Brownian Carnot engine

Journal name:
Nature Physics
Volume:
12,
Pages:
67–70
Year published:
DOI:
doi:10.1038/nphys3518
Received
Accepted
Published online

The Carnot cycle imposes a fundamental upper limit to the efficiency of a macroscopic motor operating between two thermal baths1. However, this bound needs to be reinterpreted at microscopic scales, where molecular bio-motors2 and some artificial micro-engines3, 4, 5 operate. As described by stochastic thermodynamics6, 7, energy transfers in microscopic systems are random and thermal fluctuations induce transient decreases of entropy, allowing for possible violations of the Carnot limit8. Here we report an experimental realization of a Carnot engine with a single optically trapped Brownian particle as the working substance. We present an exhaustive study of the energetics of the engine and analyse the fluctuations of the finite-time efficiency, showing that the Carnot bound can be surpassed for a small number of non-equilibrium cycles. As its macroscopic counterpart, the energetics of our Carnot device exhibits basic properties that one would expect to observe in any microscopic energy transducer operating with baths at different temperatures9, 10, 11. Our results characterize the sources of irreversibility in the engine and the statistical properties of the efficiency—an insight that could inspire new strategies in the design of efficient nano-motors.

At a glance

Figures

  1. The Brownian Carnot engine.
    Figure 1: The Brownian Carnot engine.

    a, Time evolution of the experimental protocol. bd, Thermodynamic diagrams of the engine: (1) isothermal compression (blue); (2) adiabatic compression (magenta); (3) isothermal expansion (red); (4) adiabatic expansion (green). Solid lines are the analytical values in the quasistatic limit. Filled symbols are obtained from ensemble averages over cycles of duration τ = 200ms; open symbols are obtained for τ = 30ms. The black arrow indicates the direction of the operation of the engine. b, Tpartκ diagram. c, Clapeyron diagram. The area within the cycle is equal to the mean work obtained during the cycle. d, TpartS diagram. The entropy changes only in the isothermal steps.

  2. Energetics of the Brownian Carnot engine.
    Figure 2: Energetics of the Brownian Carnot engine.

    a, Ensemble averages of stochastic work (left fenceWτright fence, blue stars) and heat (left fenceQτright fence, red pluses) transferred in one cycle as a function of the cycle duration. Green crosses are the average total energy change of the working substance left fenceΔHτright fence. Thin lines are fits to A + B/τ. b, Power output Pτ = −left fenceWτright fence/τ (black diamonds, left axis) and long-term efficiency ητ (yellow hexagons, right axis) as a function of the inverse of the cycle time. The black curve is a fit Pτ = (left fenceWright fence + Σss/τ)/τ, yielding left fenceWright fence = (−0.38 ± 0.01)kTc and Σss = (5.7 ± 0.3)kTcms with a reduced chi-square of χred2 = 1.08. The solid yellow line is a fit to ητ = (ηC + τW/τ)/(1 + τQ/τ), which yields η = (0.92 ± 0.06)ηC,τW = (−11 ± 2)ms,τQ = (−0.6 ± 6.0)ms with χred2 = 0.76. Yellow dash–dot line is the Curzon–Ahlborn efficiency , which is in excellent agreement with the location of the maximum power (vertical black dashed line). Ensemble averages are done over 50s and error bars are obtained with a statistical significance of 90%.

  3. Efficiency fluctuations at maximum power.
    Figure 3: Efficiency fluctuations at maximum power.

    Contour plot of the PDF of the efficiency ρτ=40ms, i(η) computed summing over i = 1 to400 cycles (left axis). The long-term efficiency (averaged over τexp = 50s) is shown with a vertical blue dashed line. Super Carnot efficiencies appear even far from quasistatic driving. Inset: tails of the distribution for ρτ=40ms, 10(η) (blue pluses, positive tail; red stars, negative tail). The green line is a fit to a power law to all the data shown, whose exponent is γ = (−1.9 ± 0.3).

References

  1. Carnot, S. Annales scientifiques de lÉcole Normale Supérieure Vol. 1, 393457 (Société mathématique de France, 1872).
  2. Howard, J. Mechanics of Motor Proteins and The Cytoskeleton (Sinauer Associates Sunderland, 2001).
  3. Blickle, V. & Bechinger, C. Realization of a micrometre-sized stochastic heat engine. Nature Phys. 8, 143146 (2012).
  4. Roldán, E., Martínez, I. A., Parrondo, J. M. R. & Petrov, D. Universal features in the energetics of symmetry breaking. Nature Phys. 10, 457461 (2014).
  5. Koski, J. V., Maisi, V. F., Pekola, J. P. & Averin, D. V. Experimental realization of a szilard engine with a single electron. Proc. Natl Acad. Sci. USA 111, 1378613789 (2014).
  6. Sekimoto, K. Lecture Notes in Physics Vol. 799 (Springer Verlag, 2010).
  7. Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75, 126001 (2012).
  8. Wang, G. M., Sevick, E. M., Mittag, E., Searles, D. J. & Evans, D. J. Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. Phys. Rev. Lett. 89, 050601 (2002).
  9. Polettini, M., Verley, G. & Esposito, M. Efficiency statistics at all times: Carnot limit at finite power. Phys. Rev. Lett. 114, 050601 (2015).
  10. Verley, G., Willaert, T., Van den Broeck, C. & Esposito, M. Universal theory of efficiency fluctuations. Phys. Rev. E 90, 052145 (2014).
  11. Gingrich, T. R., Rotskoff, G. M., Vaikuntanathan, S. & Geissler, P. L. Efficiency and large deviations in time-asymmetric stochastic heat engines. New J. Phys. 16, 102003 (2014).
  12. Toyabe, S., Sagawa, T., Ueda, M., Muneyuki, E. & Sano, M. Experimental demonstration of information-to-energy conversion and validation of the generalized jarzynski equality. Nature Phys. 6, 988992 (2010).
  13. Roldán, E., Martínez, I. A., Dinis, L. & Rica, R. A. Measuring kinetic energy changes in the mesoscale with low acquisition rates. Appl. Phys. Lett. 104, 234103 (2014).
  14. Martínez, I. A., Roldán, É, Dinis, L., Petrov, D. & Rica, R. A. Adiabatic processes realized with a trapped Brownian particle. Phys. Rev. Lett. 114, 120601 (2015).
  15. Crooks, G. E. & Jarzynski, C. Work distribution for the adiabatic compression of a dilute and interacting classical gas. Phys. Rev. E 75, 021116 (2007).
  16. Martínez, I. A., Roldán, É, Parrondo, J. M. R. & Petrov, D. Effective heating to several thousand kelvins of an optically trapped sphere in a liquid. Phys. Rev. E 87, 032159 (2013).
  17. Mestres, P., Martinez, I. A., Ortiz-Ambriz, A., Rica, R. A. & Roldan, E. Realization of nonequilibrium thermodynamic processes using external colored noise. Phys. Rev. E 90, 032116 (2014).
  18. Bérut, A., Petrosyan, A. & Ciliberto, S. Energy flow between two hydrodynamically coupled particles kept at different effective temperatures. Europhys. Lett. 107, 60004 (2014).
  19. Gieseler, J., Deutsch, B., Quidant, R. & Novotny, L. Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. Phys. Rev. Lett. 109, 103603 (2012).
  20. Feynman, R., Leighton, R. & Sands, M. The Feynman Lectures on Physics 2nd edn, Vol. 1 (Addison-Wesley, 1963).
  21. Curzon, F. & Ahlborn, B. Efficiency of a Carnot engine at maximum power output. Am. J. Phys. 43, 2224 (1975).
  22. Ouerdane, H., Apertet, Y., Goupil, C. & Lecoeur, P. Continuity and boundary conditions in thermodynamics: From Carnots efficiency to efficiencies at maximum power. Eur. Phys. J. Spec. Top. 224, 839864 (2015).
  23. Sekimoto, K. & Sasa, S.-i. Complementarity relation for irreversible process derived from stochastic energetics. J. Phys. Soc. Jpn 66, 33263328 (1997).
  24. Bonanca, M. V. S. & Deffner, S. Optimal driving of isothermal processes close to equilibrium. J. Chem. Phys. 140, 244119 (2014).
  25. Esposito, M., Kawai, R., Lindenberg, K. & Van den Broeck, C. Efficiency at maximum power of low-dissipation Carnot engines. Phys. Rev. Lett. 105, 150603 (2010).
  26. Verley, G., Esposito, M., Willaert, T. & den Broeck, C. V. The unlikely Carnot efficiency. Nature Commun. 5, 5721 (2014).
  27. Proesmans, K., Cleuren, B. & Van den Broeck, C. Stochastic efficiency for effusion as a thermal engine. Europhys. Lett. 109, 20004 (2015).
  28. Sarikaya, M., Tamerler, C., Jen, A. K.-Y., Schulten, K. & Baneyx, F. Molecular biomimetics: Nanotechnology through biology. Nature Mater. 2, 577585 (2003).
  29. Douglas, S. M., Bachelet, I. & Church, G. M. A logic-gated nanorobot for targeted transport of molecular payloads. Science 335, 831834 (2012).
  30. Roßnagel, J., Abah, O., Schmidt-Kaler, F., Singer, K. & Lutz, E. Nanoscale heat engine beyond the Carnot limit. Phys. Rev. Lett. 112, 030602 (2014).
  31. Gieseler, J., Quidant, R., Dellago, C. & Novotny, L. Dynamic relaxation of a levitated nanoparticle from a non-equilibrium steady state. Nature Nanotech. 9, 358364 (2014).
  32. Millen, J., Deesuwan, T., Barker, P. & Anders, J. Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere. Nature Nanotech. 9, 425429 (2014).
  33. Sekimoto, K., Takagi, F. & Hondou, T. Carnots cycle for small systems: Irreversibility and cost of operations. Phys. Rev. E 62, 77597768 (2000).

Download references

Author information

  1. These authors contributed equally to this work.

    • I. A. Martínez &
    • É. Roldán

Affiliations

  1. ICFO-Institut de Ciències Fotòniques, Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

    • I. A. Martínez,
    • É. Roldán,
    • D. Petrov &
    • R. A. Rica
  2. Laboratoire de Physique, École Normale Supérieure, CNRS UMR5672 46 Allée dItalie, 69364 Lyon, France

    • I. A. Martínez
  3. Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38 01187 Dresden, Germany

    • É. Roldán
  4. GISC-Grupo Interdisciplinar de Sistemas Complejos, 28040 Madrid, Spain

    • É. Roldán,
    • L. Dinis &
    • J. M. R. Parrondo
  5. Departamento de Fisica Atómica, Molecular y Nuclear, Universidad Complutense Madrid, 28040 Madrid, Spain

    • L. Dinis &
    • J. M. R. Parrondo

Contributions

I.A.M. designed the experiment, obtained all experimental data and analysed experimental data. É.R. designed the experiment, analysed experimental data and supported theoretical aspects. L.D. supported theoretical aspects. D.P. proposed and established the project, and supervised the experiment. J.M.R.P. proposed and established the project and developed its theoretical aspects. R.A.R. supported and supervised the experiment. All authors discussed the results and wrote the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary information (423,544 KB)

    Supplementary information

Additional data