We consider a class of stochastic dynamical systems, called piecewise deterministic Markov processes, with states (x, sigma) is an element of Omega x Gamma, Omega being a region in R(d) or the d-dimensional torus, Gamma being a finite set. The continuous variable x follows a piecewise deterministic dynamics, the discrete variable sigma evolves by a stochastic jump dynamics and the two resulting evolutions are fully-coupled. We study stationarity, reversibility and time-reversal symmetries of the process. Increasing the frequency of the sigma-jumps, the system behaves asymptotically as deterministic and we investigate the structure of its fluctuations (i.e. deviations from the asymptotic behavior), recovering in a non Markovian frame results obtained by Bertini et al. (Phys. Rev. Lett. 87(4): 040601, 2001; J. Stat. Phys. 107(3-4): 635-675, 2002; J. Stat. Mech. P07014, 2007; Preprint available online at http://www.arxiv.org/abs/0807.4457, 2008), in the context of Markovian stochastic interacting particle systems. Finally, we discuss a Gallavotti-Cohen-type symmetry relation with involution map different from time-reversal

Temperature is a well-defined quantity for systems in equilibrium. For glassy systems, it has been extended to the non-equilibrium regime, showing up as an e ective quantity in a modified version of the fluctuation-dissipation theorem. However, experimental evidence supporting this definition remains scarce. Here, we present the first direct experimental demonstration of the e ective temperature by measuring correlations and responses in single molecules in non-equilibrium steady states generated under external random forces. We combine experiment, analytical theory and simulations for systems with di erent levels of complexity, ranging from a single bead in an optical trap to two-state and multiple-state DNA hairpins. From these data, we extract a unifying picture for the existence of an e ective temperature based on the relative order of various timescales characterizing intrinsic relaxation and external driving. Our study thus introduces driven small systems as a fertile ground to address fundamental concepts in statistical physics, condensed-matter physics and biophysics. F rom living cells to stars, natural processes occur under non-equilibrium conditions. Non-equilibrium systems have been classified, phenomenological patterns identified and theoretical frameworks developed, yet our current understanding of the fundamentals of the non-equilibrium problem still remains incomplete, undoubtedly far beyond what we know for equilibrium systems. Major advances in the field are the understanding of linear irreversible thermodynamics and, more recently, the development of fluctuation theorems 1-4 and macroscopic fluctuation theories 5,6 .Temperature, the measure of warmth or coldness of a system, is a genuine statistical concept. Related to erratic, agitated and unpredictable molecular motion, it is quantified through the average kinetic energy of molecules and ultimately grounded by the atomistic character of matter. Brownian motion, the shaken motion of the grains of pollen (diffusion) observed by Robert Brown in 1827, results from the bombardment of molecules experienced by the grains. The larger the average kinetic energy of such tiny colliding objects, the higher the temperature of the system. However, thermal forces are not the only way matter can be shaken, this can be also achieved by the slow release of accumulated stress in slowly relaxing systems such as structural glasses and granular media 7 , by injecting energy (for example, through external gradients) to steady-state systems 8 or through chemical reactions in self-propelled particles in active matter 9 . Can the erratic motion caused by such forces of non-thermal origin still be described in terms of the equilibriumbased concept of temperature? Spin-glass theories applied to systems exhibiting slow relaxation to equilibrium and ageing have persistently shown how the observed non-equilibrium behaviour can be described in terms of a non-equilibrium parameter, also called effective temperature. The effective temperature quantifies violations of the fluc...

The relation between entropy and information dates back to the classical Maxwell demon (MD) paradox [1], a thought experiment proposed in 1867 by J. C. Maxwell to violate the second law of thermodynamics. A variant of the classical MD is the Szilard engine proposed by L. Szilard in 1926 in which the demon observes, at a given time, the compartment occupied by a single molecule in a vessel and extracts work by operating a pulley device. Here we introduce the Continuous Maxwell Demon (CMD), a device capable of extracting arbitrarily large amounts of work per cycle by repeated measurements of the state of a system, and experimentally test it in single DNA hairpin pulling experiments. In the CMD the demon monitors the state of the DNA hairpin (folded or unfolded) by observing it at equally spaced time intervals but extracts work only when the molecule changes state. We demonstrate that the average maximum work per cycle that can be extracted by the CMD is limited by the information-content of the stored sequences, in agreement with the second law. Work extraction efficiency is found to be maximal in the large information-content limit where work extraction is fueled by rare events.In the Szilard engine the demon performs a one-bit measurement by observing, at a given time, the compartment ( ! , " ) occupied by a single molecule in a vessel of volume at temperature ( Figure 1A). The engine operates as follows: if the molecule occupies the left compartment ( ! ), a pulley device extracts the mean work ! = − # log ! where ! = ! / is the probability of the molecule observed in the left compartment; if it occupies the right compartment ( " = − ! ) the mean extracted work equals " = − # log " with " = 1 − ! . The average work per cycle that can be extracted in the classical MD equals $%&It is maximal for ! = " = 1/2, $%& '( ≤ ) = # log 2, ) being the Landauer limit. The resolution of the MD paradox, i.e. the fact that the engine can fully convert heat into work without any other change, came from the thermodynamics of data processing. Half a century ago it was shown that any irreversible logical operation, such as bit erasure, requires energy consumption typically on the order of k B T [2,3]. In 2 general, $%& '( equals the information-content I of one bit, = − ! log ! − " log " , restoring the second-law inequality,with the mean extracted work. Subsequent developments in experimental physics, often in combination with the theory of fluctuation theorems and information feedback [4][5][6][7][8] have provided experimental realizations and models of the MD that have tested Eq.2 and the Landauer limit [9][10][11][12][13][14][15][16].Here we introduce the continuous MD (CMD), a conceptually new information-toenergy conversion device that takes advantage of extracting work from rare events. The CMD is exemplified in Figure 1B. The demon monitors the motion of the molecule by observing it at equally spaced time intervals but extracts work only when the molecule changes compartment. A work-extraction cycle starts with an initial o...