Class numbers in the imaginary quadratic field and the 1/f noise of an electron gas

Abstract

Partition functions $Z(x)$ of statistical mechanics are generally approximated by integrals. The approximation fails in small cavities or at very low temperature, when the ratio $x$ between the energy quantum and thermal energy is larger or equal to unity. In addition, the exact calculation, which is based on number theoretical concepts, shows excess low frequency noise in thermodynamical quantities, that the continuous approximation fails to predict. It is first shown that Riemann zeta function is essentially the Mellin transform of the partition function $Z(x)$ of the non degenerate (one dimensional) perfect gas. Inverting the transform leads to the conventional perfect gas law. The degeneracy has two aspects. One is related to the wave nature of particles: this is accounted for from quantum statistics, when the de Broglie wavelength exceeds the mean distance between particles. We emphasize here the second aspect which is related to the degeneracy of energy levels. It is given by the number of solutions $r_3(p)$ of the three squares diophantine equation, a highly discontinuous arithmetical function. In the conventional approach the density of states is proportional to the square root of energy, that is $r_3(p)\simeq 2 πp^{1/2}$. We found that the exact density of states relates to the class number in the quadratic field $Q(\sqrt{-p})$. One finds $1/f$ noise around the mean value.