Low-frequency fluctuations with $1/f^\alpha$ power spectrum in transient modes of water boiling on a wire heater

The investigation of transient modes of water boiling on a wire heater reveals the presence of random vibrations with the frequency dependence of power spectra of $S \sim f^{-\alpha}$, where the exponent $\alpha$ has values in the range $0.8 \le \alpha < 2$. Large-scale low-frequency fluctuations exhibiting the property of scale invariance, the duration of which is distributed by the power law $P \sim \tau^{-\beta}$, are present in experimental realizations of random processes describing thermal fluctuations. The properties of such fluctuation processes are described using two nonlinear stochastic differential equations which describe the interaction between different phase transitions. Relations of dynamic scaling are determined between the critical exponents which define the frequency dependence of the power spectra of fluctuations $\alpha$ and of the distribution function of the amplitudes of extreme low-frequency fluctuations $\beta$. It is demonstrated that the critical exponents are related by the relation $\alpha + \beta = 2$ both in the experiments and in the theoretical model of interacting phase transitions. The power spectra of fluctuations are determined in the experiments with greater simplicity and accuracy than the distribution function of extreme amplitudes. In the cases where only the spectral dependence of power spectra of fluctuations is known, the correlations between the exponents enable one to obtain information about the distribution of large-scale surges and estimate dangerous amplitudes.