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

Journal de théorie des nombres de Bordeaux (2002)

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

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topPlanat, 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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- 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.
- [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
- [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 A186 (1992), 441-481. MR1176719
- [13] A. Pais, "Subtle is the Lord... " The Science and the Life of Albert Einstein, Oxford Univ. Press, Cambridge, 1982. Zbl0525.01017MR690419
- [14] H. Rademacher, Topics in Analytic Number Theory. Springer Verlag, New York, 1973. Zbl0253.10002MR364103
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- [16] A. Weil, Elliptic functions according to Eisenstein and Kronecker. Springer Verlag, Berlin, 1976. Zbl0318.33004MR562289

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