Abstract
We study entropic functionals/fluctuations of the XY chain with Hamiltonian
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−1 u x−1 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 Schrödinger 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 Schrödinger operators.
Article PDF
Similar content being viewed by others
References
Araki, H.: Relative entropy of states of von Neumann algebras. Publ. Res. Inst. Math. Sci. Kyoto Univ. 11, 809 (1975/1976)
Araki, H.: Relative entropy of states of von Neumann algebras II. Publ. Res. Inst. Math. Sci. Kyoto Univ. 13, 173 (1977/1978)
Araki H.: On the XY-model on two-sided infinite chain. Publ. Res. Inst. Math. Sci. Kyoto Univ. 20, 277 (1984)
Araki H.: Master symmetries of the XY model. Commun. Math. Phys. 132, 155 (1990)
Araki H.: On an inequality of Lieb and Thirring. Lett. Math. Phys. 19, 167 (1990)
Avila, A.: The absolutely continuous spectrum of the almost Mathieu operator (Preprint)
Aschbacher W., Barbaroux J.-M.: Out of equilibrium correlations in the XY chain. Lett. Math. Phys. 77, 11 (2007)
Araki H., Barouch E.: On the dynamics and ergodic properties of the XY-model. J. Stat. Phys. 31, 327 (1983)
Araki H., Ho T.G.: Asymptotic time evolution of a partitioned infinite two-sided isotropic XY chain. Proc. Steklov Inst. Math. 228, 191 (2000)
Aschbacher W., Pillet C-A.: Non-equilibrium steady states of the XY chain. J. Stat. Phys. 112, 1153 (2003)
Aschbacher, W., Jakšić, 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)
Aschbacher W., Jakšić V., Pautrat Y., Pillet C.-A.: Transport properties of quasi-free fermions. J. Math. Phys. 48, 032101 (2007)
Billingsley P.: Probability and Measure. 2nd edn. Wiley, New York (1986)
Barouch E., McCoy B.M.: Statistical mechanics of the XY model II. Spin-correlation functions. Phys. Rev. A 3, 786 (1971)
Baez J.C., Segal I.E., Zhou Z.: Introduction to Algebraic and Constructive Quantum Field Theory. Princeton University Press, Princeton (1991)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 1. Springer, Berlin (1987)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 2. Springer, Berlin (1996)
Bryc W.: A remark on the connection between the large deviation principle and the central limit theorem. Stat. Prob. Lett. 18, 253 (1993)
Cohen E.G.D., Gallavotti G.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694 (1995)
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. 121, 8179–8182 (2004)
Dereziński J., De Roeck W., Maes C.: Fluctuations of quantum currents and unravelings of master equations. J. Stat. Phys. 131, 341 (2008)
Hiai F., Petz D.: The Golden–Thompson trace inequality is complemented. Lin. Alg. Appl. 181, 153 (1993)
Hume L., Robinson D.W.: Return to equilibrium in the XY model. J. Stat. Phys. 44, 829 (1986)
Evans D.J., Searles D.J.: Equilibrium microstates which generate second law violating steady states. Phys Rev. E 50, 1645 (1994)
Israel, R.: Convexity in the Theory of Lattice Gases. Princeton Series in Physics. Princeton University Press, Princeton (1979)
Jakšić, 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)
Jakšić, 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)
Jakšić, V., Ogata, Y., Pillet, C.-A.: Entropic fluctuations in statistical mechanics II. Quantum dynamical systems (In preparation)
Jakšić V., Ogata Y., Pautrat Y., Pillet C.-A.: Entropic fluctuations in quantum statistical mechanics–an introduction. In: Fröhlich, J., Salmhofer, M., Mastropietro, V., De Roeck, W., Cugliandolo, L.F. (eds) Quantum Theory from Small to Large Scales, Oxford University Press, Oxford (2012)
Jakšić V., Pillet C.-A.: On entropy production in quantum statistical mechanics. Commun. Math. Phys. 217, 285 (2001)
Jakšić V., Pillet C.-A.: Mathematical theory of non-equilibrium quantum statistical mechanics. J. Stat. Phys. 108, 787 (2002)
Jakšić, V., Pillet, C.-A.: Entropic functionals in quantum statistical mechanics. In: Proceedings of the XVIIth International Congress of Mathematical Physics. Aalborg, Denmark (2012)
Jakšić V., Pillet C.-A, Rey-Bellet L.: Entropic fluctuations in statistical mechanics I. Classical dynamical systems. Nonlinearity 24, 699 (2011)
Jordan P., Wigner E.: Pauli’s equivalence prohibition. Z. Physik 47, 631 (1928)
Kurchan, J.: A quantum fluctuation theorem. Arxiv preprint cond-mat/0007360 (2000)
Landon, B.: Master’s thesis, McGill University (In preparation)
Levitov L.S., Lesovik G.B.: Charge distribution in quantum shot noise. JETP Lett. 58, 230 (1993)
Lieb E., Schultz T., Mattis D.: Two solvable models of an antiferromagnetic chain. Ann. Phys. 16, 407 (1961)
Lieb E., Thirring W.: Inequalities for the moments of the eigenvalues of the Schrödinger 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)
Matsui T.: On conservation laws of the XY model. Math. Phys. Stud. 16, 197 (1993)
McCoy B.M.: Spin correlation functions of the XY model. Phys. Rev 173, 531 (1968)
Ogata Y., Matsui T.: Variational principle for non-equilibrium steady states of XX model. Rev. Math. Phys. 15, 905 (2003)
Ohya M., Petz D.: Quantum Entropy and Its Use. Springer, Berlin (2004)
Remling C.: The absolutely continuous spectrum of Jacobi matrices. Ann. Math. 174, 125 (2011)
de Roeck W.: Large deviation generating function for currents in the Pauli–Fierz model. Rev. Math. Phys. 21, 549 (2009)
Reed M., Simon B.: Methods of Modern Mathematical Physics, III. Scattering Theory. Academic Press, London (1978)
Rondoni L., Meḿyi ja-Monasterio C.: Fluctuations in non-equlibrium statistical mechanics: models, mathematical theory, physical mechanisms. Nonlinearity 20, 1 (2007)
Ruelle D.: Statistical Mechanics. Rigorous Result. Benjamin, New York (1969)
Ruelle D.: Entropy production in quantum spin systems. Commun. Math. Phys. 224, 3 (2001)
Simon B.: The statistical mechanics of lattice gases, I. Princeton University Press, Princeton (1993)
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. 17, 100 (2003)
Teschl, G.: Jacobi Operators and Completely Integrable Nonlinear Lattices. Mathematical Surveys and Monographs, vol. 72. AMS, Providence (1991)
Nenciu G.: Independent electrons model for open quantum systems: Landauer–Büttiker formula and strict positivity of the entropy production. J. Math. Phys. 48, 033302 (2007)
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. 226, 567 (2002)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jan Derezinski.
Rights and permissions
About this article
Cite this article
Jakšić, V., Landon, B. & Pillet, CA. Entropic Fluctuations in XY Chains and Reflectionless Jacobi Matrices. Ann. Henri Poincaré 14, 1775–1800 (2013). https://doi.org/10.1007/s00023-013-0231-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0231-2