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f fluctuations under acoustic cavitation of liquids
ARTICLE IN PRESS
Physica A 364 (2006) 63–69 www.elsevier.com/locate/physa
1=f fluctuations under acoustic cavitation of liquids V.N. Skokov, V.P. Koverda, A.V. Reshetnikov, A.V. Vinogradov Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, 620016 Ekaterinburg, Russia Received 26 May 2005; received in revised form 9 September 2005 Available online 18 October 2005
Abstract A regime of a generation of fluctuations with 1=f power spectra under cavitation of liquids in an ultrasonic field has been found experimentally. The acoustic cavitation is accompanied by the formation of various spatial structures and scaleinvariant distribution functions of fluctuations. It has been shown that local fluctuations can have non-Gaussian distribution. A two-dimensional distributed parameter system of stochastic equations describing interacting nonequilibrium phase transitions has been investigated numerically. It has been shown that the system in a wide range of changing initial conditions and the intensity of external noise is characterized by the 1=f behavior of power spectra and scale-invariant distribution functions of fluctuations. The results of numerical simulation agree qualitatively with the experimental data. r 2005 Elsevier B.V. All rights reserved. Keywords: Self-organized criticality; Acoustic cavitation; 1=f fluctuations; Noise; Power spectra; Nonequilibrium phase transitions
1. Introduction The propagation of high-intensity sound waves in a liquid gives rise to acoustic cavitation. The complicated character of interaction of forming vapor-gas bubbles between each other and with an acoustic field may result in formation of various spatial structures. Dendritic structures of vapor-gas bubbles which look like fractal clusters have been found experimentally at the origination of standing waves in an ultrasonic field. Such structures were denoted as Acoustic Lichtenberg Figures [1,2]. Ref. [2] suggests a theoretical model by which in a system of cavitation bubbles in an acoustic field there arises an instability leading to spatial self- organization. If the dimensions of ultrasonic radiator are commensurable with the sound-wave length, quasi-two-dimensional clusters may form in the vicinity of the radiator surface [3]. The complicated character of interaction of cavitation cavities between one another and with acoustic waves in an experimental cell may result in the formation of bistability and transitions between stationary states [4,5].
Corresponding author. Fax: +7 3432 678 800.
E-mail address: [email protected] (V.N. Skokov). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.09.045
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The formation of cavitation clouds may be regarded as nonequilibrium phase transitions in a complicated system of interacting cavitation cavities and acoustic waves. In an acoustic field there forms a stationary random process with nonequilibrium phase transitions, whose power spectrum may have the 1=f form [6]. Random processes with the power spectrum inversely proportional to frequency are characterized by the scale-invariant distribution of fluctuations. The scale invariance may be connected with the critical behavior or self-organization in complicated systems [7]. There are a lot of attempts to explain a possible mechanism of generation of scale-invariant fluctuations on the basis of the conception of self-organized criticality [7,8], which is used for describing complicated systems with developed fluctuations. Investigations of random processes in crisis regimes of liquid boiling have shown that fluctuations with a 1=f spectrum and self-organization of a critical state may arise as a result of interaction of subcritical and supercritical nonequilibrium phase transitions in the presence of white noise [9–11]. The extended critical behavior of fluctuations in this case is characterized by self-similar distribution of the probability density, that does not vary with time [11]. This paper presents the results of experimental study of fluctuations under the liquid cavitation in an ultrasonic field. Also, it generalizes the previously suggested theoretical model of 1=f fluctuations at nonequilibrium phase transitions to two-dimensional space-distributed system [12,13]. 2. Experiment Experiments were carried out with the use of a source of ultrasonic vibrations with a frequency of 22 kHz. The radiator was placed in an optical cell with water or glycerin. The diameter of the radiator was about 1.5 cm. Cavitation was caused by an increase in the radiator power. With changed radiation intensity in the cell resonance phenomena were observed which led to a change in the pattern created by interacting cavitation bubbles. At a low radiator power, separate cavitation centers originated at its surface (Fig. 1a). As a result of mutual attraction, bubbles lined up in chains. The number of cavitation centers increased with increasing power. The cooperative interaction of bubbles in the vicinity of the radiator surface resulted in the formation of aggregate reminiscent of fractal clusters (Fig. 1b). A vapor-gas flow was directed from the periphery to the center of a cluster. Separate clusters were able to break away from the surface and pass into the liquid volume. In experiments with glycerin, forming aggregates were more long-lived and have a more contrasting appearance. With a further increase in the power, the interacting cavitation centers formed a critically fluctuating surface (Fig. 1c). The dynamics of fluctuations in a cavitation cloud was investigated by the method of laser photometry. A laser beam was passed through the optical cell with the liquid under investigation. The diameter of the beam was about 1 mm. The intensity of the passed laser radiation was registered with a photodiode, assigned a numerical value and stored in a computer. Power spectra were determined from time series by Fast Fourier Transform method. To investigate the spectra of a random process under cavitation the laser beam was passed through different parts of the cavitation region. The results obtained depended slightly on the part of the cavitation cloud into which the beam was directed. In the initial stage of cavitation the power spectrum of the photocurrent fluctuations as well as the spectrum of acoustic emission, in the low-frequency region, had the form of white-noise spectrum. With an increase in the radiator power and a certain variation of the frequency the fluctuation intensity increased abruptly, and transitions between two levels of the oscillations were observed. Fig. 2 shows the power spectrum for fluctuations in the indicated regime. From the figure it is seen that the 1=f behavior is traced over more than four orders of the power change. Fig. 3a (1) gives an experimental time series and Fig. 3b (1) the fluctuation distribution function. The bimodal character of the distribution function showed itself clearly under the scale transformations of fluctuations. Roughened time series were created from those measured experimentally by means of averaging by a certain time scale t in accordance with the formula yðtÞ j ¼
1 t
tðjþ1Þ1 X i¼tj
xi ;
0pjpN=t,
(1)
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Fig. 1. Structures of cavitation clouds in the vicinity of the radiator surface at different ultrasonic power (see text).
where the stochastic variable xi denotes the photocurrent intensity. The used roughening procedure is a digital smoothing and it was used for multiscale analysis of time series [14]. The sequence of the roughened time series at different values of t and the corresponding distribution functions are presented in Fig. 3 (2–4). From the figure it can be seen that with increasing coefficient of scale transformations the roughened time series and their distribution functions become the same (cease to depend on t). In other words, the distribution of fluctuations becomes scale-invariant. It should be noted that the mere presence of short-wave high-amplitude surges does not affect the scale invariance of the 1=f behavior of power spectra as they involve a very small energy. The bimodal character of distribution functions was apparently connected with the fact that the part of cavitation cloud passed by laser beam (diameter about 1 mm) was a single source or several coherent sources of 1=f fluctuations. To observe more number of sources we used a magnification of the size of probed region with spreading and converging lenses (so that integral intensity of the beam was saved). In this case a decrease of fluctuation amplitude was observed under conservation of 1=f behavior of the spectrum. The bimodal character of distribution functions under a magnification in cross-section of laser beam was manifested much more faintly. It was connected with the fact that the increased beam diameter involved a number of single sources of 1=f fluctuations. In the case that a great number of interacting cavitation centers is observed, a simplified description of the fluctuation dynamics is possible. For this purpose a version of the self-organized criticality conception at interacting nonequilibrium phase transitions can be used [10–13].
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Fig. 2. The power spectra of fluctuations under acoustic cavitation (dashed line corresponds to 1=f dependence).
Fig. 3. The sequence of the roughened time series (a) and corresponding sequence of the distribution functions of fluctuations (b) at different values of coefficient of scale transformations t: 1–1, 2–30, 3–100, 4–200.
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3. The theoretical model In the case of a space extended system, we assume that the process is described by two stochastic equations of reaction–diffusion type [12,13]: qf ¼ D1 r2 f þ Q1 ðf; cÞ þ G1 ðx; tÞ, qt qc ¼ D2 r2 c þ Q2 ðf; cÞ þ G2 ðx; tÞ, ð2Þ qt where f; c are the dynamic variables (order parameters), D1 ; D2 are diffusion coefficients, G1 ; G2 are the Gaussian d-correlated noises with the identical variants. System (2) is sufficiently general and describes nonequilibrium phase transitions in many physical and chemical systems. We consider a case in which the typical space scales of the order parameter change are quite different, that is, D1 =D2 b1. The source functions characterizing the interaction of the order parameters are approximated with the expressions: Q1 ¼ fc2 þ c; Q2 ¼ f2 c þ f. In this case, the system of stochastic equations takes the following form: qf ¼ D1 r2 f fc2 þ c þ G1 ðx; tÞ, qt qc ¼ f2 c þ f þ G2 ðx; tÞ. qt System (3) describes random walks for the potential: F ¼ 12 f2 c2 fc þ 12 D1 ðrfÞ2 .
ð3Þ
(4)
In order to gain a better understanding of the physical meaning of potential (4), we perform a linear transformation: f ¼ Z y; c ¼ Z þ y, which corresponds to the turn of the phase plane. For the new variables, the expression of the potential takes the form F ¼ 12 Z4 Z2 þ 12 y4 þ y2 Z2 y2 þ 12 D1 ðrZÞ2 þ 12 D1 ðryÞ2 D1 rZry.
(5)
From the structure of expression (5) one can see that the potential FðZ; yÞ corresponds to the interacting of a subcritical phase transition with the order parameter Z and a supercritical phase transition with the order parameter y. As was shown in Refs. [10,12,13] in the cases of lumped and one-dimensional systems Eqs. (3) generate stationary stochastic processes fðtÞ and cðtÞ, whose power spectra have frequency dependences 1=f and 1=f 2 , respectively. It is interesting to consider a behavior of Eqs. (3) in two-dimensional case and to investigate power spectra and scale invariant properties of spatial structures. In the case of a two-dimensional system the stochastic equations (3) may be written as follows: 2 qf q f q2 f ¼D þ fc2 þ c þ G1 ðx; y; tÞ, qt qx2 qy2 qc ¼ cf2 þ f þ G2 ðx; y; tÞ. ð6Þ qt For numerical integration of Eqs. (6) use was made of the Eulerian difference scheme with periodic boundary and different initial conditions. The functions G1 ðx; y; tÞ and G2 ðx; y; tÞ were approximated by sets of normally distributed random numbers. The power spectra of the stochastic variable f in a wide range of variations of the governing parameters were of the 1=f form, accordingly the power spectra of the stochastic variable c had the 1=f 2 form. However, the power spectra of the variable 1=c were inversely proportional to frequency to the first power, i.e., 1=f . The distribution function of the variables in roughening the temporal and the spatial scale becomes scaleinvariant. This result is similar to the ones of investigating a lumped parameter system [9–11] and onedimensional distributed parameter systems [12,13]. The peculiarities of the investigation of a two-dimensional system manifest themselves in the possibility of obtaining spatial structures. Fig. 4a presents the spatial configuration of the stochastic variable f after 8192 integration steps Dt (Dt ¼ 0:1 is the number of steps in
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Fig. 4. The spatial configuration of the roughened stochastic variable f: (a) homogeneous initial conditions; (b) the growth of a cluster after setting the spatial inhomogeneity of the intensity of a random field.
the space 1024 1024) at homogeneous initial conditions. The pattern in Fig. 4 has been roughened over small-scale spatial fluctuations. The dark regions correspond to positive values of the variable f, the light ones to negative values. From the figure it is seen that in the system there forms a self-similar spatial structure, which corresponds to a critical state. The structures obtained are a result of the system evolution, and their form had a weak dependence on the initial conditions and variations in the intensity of the external noise in a wide range. In setting the spatial inhomogeneity of the intensity of a random field the growth of a cluster is observed (Fig. 4b). It should be noted that unlike other different diffusion models system (3) gives 1=f behavior of power spectra. At a high density of cavitation centers the pattern is not very sensitive to the details of interaction, therefore it is possible to use a simplified description of a state with system (3). At a lower density of cavitation centers the character of the bubbles interaction becomes more important. The system evolution in this case should be described with invoking hydrodynamics and nonlinear dynamics [1–5].
4. Conclusion Thus, the acoustic cavitation of liquids is accompanied by the formation of various spatial structures, the 1=f behavior of power spectra, a scale-invariant distribution function of fluctuations. The experimental results agree qualitatively with numerical simulation in the framework of the theory of 1=f fluctuations at nonequilibrium phase transitions in a space-distributed system. The results obtained have no need of a fine tuning of the controlling parameters and bear witness to the regime of self-organized criticality. Interacting simultaneous nonequlibrium phase transitions may be the underlying mechanism of 1=f noise. Nonequlibrium phase transitions are of frequent occurrence phenomena. That may be a reason why 1=f noise is so ubiquitous.
Acknowledgements The work was supported by the Russian Foundation for Basic Research (Grant no. 03-02-16215) and the Program of Basic Research of the Russian Academy of Sciences.
ARTICLE IN PRESS V.N. Skokov et al. / Physica A 364 (2006) 63–69
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
W. Lauterborn, E. Schmitz, A. Judt, Int. J. Bifurcation Chaos 3 (1993) 635. I. Akhatov, U. Parlitz, W. Lauterborn, Phys. Rev. E 54 (1996) 4990. A. Mousatov, C. Granger, B. Dubis, Ultrason. Sonochemisty 10 (2003) 191. A.O. Maximov, E.V. Sosedko, Tech. Phys. Lett. 29 (2003) 102. G.N. Sankin, V.S. Teslenko, Dokl. Phys. 48 (2003) 665. V.P. Koverda, V.N. Skokov, A.V. Reshetnikov, A.V. Vinogradov, Tech. Phys. Lett. 30 (2004) 944. H.J. Jensen, Self-organized Criticality, Cambridge University Press, New York, 1998. P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A 38 (1988) 364. V.P. Koverda, V.N. Skokov, V.P. Skripov, JETP 86 (1998) 953. V.P. Koverda, V.N. Skokov, Physica A 262 (1999) 376. V.P. Koverda, V.N. Skokov, Physica A 346 (2005) 203. V.N. Skokov, V.P. Koverda, A.V. Reshetnikov, JETP 89 (2001) 535. V.N. Skokov, A.V. Reshetnikov, V.P. Koverda, A.V. Vinogradov, Physica A 293 (2001) 1. M. Costa, A.L. Goldberger, C.-P. Peng, Phys. Rev. Lett. 89 (2002) 068102-1-4.
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Physica A 364 (2006) 63–69 www.elsevier.com/locate/physa
1=f fluctuations under acoustic cavitation of liquids V.N. Skokov, V.P. Koverda, A.V. Reshetnikov, A.V. Vinogradov Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, 620016 Ekaterinburg, Russia Received 26 May 2005; received in revised form 9 September 2005 Available online 18 October 2005
Abstract A regime of a generation of fluctuations with 1=f power spectra under cavitation of liquids in an ultrasonic field has been found experimentally. The acoustic cavitation is accompanied by the formation of various spatial structures and scaleinvariant distribution functions of fluctuations. It has been shown that local fluctuations can have non-Gaussian distribution. A two-dimensional distributed parameter system of stochastic equations describing interacting nonequilibrium phase transitions has been investigated numerically. It has been shown that the system in a wide range of changing initial conditions and the intensity of external noise is characterized by the 1=f behavior of power spectra and scale-invariant distribution functions of fluctuations. The results of numerical simulation agree qualitatively with the experimental data. r 2005 Elsevier B.V. All rights reserved. Keywords: Self-organized criticality; Acoustic cavitation; 1=f fluctuations; Noise; Power spectra; Nonequilibrium phase transitions
1. Introduction The propagation of high-intensity sound waves in a liquid gives rise to acoustic cavitation. The complicated character of interaction of forming vapor-gas bubbles between each other and with an acoustic field may result in formation of various spatial structures. Dendritic structures of vapor-gas bubbles which look like fractal clusters have been found experimentally at the origination of standing waves in an ultrasonic field. Such structures were denoted as Acoustic Lichtenberg Figures [1,2]. Ref. [2] suggests a theoretical model by which in a system of cavitation bubbles in an acoustic field there arises an instability leading to spatial self- organization. If the dimensions of ultrasonic radiator are commensurable with the sound-wave length, quasi-two-dimensional clusters may form in the vicinity of the radiator surface [3]. The complicated character of interaction of cavitation cavities between one another and with acoustic waves in an experimental cell may result in the formation of bistability and transitions between stationary states [4,5].
Corresponding author. Fax: +7 3432 678 800.
E-mail address: [email protected] (V.N. Skokov). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.09.045
ARTICLE IN PRESS V.N. Skokov et al. / Physica A 364 (2006) 63–69
64
The formation of cavitation clouds may be regarded as nonequilibrium phase transitions in a complicated system of interacting cavitation cavities and acoustic waves. In an acoustic field there forms a stationary random process with nonequilibrium phase transitions, whose power spectrum may have the 1=f form [6]. Random processes with the power spectrum inversely proportional to frequency are characterized by the scale-invariant distribution of fluctuations. The scale invariance may be connected with the critical behavior or self-organization in complicated systems [7]. There are a lot of attempts to explain a possible mechanism of generation of scale-invariant fluctuations on the basis of the conception of self-organized criticality [7,8], which is used for describing complicated systems with developed fluctuations. Investigations of random processes in crisis regimes of liquid boiling have shown that fluctuations with a 1=f spectrum and self-organization of a critical state may arise as a result of interaction of subcritical and supercritical nonequilibrium phase transitions in the presence of white noise [9–11]. The extended critical behavior of fluctuations in this case is characterized by self-similar distribution of the probability density, that does not vary with time [11]. This paper presents the results of experimental study of fluctuations under the liquid cavitation in an ultrasonic field. Also, it generalizes the previously suggested theoretical model of 1=f fluctuations at nonequilibrium phase transitions to two-dimensional space-distributed system [12,13]. 2. Experiment Experiments were carried out with the use of a source of ultrasonic vibrations with a frequency of 22 kHz. The radiator was placed in an optical cell with water or glycerin. The diameter of the radiator was about 1.5 cm. Cavitation was caused by an increase in the radiator power. With changed radiation intensity in the cell resonance phenomena were observed which led to a change in the pattern created by interacting cavitation bubbles. At a low radiator power, separate cavitation centers originated at its surface (Fig. 1a). As a result of mutual attraction, bubbles lined up in chains. The number of cavitation centers increased with increasing power. The cooperative interaction of bubbles in the vicinity of the radiator surface resulted in the formation of aggregate reminiscent of fractal clusters (Fig. 1b). A vapor-gas flow was directed from the periphery to the center of a cluster. Separate clusters were able to break away from the surface and pass into the liquid volume. In experiments with glycerin, forming aggregates were more long-lived and have a more contrasting appearance. With a further increase in the power, the interacting cavitation centers formed a critically fluctuating surface (Fig. 1c). The dynamics of fluctuations in a cavitation cloud was investigated by the method of laser photometry. A laser beam was passed through the optical cell with the liquid under investigation. The diameter of the beam was about 1 mm. The intensity of the passed laser radiation was registered with a photodiode, assigned a numerical value and stored in a computer. Power spectra were determined from time series by Fast Fourier Transform method. To investigate the spectra of a random process under cavitation the laser beam was passed through different parts of the cavitation region. The results obtained depended slightly on the part of the cavitation cloud into which the beam was directed. In the initial stage of cavitation the power spectrum of the photocurrent fluctuations as well as the spectrum of acoustic emission, in the low-frequency region, had the form of white-noise spectrum. With an increase in the radiator power and a certain variation of the frequency the fluctuation intensity increased abruptly, and transitions between two levels of the oscillations were observed. Fig. 2 shows the power spectrum for fluctuations in the indicated regime. From the figure it is seen that the 1=f behavior is traced over more than four orders of the power change. Fig. 3a (1) gives an experimental time series and Fig. 3b (1) the fluctuation distribution function. The bimodal character of the distribution function showed itself clearly under the scale transformations of fluctuations. Roughened time series were created from those measured experimentally by means of averaging by a certain time scale t in accordance with the formula yðtÞ j ¼
1 t
tðjþ1Þ1 X i¼tj
xi ;
0pjpN=t,
(1)
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Fig. 1. Structures of cavitation clouds in the vicinity of the radiator surface at different ultrasonic power (see text).
where the stochastic variable xi denotes the photocurrent intensity. The used roughening procedure is a digital smoothing and it was used for multiscale analysis of time series [14]. The sequence of the roughened time series at different values of t and the corresponding distribution functions are presented in Fig. 3 (2–4). From the figure it can be seen that with increasing coefficient of scale transformations the roughened time series and their distribution functions become the same (cease to depend on t). In other words, the distribution of fluctuations becomes scale-invariant. It should be noted that the mere presence of short-wave high-amplitude surges does not affect the scale invariance of the 1=f behavior of power spectra as they involve a very small energy. The bimodal character of distribution functions was apparently connected with the fact that the part of cavitation cloud passed by laser beam (diameter about 1 mm) was a single source or several coherent sources of 1=f fluctuations. To observe more number of sources we used a magnification of the size of probed region with spreading and converging lenses (so that integral intensity of the beam was saved). In this case a decrease of fluctuation amplitude was observed under conservation of 1=f behavior of the spectrum. The bimodal character of distribution functions under a magnification in cross-section of laser beam was manifested much more faintly. It was connected with the fact that the increased beam diameter involved a number of single sources of 1=f fluctuations. In the case that a great number of interacting cavitation centers is observed, a simplified description of the fluctuation dynamics is possible. For this purpose a version of the self-organized criticality conception at interacting nonequilibrium phase transitions can be used [10–13].
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Fig. 2. The power spectra of fluctuations under acoustic cavitation (dashed line corresponds to 1=f dependence).
Fig. 3. The sequence of the roughened time series (a) and corresponding sequence of the distribution functions of fluctuations (b) at different values of coefficient of scale transformations t: 1–1, 2–30, 3–100, 4–200.
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3. The theoretical model In the case of a space extended system, we assume that the process is described by two stochastic equations of reaction–diffusion type [12,13]: qf ¼ D1 r2 f þ Q1 ðf; cÞ þ G1 ðx; tÞ, qt qc ¼ D2 r2 c þ Q2 ðf; cÞ þ G2 ðx; tÞ, ð2Þ qt where f; c are the dynamic variables (order parameters), D1 ; D2 are diffusion coefficients, G1 ; G2 are the Gaussian d-correlated noises with the identical variants. System (2) is sufficiently general and describes nonequilibrium phase transitions in many physical and chemical systems. We consider a case in which the typical space scales of the order parameter change are quite different, that is, D1 =D2 b1. The source functions characterizing the interaction of the order parameters are approximated with the expressions: Q1 ¼ fc2 þ c; Q2 ¼ f2 c þ f. In this case, the system of stochastic equations takes the following form: qf ¼ D1 r2 f fc2 þ c þ G1 ðx; tÞ, qt qc ¼ f2 c þ f þ G2 ðx; tÞ. qt System (3) describes random walks for the potential: F ¼ 12 f2 c2 fc þ 12 D1 ðrfÞ2 .
ð3Þ
(4)
In order to gain a better understanding of the physical meaning of potential (4), we perform a linear transformation: f ¼ Z y; c ¼ Z þ y, which corresponds to the turn of the phase plane. For the new variables, the expression of the potential takes the form F ¼ 12 Z4 Z2 þ 12 y4 þ y2 Z2 y2 þ 12 D1 ðrZÞ2 þ 12 D1 ðryÞ2 D1 rZry.
(5)
From the structure of expression (5) one can see that the potential FðZ; yÞ corresponds to the interacting of a subcritical phase transition with the order parameter Z and a supercritical phase transition with the order parameter y. As was shown in Refs. [10,12,13] in the cases of lumped and one-dimensional systems Eqs. (3) generate stationary stochastic processes fðtÞ and cðtÞ, whose power spectra have frequency dependences 1=f and 1=f 2 , respectively. It is interesting to consider a behavior of Eqs. (3) in two-dimensional case and to investigate power spectra and scale invariant properties of spatial structures. In the case of a two-dimensional system the stochastic equations (3) may be written as follows: 2 qf q f q2 f ¼D þ fc2 þ c þ G1 ðx; y; tÞ, qt qx2 qy2 qc ¼ cf2 þ f þ G2 ðx; y; tÞ. ð6Þ qt For numerical integration of Eqs. (6) use was made of the Eulerian difference scheme with periodic boundary and different initial conditions. The functions G1 ðx; y; tÞ and G2 ðx; y; tÞ were approximated by sets of normally distributed random numbers. The power spectra of the stochastic variable f in a wide range of variations of the governing parameters were of the 1=f form, accordingly the power spectra of the stochastic variable c had the 1=f 2 form. However, the power spectra of the variable 1=c were inversely proportional to frequency to the first power, i.e., 1=f . The distribution function of the variables in roughening the temporal and the spatial scale becomes scaleinvariant. This result is similar to the ones of investigating a lumped parameter system [9–11] and onedimensional distributed parameter systems [12,13]. The peculiarities of the investigation of a two-dimensional system manifest themselves in the possibility of obtaining spatial structures. Fig. 4a presents the spatial configuration of the stochastic variable f after 8192 integration steps Dt (Dt ¼ 0:1 is the number of steps in
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Fig. 4. The spatial configuration of the roughened stochastic variable f: (a) homogeneous initial conditions; (b) the growth of a cluster after setting the spatial inhomogeneity of the intensity of a random field.
the space 1024 1024) at homogeneous initial conditions. The pattern in Fig. 4 has been roughened over small-scale spatial fluctuations. The dark regions correspond to positive values of the variable f, the light ones to negative values. From the figure it is seen that in the system there forms a self-similar spatial structure, which corresponds to a critical state. The structures obtained are a result of the system evolution, and their form had a weak dependence on the initial conditions and variations in the intensity of the external noise in a wide range. In setting the spatial inhomogeneity of the intensity of a random field the growth of a cluster is observed (Fig. 4b). It should be noted that unlike other different diffusion models system (3) gives 1=f behavior of power spectra. At a high density of cavitation centers the pattern is not very sensitive to the details of interaction, therefore it is possible to use a simplified description of a state with system (3). At a lower density of cavitation centers the character of the bubbles interaction becomes more important. The system evolution in this case should be described with invoking hydrodynamics and nonlinear dynamics [1–5].
4. Conclusion Thus, the acoustic cavitation of liquids is accompanied by the formation of various spatial structures, the 1=f behavior of power spectra, a scale-invariant distribution function of fluctuations. The experimental results agree qualitatively with numerical simulation in the framework of the theory of 1=f fluctuations at nonequilibrium phase transitions in a space-distributed system. The results obtained have no need of a fine tuning of the controlling parameters and bear witness to the regime of self-organized criticality. Interacting simultaneous nonequlibrium phase transitions may be the underlying mechanism of 1=f noise. Nonequlibrium phase transitions are of frequent occurrence phenomena. That may be a reason why 1=f noise is so ubiquitous.
Acknowledgements The work was supported by the Russian Foundation for Basic Research (Grant no. 03-02-16215) and the Program of Basic Research of the Russian Academy of Sciences.
ARTICLE IN PRESS V.N. Skokov et al. / Physica A 364 (2006) 63–69
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
W. Lauterborn, E. Schmitz, A. Judt, Int. J. Bifurcation Chaos 3 (1993) 635. I. Akhatov, U. Parlitz, W. Lauterborn, Phys. Rev. E 54 (1996) 4990. A. Mousatov, C. Granger, B. Dubis, Ultrason. Sonochemisty 10 (2003) 191. A.O. Maximov, E.V. Sosedko, Tech. Phys. Lett. 29 (2003) 102. G.N. Sankin, V.S. Teslenko, Dokl. Phys. 48 (2003) 665. V.P. Koverda, V.N. Skokov, A.V. Reshetnikov, A.V. Vinogradov, Tech. Phys. Lett. 30 (2004) 944. H.J. Jensen, Self-organized Criticality, Cambridge University Press, New York, 1998. P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A 38 (1988) 364. V.P. Koverda, V.N. Skokov, V.P. Skripov, JETP 86 (1998) 953. V.P. Koverda, V.N. Skokov, Physica A 262 (1999) 376. V.P. Koverda, V.N. Skokov, Physica A 346 (2005) 203. V.N. Skokov, V.P. Koverda, A.V. Reshetnikov, JETP 89 (2001) 535. V.N. Skokov, A.V. Reshetnikov, V.P. Koverda, A.V. Vinogradov, Physica A 293 (2001) 1. M. Costa, A.L. Goldberger, C.-P. Peng, Phys. Rev. Lett. 89 (2002) 068102-1-4.
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