# Fluctuation Theorems: Information and Entropy

The Jarzynski equality, introduced in 1997, established an exact relationship that involves the work fluctuations of processes arbitrarily far from equilibrium [1]. Later, Crooks' theorem [2, 3] showed that the mentioned equality is consequence of some symmetry properties under time reversal of the microscopic evolution of the system, both Hamiltonian or stochastic (Langevin equation). These properties were lately clarified via the introduction of information theory concepts like Kullback-Leibler divergence or relative entropy [4,5,6,7]. Fluctuation theorems are important in applications [3] but also in fundamental problems like Maxwell’s Demon [8,9].

Recently, we have introduced a quantitative relationship between this relative entropy and the average entropy production along the process [4,5] based on Kullback-Leibler distance, a measure of irreversibility. This result is very significant because it relates a physical magnitude, the entropy production or the energy dissipation, with a purely statistical quantity.

This relationship has allowed us to estimate or bound the dissipation or entropy production from single stationary trajectories [10]. In particular, we are able to distinguish whether or not a system dissipates energy even if we only have partial information of the system. More surprisingly, we can predict that a system dissipates energy even if it does not exhibit any current or probability flows which may reveal that the process is out of equilibrium. The extension to continuous systems requires the use of advanced techniques of relative entropy estimation, being one of the aims of this research line.

Feedback control is of special interest in statistical mechanics since the celebrated Maxwell’s Demon can be considered a special case of feedback [8]. Our group studied feedback-controlled brownian motors [11-13], and recently, fluctuation theorems have been applied to feedback-controlled systems by one member of our group [14], who extended previous results of feedback control thermodynamics [15-17].

The extension of this study to quantum systems can provide fundamental results since feedback requires a measurement. The group of M. Ueda has derived some inequalities similar to Second Law of Thermodynamics valid for quantum systems with feedback [18,19]. It would be interesting to obtain similar inequalities for classic systems [14,17]. Fluctuation theorems have been generalized to the quantum case [20] relating irreversibility and entropy production [5] like in the classic case. Some of this extensions have included the quantum measurement in practical situations [21]. We try to include the quantum measurement to fluctuation theorems in a more general way by using solvable models of measurement [23] in order to be able to analyse the irreversibility and the energetics of the measurement and quantum control.

A particular case of Maxwell’s Demon, and maybe the most studied one, is the Szilard engine [8]. The engine shows how is it possible to extract energy from an isothermal system if we acquire information from it via a measurement. A recent result of our group [24] has proved that the measurement is unnecessary if the system has a microcanonic initial condition, that is, if the initial energy of the system is known with some accuracy. One of our aims is to extend this to larger systems and also to quantum systems.

Many groups are currently working in the mentioned topics around the world, In Europe, many of these groups are represented in the European Science Foundation program Exploring the Physics of Small Devices (EPSD).

[1] C. Jarzynski, Nonequilibrium Equality for Free Energy Differences. Physical Review Letters 78, 2690-2693 (1997).

[2] C. Jarzynski, Work Fluctuation Theorems and Single-Molecule Biophysics. Progress of Theoretical Physics Supplement 165, 1-17 (2006).

[3] D. Collin, F. Ritort, C. Jarzynski, S.B. Smith, I. Tinoco, and C. Bustamante, Verification of the Crooks Fluctuation Theorem and Recovery of Rna Folding Free Energies. Nature 437, 231-234 (2005).

[4] R. Kawai, J.M.R. Parrondo, and C. Van den Broeck, Dissipation: The Phase-Space Perspective. Physical Review Letters 98, 080602 (2007).

[5] J.M.R. Parrondo, C. Van den Broeck, and R. Kawai, Entropy Production and the Arrow of Time. New Journal of Physics 11, 073008 (2009).

[6] J. Horowitz and C. Jarzynski, Comparison of Work Fluctuation Relations. Journal of Statistical Mechanics-Theory and Experiment, P11002 (2007).

[7] J. Horowitz and C. Jarzynski, Illustrative Example of the Relationship between Dissipation and Relative Entropy. Physical Review E 79, 021106 (2009).

[8] H.S. Leff and A.F. Rex, Maxwell’s Demon 2 : Entropy, Classical and Quantum Information, Computing. (Institute of Physics, Bristol, 2003).

[9] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Experimental Demonstration of Information-to-Energy Conversion and Validation of the Generalized Jarzynski Equality. Nature Physics 6, 988-992 (2010).

[10] E. Roldan and J.M.R. Parrondo, Estimating Dissipation from Single Stationary Trajectories. Physical Review Letters 105, 150607 (2010).

[11] F.J. Cao, L. Dinis, and J.M.R. Parrondo, Feedback Control in a Collective Flashing Ratchet. Physical Review Letters 93, 040603 (2004).

[12] E.M. Craig, B.R. Long, J.M.R. Parrondo, and H. Linke, Effect of Time Delay on Feedback Control of a Flashing Ratchet. EPL 81, 10002 (2008).

[13] L. Dinis, J.M.R. Parrondo, and F.J. Cao, Closed-Loop Control Strategy with Improved Current for a Flashing Ratchet. Europhysics Letters 71, 536-541 (2005).

[14] J.M. Horowitz and S. Vaikuntanathan, Nonequilibrium Detailed Fluctuation Theorem for Repeated Discrete Feedback. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 061120 (2010).

[15] H. Touchette and S. Lloyd, Information-Theoretic Limits of Control. Physical Review Letters 84, 1156-1159 (2000).

[16] F.J. Cao and M. Feito, Thermodynamics of Feedback Controlled Systems. Physical Review E 79, 041118 (2009).

[17] T. Sagawa and M. Ueda, Generalized Jarzynski Equality under Nonequilibrium Feedback Control. Physical Review Letters 104, 090602 (2010).

[18] T. Sagawa and M. Ueda, Second Law of Thermodynamics with Discrete Quantum Feedback Control. Physical Review Letters 100, 080403 (2008).

[19] T. Sagawa and M. Ueda, Minimal Energy Cost for Thermodynamic Information Processing: Measurement and Information Erasure. Physical Review Letters 102, 250602 (2009).

[20] M. Esposito, U. Harbola, and S. Mukamel, Nonequilibrium Fluctuations, Fluctuation Theorems, and Counting Statistics in Quantum Systems. Reviews of Modern Physics 81, 1665-1702 (2009).

[21] I. Callens, W. De Roeck, T. Jacobs, C. Maes, and K. Netocny, Quantum Entropy Production as a Measure of Irreversibility. Physica D-Nonlinear Phenomena 187, 383-391 (2004).

[22] M. Campisi, P. Talkner, and P. Hanggi, Fluctuation Theorems for Continuously Monitored Quantum Fluxes. Physical Review Letters 105, 140601 (2010).

[23] W.H. Zurek, Decoherence, Einselection, and the Quantum Origins of the Classical. Reviews of Modern Physics 75, 715-775 (2003).

[24] R. Marathe and J.M.R. Parrondo, Cooling Classical Particles with a Microcanonical Szilard Engine. Physical Review Letters 104, 245704 (2010).