From Planck to Ramanujan : a quantum 1 / f noise in equilibrium

Michel Planat

Journal de théorie des nombres de Bordeaux (2002)

  • Volume: 14, Issue: 2, page 585-601
  • ISSN: 1246-7405

Abstract

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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 ˜ k T = π 2 6 x with x = h ν k T in contrast to the Planck energy E ˜ = k T . 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.

How to cite

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Planat, Michel. "From Planck to Ramanujan : a quantum $1/f$ noise in equilibrium." Journal de théorie des nombres de Bordeaux 14.2 (2002): 585-601. <http://eudml.org/doc/93865>.

@article{Planat2002,
abstract = {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 $\sigma _k(n)$ 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 $\frac\{\tilde\{E\}\}\{kT\} = \frac\{\pi ^2\}\{6x\}$ with $x = \frac\{h\nu \}\{kT\}$ in contrast to the Planck energy $\tilde\{E\} = kT$. 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/ \nu $ 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.},
author = {Planat, Michel},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {2},
pages = {585-601},
publisher = {Université Bordeaux I},
title = {From Planck to Ramanujan : a quantum $1/f$ noise in equilibrium},
url = {http://eudml.org/doc/93865},
volume = {14},
year = {2002},
}

TY - JOUR
AU - Planat, Michel
TI - From Planck to Ramanujan : a quantum $1/f$ noise in equilibrium
JO - Journal de théorie des nombres de Bordeaux
PY - 2002
PB - Université Bordeaux I
VL - 14
IS - 2
SP - 585
EP - 601
AB - 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 $\sigma _k(n)$ 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 $\frac{\tilde{E}}{kT} = \frac{\pi ^2}{6x}$ with $x = \frac{h\nu }{kT}$ in contrast to the Planck energy $\tilde{E} = kT$. 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/ \nu $ 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.
LA - eng
UR - http://eudml.org/doc/93865
ER -

References

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  8. [8] P.H. Handel, Nature of 1/f frequency fluctuations in quartz crystal resonators. Solid State Electronics22 (1979), 875-876. 
  9. [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. 
  10. See also P.H. Handel, Infrared divergences, radiative corrections, and Bremsstrahlung in the presence of a thermal-equilibrium background. Phys. Rev.A38 (1988), 3082-3085. 
  11. [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). Zbl0025.10505MR4860
  12. [11] J. Kestin, J.R. Dorfman, A Course in Statistical Thermodynamics. Academic Press, New York, 1971. 
  13. [12] B.W. Ninham, B.D. Hughes, N.E. Frankel, M.L. Glasser, Möbius, Mellin and mathematical physics. Physica A186 (1992), 441-481. MR1176719
  14. [13] A. Pais, "Subtle is the Lord... " The Science and the Life of Albert Einstein, Oxford Univ. Press, Cambridge, 1982. Zbl0525.01017MR690419
  15. [14] H. Rademacher, Topics in Analytic Number Theory. Springer Verlag, New York, 1973. Zbl0253.10002MR364103
  16. [15] A. Van Der Ziel, Noise in Measurements. John Wiley and Sons, New York, 1976. 
  17. [16] A. Weil, Elliptic functions according to Eisenstein and Kronecker. Springer Verlag, Berlin, 1976. Zbl0318.33004MR562289

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