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

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.

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.

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Planat, Michel. From Planck to Ramanujan : a quantum $1/f$ noise in equilibrium. Journal de théorie des nombres de Bordeaux, Tome 14 (2002) no. 2, pp. 585-601. http://archive.numdam.org/item/JTNB_2002__14_2_585_0/

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