Universal features in the energetics of symmetry breaking

Journal name:
Nature Physics
Volume:
10,
Pages:
457–461
Year published:
DOI:
doi:10.1038/nphys2940
Received
Accepted
Published online

Abstract

A breaking of symmetry involves an abrupt change in the set of microstates a system can explore. This change has unavoidable thermodynamic implications: a shrinkage of the microstate set results in an entropy decrease, which eventually needs to be compensated by heat dissipation and hence requires work. On the other hand, in a spontaneous symmetry breaking, the available phase-space volume changes without the need for work, yielding an apparent entropy decrease. Here we show that this entropy decrease is a key ingredient of a Szilard engine and Landauer’s principle, and perform a direct measurement of the entropy change along symmetry-breaking transitions for a Brownian particle subject to a bistable potential realized through two optical traps. The experiment confirms theoretical results based on fluctuation theorems, enables the construction of a Szilard engine extracting energy from a single thermal bath, and shows that a signature of a symmetry breaking in a system’s energetics is observable.

At a glance

Figures

  1. Experimental protocol of symmetry breaking and symmetry restoration.
    Figure 1: Experimental protocol of symmetry breaking and symmetry restoration.

    a, Positions of the fixed (F) trap (blue dashed line) and moving (M) trap (red dashed line) as functions of time during the protocol. Ensemble average position of the trapped bead after implementing the protocol cyclically for t = 2,400s over F trajectories (blue solid line) and M trajectories (red solid line). b, Spatial–temporal mapping of the potential U(x, t) obtained from the statistics of trajectories of the bead for t = 2,400s in the presence of an external force such that pF = 0.8. The colour bar on the right indicates the depth of the potential energy (in units of kT). A single trajectory of the bead when it chooses the M trap is also plotted (white line).

  2. Energetics of symmetry breaking and symmetry restoration.
    Figure 2: Energetics of symmetry breaking and symmetry restoration.

    a, Ensemble average conformational entropy production in the symmetry breaking (SB), left fenceSprodright fencei(SB) (in units of k) as a function of the probability pi of adopting instance i = fixed (F), moving (M). b, Ensemble average conformational entropy production in the symmetry restoration (SR), left fenceSprodright fencei(SR) (in units of k) as a function of . Results shown as open symbols were obtained using the fast protocol (τ2 = 2s), and results shown as filled symbols were obtained using the slow protocol (τ2 = 5.5s). Blue squares represent the ensemble averages over F trajectories, and red circles represent the averages over M trajectories. Error bars were obtained using a statistical significance of 90%.

  3. Experimental realization of the Szilard engine.
    Figure 3: Experimental realization of the Szilard engine.

    The average heat (solid lines, in units of kT), the Shannon entropy of the trajectory distribution (dashed lines, in units of k) and the average entropy production (dotted lines, in units of k) as functions of time. The upper plot (blue curves) corresponds to averages over trajectories that end in the fixed (F) trap, the middle plot (red curves) to averages over trajectories that end in the moving (M) trap, and the lower plot shows averages over all trajectories (green curves). The feedback protocol is indicated by the arrows. The vertical black line separates the symmetry breaking and the symmetry restoration processes. The symmetry breaking is created with an external voltage Vext = V0 that induces probabilities pF = 0.35, pM = 0.65. When the particle chooses the fixed trap (blue) the symmetry is restored changing the voltage to Vext = VF, and so biasing the potential towards the fixed trap ( ). When the particle ends in the moving trap (red), the symmetry is restored at a voltage Vext = VM, biasing the potential towards the moving trap ( ). We also indicate in the bottom figure the value of the relative entropy (horizontal black line, lower plot).

References

  1. Leff, H. S. & Rex, A. F. Maxwell’s Demon. Entropy, Information, Computing (Adam Hilger, 1990).
  2. Parrondo, J. M. R. The Szilard engine revisited: Entropy, macroscopic randomness, and symmetry breaking phase transitions. Chaos 11, 725733 (2001).
  3. Marathe, R. & Parrondo, J. M. R. Cooling classical particles with a microcanonical Szilard engine. Phys. Rev. Lett. 104, 245704 (2010).
  4. 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).
  5. Bérut, A. et al. Experimental verification of Landauer’s principle linking information and thermodynamics. Nature 483, 187189 (2012).
  6. Maragakis, P., Spichty, M. & Karplus, M. A differential fluctuation theorem. J. Phys. Chem. B 112, 61686174 (2008).
  7. Junier, I., Mossa, A., Manosas, M. & Ritort, F. Recovery of free energy branches in single molecule experiments. Phys. Rev. Lett. 102, 070602 (2009).
  8. Alemany, A., Mossa, A., Junier, I. & Ritort, F. Experimental free-energy measurements of kinetic molecular states using fluctuation theorems. Nature Phys. 8, 688694 (2012).
  9. Horowitz, J. M. & Parrondo, J. M. R. Optimizing non-ergodic feedback engines. Acta. Phys. Pol. B 44, 803814 (2013).
  10. Sagawa, T. & Ueda, M. Minimal energy cost for thermodynamic information processing: Measurement and information erasure. Phys. Rev. Lett. 102, 250602 (2009).
  11. Horowitz, J. M., Sagawa, T. & Parrondo, J. M. R. Imitating chemical motors with optimal information motors. Phys. Rev. Lett. 111, 010602 (2013).
  12. Esposito, M. & Van den Broeck, C. Second law and Landauer principle far from equilibrium. Europhys. Lett. 95, 40004 (2011).
  13. Horowitz, J. M. & Vaikuntanathan, S. Nonequilibrium detailed fluctuation theorem for repeated discrete feedback. Phys. Rev. E 82, 061120 (2010).
  14. Horowitz, J. M. & Parrondo, J. M. R. Designing optimal discrete-feedback thermodynamic engines. New J. Phys. 13, 123019 (2011).
  15. Kawai, R., Parrondo, J. M. R. & den Broeck, C. V. Dissipation: The phase-space perspective. Phys. Rev. Lett. 98, 080602 (2007).
  16. Dunkel, J. & Hilbert, S. Phase transitions in small systems: Microcanonical vs. canonical ensembles. Physica A 370, 390406 (2006).
  17. Vaikuntanathan, S. & Jarzynski, C. Modeling Maxwell’s demon with a microcanonical Szilard engine. Phys. Rev. E 83, 061120 (2011).
  18. Martinez, I. A., Roldán, E., Parrondo, J. M. R. & Petrov, D. Effective heating to several thousand kelvin of an optically trapped sphere in a liquid. Phys. Rev. E 87, 032159 (2013).
  19. Hänggi, P., Talkner, P. & Borkovec, M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 62, 251342 (1990).
  20. Sekimoto, K. Langevin equation and thermodynamics. Prog. Theor. Phys. Suppl. 130, 1727 (1998).
  21. Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75, 126001 (2012).
  22. Mandal, D. & Jarzynski, C. Work and information processing in a solvable model of Maxwell’s demon. Proc. Natl Acad. Sci. USA 109, 1164111645 (2012).
  23. Blickle, V. & Bechinger, C. Realization of a micrometre-sized stochastic heat engine. Nature Phys. 8, 143146 (2011).

Download references

Author information

  1. These authors contributed equally to this work.

    • É. Roldán &
    • I. A. Martínez

Affiliations

  1. Departamento de Física Atómica, Molecular y Nuclear and GISC, Universidad Complutense de Madrid, 28040 Madrid, Spain

    • É. Roldán &
    • J. M. R. Parrondo
  2. ICFO—Institut de Ciències Fotòniques, Mediterranean Technology Park, Av. Carl Friedrich Gauss, 3 08860 Castelldefels (Barcelona), Spain

    • I. A. Martínez &
    • D. Petrov
  3. ICREA—Institució Catalana de Recerca i Estudis Avançats, Passeig Lluís Companys, 23, 08010 Barcelona, Spain

    • D. Petrov
  4. Deceased.

    • D. Petrov

Contributions

E.R. analysed experimental data, supported theoretical aspects and the design of the experiment, and performed computer simulations. I.A.M. designed the experiment, and obtained all experimental data. J.M.R.P. proposed and established the project, and developed its theoretical aspects. D.P. designed and supervised the experiment. All authors wrote the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (746KB)

    Supplementary Information

Additional data