[Théorème de fluctuations quantiques : peut-on aller du micro au méso ?]
Des extensions quantiques du théorème de fluctuations (TF) de Gallavotti et Cohen pour la production d'entropie ont été discutées par plusieurs auteurs. Il y a une séparation pratique entre les formes microscopiques du TF et les formes mésoscopiques (c'est-à-dire non purement hamiltoniennes) pour les systèmes ouverts. Dans un schéma microscopique, il est facile d'énoncer et de démontrer le TF. Dans un schéma mésoscopique, il est difficile d'identifier les fluctuations de la production d'entropie. (Cette difficulté est absente dans le cas classique.) Nous discutons un modèle mésoscopique particulier : une équation maîtresse de Lindblad pour laquelle nous énonçons le TF et, ce qui est plus important, nous le connectons rigoureusement avec le TF microscopique. Nous remarquons aussi que le TF est satisfait par la formule de Lesovik–Levitov pour la statistique du transport de charges.
Quantum extensions of the Gallavotti–Cohen fluctuation theorem (FT) for the entropy production have been discussed by several authors. There is a practical gap between microscopic forms of FT and mesoscopic (i.e. not purely Hamiltonian) forms for open systems. In a microscopic setup, it is easy to state and to prove FT. In a mesoscopic setup, it is difficult to identify fluctuations of the entropy production. (This difficulty is absent in the classical case.) We discuss a particular mesoscopic model: a Lindblad master equation, in which we state FT and, more importantly, connect it rigorously with the underlying microscopic FT. We also remark that FT is satisfied by the Lesovik–Levitov formula for statistics of charge transport.
Publié le :
Mot clés : Théorème de fluctuations quantiques, Production d'entropie
@article{CRPHYS_2007__8_5-6_674_0,
author = {Wojciech De Roeck},
title = {Quantum fluctuation theorem: can we go from micro to meso?},
journal = {Comptes Rendus. Physique},
pages = {674--683},
publisher = {Elsevier},
volume = {8},
number = {5-6},
year = {2007},
doi = {10.1016/j.crhy.2007.05.014},
language = {en},
}
Wojciech De Roeck. Quantum fluctuation theorem: can we go from micro to meso?. Comptes Rendus. Physique, Volume 8 (2007) no. 5-6, pp. 674-683. doi : 10.1016/j.crhy.2007.05.014. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2007.05.014/
[1] Probability of second law violations in steady flows, Phys. Rev. Lett., Volume 71 (1993), pp. 2401-2404
[2] Dynamical ensembles in nonequilibrium statistical mechanics, Phys. Rev. Lett., Volume 74 (1995), pp. 2694-2697
[3] Note on two theorems in nonequilibrium statistical mechanics, J. Stat. Phys., Volume 96 (1999), pp. 1343-1349
[4] On the origin and the use of fluctuation relations for the entropy (J. Dalibard; B. Duplantier; V. Rivasseau, eds.), Poincaré Seminar, Birkhäuser, Basel, 2003, pp. 145-191
[5] Time-reversal and entropy, J. Stat. Phys., Volume 111 (2003), pp. 1219-1244
[6] Fluctuation theorem for stochastic dynamics, J. Phys. A, Volume 31 (1998) no. 16, pp. 3719-3729
[7] A Gallavotti–Cohen type symmetry in the large deviations functional of stochastic dynamics, J. Stat. Phys., Volume 95 (1999), pp. 333-365
[8] Path ensemble averages in systems driven far from equilibrium, Phys. Rev. E, Volume 61 (2005), pp. 2361-2366
[9] Classical and quantum fluctuation theorems for heat exchange, Phys. Rev. Lett., Volume 92 (2004), p. 230602
[10] Quantum fluctuation theorem | arXiv
[11] Jarzynski relations for quantum systems and some applications (Technical note) | arXiv
[12] A quantum version of free energy—irreversible work relations, Phys. Rev. E, Volume 69 (2004) no. 2, p. 026115
[13] Fluctuation theorem for counting-statistics in electron transport through quantum junctions | arXiv
[14] Fluctuations of the dissipated heat in a quantum stochastic model, Rev. Math. Phys., Volume 18 (2006), pp. 619-653
[15] Quantum extension of the Jarzynski relation: Analogy with stochastic dephasing, Phys. Rev. Lett., Volume 90 (2003), p. 170604
[16] Phase space approach to theories of quantum dissipation, J. Chem. Phys., Volume 107 (1997), p. 5236
[17] Open quantum Markovian systems and microreversibility, Z. Physik, Volume 258 (1973), pp. 409-422
[18] On the detailed balance condition for non-Hamiltonian systems, Rep. Math. Phys., Volume 10 (1976) no. 2, pp. 249-258
[19] Unravelings of master equations and fluctuations of currents | arXiv
[20] Operator Algebras and Quantum Statistical Mechanics 2, Springer, Berlin, 1996
[21] Introduction to representations of canonical commutation and anticommutation relations (J. Derezinski; H. Siedentop, eds.), Large Coulomb Systems – QED, Springer, 2006
[22] The Green–Kubo formula and the Onsager reciprocity relations in quantum statistical mechanics, Comm. Math. Phys., Volume 265 (2006), p. 721
[23] Fluctuations of work from quantum subensembles: The case against quantum work-fluctuation theorems, Phys. Rev. E, Volume 71 (2005), p. 066102
[24] Fluctuation theorems Work is not an observable | arXiv
[25] Quantum correction of fluctuation theorem | arXiv
[26] Fluctuation theorem, nonequilibrium steady states and Maclennan–Zubarev ensembles of -asymptotic abelian dynamical systems, Quantum Prob. White Noise Anal., Volume 17 (2003), pp. 100-119
[27] Full counting statistics: An elementary derivation of Levitov's formula (Yu.V. Nazarov; Ya.M. Blanter, eds.), Quantum Noise, Kluwer, 2003
[28] Electron counting statistics and coherent states of electric current, J. Math. Phys., Volume 37 (1996), pp. 4845-4866
[29] Fredholm determinants and the statistics of charge transport | arXiv
[30] Invitation to quantum dynamical semigroups (P. Garbaczewski; R. Olkiewicz, eds.), Dynamics of Dissipation, Lecture Notes in Physics, Springer, 2002
[31] The Theory of Open Quantum Systems, Oxford Univ. Press, USA, 2002
[32] Markovian master equations, Comm. Math. Phys., Volume 39 (1974), pp. 91-110
[33] An Open Systems Approach to Quantum Optics, Springer, Berlin, 1993
[34] Self-consistency of thermal jump trajectories (M.O. Scully; H.J. Carmichael; R.J. Glauber, eds.), Directions in Quantum Optics, Lecture Notes in Physics, vol. 561, Springer, Berlin, 1999, p. 333
[35] Large deviations for currents in the spin-boson model | arXiv
[36] Stochastic limit for Pauli–Fierz operators | arXiv
[37] Convergence of multitime correlation functions in the weak and singular coupling limits, J. Math. Phys., Volume 24 (1983) no. 2, pp. 311-315
[38] Weak coupling limit as a quantum functional central limit theorem, Comm. Math. Phys., Volume 131 (1990), pp. 537-570
Cité par Sources :
Commentaires - Politique
Network and thermodynamic conditions for a single macroscopic current fluctuation theorem
David Andrieux; Pierre Gaspard
C. R. Phys (2007)