We report an experimental and theoretical analysis of the energy exchanged between two conductors kept at different temperature and coupled by the electric thermal noise. Experimentally we determine, as functions of the temperature difference, the heat flux, the out-of-equilibrium variance and a conservation law for the fluctuating entropy, which we justify theoretically. The system is ruled by the same equations as two Brownian particles kept at different temperatures and coupled by an elastic force. Our results set strong constrains on the energy exchanged between coupled nano-systems held at different temperatures.The fluctuations of thermodynamics variables play an important role in understanding the out-of-equilibrium dynamics of small systems [1, 2], such as Brownian particles [3][4][5][6][7], molecular motors [8] and other small devices [9]. The statistical properties of work, heat and entropy, have been analyzed, within the context of the fluctuation theorem [10] and stochastic thermodynamics [1, 2], in several experiments on systems in contact with a single heat bath and driven out-of-equilibrium by external forces or fields [3][4][5][6][7][8][9]. In contrast, the important case in which the system is driven out-of-equilibrium by a temperature difference and energy exchange is produced only by the thermal noise has been analyzed only theoretically on model systems [11][12][13][14][15][16][17][18][19] but never in an experiment because of the intrinsic difficulties of dealing with large temperature differences in small systems.We report here an experimental and theoretical analysis of the statistical properties of the energy exchanged between two conductors kept at different temperature and coupled by the electric thermal noise, as depicted in fig. 1a. This system is inspired by the proof developed by Nyquist [20] in order to give a theoretical explanation of the measurements of Johnson [21] on the thermal noise voltage in conductors. In his proof, assuming thermal equilibrium between the two conductors, he deduces the Nyquist noise spectral density. At that time, well before Fluctuation Dissipation Theorem (FDT), this was the second example, after the Einstein relation for Brownian motion, relating the dissipation of a system to the amplitude of the thermal noise. In this letter we analyze the consequences of removing the Nyquist's equilibrium conditions and we study the statistical properties of the energy exchanged between the two conductors kept at different temperature. This system is probably among the simplest examples where recent ideas of stochastic thermodynamics can be tested but in spite of its simplicity the explanation of the observations is far from trivial. We measure experimentally the heat flowing between the two heath baths, and show that the fluctuating entropy exhibits a conservation law. This system is very general because is ruled by the same equations of two Brownian particles kept at different temperatures and coupled by an elastic force [13,19]. Thus it gives more insight into t...

We analyze the equations governing the evolution of distributions of the work and the heat exchanged with the environment by a manipulated stochastic system, by means of a compact and general derivation. We obtain explicit solutions for these equations for the case of a dragged Brownian particle in a harmonic potential. We successfully compare the resulting predictions with the outcomes of experiments, consisting in dragging a micron-sized colloidal particle through water with a laser trap. The study of the physics of small systems has recently received a boost by the possibility of manipulating nanosystems and biomolecules. The fluctuations of the work and heat that these small systems exchange with the environment while being manipulated can be of the order or even larger than the thermal energy, leading to "transient" violations of the second principle of thermodynamics. The distributions of heat and work have been experimentally studied for a few brownian systems [1,2,3,4]. The probability distribution function (PDF) of the work done on a Brownian particle dragged by a moving quadratic potential was derived in [5,6]. The distribution turns out to be gaussian, what has been taken as an ansatz in [6] and confirmed in [7] by means of a rather involved path integral calculation. On the other hand, obtaining the PDF of the transferred heat represents a much more difficult task: the Fourier transform of this function was obtained in refs. [6,7] by exploiting the energy balance and the gaussian ansatz for the work PDF, valid when the potential is quadratic.In the present paper we derive in a simple way the differential equations governing the evolution of the PDFs of the work and heat exchanged by a brownian particle, valid for any choice of the potential acting on the particle. The solutions of these equation turn out to fulfill the well-known fluctuation relations. We evaluate the solution of these equations for a moving harmonic potential. We then experimentally study the work and the heat exchanged by a colloidal particle dragged through water by an optical trap. The PDF's predicted by our equations result in an excellent agreement with the experimental data. We were inspired by the experiment of Wang et al. [1] where the work done on a similar system was measured. However, in that experiment, only the performed work, and not the heat transferred, was sampled. Moreover, the expected gaussian distribution of the performed work was not verified, and a detailed comparison with the theoretical predictions was not attempted. However in a subsequent paper [8], the authors stressed that the PDF of the work has to be gaussian in their experimental conditions. Let us consider a Brownian particle in the overdamped regime, driven by a time-dependent potential U (x, X(t)), where X is an externally controlled parameter, that varies according to a fixed protocol X(t). The Langevin equation is given bywhere f (t)f (t ′ ) = (2Γ/β)δ(t − t ′ ) and the prime denotes derivative with respect to x. We have defined β = (k B T ) −1 , ...

We study both experimentally and theoretically the statistical properties of the energy exchanged between two electrical conductors, kept at different temperature by two different heat reservoirs, and coupled by the electric thermal noise. Such a system is ruled by the same equations as two Brownian particles kept at different temperatures and coupled by an elastic force. We measure the heat flowing between the two reservoirs, the thermodynamic work done by one part of the system on the other, and we show that these quantities exhibit a long time fluctuation theorem. Furthermore, we evaluate the fluctuating entropy, which satisfies a conservation law. These experimental results are fully justified by the theoretically analysis. Our results give more insight into the energy transfer in the famous Feymann ratchet widely studied theoretically but never in an experiment. 05.70.Ln

The mechanical unfolding of proteins is studied by extending the Wako-Saitô-Muñoz-Eaton model. This model is generalized by including an external force, and its thermodynamics turns out to be exactly solvable. We consider two molecules, the 27th immunoglobulin domain of titin and protein PIN1. We determine equilibrium force-extension curves for the titin and study the mechanical unfolding of this molecule, finding good agreement with experiments. By using an extended form of the Jarzynski equality, we compute the free energy landscape of the PIN1 as a function of the molecule length.