Realization of a micrometre-sized stochastic heat engine

Abstract

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 ), ...

Stochastic thermodynamics under coarse graining

“…They coincided with an unprecedented development in the experimental techniques used to manipulate small systems and triggered a great deal of experimental studies in a variety of contexts, such as single molecule stretching experiments [35,[38][39][40], nanomechanical oscillator work measurements [41], spectroscopic measurement of trajectory entropies [42,43], and electronic current fluctuations in full counting statistics experiments [44]. Since stochastic thermodynamics combines kinetics and thermodynamics, it has also proved extremely useful to describe the finite-time thermodynamics (e.g., efficiency at finite power) of various nanodevices operating as thermodynamic machines [45][46][47][48][49][50][51][52][53][54]. Overall, this theory is becoming a fundamental tool for the study of nanosciences.…”

Geometric Heat Engines Featuring Power that Grows with Efficiency

“…This model consists of an overdamped Brownian particle confined to one spatial dimension, in a time dependent harmonic potential V (x, t) = Λ(t) 2 x 2 , coupled to a heat bath with a time dependent inverse temperature β(t). This model was suggested in [16] and experimentally realized in [22]. Both Λ(t) and β(t) are periodic with a cycle time τ 1 .…”

“…On the other hand, driving the engine faster increases the power at the expense of efficiency, until eventually, at fast enough driving, the dissipation rate becomes significant and causes a decrease in power. Yet, as we demonstrate, this behavior is not universal, and there is no inherent trade-off between power and efficiency as a function of the cycle time, although such a trade-off always exists in cycles that exactly attain the Carnot bound.Here we analyze a class of cyclic heat engine models, referred to as geometric heat engines, which includes the paradigmatic examples of a Brownian particle in a parabolic potential [12,13,16,22] and the two-state Markovian engines [23], but is not limited to these models. In this class, the work and heat can be interpreted as areas in state space (the space of all the possible states of the engine) defined by the periodic trajectory of the engine's state.…”

Anomalous Brownian refrigerator

“…This is the reverse of the protocol that has been studied in [38] in context of single particle stochastic heat engine. There are other important protocols, for example Stirling-type protocol used in earlier studies [19], where the adiabatic steps are absent. The comparative study between stochastic thermodynamics of micro machines driven by different protocols will be focused elsewhere.…”