平衡态中的量子$1/f$噪声：从普朗克到拉马努金
Quantum $1/f$ Noise in Equilibrium: from Planck to Ramanujan
摘要：
我们描述了一个新的无质量热玻色子模型，预测在低频双曲线波动谱。结果表明，对于无限制分区$p(n)$，每个模式的配分函数是欧拉母函数.热力学量具有很强的算术结构：它们由级数给出，傅里叶系数等于除数幂的和函数$σ_k(n)$，其中$k=-1$表示自由能，$k=0$表示粒子数，$k=1$表示内能。使用梅林变换方法计算低频贡献。特别地，与普朗克能量$\tilde{E}=kT$相反，每个模式的内能发散为$\frac{\tilde{E}}{kT}=\frac{π^2}{6 x}$与$x=\frac{h ν}{kT}$。应用该理论计算了黑体辐射修正和德拜固体修正。分数能量波动被发现在低频范围内显示出$1/ν$功率谱。石英晶体谐振器中频率波动的满意模型如下。一个草图的整个拉马努金-拉德马赫理论的分区提醒以及。
英文摘要：
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.
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