Towards a large deviation theory for strongly correlated systems

Abstract A large-deviation connection of statistical mechanics is provided by N independent binary variables, the ( N → ∞ ) limit yielding Gaussian distributions. The probability of n ≠ N / 2 out of N throws is governed by e − N r , r related to the entropy. Large deviations for a strong correlated model characterized by indices ( Q , γ ) are studied, the ( N → ∞ ) limit yielding Q -Gaussians ( Q → 1 recovers a Gaussian). Its large deviations are governed by e q − N r q ( ∝ 1 / N 1 / ( q − 1 ) , q > 1 ), q = ( Q − 1 ) / ( γ [ 3 − Q ] ) + 1 . This illustration opens the door towards a large-deviation foundation of nonextensive statistical mechanics.

[1]  Constantino Tsallis,et al.  Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Bin Liu,et al.  Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma. , 2008, Physical review letters.

[3]  Constantino Tsallis,et al.  Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[4]  C. Tsallis,et al.  q-Moments remove the degeneracy associated with the inversion of the q-Fourier transform , 2011 .

[5]  Jr.,et al.  Measurement of neutral mesons in p+p collisions at sqrt(s) = 200 GeV and scaling properties of hadron production , 2010, 1005.3674.

[6]  R. DeVoe,et al.  Power-law distributions for a trapped ion interacting with a classical buffer gas. , 2009, Physical review letters.

[7]  Andrea Benaglia,et al.  Transverse-Momentum and Pseudorapidity Distributions of Charged Hadrons in pp Collisions at root s=7 TeV , 2010 .

[8]  C. Tsallis,et al.  Nonlinear relativistic and quantum equations with a common type of solution. , 2011, Physical review letters.

[9]  Jungyeon Cho,et al.  SIMULATIONS OF ELECTRON MAGNETOHYDRODYNAMIC TURBULENCE , 2009, 0904.0661.

[10]  C. Tsallis.  Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World , 2009 .

[11]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[12]  Constantino Tsallis,et al.  A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model , 2010, 1003.2232.

[13]  L. Burlaga,et al.  COMPRESSIBLE “TURBULENCE” OBSERVED IN THE HELIOSHEATH BY VOYAGER 2 , 2009 .

[14]  Mario H. Acuna,et al.  Tsallis Statistics of the Magnetic Field in the Heliosheath , 2006 .

[15]  C. Tsallis.  Generalized entropy-based criterion for consistent testing , 1998 .

[16]  Constantino Tsallis,et al.  On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics , 2008 .

[17]  A A Moreira,et al.  Thermostatistics of overdamped motion of interacting particles. , 2010, Physical review letters.

[18]  A note on q-weak law of large numbers , 2009 .

[19]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[20]  Yan Levin,et al.  Comment on "Thermostatistics of overdamped motion of interacting particles". , 2011, Physical review letters.

[21]  Constantino Tsallis,et al.  A generalized nonlinear Schrödinger equation: Classical field-theoretic approach , 2012 .

[22]  A. Saguia,et al.  Nonadditive entropy for random quantum spin-S chains , 2009, 0912.3718.

[23]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[24]  S. Bablok,et al.  Production of pions, kaons and protons in pp collisions at $\sqrt{s}= 900~\mathrm{GeV}$ with ALICE at the LHC , 2011 .

[25]  Y. Wang,et al.  Transverse momentum spectra of charged particles in proton-proton collisions at $\sqrt{s} = 900$~GeV with ALICE at the LHC , 2010 .

[26]  T. J. Humanic,et al.  Strange particle production in proton–proton collisions at √s = 0.9 TeV with ALICE at the LHC , 2010, 1012.3257.

[27]  A. Lazarian,et al.  TSALLIS STATISTICS AS A TOOL FOR STUDYING INTERSTELLAR TURBULENCE , 2009, 0905.2581.

[28]  C. Tsallis,et al.  q-Generalization of the inverse Fourier transform , 2010, 1010.6275.

[29]  J. Gibbs.  Elementary Principles in Statistical Mechanics: Developed with Especial Reference to the Rational Foundation of Thermodynamics , 1902 .

[30]  Constantino Tsallis,et al.  Generalization of symmetric alpha-stable Lévy distributions for q>1. , 2009, Journal of mathematical physics.

[31]  M. K. Lee,et al.  Nuclear modification factors of phi mesons in d plus Au, Cu plus Cu, and Au plus Au collisions at root s(NN)=200 GeV , 2010, 1004.3532.

[32]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[33]  H. Hilhorst,et al.  Note on a q-modified central limit theorem , 2010, 1008.4259.

[34]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[35]  Stefan Thurner,et al.  Limit distributions of scale-invariant probabilistic models of correlated random variables with the q-Gaussian as an explicit example , 2009, 0908.4438.

[36]  L. Burlaga,et al.  SECTORS AND LARGE-SCALE MAGNETIC FIELD STRENGTH FLUCTUATIONS IN THE HELIOSHEATH NEAR 110 AU: VOYAGER 1, 2009 , 2010 .

[37]  L. Burlaga,et al.  TRANSITION FROM THE SECTOR ZONE TO THE UNIPOLAR ZONE IN THE HELIOSHEATH: VOYAGER 2 MAGNETIC FIELD OBSERVATIONS , 2011 .

[38]  R. Cywinski,et al.  Generalized spin-glass relaxation. , 2009, Physical review letters.

[39]  Zebo Tang,et al.  Examination of the species and beam energy dependence of particle spectra using Tsallis statistics , 2009, 0912.0993.

[40]  S. Braunstein.  GEOMETRY OF QUANTUM INFERENCE , 1996 .

[41]  C. Tsallis,et al.  Strictly and asymptotically scale invariant probabilistic models of N correlated binary random variables having q-Gaussians as N → ∞ limiting distributions , 2008, 0804.1488.

[42]  Farid Ould-Saada,et al.  Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC , 2011 .

[43]  H. Touchette.  The large deviation approach to statistical mechanics , 2008, 0804.0327.

[44]  Jr.,et al.  Identified charged hadron production in p + p collisions at √ s = 200 and 62.4 GeV , 2011, 1102.0753.

[45]  Abdel Nasser Tawfik,et al.  Antiproton-to-proton ratios for ALICE heavy-ion collisions , 2010, 1011.5612.

[46]  P. Douglas,et al.  Tunable Tsallis distributions in dissipative optical lattices. , 2006, Physical review letters.

[47]  M. Acuna,et al.  MAGNETIC FIELD STRENGTH FLUCTUATIONS AND TEMPERATURE IN THE HELIOSHEATH , 2009 .

[48]  K. Duffy,et al.  THE LARGE DEVIATIONS OF ESTIMATING RATE-FUNCTIONS , 2005 .

[49]  Evaldo M. F. Curado,et al.  Andrade et al. Reply , 2011 .