Universal energy fluctuations in thermally isolated driven systems

Journal name:
Nature Physics
Volume:
7,
Pages:
913–917
Year published:
DOI:
doi:10.1038/nphys2057
Received
Accepted
Published online

Abstract

When an isolated system is brought in contact with a heat bath, its final energy is random and follows the Gibbs distribution—this finding is a cornerstone of statistical physics. The system’s energy can also be changed by performing non-adiabatic work using a cyclic process. Almost nothing is known about the resulting energy distribution in this set-up, which is in particular relevant to recent experimental progress in cold atoms, ion traps, superconducting qubits and other systems. Here we show that when the non-adiabatic process consists of many repeated cyclic processes, the resulting energy distribution is universal and different from the Gibbs ensemble. We predict the existence of two qualitatively different regimes with a continuous second-order-like transition between them. We illustrate our approach by performing explicit calculations for both interacting and non-interacting systems.

At a glance

Figures

  1. Two methods for changing the energy of a system.
    Figure 1: Two methods for changing the energy of a system.

    Schematic comparison between the usual thermal heating (traditional oven, top) and an energy increase due to non-adiabatic work (microwave oven, bottom). In the right column for each case we present a schematic picture of the resulting energy distribution.

  2. A particle in a driven chaotic cavity.
    Figure 2: A particle in a driven chaotic cavity.

    A single particle is bouncing in a deforming cavity of constant volume. The driving protocol consists in repeatedly deforming the cavity between the two shapes shown.

References

  1. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885964 (2008).
  2. Blatt, R. & Wineland, D. J. Entangled states of trapped atomic ions. Nature 453, 10081015 (2008).
  3. Petta, J. R. et al. Dynamic nuclear polarization with single electron spins. Phys. Rev. Lett. 100, 067601 (2008).
  4. Majer, J. et al. Coupling superconducting qubits via a cavity bus. Nature 449, 443447 (2007).
  5. Polkovnikov, A., Sengupta, K., Silva, A. & Vengalattore, M. Rev. Mod. Phys. (in the press); preprint at http://arxiv.org/abs/1007.5331.
  6. Dziarmaga, J. Dynamics of a quantum phase transition and relaxation to a steady state. Adv. Phys. 59, 10631189 (2010).
  7. Dalla Torre, E. G., Demler, E., Giamarchi, T. & Altman, E. Quantum critical states and phase transitions in the presence of non-equilibrium noise. Nature Phys. 6, 806810 (2010).
  8. Reif, F. Fundamentals of Statistical and Thermal Physics (Waveland Pr., 2008).
  9. Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 26902693 (1997).
  10. Crooks, G. E. Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems. J. Stat. Phys. 90, 14811487 (1998).
  11. Campisi, M., Hanggi, P. & Talkner, P. Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys. 83, 771792 (2011).
  12. Merhav, N. & Kafri, Y. Statistical properties of entropy production derived from fluctuation theorems. J. Stat. Mech.: Theor. Exp.P12022 (2010).
  13. Ashcroft, N. W. & Mermin, N. D. Solid State Physics (Brooks Cole, 1976).
  14. Jarzynski, C. Diffusion equation for energy in ergodic adiabatic ensembles. Phys. Rev. A 46, 74987509 (1992).
  15. Ott, E. Goodness of Ergodic adiabatic invariants. Phys. Rev. Lett. 42, 16281631 (1979).
  16. Cohen, D. Chaos and energy spreading for time-dependent Hamiltonians, and the various regimes in the theory of quantum dissipation. Ann. Phys. (NY) 283, 175231 (2000).
  17. Silva, A. Statistics of the work done on a quantum critical system by quenching a control parameter. Phys. Rev. Lett. 101, 120603 (2008).
  18. Santos, L., Polkovnokov, A. & Rigol, M. Entropy of isolated quantum systems after a quench. Phys. Rev. Lett. 107, 040601 (2011).
  19. Jarzynski, C. & Świa¸tecki, W. J. A universal asymptotic velocity distribution for independent particles in a time-dependent irregular container. Nucl. Phys. A 552, 19 (1993).
  20. Jarzynski, C. Energy diffusion in a chaotic adiabatic billiard gas. Phys. Rev. E 48, 43404350 (1993).
  21. Blocki, J., Brut, F. & Swiatecki, W. J. A numerical verification of the prediction of an exponential velocity spectrum for a gas of particles in a time-dependent potential well. Nucl. Phys. A 554, 107117 (1993).
  22. Blocki, J., Skalski, J. & Swiatecki, W. J. The excitation of an independent-particle gasclassical or quantalby a time-dependent potential well. Nucl. Phys. A 594, 137155 (1995).
  23. D’Alessio, L. & Krapivsky, P. L. Light impurity in an equilibrium gas. Phys. Rev. E 83, 011107 (2011).
  24. Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854858 (2008).
  25. Polkovnikov, A. Microscopic expression for heat in the adiabatic basis. Phys. Rev. Lett. 101, 220402 (2008).

Download references

Author information

Affiliations

  1. Department of Physics, Technion, Haifa 32000, Israel

    • Guy Bunin &
    • Yariv Kafri
  2. Department of Physics, Boston University, Boston, Massachusetts 02215, USA

    • Luca D’Alessio &
    • Anatoli Polkovnikov

Contributions

G.B, L.D, Y.K and A.P contributed equally to this project.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (410k)

    Supplementary Information

Additional data