The conversion of energy into mechanical work is essential for almost any industrial process. The original description of classical heat engines by Sadi Carnot in 1824 has largely shaped our understanding of work and heat exchange during macroscopic thermodynamic processes 1 . Equipped with our present-day ability to design and control mechanical devices at micro-and nanometre length scales, we are now in a position to explore the limitations of classical thermodynamics, arising on scales for which thermal fluctuations are important 2-5 . Here we demonstrate the experimental realization of a microscopic heat engine, comprising a single colloidal particle subject to a time-dependent optical laser trap. The work associated with the system is a fluctuating quantity, and depends strongly on the cycle duration time, τ, which in turn determines the efficiency of our heat engine. Our experiments offer a rare insight into the conversion of thermal to mechanical energy on a microscopic level, and pave the way for the design of future micromechanical machines.Macroscopic heat engines operating periodically between two heat baths are described well by the laws of thermodynamics, owing to the large number of internal degrees of freedom, which render fluctuations negligible. In contrast, fluctuations become visible when the typical energy scales of engines are reduced by more than twenty orders of magnitude, down to values around k B T . This regime can be achieved when the typical system dimensions are scaled down from metres to micrometres 2 . Such conditions are typically met for biomolecules 6,7 and other microelectromechanical systems (MEMS; refs 8-10) that can perform translational or rotational motion as a result of chemical reactions and electrical fields. Also, several types of Brownian motor have been discussed that are able to extract useful work by rectification of thermal noise [11][12][13] . Despite its conceptual simplicity, no attempt has been made to realize a microscopic heat engine that extracts energy by cyclically working between two heat baths in a regime where fluctuations dominate.Here we present an experimental realization of a Stirling engine, where a single Brownian particle is subjected to a time-dependent optical trap and periodically coupled to different heat baths 14 . The particle and the trapping potential replace the working gas and the piston of its macroscopic counterpart. As for conventional heat engines where the periodic coupling of the working gas to a cold and a hot heat bath changes volume and pressure inside a piston, the fluctuations of a colloidal particle subjected to an external optical potential vary on changing the temperature of the solvent.As a Brownian particle we used a single melamine bead of diameter 2.94 μm suspended in water and confined in a vitreous sample cell 4 μm in height. Using a highly focused infrared laser beam we exerted a parabolic trapping potential U (R,k) = (1/2)k(t )R 2 on the particle, where R is its radial distance from the trapping centre and k(t ), ...

We study the motion of an overdamped colloidal particle in a time-dependent nonharmonic potential. We demonstrate the first lawlike balance between applied work, exchanged heat, and internal energy on the level of a single trajectory. The observed distribution of applied work is distinctly non-Gaussian in good agreement with numerical calculations. Both the Jarzynski relation and a detailed fluctuation theorem are verified with good accuracy. DOI: 10.1103/PhysRevLett.96.070603 PACS numbers: 05.40.ÿa, 05.70.ÿa, 82.70.Dd Since more than a century, the first law relating the work applied to a system with both the exchanged heat and an increase in internal energy is one of the cornerstones of macroscopic physics. Its consistent formulation for a mesoscopic system like a driven colloidal particle, however, was suggested only about a decade ago [1]. Since on these scales thermal fluctuations are relevant, probability distributions for work, heat, and internal energy replace the sharp values of their macroscopic counterparts. Various theoretical relations like the fluctuation theorem [2,3], the Jarzynski relation [4], and the Hatano-Sasa relation [5] involving these distributions in different settings extend the second law to the mesoscopic realm at least as long as the notion of a constant temperature of the ambient heat bath remains meaningful [for a review, see [6]]. Such theorems have been tested experimentally using both biomolecules manipulated mechanically [7,8] as well as colloidal particles in time dependent laser traps [9][10][11]. Common to all colloidal experiments, so far, is that these laser traps generate a harmonic potential albeit with a timedependent center or ''spring constant.'' Consequently, often the interesting distributions are Gaussian even though for certain quantities non-Gaussian distributions can occur [10,12].In this Letter, we study the thermodynamics of single colloidal trajectories in a time-dependent nonharmonic potential which, generically, gives rise to non-Gaussian distributions. Only for very short or very long trajectories, one expects Gaussian distributions even in this nonharmonic case [13]. In particular, we identify applied work, exchanged heat, and change in internal energy along a single trajectory and thus test the consistency of these notions on this level, or, put differently, illustrate the validity of the first law. We measure the distribution of work in the non-Gaussian regime and compare it to theoretical prediction. Such a comparison does not involve a single fit parameter since all quantities are measured independently, which is another advantage of colloidal systems. Finally, we test the Jarzynski relation which expresses the free energy difference between two equilibrium states in terms of the nonequilibrium work spent in the transition between the two states. Such an illustration of the Jarzynski relation in the non-Gaussian regime comes timely given ongoing theoretical criticism of its validity [14,15]. In our study, particle trajectories were determined usi...

The Einstein relation connecting the diffusion constant and the mobility is violated beyond the linear response regime. For a colloidal particle driven along a periodic potential imposed by laser traps, we test the recent theoretical generalization of the Einstein relation to the nonequilibrium regime which involves an integral over measurable velocity correlation functions. DOI: 10.1103/PhysRevLett.98.210601 PACS numbers: 05.40.ÿa, 82.70.Dd A comprehensive theory of systems driven out of equilibrium is still lacking quite in contrast to the universal description of equilibrium systems by the GibbsBoltzmann distribution. Linear response theory provides exact relations valid, however, only for small deviations from equilibrium [1]. The arguably most famous linear response relation is the Einstein relationinvolving the diffusion constant D, the mobility , and the thermal energy k B T [2]. In his original derivation for a suspension in a force field, Einstein balances the diffusive current with a linear drift. The Einstein relation embodies a deep connection between fluctuations causing diffusion and dissipation responsible for friction expressed by a finite mobility.In the present Letter, we report on the extension of the classical Einstein relation beyond the linear response regime using a driven colloidal particle as a paradigmatic system. Our previous theoretical work [3] and its present experimental test thus introduce a third type of exact relation valid for and relevant to small driven systems coupled to a heat bath of constant temperature T. The previously discovered exact relations comprise, first, the fluctuation theorem [4,5] which quantifies the steady state probability of observing trajectories of negative entropy production. Second, the Jarzynski relation [6] expresses the free energy difference between different equilibrium states by a nonlinear average of the work spent in driving such a transition [7]. Both the fluctuation theorem and the Jarzynski relation as well as their theoretical extensions [8][9][10] have been tested in various experimental systems such as micromechanically manipulated biomolecules [11,12], colloids in time-dependent laser traps [13][14][15], Rayleigh-Benard convection [16], mechanical oscillators [17], and optically driven single two-level systems [18]. Such exact relations (and the study of their limitations) are fundamentally important since they provide the first elements of a future more comprehensive theory of nonequilibrium systems.For a nonequilibrium extension of the Einstein relation (1), consider the overdamped motion xt of a particle moving along a periodic one-dimensional potential Vx governed by the Langevin equationwith F ÿ@V=@x f and f a nonconservative force. The friction coefficient determines the correlations htt 0 i 2k B T=t ÿ t 0 of the white noise . Therefore, Eq. (2) describes a colloidal bead driven to nonequilibrium under the assumption that the fluctuating forces arising from the heat bath are not affected by the driving.For the crucial quantiti...

The validity of the fluctuation theorem for entropy production as deduced from the observation of trajectories implicitly requires that all slow degrees of freedom are accessible. We experimentally investigate the role of hidden slow degrees of freedom in a system of two magnetically coupled driven colloidal particles. The apparent entropy production based on the observation of just one particle obeys a fluctuation theorem-like symmetry with a slope of 1 in the short time limit. For longer times, we find a constant slope, but different from 1. We present theoretical arguments for a generic linear behavior both for small and large apparent entropy production but not necessarily throughout. By fine-tuning experimental parameters, such an intermediate nonlinear behavior can indeed be recovered in our system as well.