AI 中文总结
研究1/f噪声出现条件，通过分析扩散方程找到例子，得出其出现的充要条件，即由两个常数频率表征且特定频率范围无其他常数参数，该条件适用于任何系统，解释了1/f噪声普遍性及部分其他噪声系统。
AI 中文摘要
虽然1/f噪声普遍存在且在各种系统中都能发现，但其物理机制仍不确定。通过对普通扩散方程的分析研究，我们找到了1/f噪声的另一个例子。该例子的公式与流体湍流中关于标度的现有知识，暗示了任何平稳1/f噪声出现的充要条件。即噪声需由两个常数频率$f_{\rm low} \ll f_{\rm high}$表征。对于从$f = f_{\rm low}$到$f_{\rm high}$的频率范围，除噪声平均幅度外，还需无其他常数参数。在$f_{\rm low} \ll f \ll f_{\rm high}$时，噪声渐近地按1/f标度。我们的条件具有统计性且简单，适用于任何系统，解释了1/f噪声的普遍性，还可通过类似于湍流的间歇性应用于一些具有$\alpha \ne 1.0$的1/f$^{\alpha}$噪声的系统。
英文摘要
While $1/f$ noise is ubiquitous and has been found in various systems, its physics remains uncertain. From an analytical study of an ordinary diffusion equation, we find an additional example of the $1/f$ noise. The formula for this example, together with existing knowledge about scaling in fluid turbulence, implies a necessary and sufficient condition for the occurrence of any stationary $1/f$ noise. That is, the noise needs to be characterized by two constant frequencies of $f_{\rm low} \ll f_{\rm high}$. For a frequency range from $f = f_{\rm low}$ to $f_{\rm high}$, it is further needed that, except for the mean amplitude of the noise, there is no other constant parameter. Then, at $f_{\rm low} \ll f \ll f_{\rm high}$, the noise scales asymptotically as $1/f$. Being statistical and simple, our condition applies to any system and hence explains the ubiquity of the $1/f$ noise. It is also applicable to some systems with noise of $α\ne 1.0$ for $1/f^α$, via intermittency analogous to that of the turbulence.
Comments7 pages, to appear in Physical Review E
Journal refPRE, 114, 024106 [2026]
