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f spectrum
Accepted Manuscript Oscillations and waves in a spatially distributed system with a 1/f spectrum V.P. Koverda, V.N. Skokov
PII: DOI: Reference:
S0378-4371(17)30957-3 https://doi.org/10.1016/j.physa.2017.09.065 PHYSA 18670
To appear in:
Physica A
Received date : 26 May 2017 Revised date : 12 September 2017 Please cite this article as: V.P. Koverda, V.N. Skokov, Oscillations and waves in a spatially distributed system with a 1/f spectrum, Physica A (2017), https://doi.org/10.1016/j.physa.2017.09.065 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
is observed. f and spatial 1/k power spectra correspond to the maximum entropy. In a spatial distributed system there arises spatio-temporal stochastic resonance.
*Manuscript Click here to view linked References
Oscillations and W aves in a Spatially Distributed System with a 1/ f Spectrum V.P. Koverda1 and V.N. Skokov Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, 620016, Ekaterinburg, Russia A spatially distributed system with a 1/f power spectrum is described by two nonlinear stochastic equations. Conditions for the formation of auto-oscillations have been found using numerical methods. The formation of a 1/f and 1/ k spectrum simultaneously with the formation and motion of waves under the action of white noise has been demonstrated. The large extreme fluctuations with 1/f and 1/k spectra correspond to the maximum entropy, which points to the stability of such processes. It is shown that on the background of formation and motion of waves at an external periodic action there appears spatio-temporal stochastic resonance, at which one can observe the expansion of the region of periodic pulsations under the action of white noise.
PACS: 05.40.-a; 05.40.Gg; 05.40.Jc; 05.40.Ca; 05.65.+b; 05.70.Fh
Keywords: nonlinear stochastic equations, 1/f noise, nonequilibrium phase transitions, spatiotemporal stochastic resonance, maximum entropy.
1
Corresponding author. e-mail address: [email protected]
1. Introduction Fluctuation phenomena are at the basis of many physical processes. In equilibrium macroscopic systems with a large number of structural elements the probability of large fluctuations is low and, as a rule, they have no pronounced effect on the system behavior. At the same time, it is well known that many processes are accompanied by considerable deviations from average values (e.g. [1-7]). Such macroscopic fluctuations may arise in complex statistical systems which are far from thermodynamic equilibrium and show up as extreme pulsations of the process parameters. In [8-11] extreme pulsations are revealed in critical and transient processes of heat and mass transfer, such as boiling crisis, explosive boiling-up in liquid jets and ultrasonic induced cavitations. A characteristic feature of large fluctuations is a scale invariant probability density
has a power form, as distinct from an exponential relaxation of fluctuations in equilibrium systems. Such a situation is observed at the thermodynamic critical point of a liquid vapor phase transition. The scale invariance of fluctuations of thermodynamic quantities close to the critical point is determined by the conditions of convergence of properties of different phases and requires a fine tuning and large relaxation times [12, 13]. As distinct from the thermodynamic critical point, nonequilibrium processes with large fluctuations show their stable spatial and temporal scale invariance without a fine tuning of the parameters. Therefore the origin of extreme large fluctuations is often connected with the concept of self-organized criticality [1, 14], which describes the avalanche dynamics and is used for demonstrating a critical behavior in a large number of computer models. Another distinctive feature of large fluctuations is the 1/f behavior of power spectra. The energy of fluctuations with a 1/ f power spectrum accumulates on low frequencies, therefore large scale surges may be observed in a system [1, 3, 15]. Random processes with extremely large
2
fluctuations are often described with the use of analogies with models of self-organized criticality [1, 14], turbulent flows [16], and fractional integration of white noise [17]. In our previous works it was shown that random processes with extreme large fluctuations may arise in interacting phase transitions under the action of white noise [10, 18]. In this case, on the achievement of the critical value of the noise intensity, one can observe in the system a noise induced transition, and the power spectrum of fluctuations takes the 1/f type. To the state of a system with a 1/f power spectrum corresponds the maximum entropy, which points to the stability of random processes with such a spectrum [19]. At an external periodic action the stochastic resonance may arise in a system, which shows up in an increase of sensitivity of a nonlinear system to an external periodic action in a presence of a noise [20-24]. In this case the amplitude of periodic oscillations increases many times with an addition of a noise. In [25] it is shown that a combined action of a noise and a periodic force on coupled phase transitions may lead to stochastic resonance. Stochastic resonance increases with a decrease in the frequency of the periodic force. The present paper examines the variant of a system of two nonlinear stochastic equations simulating coupled phase transitions in a spatially distributed system and in which it is possible to observe the spontaneous formation and propagation of waves simultaneously with the formation of the temporal 1/f and spatial 1/k power spectra. Section 1 introduces the variant of the spatial distributed system in which nonlinear effects connected with extreme large fluctuations are considered. Section 2 shows the conditions of formation and growth of waves in the system. The stability of 1/f noise and 1/k spectra is analyzed in section 3. Section 4 describes the properties of spatial- temporal stochastic resonance. 1. L arge fluctuations with a 1/f power spectrum at coupled phase transitions In consideration of random processes which describe coupled phase transitions with extremely large fluctuations use is made of a system of nonlinear stochastic equations [10, 18]:
3
d 2 1 ( t ), dt d 2 2 2 ( t ), dt
where
and
(1)
are dynamic variables. 1 and 2 are Gaussian white noises with
k ( t ) 0 , k (t )l (t ) 2 k ,l (t t ), k , l 1,2 and is their amplitude. The criticality of system (1) correspond to that level of white noise which correspond to a noise-induced transition [18]. The power spectrum S of the random process ( t ) described by Eqs. (1) has a 1/f type in a wide range of white noise intensity, i.e. S is proportional to 1 f , and the power spectrum S of the random process ( t ) is proportional to 1 f 2 . The lumped system (1) may be generalized for the spatially distributed case [26]:
2 2 D 2 1 ( x, t ),
t
x
2 2 ( x, t ),
t
where k ( x, t ) 0 , k ( x, t )l ( x , t ) 2 k ,l ( x x ) ( t t ), k , l 1,2 and
(2)
D - is the
diffusion coefficient. The statistics of fluctuations and the scaling relations demonstrated by the spatially distributed system (2) are similar to those of the lumped system (1). In the spatially distributed system (2), on the achievement of a sufficiently high intensity of white noise, one can observe noise-induced synchronization which is look like a critical phase transition [27]. For a demonstration and a consideration of other nonlinear effects connected with extreme fluctuations in hydrodynamics and thermal physics, such as a spontaneous formation of waves under the action of white noise, and spatio-temporal stochastic resonance, the different generalization of a lumped system has been taken in the present paper:
2
2 D 2 v 1 ( x, t ),
t
x
x
2
2 2 D 2 v 2 ( x, t ),
t
x
x
(3)
4
where the designations of system (2) have been retained, and v is the drift velocity. The diffusion coefficient and also the drift velocity have been taken equal in one and the other equation of system (3). Calculation show that at different values of the diffusion coefficient and the drift velocity in the equations of system (3) the stationary solution is determined by the largest D and
v . In this case, only the time of establishment of a stationary state increases. Therefore, we have taken
diffusion coefficients and drift velocities equal in both equations. Qualitatively this
approximation does not influence to observed nonlinear phenomena. The deterministic parts of the equations of system (3) differ only in the coefficient 2 before the linear term in the second equation. This makes the second equation of system (3) the governing, as in the lumped system (1) [19]. 2. W ave formation and propagation For numerical integration the modified Euler scheme is used [26, 28] and system (3) is written as follows:
D D i , j 1 t i , j 1 t i , j t v i , j t v i , j 1 t )[1 ( i , j )2 t D t ]1 2 2 0.5 e ( pi ) j ( t ) ,
i 1, j ( i , j
D D i 1, j ( i , j i , j 1 t i , j 1 t 2 i , j t v i , j t v i , j 1 t )[1 ( i , j )2 t D t ]1 2 2 0.5 e ( qi ) j ( t ) ,
(4)
where pi and qi are sequences of Gaussian random numbers with a zero mean and a unit standard deviation. These Gaussian random numbers simulate white noise. The variance of the random numbers e 2 relates to the white noise intensity 2 by the following relations:
e 2 2t . The spatial sample x is expressed in terms of the time integration step t and included in the diffusion coefficient D and the drift velocity v . Therefore, the diffusion coefficient is proportional to the square of the integration step, i.e. D ~ ( t ) 2 , and the drift velocity is proportional to the first power of the integration step, i.e. v ~ t . This should be taken into account in changing the integration step t . 5
In numerical calculations use was made of periodic boundary conditions. In the spatially distributed system (3) there are first temporal and the second spatial derivative, therefore for the achievement of a stability solution in numerical integration of system (4) there must be more temporal points than spatial one. Integration was carried out for a system containing 128 space points at 4096 time integration steps t 0.1 . The diffusion coefficient D and the drift velocity
v were in different calculations in the range from 0.3 to 1. If
we
choose,
as
the
initial
conditions,
0, j A cos(2 k0 jK 1 )
and
0, j A cos(2 k0 jK 1 ) , then, irrespective of the value of the amplitude A , the wave length adjust to the wavenumber k0 given by the initial conditions. This is observed both in the absence of white noise and at its low intensities at the expense of synchronization, which may arise in nonlinear systems. The value of k0 determines the initial number of waves. The wave propagation in the absence of noise is demonstrated in Fig 1. This figure shows time and space series of the variables and obtained by the numerical integration in the absence of noise. To make it more obvious, the contour plots are shown in the figures. Fig. 2 shows the phase trajectory in the plane at one of a spatial point in the absence of noise. For initial conditions close to zero and when the intensity of noise increases, one can observe a spontaneous formation and the growth of waves. This process is shown in Fig.3. The number of waves is limited, for the calculation parameters chosen the number of waves is from 4 to 6. With a further increase in the white noise intensity the number of waves begins to decrease at the cost of their unification into larger ones. In this case the maximum of a power spectrum (both temporal and spatial) determined by the number of waves in the system increases and moves into the region of low frequencies. This continues until one wave remains. The existing value of the noise amplitude is in this case e 1 . Figure 4 shows the dependence of the maximum of the power spectrum on the noise amplitude for different initial numbers of waves. The nonmonotonic of the dependence in Fig.4 is caused by the spontaneous processes of wave 6
formation and coalescence of waves. The process of wave coalescence and, accordingly, the decrease of the characteristic frequency corresponding to the spectrum maximum can be seen well in the upper insert (Fig.4). This plot shows a step-like decrease in the frequency with increasing noise amplitude, which corresponds to the enlargements of the waves. 3. M aximum entropy and 1/ f spectra On the achievement by the white noise intensity of a critical value ( e 1 ) the frequency power spectra of variables and become inversely proportional to the frequency and the square of frequency, i.e. S ( f ) ~ 1 / f and S ( f ) ~ 1 / f 2 , as in the lumped system (1). The spatial power spectra also obey similar dependences: S ( k ) ~ 1 / k and S ( k ) ~ 1 / k 2 , where we have an inversely proportional dependence on the wave number and the wave number squared. Fig.5 shows such dependences of temporal and spatial power spectra of the variables and obtained from numerical solutions (4) by FFT method. The probability density function P ( ) found from the solution of system (3) is shown in Fig.6 (curve1). Careful analysis shows that the probability density function P ( ) decreases at large by the power law [19]. The probability density function P ( ) has two maxima (curve 2 in Fig.6), but at large values of the probability density function decreases in the same way as for the Gaussian function. The problem of stability of extreme low-frequency fluctuations is of significance. As a criterion of the process stability may be chosen the maximum entropy principle. The solutions of stochastic equations are used to find the probability density functions of variables from which the Shannon informational entropy is determined [29]:
H Pn log( Pn ).
(5)
n
Probability density functions should be normalized:
n Pn 1 .
The index n refers to the
sequence of sampling of the argument of the probability density functions. The Shannon entropy 7
has essential properties of the Gibbs statistical entropy, but is also valid not only for physical systems, but also for a wider class of social, biological, communication systems studied by statistical methods [30]. In statistical mechanics the Gibbs Shannon maximum entropy principle is used for the Gaussian fluctuations. The Shannon informational entropy is believed to be an inapplicable for random processes with a 1/ f power spectrum owing to the nonintegrability of the formula (5), as a probability density functions of 1/ f [30, 31]. To solve Tsallis [32] has suggested deforming the logarithmic function in the expression for the entropy. The Tsallis entropy is determined by the expression
HT
1 q P n 1 . 1 q n
(6)
The use of the entropy introduced by Renyi [33] has also been suggested for power probability density functions:
HR
1 ln P nq . 1 q n
(7)
Both entropies H T and H R contain the dependence on the parameter q , whose value determines the concrete position of the maximum entropy. In our opinion, a complex system cannot be characterized by a single distribution function of variables. A sufficiently detailed description of a complex system have to contain a system of nonlinear stochastic equations, which are in a master-slave hierarchy. It can be shown by analogy with system (1) that the second equation of system (3) is the governing one [19]. The Gaussian character of the function P ( ) at large values makes it possible to use this function for finding the Gibbs Shannon entropy (5). It was been found that the entropy H for the probability density function P ( 2 ) of the square of the variable calculated by formula (5) has a maximum. Fig.7 shows the calculated dependence of the Gibbs Shannon entropy H on the noise amplitude e (curve 1). The 8
position of the maximum of the dependence H ( e ) in Fig.7 corresponds to the critical amplitude of white noise, at which a noise-induced transition is observed in the system, and the temporal and spatial power spectra are respectively look like 1/f and 1/ k. A further increase in the white noise intensity leads to a decrease in the entropy and the destruction of the 1/ f and 1/k spectra. The value of the parameter q in the expression of Tsallis (6) and Renyi (7) may be determined from the condition that the position of maximum Gibbs Shannon entropy H ( e ) for governing equation and the distribution of coincides with the position of the maximum entropies of Tsallis H T ( e ) and Renyi H R ( e ) for the distribution of . The value of q 0.6 has been found, at which the position of the maximum entropy of Tsallis coincides with the position of the maximum Gibbs Shannon entropy. For the Renyi entropy the value of the parameter q found is also 0.6. The value of q , as well as the position of maximum entropy, does not depend on the integration step in expression (6) and (7). The results of calculations of
H T ( e ) and H R ( e ) depending on the noise amplitude are presented in Fig.7 (curves 2 and 3). Thus, if a distribution function decreases by the law of Gaussian distribution, it may be used in calculating the Gibbs Shannon entropy, whose maximum determines a stable process. For a di formulas of Tsallis (6) and Renyi (7), if it is possible to determine the value of the parameter q . 4. Spatio-temporal stochastic resonance External actions on a system lead to the complication of the process. In nonlinear systems, simulating coupled phase transitions, along with 1/f fluctuations, stochastic resonance may be observed, at which in the presence of noise the amplitude of the periodic force increases many times [25]. In the system (3) along with a stochastic resonance described in [25] there may arise a spatio-temporal stochastic resonance. spatio-temporal stochastic resonance is important for the pattern formation in nonlinear systems [34-36]. 9
At an external periodic action system (3) is rewritten as follows: 2 2 D 2 v 1 ( x, t ) A sin( t ), t x x 2 2 2 D 2 v 2 ( x, t ) A sin( t ), t x x
(8)
where A and are the amplitude and the frequency of an external periodic force. In the region of low frequencies of an external action in system (8), as in the lumped system [25] one can observed a nonmonotonic dependence of the maximum of spectral density
S max of the variables and on the white noise intensity. An increase in the power of the periodic signal in the presence of noise points to stochastic resonance in the system. Along with the temporal stochastic resonance in (8) the stochastic spatio-temporal resonance takes place. The spatio-temporal stochastic resonance is most clearly demonstrated by the effect of white noise on the expansion of the periodic pattern region. Fig.8 (1, 2) shows spatio-temporal pattern of the variables and in the absence of the noise. It is seen that at given parameters of the frequency and the amplitude of an external action the oscillations and the spatio-temporal pattern are absent at large times. The inclusion of white noise at a certain intensity ( e 0.2 0.3 ) and invariability of the other parameters increases the amplitude of oscillations. In this case the periodic oscillations are restored in the whole range of time. Such a restoration of oscillations is shown in Fig.8 (3, 4). Fig.9 shows the dependences of the maximum temporal and spatial spectra at the frequency of the external periodic action on the white noise amplitude e . The nonmonotonic dependence of the maximum of spectral density characterizing the power of the spatio temporal oscillations at the frequency of the external periodic action is connected with the stochastic spatio temporal resonance in the system. Conclusion A spatially distributed system, simulating coupled interacting phase transitions, is described by two nonlinear stochastic equations. In this system it is possible to observe a
10
spontaneous formation and propagation of waves under the action of white noise. If noise intensity is critical value, against the background of formation and motion of waves emerge 1/ f and 1/k power spectra of extreme large fluctuations. The distribution of extreme fluctuations corresponds to the maximum entropy, which points to their stability. At an external periodic force in a system one can observe spatio-temporal resonance, at which, along with an increase in the amplitude, an expansion of the region of the periodic oscillations and spatio temporal pattern. A cknowledgements The work was partially supported by the Russian Foundation for Basic Research (Grant no. 15-08-02210a) and the Research Program of the Ural Branch of the Russian Academy of Sciences (no 15-1-2-7). References 1.
P. Bak, How Nature Works. The Science of Self-Organized Criticality. Berlin: Springer, 1996.
2.
B. Mandelbrot, The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982.
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Yu.L. Klimontivich, Statistical Theory of Open Systems. Dordrecht: Kluwer, 1995.
4.
R. N. Mantegna and H. E. Stanley, Nature 46 (1995) 376.
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L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim Rev. Mod. Phys. 87 (2015) 593.
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H. Touchette, Phys. Rep. 478 (2009) 1.
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F. J. Corberi, Phys. A 48 (2015) 465003.
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V.N. Skokov, V.P. Koverda, V.P. Skripov, Cryogenics 37 (1997) 263.
9.
V.N. Skokov, V.P. Koverda, A.V. Reshetnikov, JETP Lett. 69 (1999) 636.
10. V.N. Skokov, V.P. Koverda, A.V. Reshetnikov, V.P. Skripov, N.A. Mazheiko , A.V. Vinogradov, Int. J. Heat Mass Transfer 46 (2003) 1879.
11
11. V.N. Skokov, V.P. Koverda, A.V. Reshetnikov, and A.V. Vinogradov, Physica A 364 (2006) 63. 12. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena Oxford, New York and Oxford, 1971. 13. H. E. Stanley, Rev. Mod. Phys. 71 (1999) 358. 14. P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A 38 (1988) 364. 15. M.B. Weissman, Rev. Mod. Phys.. 60 (1988) 537. 16. A.N. Kolmogorov, Proceedings of the USSR Academy of Sciences 30(1941) 299. 17. B.B. Mandelbrot., J.W. Van Ness. SLAM Rev. 10 (1968) 422. 18. V.P. Koverda, V.N. Skokov, Physica A 346 (2005) 203. 19. V.P. Koverda, V.N. Skokov, Physica A 391 (2012) 21. 20. R. Benzi, S. Sutera, A. Vulpiani, J. Phys. A 14 (1981) L453. 21. V.S. Anishchenko, A.B. Neiman, F. Moss, L. Schimansky-Geier, Uspekhi fizicheskikh nauk 169 (1999) 7. 22. Lindner B., Garcia-Ojalvo J., Neiman A., Schimansky-Geier L., Phys. Rep. 392 (2004) 321. 23. J.J. Collins, C.C. Chow, T.T Imhoff, Nature 376 (1995) 236. 24. D. Nozaki, J.J. Collins, Y. Yamamato, Phys. Rev. E 60 (1999) 4637. 25. V.P. Koverda , V.N. Skokov, Physica A 393 (2014) 173. 26. V.P. Koverda, V.N. Skokov, Physica A 390 (2011) 2468. 27. V.P. Koverda, V.N. Skokov, Physica A 452 (2016) 2126. 28. V.P. Koverda, V.N. Skokov, Physica A 388 (2009) 1804. 29. C. Shannon, Bell Syst. Tech. 27 (1948) 379. 30. A.G. Bashkirov, Theoret. Math. Phys. 149 (2006) 1559. 31. E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 32 (1983) 209. 32. C. Tsallis, J. Stat. Phys. 52 (1988) 479. 33. A. Renyi, Probability Theory. Amsterdam: North-Holland, 1970.
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34. F. Sagués, J.M. Sancho, J. García-Ojalvo, Rev. Mod. Phys. 79 (2007) 829. 35. J.M.R. Parrondo, C. van den Broeck, J. Buceta, F.J. de la Rubia, Physica A 224 (1996) 153. 36. J.M.G. Vilar, J.M. Rubi, Phys. Rev. Lett. 78 (1997) 2886.
13
C aptions to the F igures Fig.1. Spatio-temporal oscillations of variables (1) and (2) and wave propagation in the absence of noise. Fig.2. Phase trajectory in the plane at one of a spatial point in the absence of noise. Fig.3. Spontaneous formation and growth of waves with increasing amplitude of white noise: 1
e 0.2 ; 2 e 0.25 ; 3 e 0.3 . Fig.4. Dependence of the maximum of power spectra of the variable on the noise amplitude
e for different initial numbers of waves. Numbers show the initial numbers of waves. In the upper insert it is shown the dependence of the frequency corresponding to the maximum of the power spectrum of the variable on the noise amplitude e . Fig.5. Temporal and spatial spectra of variables of system (3): 1 temporal spectrum S ( f ) ; 2 temporal spectrum S ( f ) ; 3 spatial spectrum S ( k ) ; 4 spatial spectrum S ( k ) ; dash lines show the dependences ~ 1 f and ~ 1 k . Fig.6. Probability density functions for the variables: 1 P ( ) ; 2 P ( ) . Fig.7. Dependence of entropy on the noise amplitude: 1 Gibbs Shannon entropy H ( e ) ; 2 Tsallis entropy H T ( e ) for q 0.6 ; 3 Renyi entropy H R ( e ) for q 0.6 . Fig.8. Spatio-temporal pattern of the oscillations of the variable and : 1, 2 in the absence of white noise; 3, 4 under the action of white noise with the amplitude e 0.3 ; the contour plots are shown in the upper inserts. Fig.9. Dependencies of the maximum temporal and spatial power spectra at the frequency of the external periodic action on the white noise amplitude e : 1 maximum temporal spectra of
; 2 maximum temporal spectra of ; 3 maximum spatial spectra of ; 4 maximum spatial spectra of .
14
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PII: DOI: Reference:
S0378-4371(17)30957-3 https://doi.org/10.1016/j.physa.2017.09.065 PHYSA 18670
To appear in:
Physica A
Received date : 26 May 2017 Revised date : 12 September 2017 Please cite this article as: V.P. Koverda, V.N. Skokov, Oscillations and waves in a spatially distributed system with a 1/f spectrum, Physica A (2017), https://doi.org/10.1016/j.physa.2017.09.065 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
is observed. f and spatial 1/k power spectra correspond to the maximum entropy. In a spatial distributed system there arises spatio-temporal stochastic resonance.
*Manuscript Click here to view linked References
Oscillations and W aves in a Spatially Distributed System with a 1/ f Spectrum V.P. Koverda1 and V.N. Skokov Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, 620016, Ekaterinburg, Russia A spatially distributed system with a 1/f power spectrum is described by two nonlinear stochastic equations. Conditions for the formation of auto-oscillations have been found using numerical methods. The formation of a 1/f and 1/ k spectrum simultaneously with the formation and motion of waves under the action of white noise has been demonstrated. The large extreme fluctuations with 1/f and 1/k spectra correspond to the maximum entropy, which points to the stability of such processes. It is shown that on the background of formation and motion of waves at an external periodic action there appears spatio-temporal stochastic resonance, at which one can observe the expansion of the region of periodic pulsations under the action of white noise.
PACS: 05.40.-a; 05.40.Gg; 05.40.Jc; 05.40.Ca; 05.65.+b; 05.70.Fh
Keywords: nonlinear stochastic equations, 1/f noise, nonequilibrium phase transitions, spatiotemporal stochastic resonance, maximum entropy.
1
Corresponding author. e-mail address: [email protected]
1. Introduction Fluctuation phenomena are at the basis of many physical processes. In equilibrium macroscopic systems with a large number of structural elements the probability of large fluctuations is low and, as a rule, they have no pronounced effect on the system behavior. At the same time, it is well known that many processes are accompanied by considerable deviations from average values (e.g. [1-7]). Such macroscopic fluctuations may arise in complex statistical systems which are far from thermodynamic equilibrium and show up as extreme pulsations of the process parameters. In [8-11] extreme pulsations are revealed in critical and transient processes of heat and mass transfer, such as boiling crisis, explosive boiling-up in liquid jets and ultrasonic induced cavitations. A characteristic feature of large fluctuations is a scale invariant probability density
has a power form, as distinct from an exponential relaxation of fluctuations in equilibrium systems. Such a situation is observed at the thermodynamic critical point of a liquid vapor phase transition. The scale invariance of fluctuations of thermodynamic quantities close to the critical point is determined by the conditions of convergence of properties of different phases and requires a fine tuning and large relaxation times [12, 13]. As distinct from the thermodynamic critical point, nonequilibrium processes with large fluctuations show their stable spatial and temporal scale invariance without a fine tuning of the parameters. Therefore the origin of extreme large fluctuations is often connected with the concept of self-organized criticality [1, 14], which describes the avalanche dynamics and is used for demonstrating a critical behavior in a large number of computer models. Another distinctive feature of large fluctuations is the 1/f behavior of power spectra. The energy of fluctuations with a 1/ f power spectrum accumulates on low frequencies, therefore large scale surges may be observed in a system [1, 3, 15]. Random processes with extremely large
2
fluctuations are often described with the use of analogies with models of self-organized criticality [1, 14], turbulent flows [16], and fractional integration of white noise [17]. In our previous works it was shown that random processes with extreme large fluctuations may arise in interacting phase transitions under the action of white noise [10, 18]. In this case, on the achievement of the critical value of the noise intensity, one can observe in the system a noise induced transition, and the power spectrum of fluctuations takes the 1/f type. To the state of a system with a 1/f power spectrum corresponds the maximum entropy, which points to the stability of random processes with such a spectrum [19]. At an external periodic action the stochastic resonance may arise in a system, which shows up in an increase of sensitivity of a nonlinear system to an external periodic action in a presence of a noise [20-24]. In this case the amplitude of periodic oscillations increases many times with an addition of a noise. In [25] it is shown that a combined action of a noise and a periodic force on coupled phase transitions may lead to stochastic resonance. Stochastic resonance increases with a decrease in the frequency of the periodic force. The present paper examines the variant of a system of two nonlinear stochastic equations simulating coupled phase transitions in a spatially distributed system and in which it is possible to observe the spontaneous formation and propagation of waves simultaneously with the formation of the temporal 1/f and spatial 1/k power spectra. Section 1 introduces the variant of the spatial distributed system in which nonlinear effects connected with extreme large fluctuations are considered. Section 2 shows the conditions of formation and growth of waves in the system. The stability of 1/f noise and 1/k spectra is analyzed in section 3. Section 4 describes the properties of spatial- temporal stochastic resonance. 1. L arge fluctuations with a 1/f power spectrum at coupled phase transitions In consideration of random processes which describe coupled phase transitions with extremely large fluctuations use is made of a system of nonlinear stochastic equations [10, 18]:
3
d 2 1 ( t ), dt d 2 2 2 ( t ), dt
where
and
(1)
are dynamic variables. 1 and 2 are Gaussian white noises with
k ( t ) 0 , k (t )l (t ) 2 k ,l (t t ), k , l 1,2 and is their amplitude. The criticality of system (1) correspond to that level of white noise which correspond to a noise-induced transition [18]. The power spectrum S of the random process ( t ) described by Eqs. (1) has a 1/f type in a wide range of white noise intensity, i.e. S is proportional to 1 f , and the power spectrum S of the random process ( t ) is proportional to 1 f 2 . The lumped system (1) may be generalized for the spatially distributed case [26]:
2 2 D 2 1 ( x, t ),
t
x
2 2 ( x, t ),
t
where k ( x, t ) 0 , k ( x, t )l ( x , t ) 2 k ,l ( x x ) ( t t ), k , l 1,2 and
(2)
D - is the
diffusion coefficient. The statistics of fluctuations and the scaling relations demonstrated by the spatially distributed system (2) are similar to those of the lumped system (1). In the spatially distributed system (2), on the achievement of a sufficiently high intensity of white noise, one can observe noise-induced synchronization which is look like a critical phase transition [27]. For a demonstration and a consideration of other nonlinear effects connected with extreme fluctuations in hydrodynamics and thermal physics, such as a spontaneous formation of waves under the action of white noise, and spatio-temporal stochastic resonance, the different generalization of a lumped system has been taken in the present paper:
2
2 D 2 v 1 ( x, t ),
t
x
x
2
2 2 D 2 v 2 ( x, t ),
t
x
x
(3)
4
where the designations of system (2) have been retained, and v is the drift velocity. The diffusion coefficient and also the drift velocity have been taken equal in one and the other equation of system (3). Calculation show that at different values of the diffusion coefficient and the drift velocity in the equations of system (3) the stationary solution is determined by the largest D and
v . In this case, only the time of establishment of a stationary state increases. Therefore, we have taken
diffusion coefficients and drift velocities equal in both equations. Qualitatively this
approximation does not influence to observed nonlinear phenomena. The deterministic parts of the equations of system (3) differ only in the coefficient 2 before the linear term in the second equation. This makes the second equation of system (3) the governing, as in the lumped system (1) [19]. 2. W ave formation and propagation For numerical integration the modified Euler scheme is used [26, 28] and system (3) is written as follows:
D D i , j 1 t i , j 1 t i , j t v i , j t v i , j 1 t )[1 ( i , j )2 t D t ]1 2 2 0.5 e ( pi ) j ( t ) ,
i 1, j ( i , j
D D i 1, j ( i , j i , j 1 t i , j 1 t 2 i , j t v i , j t v i , j 1 t )[1 ( i , j )2 t D t ]1 2 2 0.5 e ( qi ) j ( t ) ,
(4)
where pi and qi are sequences of Gaussian random numbers with a zero mean and a unit standard deviation. These Gaussian random numbers simulate white noise. The variance of the random numbers e 2 relates to the white noise intensity 2 by the following relations:
e 2 2t . The spatial sample x is expressed in terms of the time integration step t and included in the diffusion coefficient D and the drift velocity v . Therefore, the diffusion coefficient is proportional to the square of the integration step, i.e. D ~ ( t ) 2 , and the drift velocity is proportional to the first power of the integration step, i.e. v ~ t . This should be taken into account in changing the integration step t . 5
In numerical calculations use was made of periodic boundary conditions. In the spatially distributed system (3) there are first temporal and the second spatial derivative, therefore for the achievement of a stability solution in numerical integration of system (4) there must be more temporal points than spatial one. Integration was carried out for a system containing 128 space points at 4096 time integration steps t 0.1 . The diffusion coefficient D and the drift velocity
v were in different calculations in the range from 0.3 to 1. If
we
choose,
as
the
initial
conditions,
0, j A cos(2 k0 jK 1 )
and
0, j A cos(2 k0 jK 1 ) , then, irrespective of the value of the amplitude A , the wave length adjust to the wavenumber k0 given by the initial conditions. This is observed both in the absence of white noise and at its low intensities at the expense of synchronization, which may arise in nonlinear systems. The value of k0 determines the initial number of waves. The wave propagation in the absence of noise is demonstrated in Fig 1. This figure shows time and space series of the variables and obtained by the numerical integration in the absence of noise. To make it more obvious, the contour plots are shown in the figures. Fig. 2 shows the phase trajectory in the plane at one of a spatial point in the absence of noise. For initial conditions close to zero and when the intensity of noise increases, one can observe a spontaneous formation and the growth of waves. This process is shown in Fig.3. The number of waves is limited, for the calculation parameters chosen the number of waves is from 4 to 6. With a further increase in the white noise intensity the number of waves begins to decrease at the cost of their unification into larger ones. In this case the maximum of a power spectrum (both temporal and spatial) determined by the number of waves in the system increases and moves into the region of low frequencies. This continues until one wave remains. The existing value of the noise amplitude is in this case e 1 . Figure 4 shows the dependence of the maximum of the power spectrum on the noise amplitude for different initial numbers of waves. The nonmonotonic of the dependence in Fig.4 is caused by the spontaneous processes of wave 6
formation and coalescence of waves. The process of wave coalescence and, accordingly, the decrease of the characteristic frequency corresponding to the spectrum maximum can be seen well in the upper insert (Fig.4). This plot shows a step-like decrease in the frequency with increasing noise amplitude, which corresponds to the enlargements of the waves. 3. M aximum entropy and 1/ f spectra On the achievement by the white noise intensity of a critical value ( e 1 ) the frequency power spectra of variables and become inversely proportional to the frequency and the square of frequency, i.e. S ( f ) ~ 1 / f and S ( f ) ~ 1 / f 2 , as in the lumped system (1). The spatial power spectra also obey similar dependences: S ( k ) ~ 1 / k and S ( k ) ~ 1 / k 2 , where we have an inversely proportional dependence on the wave number and the wave number squared. Fig.5 shows such dependences of temporal and spatial power spectra of the variables and obtained from numerical solutions (4) by FFT method. The probability density function P ( ) found from the solution of system (3) is shown in Fig.6 (curve1). Careful analysis shows that the probability density function P ( ) decreases at large by the power law [19]. The probability density function P ( ) has two maxima (curve 2 in Fig.6), but at large values of the probability density function decreases in the same way as for the Gaussian function. The problem of stability of extreme low-frequency fluctuations is of significance. As a criterion of the process stability may be chosen the maximum entropy principle. The solutions of stochastic equations are used to find the probability density functions of variables from which the Shannon informational entropy is determined [29]:
H Pn log( Pn ).
(5)
n
Probability density functions should be normalized:
n Pn 1 .
The index n refers to the
sequence of sampling of the argument of the probability density functions. The Shannon entropy 7
has essential properties of the Gibbs statistical entropy, but is also valid not only for physical systems, but also for a wider class of social, biological, communication systems studied by statistical methods [30]. In statistical mechanics the Gibbs Shannon maximum entropy principle is used for the Gaussian fluctuations. The Shannon informational entropy is believed to be an inapplicable for random processes with a 1/ f power spectrum owing to the nonintegrability of the formula (5), as a probability density functions of 1/ f [30, 31]. To solve Tsallis [32] has suggested deforming the logarithmic function in the expression for the entropy. The Tsallis entropy is determined by the expression
HT
1 q P n 1 . 1 q n
(6)
The use of the entropy introduced by Renyi [33] has also been suggested for power probability density functions:
HR
1 ln P nq . 1 q n
(7)
Both entropies H T and H R contain the dependence on the parameter q , whose value determines the concrete position of the maximum entropy. In our opinion, a complex system cannot be characterized by a single distribution function of variables. A sufficiently detailed description of a complex system have to contain a system of nonlinear stochastic equations, which are in a master-slave hierarchy. It can be shown by analogy with system (1) that the second equation of system (3) is the governing one [19]. The Gaussian character of the function P ( ) at large values makes it possible to use this function for finding the Gibbs Shannon entropy (5). It was been found that the entropy H for the probability density function P ( 2 ) of the square of the variable calculated by formula (5) has a maximum. Fig.7 shows the calculated dependence of the Gibbs Shannon entropy H on the noise amplitude e (curve 1). The 8
position of the maximum of the dependence H ( e ) in Fig.7 corresponds to the critical amplitude of white noise, at which a noise-induced transition is observed in the system, and the temporal and spatial power spectra are respectively look like 1/f and 1/ k. A further increase in the white noise intensity leads to a decrease in the entropy and the destruction of the 1/ f and 1/k spectra. The value of the parameter q in the expression of Tsallis (6) and Renyi (7) may be determined from the condition that the position of maximum Gibbs Shannon entropy H ( e ) for governing equation and the distribution of coincides with the position of the maximum entropies of Tsallis H T ( e ) and Renyi H R ( e ) for the distribution of . The value of q 0.6 has been found, at which the position of the maximum entropy of Tsallis coincides with the position of the maximum Gibbs Shannon entropy. For the Renyi entropy the value of the parameter q found is also 0.6. The value of q , as well as the position of maximum entropy, does not depend on the integration step in expression (6) and (7). The results of calculations of
H T ( e ) and H R ( e ) depending on the noise amplitude are presented in Fig.7 (curves 2 and 3). Thus, if a distribution function decreases by the law of Gaussian distribution, it may be used in calculating the Gibbs Shannon entropy, whose maximum determines a stable process. For a di formulas of Tsallis (6) and Renyi (7), if it is possible to determine the value of the parameter q . 4. Spatio-temporal stochastic resonance External actions on a system lead to the complication of the process. In nonlinear systems, simulating coupled phase transitions, along with 1/f fluctuations, stochastic resonance may be observed, at which in the presence of noise the amplitude of the periodic force increases many times [25]. In the system (3) along with a stochastic resonance described in [25] there may arise a spatio-temporal stochastic resonance. spatio-temporal stochastic resonance is important for the pattern formation in nonlinear systems [34-36]. 9
At an external periodic action system (3) is rewritten as follows: 2 2 D 2 v 1 ( x, t ) A sin( t ), t x x 2 2 2 D 2 v 2 ( x, t ) A sin( t ), t x x
(8)
where A and are the amplitude and the frequency of an external periodic force. In the region of low frequencies of an external action in system (8), as in the lumped system [25] one can observed a nonmonotonic dependence of the maximum of spectral density
S max of the variables and on the white noise intensity. An increase in the power of the periodic signal in the presence of noise points to stochastic resonance in the system. Along with the temporal stochastic resonance in (8) the stochastic spatio-temporal resonance takes place. The spatio-temporal stochastic resonance is most clearly demonstrated by the effect of white noise on the expansion of the periodic pattern region. Fig.8 (1, 2) shows spatio-temporal pattern of the variables and in the absence of the noise. It is seen that at given parameters of the frequency and the amplitude of an external action the oscillations and the spatio-temporal pattern are absent at large times. The inclusion of white noise at a certain intensity ( e 0.2 0.3 ) and invariability of the other parameters increases the amplitude of oscillations. In this case the periodic oscillations are restored in the whole range of time. Such a restoration of oscillations is shown in Fig.8 (3, 4). Fig.9 shows the dependences of the maximum temporal and spatial spectra at the frequency of the external periodic action on the white noise amplitude e . The nonmonotonic dependence of the maximum of spectral density characterizing the power of the spatio temporal oscillations at the frequency of the external periodic action is connected with the stochastic spatio temporal resonance in the system. Conclusion A spatially distributed system, simulating coupled interacting phase transitions, is described by two nonlinear stochastic equations. In this system it is possible to observe a
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spontaneous formation and propagation of waves under the action of white noise. If noise intensity is critical value, against the background of formation and motion of waves emerge 1/ f and 1/k power spectra of extreme large fluctuations. The distribution of extreme fluctuations corresponds to the maximum entropy, which points to their stability. At an external periodic force in a system one can observe spatio-temporal resonance, at which, along with an increase in the amplitude, an expansion of the region of the periodic oscillations and spatio temporal pattern. A cknowledgements The work was partially supported by the Russian Foundation for Basic Research (Grant no. 15-08-02210a) and the Research Program of the Ural Branch of the Russian Academy of Sciences (no 15-1-2-7). References 1.
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11. V.N. Skokov, V.P. Koverda, A.V. Reshetnikov, and A.V. Vinogradov, Physica A 364 (2006) 63. 12. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena Oxford, New York and Oxford, 1971. 13. H. E. Stanley, Rev. Mod. Phys. 71 (1999) 358. 14. P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A 38 (1988) 364. 15. M.B. Weissman, Rev. Mod. Phys.. 60 (1988) 537. 16. A.N. Kolmogorov, Proceedings of the USSR Academy of Sciences 30(1941) 299. 17. B.B. Mandelbrot., J.W. Van Ness. SLAM Rev. 10 (1968) 422. 18. V.P. Koverda, V.N. Skokov, Physica A 346 (2005) 203. 19. V.P. Koverda, V.N. Skokov, Physica A 391 (2012) 21. 20. R. Benzi, S. Sutera, A. Vulpiani, J. Phys. A 14 (1981) L453. 21. V.S. Anishchenko, A.B. Neiman, F. Moss, L. Schimansky-Geier, Uspekhi fizicheskikh nauk 169 (1999) 7. 22. Lindner B., Garcia-Ojalvo J., Neiman A., Schimansky-Geier L., Phys. Rep. 392 (2004) 321. 23. J.J. Collins, C.C. Chow, T.T Imhoff, Nature 376 (1995) 236. 24. D. Nozaki, J.J. Collins, Y. Yamamato, Phys. Rev. E 60 (1999) 4637. 25. V.P. Koverda , V.N. Skokov, Physica A 393 (2014) 173. 26. V.P. Koverda, V.N. Skokov, Physica A 390 (2011) 2468. 27. V.P. Koverda, V.N. Skokov, Physica A 452 (2016) 2126. 28. V.P. Koverda, V.N. Skokov, Physica A 388 (2009) 1804. 29. C. Shannon, Bell Syst. Tech. 27 (1948) 379. 30. A.G. Bashkirov, Theoret. Math. Phys. 149 (2006) 1559. 31. E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 32 (1983) 209. 32. C. Tsallis, J. Stat. Phys. 52 (1988) 479. 33. A. Renyi, Probability Theory. Amsterdam: North-Holland, 1970.
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C aptions to the F igures Fig.1. Spatio-temporal oscillations of variables (1) and (2) and wave propagation in the absence of noise. Fig.2. Phase trajectory in the plane at one of a spatial point in the absence of noise. Fig.3. Spontaneous formation and growth of waves with increasing amplitude of white noise: 1
e 0.2 ; 2 e 0.25 ; 3 e 0.3 . Fig.4. Dependence of the maximum of power spectra of the variable on the noise amplitude
e for different initial numbers of waves. Numbers show the initial numbers of waves. In the upper insert it is shown the dependence of the frequency corresponding to the maximum of the power spectrum of the variable on the noise amplitude e . Fig.5. Temporal and spatial spectra of variables of system (3): 1 temporal spectrum S ( f ) ; 2 temporal spectrum S ( f ) ; 3 spatial spectrum S ( k ) ; 4 spatial spectrum S ( k ) ; dash lines show the dependences ~ 1 f and ~ 1 k . Fig.6. Probability density functions for the variables: 1 P ( ) ; 2 P ( ) . Fig.7. Dependence of entropy on the noise amplitude: 1 Gibbs Shannon entropy H ( e ) ; 2 Tsallis entropy H T ( e ) for q 0.6 ; 3 Renyi entropy H R ( e ) for q 0.6 . Fig.8. Spatio-temporal pattern of the oscillations of the variable and : 1, 2 in the absence of white noise; 3, 4 under the action of white noise with the amplitude e 0.3 ; the contour plots are shown in the upper inserts. Fig.9. Dependencies of the maximum temporal and spatial power spectra at the frequency of the external periodic action on the white noise amplitude e : 1 maximum temporal spectra of
; 2 maximum temporal spectra of ; 3 maximum spatial spectra of ; 4 maximum spatial spectra of .
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