From Planck to Ramanujan : a quantum 1/f noise in equilibrium
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 585-601

We describe a new model of massless thermal bosons which predicts an hyperbolic fluctuation spectrum at low frequencies. It is found that the partition function per mode is the Euler generating function for unrestricted partitions p(n). Thermodynamical quantities carry a strong arithmetical structure : they are given by series with Fourier coefficients equal to summatory functions σ k (n) of the power of divisors, with k=-1 for the free energy, k=0 for the number of particles and k=1 for the internal energy. Low frequency contributions are calculated using Mellin transform methods. In particular the internal energy per mode diverges as E ˜ kT=π 2 6x with x=hν kT in contrast to the Planck energy E ˜=kT. The theory is applied to calculate corrections in black body radiation and in the Debye solid. Fractional energy fluctuations are found to show a 1/ν power spectrum in the low frequency range. A satisfactory model of frequency fluctuations in a quartz crystal resonator follows. A sketch of the whole Ramanujan-Rademacher theory of partitions is reminded as well.

J'introduis un nouveau modèle de bosons thermiques sans masse ; il prédit, pour les fluctuations, un spectre hyperbolique aux basses fréquences. On trouve que la fonction de partition par mode est la fonction génératrice d'Euler pour le nombre de partitions p(n). Les quantités thermodynamiques ont une structure arithmétique profonde : elles sont données par des séries, dont les coefficients de Fourier sont les fonctions sommatoires σ k (n) des puissances des diviseurs de n, avec k=-1 pour l'énergie libre, k=0 pour le nombre de particules, et k=1 pour l'énergie interne. Les contributions de basse fréquence sont calculées par l'usage de transformées de Mellin. En particulier, l'énergie interne par mode diverge comme E ˜ k T = π 2 6 x avec x=hν kT, au contraire de l'énergie de Planck E ˜=kT. La théorie est appliquée à la correction de la loi de rayonnement du corps noir et au solide de Debye. Les fluctuations fractionnaires de l'énergie présentent un spectre en 1/ν aux basse fréquences. On en déduit un modèle satisfaisant pour les fluctuations d'un résonateur à quartz. On rappelle aussi les résultats essentiels de la théorie mathématique des partitions de Ramanujan-Rademacher.

@article{JTNB_2002__14_2_585_0,
     author = {Planat, Michel},
     title = {From Planck to Ramanujan : a quantum $1/f$ noise in equilibrium},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {585-601},
     zbl = {02184601},
     mrnumber = {2040695},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_585_0}
}
Planat, Michel. From Planck to Ramanujan : a quantum $1/f$ noise in equilibrium. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 585-601. http://www.numdam.org/item/JTNB_2002__14_2_585_0/

[1] D.A. Abbott, B.R. Davis, N.J. Phillips, K. Eshraghian, Quantum vacuum fluctuations, zero point energy and the question of observable noise. In: Unsolved Problems of Noise in Physics, Biology, Electronic Technology and Information Technology, eds Ch. R. Doering, L. B. Kiss and M. F. Shlesinger, World Scientific, 1997, 131-138.

[2] T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory (Second Edition). Springer Verlag, New York, 1990. | MR 1027834 | Zbl 0697.10023

[3] F. Bloch, A. Nordsieck, Note on the Radiation Field of the Electron. Phys. Rev. 52 (1937), 54-59. | JFM 63.1393.02 | Zbl 0017.23504

[4] H.B. Callen, T.A. Welton, Irreversibility and generalized noise. Phys. Rev. 83 (1951), 34-40. | MR 44778 | Zbl 0044.41201

[5] E. Elizalde, Ten Physical Applications of Spectral Zeta Functions. Springer Verlag Lecture Notes in Physics Vol. m35, Berlin, 1995. | MR 1448403 | Zbl 0855.00002

[6] P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: harmonic sums. Theoretical Computer Science 144 (1995), 3-58. | MR 1337752 | Zbl 0869.68057

[7] J.-J. Gagnepain, J. Uebersfeld, G. Goujon, P.H. Handel, Relation between 1/f noise and Q-factor in quartz resonators at room and low temperature, first theoretical interpretation. In: Proc. 35th Annual Symposium on Frequency Control, Philadelphia, 1981, 476-483.

[8] P.H. Handel, Nature of 1/f frequency fluctuations in quartz crystal resonators. Solid State Electronics 22 (1979), 875-876.

[9] P.H. Handel, Quantum 1/f noise in the presence of a thermal radiation backgrnund. In: Proc. II Int. Symp. on 1/f Noise, eds C. M. Van Vliet and E. R. Chenette, Orlando, 1980, 96-110.

See also P.H. Handel, Infrared divergences, radiative corrections, and Bremsstrahlung in the presence of a thermal-equilibrium background. Phys. Rev. A 38 (1988), 3082-3085.

[10] G.H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work. Cambridge Univ. Press, London, 1940 (reprinted by Chelsea, New York, 1962). | MR 4860 | Zbl 0025.10505

[11] J. Kestin, J.R. Dorfman, A Course in Statistical Thermodynamics. Academic Press, New York, 1971.

[12] B.W. Ninham, B.D. Hughes, N.E. Frankel, M.L. Glasser, Möbius, Mellin and mathematical physics. Physica A 186 (1992), 441-481. | MR 1176719

[13] A. Pais, "Subtle is the Lord... " The Science and the Life of Albert Einstein, Oxford Univ. Press, Cambridge, 1982. | MR 690419 | Zbl 0525.01017

[14] H. Rademacher, Topics in Analytic Number Theory. Springer Verlag, New York, 1973. | MR 364103 | Zbl 0253.10002

[15] A. Van Der Ziel, Noise in Measurements. John Wiley and Sons, New York, 1976.

[16] A. Weil, Elliptic functions according to Eisenstein and Kronecker. Springer Verlag, Berlin, 1976. | MR 562289 | Zbl 0318.33004