The temperature of a single chaotic eigenstate
Research on the chaotic eigenstates and models in quantum systems, including the Fermi-Pasta-Ulam model and linear mode representation. Analytical and numerical results from various studies provide insights into the behavior of energy modes in chaotic systems. Explore the border of chaos and delve into the concept of quantum chaos in deterministic systems through the spectrum and eigenfunctions analysis. Discover the onset of chaos in energy eigenstates and the connection to the statistical theory of spectra.
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The temperature of a single chaotic eigenstate F. M. Izrailev Instituto de F sica, BUAP, Puebla, M xico Michigan State University, USA In collaboration with: F. Borgonovi and F. Mattiotti (Brescia, Italy) E.Lansing, March 10, 2017
The Fermi-Pasta-Ulam (FPU) model F.E.Fermi, J.Pasta and S.Ulam, Studies of the Nonlinear Problems, I. Los Alamos Report LA-1940 (1955). Special Issue: CHAOS 15 (2005). model: ( ( ) ) 1 ( ( ) ) n ( ( ) ) 1 3 3 n = = + + + + x x x x x x x x 2 + + n n + + n n n n 1 1 model: ( ( ) ) 1 ( ( ) ) n ( ( ) ) 1 2 2 n = = + + + + x x x x x x x x 2 + + n + + n n n n n 1 1 G.P.Berman and F.M.I., CHAOS, 15 (2005) 015104
Linear mode representation kn 2 N ( ( ) ) t k ( ( ) ) t n = = = = Q x sin N N n 1 model: N + + = = 2 Q Q D 1 Q Q Q k k k ijl i j l = = l , j , i k model: = = sin 2 k N 2 N j , i k + + = = 2 Q Q C 1 Q Q k k ij i j = = Expectation of FPU: Equipartition of energy between linear modes due to a nonlinear interaction, for large E.Lansing, March 10, 2017 N 1
Numerical result of FPU 1955 Quasi-periodic oscillations of the energy in each mode Synergetic approach numerical experiments S.Ulam !! S.M.Ulam, A Collection of Mathematical Problems 1960. E.Lansing, March 10, 2017
Border of chaos 1965 E 3 Analytical result: N k k k N 3 ; E 3 2 cr 3 2 2 k k N N k N ; N N k for the border of chaos N F.M.Izrailev, B.V.Chirikov Report INP (1965); Dokl. Akad.Nauk SSSR , 166 (1966) 57 . Numerical confirmation: F.M.Izrailev, A.I.Khisamutdinov, B.V.Chirikov, Report 252 INP, 1968 (LA-4440-TR, Los Alamos, 1970) E.Lansing, March 10, 2017
Quantum chaos in deterministic systems S.W. McDonald and A.N. Kaufman, Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories , Phys. Rev. Lett. 42 (1979) 1189. G.Casati, I.Guarneri, F.Valz-Gris, On the connection between quantization of nonintegrable systems and statistical theory of spectra , Lett. Nuovo Cimento 28 (1980) 279. M.V. Berry, Quantizing a Classically Ergodic System: Sinai s Billiard and the KKR Method , Annals of Physics, 131 (1981) 163. O.Bohigas, M.-J.Giannoni, C.Schmit, Characterization of Quantum Chaotic Spectra and Universality of Level Fluctuation Laws , Phys. Rev. Lett. 52 (1984) 1. E.Lansing, March 10, 2017
Chaotic eigenstates M.Shapiro and G.Goelman, Onset of Chaos in an Isolated Energy Eigenstate , Phys. Rev. Lett. 53 (1984) 1714. R.V.Jensen and R.Shankar, Phys. Rev. Lett. 54 (1985) 1879. E.Lansing, March 10, 2017
Chaotic eigenstates as the condition for thermalization L.D.Landau and E.M.Lifshitz: Statistical Physics, Vol.5 (Pergamon, Oxford, 1969) J.M.Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046. M.Srednicki, Chaos and quantum thermalization , Phys. Rev. E 50 (1994) 888. E.Lansing, March 10, 2017
Chaos and thermalization in nuclei and atoms M.Horoi, V.Zelevinsky, B.A.Brown, Phys. Rev. Lett. 74 (1995) 5194; V.Zelevinsky, M.Horoi, B.A.Brown, Phys. Lett. B 350 (1995) 141; V.Zelevinsky, B.A.Brown, M.Horoi, N.Frazier, Phys. Rep. 276 (1996) 85. V.V.Flambaum, A.A.Gribakina, G.F.Gribakin, M.G.Kozlov, Structure of compound states in the chaotic spectrum of the Ce atom: Localization properties, matrix elements, and enhancement of weak perturbations, Phys. Rev. A 50 (1994) 267. in particular, the reduced density matrix operator was analyzed numerically for individual eigenstates, and compared with analytical average over number of chaotic states E.Lansing, March 10, 2017
Chaotic dynamics of systems of interacting particles H H V = = + + 0 H - non-perturbed part (describes the non- interacting particles/quasi-particles) 0 V - interaction between particles, or, with an external field Many-body chaos how to characterize?
Chaotic eigenstates in a gold atom G.F.Gribakin, A.A.Gribakina, V.V.Flambaum, arXiv:physics/9811010; Aust. J. Phys. 52 (1999) 443. (problem of electron recombination ) E.Lansing, March 10, 2017
Thermalization in an isolated gold atom G.F.Gribakin, A.A.Gribakina, V.V.Flambaum, arXiv:physics/9811010; Aust. J. Phys. 52 (1999) 443. E.Lansing, March 10, 2017
Two-Body Interaction Model 1 m 2 + + + + + + = = + + H a a V a a a a k k k kqpr k q p r k kqpr k , q , p , r single-particle states V two-body matrix elements (random or dynamical) kqpr number of single-particle states number of particles ( quaisi-particles ) energy of single-particle states m n k M H is considered in the many-particle basis of H determines the basis in which the dynamics occurs + + = = H a a k k k 0 k 0 E.Lansing, March 10, 2017
V.V.Flambaum and F.M.I., Statistical theory of finite Fermi systems based on the structure of chaotic eigenstates , Phys. Rev. E 56 (1997) 5144; V.V.Flambaum, F.M.I., G.Casati, Phys. Rev. E 54 (1996) 2136. D - density of all many-body states i d - density of many-body states directly connected by the two-body interaction f V 1 Onset of chaos: d f ( )( ) + 8 / 4 1 1 V V d n n n m n 0 cr H j ij E.Lansing, March 10, 2017
Strength function: from Breit-Wigner to Gauss ( ) E ( ) E 2 = P C n n 0 0 2 2 H BW is characterized by half-width: n n m m 0 0 m 2 = 2 n H Gauss is characterized by its variance: n m 0 0 n 0 Transition to chaos occurs when !! E.Lansing, March 10, 2017
Thank you for your attention! E.Lansing, March 10, 2017
Transition to chaos: chaotic eigenstates E.Lansing, March 10, 2017
Structure of eigenfunctions model 1 model 2 2 2 n C n C n n E.Lansing, March 10, 2017
Emergence of chaotic states n H H - basis of - basis of 0 E.Lansing, March 10, 2017
Chaos and relaxation dynamics in 1/2-spin models model 1 integrable model 2 non-integrable 5 . 0 cr - for transition from Poisson to Wigner-Dyson L.F.Santos, F.Borgonovi, F.M.I., Phys. Rev. Lett. 108 (2012) 094102; Phys. Rev. E 85 (2012) 036209. E.Lansing, March 10, 2017
Chaos in integrable systems B.V.Chirikov, Transient Chaos in Quantum and Classical Mechanics , Foundation of Physics, Vol.16, No.1 (1986). Abstract: Bogolubov s classical example of statistical relaxation in a many-dimensional linear oscillator is discussed. The relation of the discovered relaxation mechanism to quantum dynamics as well as to some new problems in classical mechanics is considered. N.N.Bogoliubov, On Some Statistical Methods in Mathematical Physics , Academy of Sciences USSR Publishers, Kiev, 1945, p.115 (Russian); in: Selected Papers (Naukova Dumka, Kiev, 1970, Vol.2, p.77 (Russian). E.Lansing, March 10, 2017
Foundation of statistical mechanics Two mechanisms of a statistical behavior (relaxation to a steady state distribution) in classical mechanics: N Thermodynamical limit ; Exponential instability plus boundary in phase space ( ) dynamical (deterministic) chaos 0 What is common for both mechanisms? Infinite number of statistically independent frequencies in the time evolution of observables. in quantum mechanics only second mechanism B.V.Chirikov, Linear and nonlinear dynamical chaos , Open. Sys. & Informaion Dyn. 4 (1997) 241-280 . E.Lansing, March 10, 2017
Quantum chaos: Deterministic quantum systems with strong chaos in the classical limit Classical chaos Wave chaos Properties: (a) spectrum (b) eigenstates (c) dynamics Deterministic quantum systems without classical limit Disordered quantum systems Santa Barbara, August 20, 2012 E.Lansing, March 10, 2017
Strength functions (LDOS) E.Lansing, March 10, 2017
Delocalization in energy shell E.Lansing, March 10, 2017
Fermi-Dirac distribution Circles: analytical description versus numerical data, Diamonds: Fermi-Dirac with thermodynamical temperature E.Lansing, March 10, 2017
