Fluctuation, Dissipation, and Non-Boltzmann Energy Distributions

  • Tamás Sándor BiróEmail author
  • Antal Jakovác
Part of the SpringerBriefs in Physics book series (SpringerBriefs in Physics)


After reviewing the field theoretical description of the conventional thermal equilibrium and studying the linear relaxation to this equilibrium, it is interesting to consider problems arising in small systems not satisfying the pre-requisites for the Gibbs–Boltzmann treatment. In this chapter, we discuss general questions of thermal ensembles and the stochastic dynamics picture of a practically unknown environment, and at the end we sketch briefly the way the Keldysh formalism incorporates physical noise into its descriptive framework.


  1. 1.
    Y. Takahashi, H. Umezawa, Thermofield dynamics. Collective Phenom. 2(55) (1975). (Reprinted: J. Mod. Phys. B 10, 1755, 1996.)Google Scholar
  2. 2.
    H. Umezawa, Advanced Field Theory Macro and Thermal Physics (AIP Press, New York, Micro, 1995)Google Scholar
  3. 3.
    Run-Qin Yang, A complexity of quantum field theory states and application in thermofield double states. Phys. Rev. D 97, 066004 (2018)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    R.A. Fisher: The Negative Binomial Distribution, Blackwell Publishing Ltd. University College London (1941)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Joseph, Hilbe: Negative Binomial Regression (Cambridge University Press, Cambridge, 2011)Google Scholar
  6. 6.
    H. Abdel El-Shaarawi, Negative binomial Distribution—Applications, in Wiley StatsRef: Statistical Reference Online.
  7. 7.
    M. Arneodo et al., (EMC): comparison of multiplicity distributions to the negative binomial distribution in muon-proton scattering. Z. Phys. C 35, 335 (1987)ADSCrossRefGoogle Scholar
  8. 8.
    O.G. Tchikilev, Modified negative binomial description of the multiplicity distributions in lepton-nucleon scattering. Phys. Lett. B 388, 848 (1996)ADSCrossRefGoogle Scholar
  9. 9.
    A. Adare (PHENIX), Charged hadron multiplicity fluctuations in Au + Au and Cu + Cu collisions from \(\sqrt{s_{NN}}=\) 22.5 to 200 GeV. Phys. Rev. C 78, 044902 (2008)Google Scholar
  10. 10.
    ALICE Collaboration, K. Aamodt et al., Charged-particle multiplicity measurement in proton–proton collisions at \(\sqrt{s}=\) 0.9 and 2.36 TeV with ALICE at LHC, EPJ C 68, 89 (2010)Google Scholar
  11. 11.
    G. Wilk, Z. Wlodarczyk, How to retrieve additional information from the multiplicity distributions. J. Phys. G 44, 015002 (2017)ADSCrossRefGoogle Scholar
  12. 12.
    R. Hagedorn, Nuovo Cimento Suppl. 3, 147 (1965)Google Scholar
  13. 13.
    R. Hagedorn, Nuovo Cimento A 52, 64 (1967)CrossRefGoogle Scholar
  14. 14.
    R. Hagedorn, Riv. Nuovo Cimento 6, 1 (1983)MathSciNetCrossRefGoogle Scholar
  15. 15.
    J. Rafelski (Ed.), Melting Hadrons, Boiling Quarks–From Hagedorn Temperature to Ultra-Relativistic Heavy-Ion Collisions at CERN, SpringerOpen (2016)Google Scholar
  16. 16.
    P. Vilfredo, La courbe de la répartition de la richesse, (Orig. pub., in 1965 Œuvres complètes de Vilfredo Pareto, ed. by G. Busino (Librairie Droz, Geneva, 1896)Google Scholar
  17. 17.
    R. Koch, Living the 80/20 Way: Work Less, Worry Less, Succeed More, Enjoy More (Nicholas Bearley Pub, London, 2004)Google Scholar
  18. 18.
    W.J. Reed, The Pareto Zipf and other power laws. Econom. Lett. 74, 15 (2001)CrossRefGoogle Scholar
  19. 19.
    C. Tsallis, Nonadditive entropy: the concept and its use. EPJ A 40, 257 (2009)ADSCrossRefGoogle Scholar
  20. 20.
    C.-Y. Wong, G. Wilk, L.J.L. Cirto, C. Tsallis, From QCD-based hard-scattering to nonextensive statistical mechanical description of transverse momentum spectra in high-energy \(pp\) and \(p\overline{p}\) collisions, Phys. Rev. D 91, 114027 (2015)Google Scholar
  21. 21.
    M. Biyajima, T. Mizoguchi, N. Nakajima, N. Suzuki, G. Wilk, Modified Hagedorn formula including temperature fluctuation—estimation of temperatures at RHIC experiments. EPJ C 48, 597 (2006)ADSGoogle Scholar
  22. 22.
    J.M. Zhang, Y. Liu, Fermi’s golden rule: its derivation and breakdown by an ideal model. Eur. J. Phys. 37, 065406 (2016)CrossRefGoogle Scholar
  23. 23.
    R.P. Feynman, F.L. Vernon, The theory of a general quantum system interacting with a linear dissipative system. Ann. Phys. 24, 118 (1963)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    T.S. Biro, Is there a temperature? Fundamental Theories of Physics 1014 (Springer, 2011)Google Scholar
  25. 25.
    P.M. Stevenson, Gaussian effective potential: quantum mechanics. Phys. Rev. D 30, 1712 (1984)ADSCrossRefGoogle Scholar
  26. 26.
    P.M. Stevenson, Gaussian effective potential II: \(\lambda \varphi ^4\) field theory. Phys. Rev. D 32, 1389 (1985)ADSCrossRefGoogle Scholar
  27. 27.
    P.M. Stevenson, Gaussian Effective Potential III: \(\varphi ^6\) theory and bound states. Phys. Rev. D 33, 2305 (1985)ADSCrossRefGoogle Scholar
  28. 28.
    A. Einstein: Zur Theorie der Brownschen Bewegung, Ann. Phys. 17, 549, 1905; 19, 371, 1906Google Scholar
  29. 29.
    P. Langevin, Sur la théorie du mouvement brownien, Comptes Rendus Acad. Sci. (Paris) 146, 530 (1908)Google Scholar
  30. 30.
    A.D. Fokker, Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Ann. Phys. 43, 43 (1914)Google Scholar
  31. 31.
    M. Planck: Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie, Sitz. Ber. Preuss. Akad. Wiss. 324 (1917)Google Scholar
  32. 32.
    H. Risken, The Fokker-Planck Equation (Methods of Solution and Applications, Springer, New York, 1989)CrossRefGoogle Scholar
  33. 33.
    C. Greiner: Interpretation Thermischer Feldtheorie mit Hilfe von Langevin-Prozessen, (in German), Habilitation thesis, Justus-Liebig University Giessen (1999)Google Scholar
  34. 34.
    E. Cortés, B.J. West, K. Lindenberg, On the generalized Langevin equation: classical and quantum mechanical. J. Chem. Phys. 82, 2708 (1985)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    T.S. Biró, G. Purcsel, G. Györgyi, A. Jakovác, Z. Schram, Power-law tailed spectra from equilibrium. Nucl. Phys. A 774, 845 (2006)ADSCrossRefGoogle Scholar
  36. 36.
    T.S. Biro, A. Jakovac, Power-law tails from multiplicative noise. Phys. Rev. Lett. 94, 132302 (2005)ADSCrossRefGoogle Scholar
  37. 37.
    Walter Hans Schottky, Über spontane Stromschwankungen in verschiedenen Elektrizttsleitern. Annalen der Physik 57, 541 (1918)CrossRefGoogle Scholar
  38. 38.
    R. Allard, J. Faubert, D.G. Pelli (eds.), Using noise to characterize vision (Frontiers in Psychology, Frontiers Media SA, 2016)Google Scholar

Copyright information

© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.H.A.S. Wigner Research Centre for PhysicsBudapestHungary
  2. 2.Institute of PhysicsRoland Eötvös UniversityBudapestHungary

Personalised recommendations