Tsallis Distribution Decorated with Log-Periodic Oscillation

In many situations, in all branches of physics, one encounters power-like behavior of some variables which are best described by a Tsallis distribution characterized by a nonextensivity parameter $q$ and scale parameter $T$. However, there exist experimental results which can be described only by a Tsallis distributions which are additionally decorated by some log-periodic oscillating factor. We argue that such a factor can originate from allowing for a complex nonextensivity parameter $q$. The possible information conveyed by such an approach (like the occurrence of complex heat capacity, the notion of complex probability or complex multiplicative noise) will also be discussed.

[1]  G. Wilk,et al.  Log-periodic oscillations of transverse momentum distributions , 2014, 1403.3508.

[2]  G. Wilk,et al.  Relativistic Hard-Scattering and Tsallis Fits to pT Spectra in pp Collisions at the LHC , 2013, 1309.7330.

[3]  G. Wilk,et al.  System size dependence of the log-periodic oscillations of transverse momentum spectra , 2014, 1411.5148.

[4]  The Atlas Collaboration,et al.  Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC , 2010, 1012.5104.

[5]  G. Wilk,et al.  The Imprints of nonextensive statistical mechanics in high-energy collisions , 2000 .

[6]  Constantino Tsallis,et al.  Nonadditive entropy: The concept and its use , 2008, 0812.4370.

[7]  G. Barnafoldi,et al.  Microcanonical jet-fragmentation in proton–proton collisions at LHC energy , 2012, 1204.1508.

[8]  Grzegorz Wilk,et al.  Tsallis Fits to p_t Spectra for pp Collisions at LHC , 2012, 1210.3661.

[9]  Renio S. Mendes,et al.  SPECIFIC HEAT ANOMALIES ASSOCIATED WITH CANTOR-SET ENERGY SPECTRA , 1997 .

[10]  C. Michael,et al.  Consequences of momentum conservation for particle production at large transverse momentum , 1977 .

[11]  Ernesto P. Borges,et al.  Fluxes of cosmic rays: A delicately balanced anomalous thermal equilibrium , 2002 .

[12]  J. Muzy,et al.  Complex fractal dimensions describe the hierarchical structure of diffusion-limited-aggregate clusters. , 1996, Physical review letters.

[13]  Carnot cycle for interacting particles in the absence of thermal noise. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Didier Sornette,et al.  Log-periodic oscillations for biased diffusion on random lattice , 1998 .

[15]  Włodarczyk,et al.  Interpretation of the nonextensivity parameter q in some applications of tsallis statistics and Levy distributions , 2000, Physical review letters.

[16]  M. D. Azmi,et al.  Transverse momentum distributions in proton–proton collisions at LHC energies and Tsallis thermodynamics , 2014, 1401.4835.

[17]  M. Z. Mehta,et al.  Transverse-momentum and pseudorapidity distributions of charged hadrons in pp collisions at $\sqrt{s} = 0.9 $ and 2.36 TeV , 2010 .

[18]  A. Deppman,et al.  Systematic analysis of pT -distributions in p + p collisions , 2012, 1209.2367.

[19]  J. Richard,et al.  Entropy production in ac-calorimetry , 2007, 0706.4216.

[20]  P. Shukla,et al.  System size dependence of hadron $p_T$ spectra in p+p and Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV , 2013, 1310.4022.

[21]  Grzegorz Wilk,et al.  The imprints of superstatistics in multiparticle production processes , 2011, 1110.4220.

[22]  J. Cleymans,et al.  The Tsallis distribution in proton?proton collisions at $\sqrt{s}$ = 0.9 TeV at the LHC , 2011, 1110.5526.

[23]  R. Hagedorn.  Multiplicities,pT distributions and the expected hadron → quark-gluon phase transition , 1983 .

[24]  A. Deppman.  Properties of hadronic systems according to the non-extensive self-consistent thermodynamics , 2012, 1212.0379.

[25]  Tadeusz Wibig,et al.  The non-extensivity parameter of a thermodynamical model of hadronic interactions at LHC energies , 2010, 1005.5652.

[26]  ALICE Collaboration,et al.  Transverse momentum spectra of charged particles in proton-proton collisions at $\sqrt{s} = 900$ GeV with ALICE at the LHC , 2010, 1007.0719.

[27]  Constantino Tsallis,et al.  Fluxes of cosmic rays: a delicately balanced stationary state , 2002 .

[28]  M. Ausloos,et al.  How the financial crash of October 1997 could have been predicted , 1998 .

[29]  New simulations on old biased diffusion , 1999 .

[30]  Péter Ván,et al.  Quark-gluon plasma connected to finite heat bath , 2013 .

[31]  P. Beckmann,et al.  The scattering of electromagnetic waves from rough surfaces , 1963 .

[32]  G. Wilk,et al.  Nonextensive information entropy for stochastic networks , 2002 .

[33]  Grzegorz Wilk,et al.  Tsallis Fits to pT Spectra and Multiple Hard Scattering in pp Collisions at LHC , 2013 .

[34]  Nielsen,et al.  Fluctuation-dissipation theorem for frequency-dependent specific heat. , 1996, Physical review. B, Condensed matter.

[35]  Maciej Rybczyński,et al.  Tsallis statistics approach to the transverse momentum distributions in p–p collisions , 2014, 1401.5639.

[36]  P. Shukla,et al.  Hadron spectra in p+p collisions at RHIC and LHC energies , 2013, 1304.6224.

[37]  D. Sornette,et al.  Precursors, aftershocks, criticality and self-organized criticality , 1998 .

[38]  G. Wilk,et al.  Consequences of temperature fluctuations in observables measured in high-energy collisions , 2012, 1203.4452.

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

[40]  Critical behavior of an Ising system on the Sierpinski carpet: a short-time dynamics study. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  D. Sornette,et al.  SPONTANEOUS GENERATION OF DISCRETE SCALE INVARIANCE IN GROWTH MODELS , 1997 .

[42]  C. Michael.  Large transverse momentum and large mass production in hadronic interactions , 1979 .

[43]  G. Wilk,et al.  Power laws in elementary and heavy-ion collisions , 2008, 0810.2939.

[44]  G. Wilk,et al.  On possible origins of power-law distributions , 2013, 1307.7855.

[45]  C. Collaboration,et al.  Charged particle transverse momentum spectra in pp collisions at sqrt(s) = 0.9 and 7 TeV , 2011, 1104.3547.

[46]  `Real' versus `imaginary' noise in diffusion-limited reactions , 1997, cond-mat/9701069.

[47]  James S. Harris,et al.  Centrality Dependence of Charged Particle Production at Large Transverse Momentum in Pb--Pb Collisions at $\sqrt{s_{\rm{NN}}} = 2.76$ TeV , 2012 .

[48]  CONNECTION BETWEEN ENERGY SPECTRUM, SELF-SIMILARITY, AND SPECIFIC HEAT LOG-PERIODICITY , 1998, cond-mat/9803265.

[49]  T. S. Biro,et al.  New Entropy Formula with Fluctuating Reservoir , 2014, 1405.3813.

[50]  T. Tuuva,et al.  Study of high-pT charged particle suppression in PbPb compared to pp collisions at $\sqrt{s_{\mathrm{NN}}}=2.76~\mathrm{TeV}$ , 2012 .

[51]  Didier Sornette,et al.  Complex Exponents and Log-Periodic Corrections in Frustrated Systems , 1996 .

[52]  Marcel Ausloos,et al.  The crash of October 1987 seen as a phase transition: amplitude and universality , 1998 .

[53]  Christian Beck.  Non-extensive statistical mechanics and particle spectra in elementary interactions , 2000 .

[54]  Diffusion in random one-dimensional systems , 1983 .

[55]  E. Curado,et al.  A thermodynamical formalism describing mechanical interactions , 2014 .

[56]  Didier Sornette,et al.  Discrete scale invariance, complex fractal dimensions, and log‐periodic fluctuations in seismicity , 1996 .

[57]  E. Curado,et al.  A nonextensive thermodynamical equilibrium approach in e+e−→ hadrons , 1999, hep-ph/9905255.

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

[59]  Christian Beck,et al.  Dynamical Foundations of Nonextensive Statistical Mechanics , 2001, cond-mat/0105374.

[60]  A. Plastino.  From Gibbs microcanonical ensemble to Tsallis generalized canonical distribution , 1994 .

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

[62]  J. Hołyst,et al.  Log-periodic oscillations and noise-free stochastic multiresonance due to self-similarity of fractals , 2003 .

[63]  Bayesian approach to superstatistics , 2006, cond-mat/0601154.

[64]  Grzegorz Wilk,et al.  Tsallis fits to p T spectra and multiple hard scattering in p p collisions at the LHC , 2013, 1305.2627.

[65]  L. A. Rios,et al.  Comment on “Debye shielding in a nonextensive plasma” [Phys. Plasmas 18, 062102 (2011)] , 2012 .

[66]  J. Schawe.  A comparison of different evaluation methods in modulated temperature DSC , 1995 .

[67]  C. Tsallis,et al.  Possible Implication of a Single Nonextensive pT Distribution for Hadron Production in High-Energy pp Collisions , 2014, 1412.0474.

[68]  D. Sornette.  Discrete scale invariance and complex dimensions , 1997, cond-mat/9707012.

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

[70]  G. Wilk,et al.  Nonextensive thermal sources of cosmic rays , 2009, 0904.0528.

[71]  D. Sornette,et al.  Stock Market Crashes, Precursors and Replicas , 1995, cond-mat/9510036.

[72]  T. S. Biro,et al.  Generalised Tsallis Statistics in Electron-Positron Collisions , 2011, 1101.3023.

[73]  Christian Beck,et al.  Generalized statistical mechanics of cosmic rays , 2004 .

[74]  Brian Cox.  Transverse-momentum and pseudorapidity distributions of charged hadrons in pp collisions at √s = 0.9 and 2.36TeV , 2010 .

[75]  T. Wibig.  Constrains for non-standard statistical models of particle creations by identified hadron multiplicity results at LHC energies , 2013, 1304.0655.

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

[77]  Ali Abdi,et al.  On the PDF of the sum of random vectors , 2000, IEEE Trans. Commun..

[78]  Michail Zak.  Incompatible stochastic processes and complex probabilities , 1998 .

[79]  Macroscopic non-equilibrium thermodynamics in dynamic calorimetry , 2007, 0706.4143.

[80]  R. Andrade.  Detailed characterization of log-periodic oscillations for an aperiodic ising model. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[81]  Z.Wlodarczyk,et al.  Tsallis distribution with complex nonextensivity parameter q , 2014, 1403.3263.

[82]  Antal Jakovác,et al.  Power-law tails from multiplicative noise. , 2005, Physical review letters.

[83]  de Moura FA,et al.  Convergence to the critical attractor of dissipative maps: log-periodic oscillations, fractality, and nonextensivity , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[85]  A. Deppman,et al.  Nonextensivity of hadronic systems , 2012, 1210.1725.

[86]  A. Savoy-Navarro,et al.  Transverse momentum spectra for charged particles at the cern proton-antiproton collider , 1982 .

[87]  Jan Henrik Wosnitza,et al.  Can log-periodic power law structures arise from random fluctuations? , 2014 .

[88]  M. P. Almeida.  Generalized entropies from first principles , 2001 .

[89]  J. Cleymans,et al.  Relativistic thermodynamics: Transverse momentum distributions in high-energy physics , 2012, The European Physical Journal A.

[90]  Constantino Tsallis,et al.  Non-extensive thermostatistics: brief review and comments , 1995 .

[91]  Fu-Hu Liu,et al.  Formulation of transverse mass distributions in Au–Au collisions at sNN=200 GeV/nucleon , 2013, 1402.6023.

[92]  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.

[93]  Walton,et al.  Equilibrium distribution of heavy quarks in fokker-planck dynamics , 2000, Physical review letters.

[94]  On the limiting cases of nonextensive thermostatistics , 2006, cond-mat/0611068.

[95]  Sandpile model on the Sierpinski gasket fractal. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[96]  Z. Citron,et al.  and scaling properties of hadron production , 2011 .

[97]  G. Wilk,et al.  Information theory point of view on stochastic networks , 2005 .