A Framework for Sequential Measurements and General Jarzynski Equations

Heinz-Jürgen Schmidt and Jochen Gemmer

Abstract

We formulate a statistical model of two sequential measurements and prove a so-called J-equation that leads to various diversifications of the well-known Jarzynski equation including the Crooks dissipation theorem. Moreover, the J-equation entails formulations of the Second Law going back to Wolfgang Pauli. We illustrate this by an analytically solvable example of sequential discrete position–momentum measurements accompanied with the increase of Shannon entropy. The standard form of the J-equation extends the domain of applications of the standard quantum Jarzynski equation in two respects: It includes systems that are initially only in local equilibrium, and it extends this equation to the cases where the local equilibrium is described by microcanononical, canonical, or grand canonical ensembles. Moreover, the case of a periodically driven quantum system in thermal contact with a heat bath is shown to be covered by the theory presented here if the quantum system assumes a quasi-Boltzmann distribution. Finally, we shortly consider the generalised Jarzynski equation in classical statistical mechanics.

  • [1]

    C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).

  • [2]

    J. Kurchan, arXiv:0007360v2 [cond-mat.stat-mech].

  • [3]

    H. Tasaki, arXiv:0000244v2 [cond-mat.stat-mech].

  • [4]

    S. Mukamel, Phys. Rev. Lett. 90, 170604 (2003).

  • [5]

    P. Talkner, M. Morillo, J. Yi, and P. Hänggi, New J. Phys. 15, 095001 (2013).

  • [6]

    T. Schmiedl and U. Seifert, J. Chem. Phys. 126, 044101 (2007).

  • [7]

    K. Saito and Y. Utsumi, Phys. Rev. B 78, 115429 (2008).

  • [8]

    D. Andrieux, P. Gaspard, T. Monnai, and S. Tasaki, New J. Phys. 11, 043014 (2009), Erratum in: New J. Phys. 11, 109802 (2009).

  • [9]

    J. Yi, P. Talkner, and M. Campisi, Phys. Rev. E 84, 011138 (2011).

  • [10]

    M. Esposito, Phys. Rev. E 85, 041125 (2012).

  • [11]

    J. Yi, Y. W. Kim, and P. Talkner, Phys. Rev. E 85, 051107 (2012).

  • [12]

    M. Campisi, P. Hänggi, and P. Talkner, Rev. Mod. Phys. 83, 771 (2011), Erratum in: Rev. Mod. Phys. 83, 1653 (2011).

  • [13]

    P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E 75, 050102 (2007).

  • [14]

    P. Busch, P. Lahti, J.-P. Pellonpä, and K. Ylinen, Quantum Measurement, Springer-Verlag, Berlin 2016.

  • [15]

    A. J. Roncaglia, F. Cerisola, and J. P. Paz, Phys. Rev. Lett. 113, 250601 (2014).

  • [16]

    G. De Chiara, A. J. Roncaglia, F. Cerisola, and J. P. Paz, New J. Phys. 17, 035004 (2015).

  • [17]

    M. Campisi and P. Hänggi, Entropy 13, 2024 (2011).

  • [18]

    W. Pauli, in: Probleme der Moderne Physik, Arnold Sommerfeld zum 60, Geburtstag 1928. Reprinted in Collected Scientific Papers by Wolfgang Pauli, Vol. 1 (Eds. R. Kronig and V. Weisskopf), Interscience, New York 1964, p. 549.

  • [19]

    C. E. Shannon, Bell Syst. Tech. J. 27, 379 (1948).

  • [20]

    R. Serfozo, Basics of Applied Stochastic Processes, Springer-Verlag, Berlin 2009, Corrected 2nd printing 2012.

  • [21]

    V. Vedral, J. Phys. A 45, 272001 (2012).

  • [22]

    G. P. Martins, N. K. Bernandes, and M. F. Santos, Phys. Rev. A 99, 032124 (2019).

  • [23]

    M. Campisi, J. Pekola, and R. Fazio, New J. Phys. 19, 053027 (2017).

  • [24]

    H.-P. Breuer, W. Huber, and F. Petruccione, Phys. Rev. E 61, 4883 (2000).

  • [25]

    M. Langemeyer and M. Holthaus, Phys. Rev. E 89, 012101 (2014).

  • [26]

    O. R. Diermann, H. Frerichs, and M. Holthaus, Phys. Rev. E 100, 012102 (2019).

  • [27]

    H.-J. Schmidt, J. Schnack, and M. Holthaus, Phys. Rev. E 100, 042141 (2019).

  • [28]

    J. Gemmer and R. Steinigeweg, Phys. Rev. E 89, 042113 (2014).

  • [29]

    O. Penrose, Foundations of Statistical Mechanics: A Deductive Treatment, Pergamon Press, Oxford 1970.

  • [30]

    C. Tsallis, J. Stat. Mech. 52, 479 (1988).

  • [31]

    G. E. Crooks, Phys. Rev. E 60, 2721 (1999).

  • [32]

    D. Schmidtke, L. Knipschild, M. Campisi, R. Steinigeweg, and J. Gemmer, Phys. Rev. E 98, 012123 (2018).

  • [33]

    Y. Morikuni and H. Tasaki, J. Stat. Phys. 143, 1 (2011).

  • [34]

    G. Birkhoff, Tres observaciones sobre el algebra lineal, Univ. Nac. Tucumán Rev. Ser. A 5, 147 (1946).

  • [35]

    D. Šafránek, J. M. Deutsch, and A. Aguirre, Phys. Rev. A 99, 012103 (2019).

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