Energy absorption by ‘sparse’ systems: beyond linear response theory

The analysis of the response to driving in the case of weakly chaotic or weakly interacting systems should go beyond linear response theory. Due to the ‘sparsity’ of the perturbation matrix, a resistor-network picture of transitions between energy levels is essential. The Kubo formula is modified, replacing the ‘algebraic’ average over the squared matrix elements by a ‘resistor-network’ average. Consequently, the response becomes semi-linear rather than linear. Some novel results have been obtained in the context of two prototype problems: the heating rate of particles in billiards with vibrating walls; and the Ohmic Joule conductance of mesoscopic rings driven by electromotive force. The results obtained are contrasted with the ‘Wall formula’ and the ‘Drude formula’.

[1]  E. Ryabov,et al.  Intramolecular vibrational redistribution: from high-resolution spectra to real-time dynamics , 2012 .

[2]  L. Pecora,et al.  "Weak quantum chaos" and its resistor network modeling. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  L. Pecora,et al.  Quantum response of weakly chaotic systems , 2010, 1005.4207.

[4]  T. Kottos,et al.  Random-matrix modeling of semilinear response, the generalized variable-range hopping picture, and the conductance of mesoscopic rings , 2009, 0908.3991.

[5]  N. Davidson,et al.  Semilinear response for the heating rate of cold atoms in vibrating traps , 2008, 0810.0360.

[6]  Tal Peer,et al.  The mesoscopic conductance of disordered rings, its random matrix theory and the generalized variable range hopping picture , 2007, 0712.0439.

[7]  D. Cohen.  From the Kubo formula to variable-range hopping , 2006, cond-mat/0611663.

[8]  Alessandro Silva,et al.  Multiphoton processes in driven mesoscopic systems , 2006, cond-mat/0611083.

[9]  D. Cohen,et al.  The conductance of a multi-mode ballistic ring: Beyond Landauer and Kubo , 2006, cond-mat/0603484.

[10]  T. Kottos,et al.  Rate of energy absorption by a closed ballistic ring , 2005, cond-mat/0505295.

[11]  T. Prosen,et al.  Microwave control of transport through a chaotic mesoscopic dot , 2005 .

[12]  T. Prosen,et al.  Mesoscopic "Rydberg" atom in a microwave field , 2004, cond-mat/0411155.

[13]  D. Basko,et al.  Dynamic localization in quantum dots: analytical theory. , 2002, Physical review letters.

[14]  E. Heller,et al.  Rate of energy absorption for a driven chaotic cavity , 2000, nlin/0006041.

[15]  Cohen,et al.  Quantum-mechanical nonperturbative response of driven chaotic mesoscopic systems , 2000, Physical review letters.

[16]  Heller,et al.  Deformations and dilations of chaotic billiards: dissipation rate, and quasiorthogonality of the boundary wave functions , 2000, Physical review letters.

[17]  D. Cohen.  Chaos and Energy Spreading for Time-Dependent Hamiltonians, and the Various Regimes in the Theory of Quantum Dissipation , 1999, cond-mat/9902168.

[18]  D. Cohen.  Quantum Dissipation due to the Interaction with Chaotic Degrees of Freedom and the Correspondence Principle , 1998, cond-mat/9810395.

[19]  Izrailev,et al.  Wigner random banded matrices with sparse structure: Local spectral density of states. , 1996, Physical review letters.

[20]  E. Austin,et al.  A random matrix model for the non-perturbative response of a complex quantum system , 1995 .

[21]  Y. Gefen,et al.  ALMOST) EVERYTHING YOU ALWAYS WANTED TO KNOW ABOUT THE CONDUCTANCE OF MESOSCOPIC SYSTEMS , 1995 .

[22]  Jarzynski.  Thermalization of a Brownian particle via coupling to low-dimensional chaos. , 1995, Physical review letters.

[23]  T. Prosen.  Statistical Properties of Matrix Elements in a Hamilton System Between Integrability and Chaos , 1994 .

[24]  Jarzynski.  Energy diffusion in a chaotic adiabatic billiard gas. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  Imre Kondor,et al.  Sensitivity of spin-glass order to temperature changes , 1993 .

[26]  Izrailev,et al.  Two-parameter scaling in the Wigner ensemble. , 1993, Physical review letters.

[27]  T. Prosen,et al.  Energy level statistics and localization in sparsed banded random matrix ensemble , 1993 .

[28]  E. Austin,et al.  Distribution of matrix elements of a classically chaotic system , 1992 .

[29]  M. Berry,et al.  Discordance between quantum and classical correlation moments for chaotic systems , 1992 .

[30]  Wilkinson,et al.  Spectral statistics in semiclassical random-matrix ensembles. , 1991, Physical review letters.

[31]  D. Leitner,et al.  Localization and spectral statistics in a banded random matrix ensemble , 1991 .

[32]  M. Wilkinson.  Statistical aspects of dissipation by Landau-Zener transitions , 1988 .

[33]  E. Ott,et al.  The goodness of ergodic adiabatic invariants , 1987 .

[34]  Grebogi,et al.  Ergodic adiabatic invariants of chaotic systems. , 1987, Physical review letters.

[35]  Levine,et al.  Transition-strength fluctuations and the onset of chaotic motion. , 1986, Physical review letters.

[36]  Pérès,et al.  Distribution of matrix elements of chaotic systems. , 1986, Physical review. A, General physics.

[37]  Alexander L. Efros,et al.  Electronic Properties of Doped Semi-conductors , 1984 .

[38]  E. Ott.  Goodness of ergodic adiabatic invariants , 1979 .

[39]  J. Błocki,et al.  One-body dissipation and the super-viscidity of nuclei , 1978 .

[40]  S. Koonin,et al.  One-body dissipation in a linear response approach , 1977 .

[41]  D. Gross.  Theory of nuclear friction , 1975 .

[42]  Vinay Ambegaokar,et al.  Hopping Conductivity in Disordered Systems , 1971 .

[43]  N. F. Mott,et al.  Conduction in non-Crystalline systems: IV. Anderson localization in a disordered lattice , 1970 .

[44]  E. Abrahams,et al.  Impurity Conduction at Low Concentrations , 1960 .

[45]  N. Mott,et al.  Electronic Processes In Non-Crystalline Materials , 1940 .