Unfortunately, processes occurring under these optimal conditions take an infinitely long time to occur and thus lead to zero power outputs (power is an energy per unit time) and are therefore of little practical use. A more practically relevant question is to study the efficiency at maximum power. Stochastic thermodynamics describes systems by stochastic dynamics which, when ensemble averaged, lead to thermodynamics. Such a description incorporates time scales via rates in a natural way and thus provides a powerful framework to study efficiencies at maximum power. Two classes of systems can be considered using this formalism: systems maintained in nonequilibrium steady-states by external reservoirs and systems driven by external time-dependent forces. For systems belonging to the first class, older results had shown that high efficiencies are easier to reach in small systems. This is due to the possibility of realising a condition of strong coupling between the input flux and the output flux which generates the power. This occurs when the same elementary transfer process is at the origin of these two fluxes. Remarkably, the efficiency at maximum power in small systems with strong coupling displays universal features. In this project, I succeeded in identifying the analog of the condition of strong coupling for systems of the second class. I also characterised the resulting universal features on the efficiency. The key notion is that of weak dissipation which occurs when the entropy production of a periodically driven system behaves as the inverse period of the driving cycle [7]. This conclusion was reached after an in-depth study of the finite-time thermodynamics of a specific model consisting of a single level quantum dot externally driven by a time-dependent field [5, 6]. In [12], I also later studied the finite-time efficiency of an analytically solvable model of a stochastically driven two level system. I should finally mention that findings not initially envisioned in this project led me to generalise the formulation of stochastic thermodynamics to systems driven out-of-equilibrium by external reservoirs and a time-dependent force [2, 3, 4]. These results could play an important role in the study of the efficiency of nanodevices subjected to these two types of driving mechanisms.

iii) Information and stochastic thermodynamics

The pioneer works of Landauer and Bennett have shown how that information is intimately connected to thermodynamics. Stochastic thermodynamics and the recent results known as fluctuation theorems have made this connection even clearer by connecting the second law of thermodynamics to the asymmetry of the information production needed to track the trajectory during the forward evolution and the time-reversed evolution. In [9] (partly based on results derived in [8, 9]), I have proposed a nonequilibrium formulation of the Landauer principle which very clearly demonstrate that information (measured as a relative entropy between the nonequilibrium and equilibrium system probability distribution) can be used to extract more work from a system than predicted by equilibrium thermodynamics. This result could become very important for developing a consistent thermodynamic description of small systems subjected to feedback control.

iv) Modified fluctuation theorem for the electron counting statistics in quantum dots

Fluctuation theorems characterise the universal features in the nonequilibrium fluctuations of small systems. Electron counting statistics experiments have shown unexplained discrepancies between certain key features of the statistics and the theoretical prediction of fluctuation theorems. In [11], I have proposed a model to explain these discrepancies which takes into account the fact that single electron transfers are experimentally detected by a capacitive coupling to an auxiliary circuit which affects the quantum dot counting statistics. This kind of developments are important steps towards a better understanding of the thermodynamics of quantum measurement.

[1] M. Esposito, K. Lindenberg and I. Sokolov, 'On the equivalence between events and time based current statistics', EPL 89, 10008 (2010)

[2] M. Esposito and C. Van den Broeck, 'Three fundamental detailed fluctuation theorems', Phys. Rev. Lett. 104, 090601 (2010)

[3] M. Esposito and C. Van den Broeck, 'The three faces of the second law: I. Master equation formulation', Phys. Rev. E 82, 011143 (2010)

[4] C. Van den Broeck and M. Esposito, 'The three faces of the second law: II. Fokker-Planck formulation', Phys. Rev. E 82, 011144 (2010)

[5] M. Esposito, R. Kawai, K. Lindenberg and C. Van den Broeck, 'Finite time thermodynamics for a single level quantum dot', EPL 89, 20003 (2010)

[6] M. Esposito, R. Kawai, K. Lindenberg and C. Van den Broeck, 'Quantum-dot Carnot engine at maximum power', Phys. Rev. E 81, 041106 (2010)

[7] M. Esposito, R. Kawai, K. Lindenberg and C. Van den Broeck, 'Efficiency at maximum power of low dissipation Carnot engines', Phys. Rev. Lett. 105, 150603 (2010)

[8] M. Esposito and T. Monnai, 'Nonequilibrium thermodynamics and Nose-Hoover dynamics', J. Phys. Chem. B 115, 5144 (2011)

[9] M. Esposito and C. Van den Broeck, 'Second law and Landauer principle far from equilibrium', EPL 95, 40004 (2011)

[10] M. Esposito, K. Lindenberg and C. Van den Broeck, 'Entropy production as correlation between system and reservoir', New J. Phys. 12, 013013 (2010)

[11] G. Bulnes Cuetara, M. Esposito and P. Gaspard, 'Fluctuation theorems for capacitively coupled electronic currents', Phys. Rev. B 84, 165114 (2011)

[12] N. Kumar, C. Van den Broeck, M. Esposito and K. Lindenberg, 'Thermodynamics of a stochastic twin elevator', Phys. Rev. E 84, 051134 (2011)

i) Event- versus time-based current statistics

A system out of equilibrium is crossed by nonzero net currents. These currents provide sensitive information about the internal system dynamics. With the development of more sensitive scientific instruments, fluctuations of currents crossing very small devices can be measured. A current is a number of events (energy or particle transfers) in a given time interval. A lot of attention had been devoted recently to the study of current fluctuations since they are closely related to fluctuation theorems. These approaches were based on the probability that in a given time interval a certain number of events have occurred. I proposed to investigate current fluctuations in a different way than this event-based approach by considering the probability distribution of the time needed for a transfer of a given number of events to occur. I have shown in [1] that this time-based approach can lead to different statistics than the event-based approach. For normal statistics (fast decaying waiting time distribution) the two approaches lead to similar results, but in the case of anomalous statistics (slowly decaying waiting time distribution) differences arise. This suggests that experimental measurements of currents using the time-based approach could be a useful tool to identify anomalous statistics.

ii) Efficiency at maximum power

A fundamental result of thermodynamics is that best efficiencies of an engine (mechanical, thermal, chemical) can be reached when working reversibly, i.e. very close to equilibrium. Unfortunately, processes occurring under these optimal conditions take an infinitely long time to occur and thus lead to zero power outputs (power is an energy per unit time) and are therefore of little practical use. A more practically relevant question is to study the efficiency at maximum power. Stochastic thermodynamics describes systems by stochastic dynamics which, when ensemble averaged, lead to thermodynamics. Such a description incorporates time scales via rates in a natural way and thus provides a powerful framework to study efficiencies at maximum power. Two classes of systems can be considered using this formalism: systems maintained in nonequilibrium steady-states by external reservoirs and systems driven by external time-dependent forces. For systems belonging to the first class, older results had shown that high efficiencies are easier to reach in small systems. This is due to the possibility of realising a condition of strong coupling between the input flux and the output flux which generates the power. This occurs when the same elementary transfer process is at the origin of these two fluxes. Remarkably, the efficiency at maximum power in small systems with strong coupling displays universal features. In this project, I succeeded in identifying the analog of the condition of strong coupling for systems of the second class. I also characterised the resulting universal features on the efficiency. The key notion is that of weak dissipation which occurs when the entropy production of a periodically driven system behaves as the inverse period of the driving cycle [7]. This conclusion was reached after an in-depth study of the finite-time thermodynamics of a specific model consisting of a single level quantum dot externally driven by a time-dependent field [5, 6]. In [12], I also later studied the finite-time efficiency of an analytically solvable model of a stochastically driven two level system. I should finally mention that findings not initially envisioned in this project led me to generalise the formulation of stochastic thermodynamics to systems driven out-of-equilibrium by external reservoirs and a time-dependent force [2, 3, 4]. These results could play an important role in the study of the efficiency of nanodevices subjected to these two types of driving mechanisms.

iii) Information and stochastic thermodynamics

The pioneer works of Landauer and Bennett have shown how that information is intimately connected to thermodynamics. Stochastic thermodynamics and the recent results known as fluctuation theorems have made this connection even clearer by connecting the second law of thermodynamics to the asymmetry of the information production needed to track the trajectory during the forward evolution and the time-reversed evolution. In [9] (partly based on results derived in [8, 9]), I have proposed a nonequilibrium formulation of the Landauer principle which very clearly demonstrate that information (measured as a relative entropy between the nonequilibrium and equilibrium system probability distribution) can be used to extract more work from a system than predicted by equilibrium thermodynamics. This result could become very important for developing a consistent thermodynamic description of small systems subjected to feedback control.

iv) Modified fluctuation theorem for the electron counting statistics in quantum dots

Fluctuation theorems characterise the universal features in the nonequilibrium fluctuations of small systems. Electron counting statistics experiments have shown unexplained discrepancies between certain key features of the statistics and the theoretical prediction of fluctuation theorems. In [11], I have proposed a model to explain these discrepancies which takes into account the fact that single electron transfers are experimentally detected by a capacitive coupling to an auxiliary circuit which affects the quantum dot counting statistics. This kind of developments are important steps towards a better understanding of the thermodynamics of quantum measurement.

[1] M. Esposito, K. Lindenberg and I. Sokolov, 'On the equivalence between events and time based current statistics', EPL 89, 10008 (2010)

[2] M. Esposito and C. Van den Broeck, 'Three fundamental detailed fluctuation theorems', Phys. Rev. Lett. 104, 090601 (2010)

[3] M. Esposito and C. Van den Broeck, 'The three faces of the second law: I. Master equation formulation', Phys. Rev. E 82, 011143 (2010)

[4] C. Van den Broeck and M. Esposito, 'The three faces of the second law: II. Fokker-Planck formulation', Phys. Rev. E 82, 011144 (2010)

[5] M. Esposito, R. Kawai, K. Lindenberg and C. Van den Broeck, 'Finite time thermodynamics for a single level quantum dot', EPL 89, 20003 (2010)

[6] M. Esposito, R. Kawai, K. Lindenberg and C. Van den Broeck, 'Quantum-dot Carnot engine at maximum power', Phys. Rev. E 81, 041106 (2010)

[7] M. Esposito, R. Kawai, K. Lindenberg and C. Van den Broeck, 'Efficiency at maximum power of low dissipation Carnot engines', Phys. Rev. Lett. 105, 150603 (2010)

[8] M. Esposito and T. Monnai, 'Nonequilibrium thermodynamics and Nose-Hoover dynamics', J. Phys. Chem. B 115, 5144 (2011)

[9] M. Esposito and C. Van den Broeck, 'Second law and Landauer principle far from equilibrium', EPL 95, 40004 (2011)

[10] M. Esposito, K. Lindenberg and C. Van den Broeck, 'Entropy production as correlation between system and reservoir', New J. Phys. 12, 013013 (2010)

[11] G. Bulnes Cuetara, M. Esposito and P. Gaspard, 'Fluctuation theorems for capacitively coupled electronic currents', Phys. Rev. B 84, 165114 (2011)

[12] N. Kumar, C. Van den Broeck, M. Esposito and K. Lindenberg, 'Thermodynamics of a stochastic twin elevator', Phys. Rev. E 84, 051134 (2011)