Entropic Fluctuations in XY Chains and Reflectionless Jacobi Matrices

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• 作者：Vojkan Jak?i? ; Benjamin Landon ; Claude-Alain Pillet
• 刊名：Annales Henri Poincare
• 出版年：2013
• 出版时间：November 2013
• 年：2013
• 卷：14
• 期：7
• 页码：1775-1800
• 全文大小：383KB
• 参考文献：1. Araki, H.: Relative entropy of states of von Neumann algebras. Publ. Res. Inst. Math. Sci. Kyoto Univ. g class="a-plus-plus">11g>, 809 (1975/1976)
  2. Araki, H.: Relative entropy of states of von Neumann algebras II. Publ. Res. Inst. Math. Sci. Kyoto Univ. g class="a-plus-plus">13g>, 173 (1977/1978)
  3. Araki H.: On the XY-model on two-sided infinite chain. Publ. Res. Inst. Math. Sci. Kyoto Univ. g class="a-plus-plus">20g>, 277 (1984) g/10.2977/prims/1195181608">CrossRef
  4. Araki H.: Master symmetries of the XY model. Commun. Math. Phys. g class="a-plus-plus">132g>, 155 (1990) g/10.1007/BF02278005">CrossRef
  5. Araki H.: On an inequality of Lieb and Thirring. Lett. Math. Phys. g class="a-plus-plus">19g>, 167 (1990) g/10.1007/BF01045887">CrossRef
  6. Avila, A.: The absolutely continuous spectrum of the almost Mathieu operator (Preprint)
  7. Aschbacher W., Barbaroux J.-M.: Out of equilibrium correlations in the XY chain. Lett. Math. Phys. g class="a-plus-plus">77g>, 11 (2007) g/10.1007/s11005-006-0049-7">CrossRef
  8. Araki H., Barouch E.: On the dynamics and ergodic properties of the XY-model. J. Stat. Phys. g class="a-plus-plus">31g>, 327 (1983) g/10.1007/BF01011585">CrossRef
  9. Araki H., Ho T.G.: Asymptotic time evolution of a partitioned infinite two-sided isotropic XY chain. Proc. Steklov Inst. Math. g class="a-plus-plus">228g>, 191 (2000)
  10. Aschbacher W., Pillet C-A.: Non-equilibrium steady states of the XY chain. J. Stat. Phys. g class="a-plus-plus">112g>, 1153 (2003) g/10.1023/A:1024619726273">CrossRef
  11. Aschbacher, W., Jak?i?, V., Pautrat, Y., Pillet C.-A.: Topics in non-equilibrium quantum statistical mechanics. In: Attal, S., Joye, A., Pillet, C.-A. (eds.) Open Quantum System III. Recent Developments. Lecture Notes in Mathematics, vol. 1882. Springer, Berlin (2006)
  12. Aschbacher W., Jak?i? V., Pautrat Y., Pillet C.-A.: Transport properties of quasi-free fermions. J. Math. Phys. g class="a-plus-plus">48g>, 032101 (2007) g/10.1063/1.2709849">CrossRef
  13. Billingsley P.: Probability and Measure. 2nd edn. Wiley, New York (1986)
  14. Barouch E., McCoy B.M.: Statistical mechanics of the XY model II. Spin-correlation functions. Phys. Rev. A g class="a-plus-plus">3g>, 786 (1971) g/10.1103/PhysRevA.3.786">CrossRef
  15. Baez J.C., Segal I.E., Zhou Z.: Introduction to Algebraic and Constructive Quantum Field Theory. Princeton University Press, Princeton (1991)
  16. Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 1. Springer, Berlin (1987) g/10.1007/978-3-662-02520-8">CrossRef
  17. Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 2. Springer, Berlin (1996)
  18. Bryc W.: A remark on the connection between the large deviation principle and the central limit theorem. Stat. Prob. Lett. g class="a-plus-plus">18g>, 253 (1993) g/10.1016/0167-7152(93)90012-8">CrossRef
  19. Cohen E.G.D., Gallavotti G.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. g class="a-plus-plus">74g>, 2694 (1995) g/10.1103/PhysRevLett.74.3812">CrossRef
  20. Carberry D.M., Williams S.R., Wang G.M., Sevick E.M., Evans D.J.: The Kawasaki identity and the fluctuation theorem. J. Chem. Phys. g class="a-plus-plus">121g>, 8179-182 (2004) g/10.1063/1.1802211">CrossRef
  21. Dereziński J., De Roeck W., Maes C.: Fluctuations of quantum currents and unravelings of master equations. J. Stat. Phys. g class="a-plus-plus">131g>, 341 (2008) g/10.1007/s10955-008-9500-8">CrossRef
  22. Hiai F., Petz D.: The Golden–Thompson trace inequality is complemented. Lin. Alg. Appl. g class="a-plus-plus">181g>, 153 (1993) g/10.1016/0024-3795(93)90029-N">CrossRef
  23. Hume L., Robinson D.W.: Return to equilibrium in the XY model. J. Stat. Phys. g class="a-plus-plus">44g>, 829 (1986) g/10.1007/BF01011909">CrossRef
  24. Evans D.J., Searles D.J.: Equilibrium microstates which generate second law violating steady states. Phys Rev. E g class="a-plus-plus">50g>, 1645 (1994) g/10.1103/PhysRevE.50.1645">CrossRef
  25. Israel, R.: Convexity in the Theory of Lattice Gases. Princeton Series in Physics. Princeton University Press, Princeton (1979)
  26. Jak?i?, V.: Topics in spectral theory. In: Attal, S., Joye, A., Pillet, C.-A. (eds.) Open Quantum Systems I. The Hamiltonian Approach. Lecture Notes in Mathematics, vol. 1880. Springer, Berlin (2006)
  27. Jak?i?, V., Kritchevski, E., Pillet, C.-A.: Mathematical theory of the Wigner–Weisskopf atom. In: Dereziński, J., Siedentop, H. (eds.) Large Coulomb Systems. Lecture Notes in Physics, vol. 695. Springer, Berlin (2006)
  28. Jak?i?, V., Ogata, Y., Pillet, C.-A.: Entropic fluctuations in statistical mechanics II. Quantum dynamical systems (In preparation)
  29. Jak?i? V., Ogata Y., Pautrat Y., Pillet C.-A.: Entropic fluctuations in quantum statistical mechanics–an introduction. In: Fr?hlich, J., Salmhofer, M., Mastropietro, V., De Roeck, W., Cugliandolo, L.F. (eds) Quantum Theory from Small to Large Scales, Oxford University Press, Oxford (2012)
  30. Jak?i? V., Pillet C.-A.: On entropy production in quantum statistical mechanics. Commun. Math. Phys. g class="a-plus-plus">217g>, 285 (2001) g/10.1007/s002200000339">CrossRef
  31. Jak?i? V., Pillet C.-A.: Mathematical theory of non-equilibrium quantum statistical mechanics. J. Stat. Phys. g class="a-plus-plus">108g>, 787 (2002) g/10.1023/A:1019818909696">CrossRef
  32. Jak?i?, V., Pillet, C.-A.: Entropic functionals in quantum statistical mechanics. In: Proceedings of the XVIIth International Congress of Mathematical Physics. Aalborg, Denmark (2012)
  33. Jak?i? V., Pillet C.-A, Rey-Bellet L.: Entropic fluctuations in statistical mechanics I. Classical dynamical systems. Nonlinearity g class="a-plus-plus">24g>, 699 (2011) g/10.1088/0951-7715/24/3/003">CrossRef
  34. Jordan P., Wigner E.: Pauli’s equivalence prohibition. Z. Physik g class="a-plus-plus">47g>, 631 (1928) g/10.1007/BF01331938">CrossRef
  35. Kurchan, J.: A quantum fluctuation theorem. Arxiv preprint cond-mat/0007360 (2000)
  36. Landon, B.: Master’s thesis, McGill University (In preparation)
  37. Levitov L.S., Lesovik G.B.: Charge distribution in quantum shot noise. JETP Lett. g class="a-plus-plus">58g>, 230 (1993)
  38. Lieb E., Schultz T., Mattis D.: Two solvable models of an antiferromagnetic chain. Ann. Phys. g class="a-plus-plus">16g>, 407 (1961) g/10.1016/0003-4916(61)90115-4">CrossRef
  39. Lieb E., Thirring W.: Inequalities for the moments of the eigenvalues of the Schr?dinger Hamiltonian and their relation to Sobolev inequalities. In: Lieb, E., Simon, B., Wightman, A.S. (eds) Studies in Mathematical Physics, Princeton University Press, Princeton (1976)
  40. Matsui T.: On conservation laws of the XY model. Math. Phys. Stud. g class="a-plus-plus">16g>, 197 (1993)
  41. McCoy B.M.: Spin correlation functions of the XY model. Phys. Rev g class="a-plus-plus">173g>, 531 (1968) g/10.1103/PhysRev.173.531">CrossRef
  42. Ogata Y., Matsui T.: Variational principle for non-equilibrium steady states of XX model. Rev. Math. Phys. g class="a-plus-plus">15g>, 905 (2003) g/10.1142/S0129055X03001850">CrossRef
  43. Ohya M., Petz D.: Quantum Entropy and Its Use. Springer, Berlin (2004)
  44. Remling C.: The absolutely continuous spectrum of Jacobi matrices. Ann. Math. g class="a-plus-plus">174g>, 125 (2011) g/10.4007/annals.2011.174.1.4">CrossRef
  45. de Roeck W.: Large deviation generating function for currents in the Pauli–Fierz model. Rev. Math. Phys. g class="a-plus-plus">21g>, 549 (2009) g/10.1142/S0129055X09003694">CrossRef
  46. Reed M., Simon B.: Methods of Modern Mathematical Physics, III. Scattering Theory. Academic Press, London (1978)
  47. Rondoni L., Me?yi ja-Monasterio C.: Fluctuations in non-equlibrium statistical mechanics: models, mathematical theory, physical mechanisms. Nonlinearity g class="a-plus-plus">20g>, 1 (2007) g/10.1088/0951-7715/20/10/R01">CrossRef
  48. Ruelle D.: Statistical Mechanics. Rigorous Result. Benjamin, New York (1969)
  49. Ruelle D.: Entropy production in quantum spin systems. Commun. Math. Phys. g class="a-plus-plus">224g>, 3 (2001) g/10.1007/s002200100534">CrossRef
  50. Simon B.: The statistical mechanics of lattice gases, I. Princeton University Press, Princeton (1993)
  51. Tasaki S., Matsui T.: Fluctuation theorem, nonequilibrium steady states and MacLennan-Zubarev ensembles of a class of large quantum systems. Quantum Prob. White Noise Anal. g class="a-plus-plus">17g>, 100 (2003) g/10.1142/9789812704412_0006">CrossRef
  52. Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs, vol. 72. AMS, Providence (1991)
  53. Nenciu G.: Independent electrons model for open quantum systems: Landauer–Büttiker formula and strict positivity of the entropy production. J. Math. Phys. g class="a-plus-plus">48g>, 033302 (2007) g/10.1063/1.2712418">CrossRef
  54. Volberg A., Yuditskii P.: On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a finite system of intervals, or a Cantor set of positive length. Commun. Math. Phys. g class="a-plus-plus">226g>, 567 (2002) g/10.1007/s002200200623">CrossRef
  55. Yafaev, D.R.: Mathematical Scattering Theory. General Theory. Translated from Russian by J. R. Schulenberger. Translations of Mathematical Monographs, 105. American Mathematical Society, Providence (1992)
  
• 作者单位：Vojkan Jak?i? (1)
  Benjamin Landon (1)
  Claude-Alain Pillet (2) (3)
  
  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada
  2. Aix-Marseille Université, CNRS UMR 7332, CPT, 13288, Marseille, France
  3. Université du Sud Toulon-Var, CNRS UMR 7332, CPT, 83957, La Garde, France
  
• ISSN：1424-0661

文摘

We study entropic functionals/fluctuations of the XY chain with Hamiltonian $$\begin{array}{ll} \frac{1}{2} \sum\limits_{x \in \mathbb{Z}}J_x( \sigma_x^{(1)} \sigma_{x+1}^{(1)} +\sigma_x^{(2)} \sigma_{x+1}^{(2)}) + \lambda_x \sigma_x^{(3)}\end{array}$$ where initially the left (x??0)/right (x?>?0) part of the chain is in thermal equilibrium at inverse temperature β l /β r . The temperature differential results in a non-trivial energy/entropy flux across the chain. The Evans–Searles (ES) entropic functional describes fluctuations of the flux observable with respect to the initial state while the Gallavotti–Cohen (GC) functional describes these fluctuations with respect to the steady state (NESS) the chain reaches in the large time limit. We also consider the full counting statistics (FCS) of the energy/entropy flux associated with a repeated measurement protocol, the variational entropic functional (VAR) that arises as the quantization of the variational characterization of the classical Evans–Searles functional and a natural class of entropic functionals that interpolate between FCS and VAR. We compute these functionals in closed form in terms of the scattering data of the Jacobi matrix hu x ?=?J x u x+1?+?λ x u x +?J x? u x? canonically associated with the XY chain. We show that all these functionals are identical if and only if h is reflectionless (we call this phenomenon entropic identity). If h is not reflectionless, then the ES and GC functionals remain equal but differ from the FCS, VAR and interpolating functionals. Furthermore, in the non-reflectionless case, the ES/GC functional does not vanish at α?=?1 (i.e., the Kawasaki identity fails) and does not have the celebrated α ?1 ?α symmetry. The FCS, VAR and interpolating functionals always have this symmetry. In the Schr?dinger case, where J x ?=?J for all x, the entropic identity leads to some unexpected open problems in the spectral theory of one-dimensional discrete Schr?dinger operators.