Quantum $1/f$ Noise in Equilibrium: from Planck to Ramanujan

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 $σ_k(n)$ 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 $\frac{\tilde{E}}{kT}=\frac{π^2}{6 x}$ with $x=\frac{h ν}{kT}$ 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 $1/ν$ 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.