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Evolution of a stochastic system within the framework of Tsallis statistics
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Physica A 354 (2005) 262–280 www.elsevier.com/locate/physa
Evolution of a stochastic system within the framework of Tsallis statistics Dmitrii O. Kharchenko, Vasilii O. Kharchenko Sumy State University, Rimskii-Korsakov St. 2, 40007 Sumy, Ukraine Received 29 June 2004; received in revised form 21 December 2004 Available online 19 April 2005
Abstract The general approach of a nonlinear Fokker–Planck equation is applied to investigate the behavior of main statistical moments of a stochastic system. It was shown that the system described by Tsallis statistics can undergo transitions inherent to multiplicative noise-induced transitions. The scaling laws of motion are defined to identify an anomalous time behavior of the system with multiplicative noise. r 2005 Elsevier B.V. All rights reserved. PACS: 02.50.r; 05.40.a; 72.70.+m Keywords: Tsallis statistics; Noise; Statistical moments; Anomalous diffusion
1. Introduction The study of processes which obey nonextensive statistics [1] remains a cornerstone of modern statistical mechanics. There is a wide spectrum of complex systems for which the standard and powerful Boltzmann–Gibbs (BG) approach cannot give correct results [2]. Typical and well-known examples of processes where standard BG statistics arrives at difficulties to interpret the experimental data are Le´vy-type anomalous diffusion (see, for example, [3] and references therein), Corresponding author.
E-mail address: [email protected] (D.O. Kharchenko). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.01.057
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correlated-type anomalous diffusion [4], self-gravitating processes [5], and financial risks and decisions [6]. To add to this list we may mention systems with long-range microscopic memory [7], granular systems [8], solar neutrinos [9], and so on. A satisfactory description of such processes is achieved within the framework of generalized (nonextensive) statistical mechanics [2], where power-law distributions, algebraic correlations and violation of the habitual additivity (extensivity) rule for entropy functional can be introduced. The nonextensive statistics is based on a generalized entropic form which keeps all thermodynamic relations, from which BG relations can be derived in a special limit [10,11]. An evolution of systems obeying the nonextensive statistics can be described with the help of a nonlinear Fokker–Planck equation (NFPE) or fractional Fokker– Planck equation (associated with fractional order evolution equation for a physical quantity). A formalism based on the NFPE has been widely discussed since 1995, after work by Plastino and Plastino [12]. Usually, the form of NFPE is constructed in the framework of special assumptions about correlation between micro- and macro-levels in a hierarchical system. An application of the NFPE to a free particle system was discussed in Ref. [13], and a study of the self-organized criticality with the help of the NFPE of a fractional order is proposed in Ref. [14]. Recently, the generalized scheme for nonlinear kinetics was considered and the NFPE was obtained on the basis of a master equation [15,16]. As shown in Ref. [16], in the framework of such an approach one can pass on to the nonextensive statistics normally. In works that concern a study of dynamics of systems with nonextensive statistics, main attention is paid to investigate the form of the NFPE or to define main laws of nonextensive thermodynamics. However, the dynamics of the system obeying a generalized approach is an open question. Usually, one deals with a free particle system to investigate the Le´vy-type processes and to find relations between the parameter of nonextensivity q and exponent inherent to Le´vy-type diffusion. The statistical picture of the system in quadratic potential of the stochastic quantity x is proposed in Refs. [12,4], where the explicit form of the distribution was calculated. The present effort is an attempt to show a picture of the system evolution in terms of first and second moments of stochastic quantity within the framework of the generalized approach. We will investigate the behavior of main statistical moments to show in what way the passage to the generalized approach transforms the picture of the ordering of the system described by x4 -potential. We will set scaling properties of corresponding time dependencies and define the influence of multiplicative noise on fractal properties of the phase space of the system. The work is organized as follows. We represent main ideas to derive the NFPE (Section 2). Based on the form of NFPE, evolution equations for first and second statistical moments are obtained in Section 3. Here we will show the form of the corresponding Langevin equation containing terms defining the hierarchical coupling between micro- and macro-levels of the system. Two special cases inherent to Tsallis statistics are considered in Sections 3.1 and 3.2. Section 3.3 is devoted to investigating a multiplicative noise influence where the main scaling laws of motion are defined. The main results are collected in Section 4.
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2. Nonlinear Fokker–Planck equation A keypoint to the nonextensive statistics is a master equation for the probability density function p ¼ pðx; tÞ in the form Z qpðx; tÞ ¼ ½wðx0 ; x x0 Þgðp0 ; pÞ wðx; x0 xÞgðp; p0 Þ dx0 , (1) qt where wðx; x0 Þ is the transition rate which depends on the nature of interactions between the particle and the bath and is a function of starting x and arrival x0 sites. The factor gðp; p0 Þ is an arbitrary function of the particle populations of the starting and the arrival sites. This function satisfies the following conditions: gð0; p0 Þ ¼ 0 if the starting site is empty, the transition probability is equal to zero; gðp; 0Þa0 requires that in the case the arrival site is empty the transition probability depends on the population of the starting site. In the case gðp0 ; pÞ ¼ p0 and gðp; p0 Þ ¼ p we have the standard liner kinetics and the master equation corresponds to the Chapman– Kolmogorov equation. The master equation (1) yields the NFPE in the form [16] qp q qDðxÞ q q ln kðpÞ qp ¼ f ðxÞ DðxÞgðpÞ , (2) gðpÞ þ qt qx qx qx qp qx where gðpÞ ¼ gðp; pÞ and the function kðpÞ40 is defined through the condition q ln kðpÞ q gðp; p0 Þ ¼ ln 0 . qp qp gðp ; pÞ p¼p0
(3)
The first and second moments Rover wðx; x0 Þ correspond R to the drift and diffusion terms, respectively, i.e., f ðxÞ dx0 x0 wðx; x0 Þ, 2DðxÞ dx0 x0 2 wðx; x0 Þ. In the stationary equilibrium state, a solution ps ¼ pðx; 1Þ of Eq. (2) takes the form Z x f ðx0 Þ ln kðps Þ ¼ bðU ef ðxÞ mÞ; U ef ðxÞ ¼ dx0 þ ln DðxÞ , (4) Dðx0 Þ where gðpÞ remains an arbitrary function, b is the constant related to the noise R intensity and m takes care of the normalization condition p dx ¼ 1. A nonlinearity of the Fokker–Planck equation is defined through the form of the function kðpÞ: at ln kðpÞ ¼ ln p we pass to the BG statistics; a q-deformation of the logarithm ln k ! lnq k promotes Tsallis statistics, where lnq k ¼ ðk1q 1Þ=ð1 qÞ.
3. Equations for averages The aim of this paper is to consider the evolution of a system described by Eq. (2). To this end, we explore the behavior of the first moment hxi being an order parameter Z, a variance hðdxÞ2 i (dx ¼ hxi x) which plays the role of a one-time correlation function S and a two-time correlator (Green function) Gðt; t0 Þ ¼ hxðtÞxðt0 Þi.
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To derive corresponding evolution equations we use the standard technique. Let us multiply Eq. (2) by xn (n ¼ 1; 2) and integrate over x. As a result, we arrive at an evolution equation for the first and second moments in the form d dDðxÞ q q ln kðpÞ hxi ¼ f ðxÞ þ DðxÞp , dt dx qx qp d 2 dDðxÞ q q ln kðpÞ hx i ¼ 2 x f ðxÞ þ DðxÞp dt dx qx qp q ln kðpÞ þ 2 DðxÞp . ð5Þ qp The obtained equations and Eq. (2) allow us to write down an effective Langevin equation as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx dDðxÞ q q ln kðpÞ q ln kðpÞ ¼ f ðxÞ þ DðxÞp þ DðxÞp xðtÞ , (6) dt dx qx qp qp where the generalized Langevin source xðtÞ is defined as a white noise with standard properties: hxðtÞi ¼ 0, hxðtÞxðt0 Þi ¼ 2dðt t0 Þ. In the case ln kðpÞ ¼ ln p we pass to the linear Fokker–Planck equation and arrive at the standard Langevin equation in Itoˆ interpretation with a multiplicative noise of pffiffiffiffiffiffiffiffiffiffi amplitude DðxÞ. In such a case, the multiplicative noise appearance is induced by the influence of environment. At ln kðpÞa ln p the multiplicative character of the noise in Eq. (6) is related to the appearance of terms with p-dependence. Such dependencies of drift and diffusion terms at D ¼ const can be explained as a twosided feedback between different (micro- and macro-) hierarchical levels of the system [13]. Therefore, the function kðpÞ governs a procedure of the hierarchical coupling of the system states. A solution of the problem related to a system with multiplicative noise within the framework of linear kinetics (ln kðpÞ ¼ ln p) was proposed in Ref. [17], where an arbitrary multiplicative noise (in a power-law form) is considered. It was shown that the multiplicative noise governs the system dynamics (it changes the bifurcation values of the system parameters): the power-law dependence of the amplitude of multiplicative noise leads to an anomalous behavior of the system (its phase space acquires fractal properties) [18]. Systems with the nonlinear kinetics ln kðpÞa ln p are formed hierarchically and, therefore, should have fractal properties. A typical and well-known example is a motion of the ‘‘particle’’ on a hierarchical tree in an ultrametric space [19]). Unfortunately, a description of such objects is carried out on the basis of the macroscopic approach, where the NFPE is postulated initially. Using a theory of diffusion processes, one can derive the corresponding Fokker–Planck equation and write down the effective Langevin equation containing terms that depend on the probability density function p. Initially, such a kind of problem was discussed in Refs. [20,21]. Next, fixing ln½kðpÞ through a q-deformed logarithm, we consider the system with the Tsallis statistics for both cases of kðpÞ a linear form of kðpÞ ¼ p (Section 3.1) and a power-law form of kðpÞ ¼ p1=q (Section 3.2). In the framework of this approach we
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focus on properties of the ordering process. Considering the model of the system with D ¼ const and x4 -potential V ðxÞ ¼
e 2 x4 x þ , 2 4
(7)
where e ¼ ðT c TÞ=T c is a control parameter acting as a temperature counted from a critical value T c , we discuss the dynamics of the main statistical moments. 3.1. Linear form: kðpÞ ¼ p Assuming ln½kðpÞ ¼ lnq p , we arrive at the NFPE in the form qp q qDðxÞ qDðxÞp1q q2 ¼ f ðxÞ þ p þ 2 DðxÞp2q . qt qx qx qx qx
(8)
(9)
Next, we use the Fokker–Planck equation (9) to find the long-time scaling behavior of processes. To this end, we focus on a free particle system supposing f ðxÞ ¼ 0 and D ¼ const. In such a case we arrive at equivalent equations for the stochastic process xðtÞ and probability density function as follows: sffiffiffiffiffiffiffiffiffiffiffi dx 2D ð1qÞ=2 ¼ p xðtÞ , (10) dt 2q qp D q2 2q ¼ p . qt 2 q qx2
(11)
The nonlinear diffusion equation (11) can be solved in an automodel regime: xðtÞ ¼ yaðtÞ, pðx; tÞ ¼ ðaðtÞÞ1 jðx=aðtÞÞ. Such an assumption allows one to identify a width of the diffusion burst which increases as tm . In general, the exponent m defines anomalous scaling in time which is reduced to the Hurst exponent H ¼ 1=2 of the diffusion process in BG statistics. Therefore, instead of Eq. (11) we get da d d2 j y j ¼ aq2 D 2 j2q . (12) dt dy dy _ 2q ¼ const ¼ n, Eq. (12) reads Assuming aa nyj ¼
D d 2q j . 2 q dy
The solution of Eq. (13) has the form of Tsallis distribution 1=ð1qÞ ð1 qÞn ðy y0 Þ2 , jðyÞ ¼ jðy0 Þ1q 2D
(13)
(14)
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where conditions jðy ! 1Þ ¼ 0 and ðdjðyÞ=dyÞy!1 ¼ 0 are used. The time dependence aðtÞ is in the form aðtÞ ¼ ½aðt0 Þ3q þ nð3 qÞðt t0 Þ1=ð3qÞ .
(15)
R Making use of such a scaling procedure, we define hxi ¼ xpðxÞ dx and set hx2 ðctÞi ¼ c2m hx2 ðtÞi. Hence, we obtain an expression for the exponent m in the form m ¼ ð3 qÞ1 [13]. The scaling law can be obtained from the solution of Eq. (10), where for the terms up to the first order in dt, we have 0
Z tZ
t0
pð1qÞ=2 ðxðtÞ; tÞpð1qÞ=2 ðxðt0 Þ; t0 ÞhdW ðtÞ dW ðt0 Þi
hxðtÞxðt Þi ¼ D 0
0
¼ hx2 ðtÞit2=ð3qÞ ,
ð16Þ
where t ¼ minðt; t0 Þ and the definition of the white noise xðtÞ ¼ dW ðtÞ=dt is used; W ðtÞ is the Wiener process. Considering the dynamics of the statistical modes, we use the deterministic force f ðxÞ ¼ qV =qx and construct a close loop system for the order parameter Z ¼ hxi and one-time autocorrelation function S ¼ hðZ xÞ2 i. Deriving these equations, we express averages with p-dependencies through the form of the stationary distribution function pðxÞ. A stationary solution of Eq. (9) is the Tsallis distribution: pðxÞ ¼ Z1 ½1 bð1 qÞV ðxÞ1=ð1qÞ ,
(17)
where Z is the normalization constant, b ¼ Z1q =D. Taking into account Eq. (7), evolution equations for the order parameter and autocorrelation function read as dZ ¼ Zðe Z2 3SÞ , dt dS 1q 1 4 2b1 D 2 2 2 2 ¼ 2Sðe 3ðS þ Z ÞÞ þ eðS þ Z Þ ½Z þ 6SZ þ 3S þ , dt 2q 2 2q ð18Þ where t ¼ ð2 qÞt. An equation for the Green function Gðt; t0 Þ ¼ hxðtÞxðt0 Þi is in the form q G ¼ ðe 3ðS þ Z2 ÞÞG . (19) qt Consider a stationary picture of the system behavior. Assuming dZ=dt ¼ dS=dt ¼ 0, we find that the first solution of the order parameter equation defines the disordered state being the attractive point C 0 ðZ0 ; S 0 Þ with coordinates 1=2 ! ð5 3qÞe 12ð9 5qÞb1 D 1þ 1þ Z0 ¼ 0; S 0 ¼ (20) 3ð9 5qÞ ð5 3qÞ2 e2
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on a phase plane ðZ; SÞ. The ordered state appears if we take D and e inside a domain bounded by a solution of the phase diagram equation 24b1 D ð9 5qÞ ¼ 4ð5 3qÞ2 6ð9 5qÞð1 qÞ . e2
(21)
Solutions of Eq. (21) are shown in Fig. 1. It is seen that a decrease in the nonlinearity index q shifts the bifurcation point toward large noise intensity D (Fig. 1a). Formally, such a dependence shows that in the case qa1 the system behaves as a system with a multiplicative noise [17]. Indeed, here a decrease in the index q promotes the dependence of the bifurcation point through the noise intensity D. Fig. 1b shows the influence of the noise intensity on the bifurcation point
1 1.6
0.8
1 2
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5 0.4
5 6 7
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q
1.2
0.8
0.4
1
3
2
0.0 0.0 (c)
0.2
0.4
0.6
0.8
D
Fig. 1. Phase diagrams for the system with kðpÞ ¼ p: (a) e vs. D (curves 1–7 correspond to q ¼ 0:01; 0:3; 0:6; 0:8; 0:99; 1:3; 1:6); (b) e vs. q (curves 1–5 correspond to D ¼ 0:02; 0:05; 0:1; 0:2; 0:4); (c) D vs. q (curves 1–3 correspond to e ¼ 0:3; 0:6; 0:99. A domain of ordered state is situated below the corresponding curves in a and c plates.
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with variation of the index q. Here the ordered state with Za0 is realized at large e if we increase D. The critical noise intensity increases with an increase in e (Fig. 1c). According to the obtained phase diagrams, we find that the ordered state is characterized by dependencies shown in Fig. 2. It is seen that the ordered state is realized in a discontinuous manner inherent to the first-order phase transition. Such a picture of the order parameter (variance) dependence can be explained introducing the correlation effects, i.e. Sa0. It is well known that on the macroscopic level we can neglect correlations and arrive at the standard Landau picture of phase transitions of the second order, where Z_ ¼ hf ðxÞi ¼ f ðZÞ, Z0 / e1=2 . The ordered state is characterized by two stationary points on the phase plane ðZ; SÞ with coordinates: 1=2 Z , 0 ¼ ½e 3S
S 0
1=2 ! ð5 3qÞe 6ð9 5qÞ 1 2 1 1 ¼ ðb D þ e ð1 qÞ=4Þ . 3ð9 5qÞ ð5 3qÞ2 e2
ð22Þ
þ þ Here upper and lower signs define the saddle CðZ 0 ; S 0 Þ and node NðZ0 ; S 0 Þ on the phase plane (see Fig. 3.). Dotted lines in Fig. 2 display the position of saddle S and solid lines represent the attractive point N position. As seen from Fig. 2, with an increase in q the ordered state is characterized by small values of the order parameter. The system behavior does not change essentially if we take qo1 or q41. To consider dynamics of the system we present solutions of Eq. (18) in Fig. 3. Here at small temperatures the phase portrait is characterized by the single point C 0 defining the disordered state (Fig. 3a). When e increases two additional points þ þ CðZ 0 ; S 0 Þ and NðZ0 ; S 0 Þ appear (Fig. 3b). In the vicinity of the point C the slowing down of the order parameter ZðtÞ and one-time correlator SðtÞ occurs (Fig. 4). Analysis of the time dependencies is performed with the help of the Lyapunov’s exponents method. Here for the system evolving in time t we find lq ¼ ð2 qÞlðeÞ, where lðeÞ is defined for the system evolving in time tð2 qÞ. It is seen that points N and C are node and saddle if qo2. The system slowly attains these points if q takes large magnitudes. Let us take a look at the correlation processes in the vicinity of points C 0 and N (see Fig. 5). Solutions of Eq. (19) are shown in Fig. 5 at different values of e and q at identical initial conditions. It is seen that at eoe0 the monotonic decrease appears at large initial magnitude Zð0Þ (Fig. 5a). In case e4e0 , a maximum of dependence disappears at condition Zð0ÞoZc corresponding to the domain of the ordered state (Fig. 5b).
3.2. Power-law form: kðpÞ ¼ p1=q Assuming ln½kðpÞ ¼ lnq ½p1=q ,
(23)
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2 3
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(a)
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5
0.1
ε0
0.0 0.0
(b)
4
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0.2
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ε
0.6
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1.0
Fig. 2. Stationary values of order parameter (a) and autocorrelator (b) vs. control parameter e (curves 1, 2 are plotted at D ¼ 0:2 and q ¼ 0:4; 0:7; curve 3 corresponds to D ¼ 0:05, q ¼ 1; curves 4, 5 are plotted at D ¼ 0:05, q ¼ 1:2; 0:7). The bifurcation temperature is denoted as e0 .
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S
0.6
0.4 C0 0.2
0.0 0.0
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(a)
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C0
0.2
C
0.1 N
0.0 0.0 (b)
0.2
0.4
η
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0.8
Fig. 3. Phase portraits at D ¼ 0:2: (a) e ¼ 0:4, q ¼ 0:6; (b) e ¼ 0:7, q ¼ 0:3.
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0.8 3
0.6 η
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(a)
15 τ
20
25
0.3
1
S
0.2
2
0.1 3
0.0 0.0
(b)
5
10 τ
15
Fig. 4. Time dependencies of the order parameter (a) and the autocorrelator (b) corresponding to different trajectories on the phase portrait in Fig. 3 at q ¼ 0:3, e ¼ 0:7, D ¼ 0:2 (curves 1, 2, 3 correspond to Sð0Þ ¼ 0, Zð0Þ ¼ 0:215; 0:225; 0:99).
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0.10
G(t,0)
0.08
0.06 1 0.04 2 0.02 3 0.00 2
0.0
4
(a)
6 τ
8
10
0.12
G(τ ,0)
1 2 0.08
0.04 3
0.00 0.0
(b)
2
τ
4
6
Fig. 5. Correlation functions in the vicinity of points C 0 , N, at D ¼ 0:2: (a) e ¼ 0:4, q ¼ 0:6 (curves 1, 2, 3 correspond to (Sð0Þ ¼ 0, Zð0Þ ¼ 0:2), (Sð0Þ ¼ 0:6, Z ¼ 0:4), (Sð0Þ ¼ 0:1, Z ¼ 0:8); (b) e ¼ 0:4, q ¼ 0:6 (curves 1, 2, 3 correspond to (Sð0Þ ¼ 0, Zð0Þ ¼ 0:215), (Sð0Þ ¼ 0, Z ¼ 0:225), (Sð0Þ ¼ 0:1, Z ¼ 0:99).
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the NFPE can be written as follows: qp q qDðxÞ 1 qDðxÞpð1qÞ=q 1 q2 ¼ f ðxÞ þ DðxÞp1=q . pþ qt qx qx q q qx2 qx
(24)
As in the previous subsection, we investigate the free particle system at first. In such a case, the diffusion equation reads qp q2 ¼ D 2 p1=q . qt qx
(25)
Using the results of the previous subsection, we can find solutions in the automodel regime for both the function aðtÞ and the spatial component jðyÞ: aðtÞ ¼ ½aðt0 Þ1þ1=q þ nð1 þ 1=qÞðt t0 Þq=ð1qÞ ,
(26)
q=ð1qÞ nð1 qÞ 1=qþ1 2 ðy y0 Þ jðyÞ ¼ jðy0 Þ . 2D
(27)
It is easy to see that the correlation function hxðtÞxðt0 Þiq is the second-order moment defined through the probability density function1 Z t hxðtÞxðt0 Þiq ¼ p1=q1 ðxðtÞ; tÞ dt ¼ hx2 ðtÞit2q=ð1þqÞ . (28) 0
Hence, for the exponent m we have m ¼ q=ð1 þ qÞ .
(29)
It differs from the expression obtained in the linear case of kðpÞ. Here one has m ¼ 0 at q ¼ 0, whereas m ¼ H ¼ 1=2 at q ¼ 1. Considering the evolution of the system in x4 -potential, we write down equations for the order parameter Z ¼ hxiq and autocorrelation function S ¼ hðdxÞ2 iq in the form dZ ¼ Zðe Z2 3SÞ , dt dS ¼ 2Sðe 3ðS þ Z2 ÞÞ þ ð1 qÞ eðS þ Z2 Þ 12½Z4 þ 6SZ2 þ 3S 2 þ 2b1 q D. dt ð30Þ An evolution equation for the Green function coincides with Eq. (19). In these equations we use the renormalized time t ¼ t=q, and the constant bq is defined from the normalization condition. R Here the averages are defined through the standard scheme hxn i ¼ xn p dx. Due to the construction k ¼ p1=q , the probability density p should be considered as an escort probability density. Indeed, R introducing f ðxÞ ¼ p1=q ðxÞ, Raverages should be meant as q-averages, i.e., hxn iq ¼ xn f q ðxÞ dx whereas the normalization condition is f q ðxÞ dx ¼ 1 (see Ref. [16]). 1
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Here the disordered state is characterized by a single stable node C 0 with coordinates: 1=2 ! ð3 qÞe 12ð5 qÞb1 D 1þ 1þ . (31) Z0 ¼ 0; S 0 ¼ 3ð5 qÞ ð3 qÞ2 e2 The ordered state appears when the bifurcation line obtained as a solution of the equation 24b1 D ð5 qÞ ¼ ðq 3Þ2 6ð5 qÞð1 qÞ e2
(32)
1
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1 0.20 2 0.15
disorder
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q
D
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0.0 0.0
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q
1.2 1
2
3
0.8
0.4
0.0 0.00
(c)
0.05
0.10
0.15
0.20
D
Fig. 6. Phase diagrams for the system with kðpÞ ¼ p1=q : (a) e vs. D (curves 1–6 correspond to q ¼ 0:01, 0.1, 0.6, 0.99, 1.3, 1.5); (b) e vs. q (curves 1–3 correspond to D ¼0.01, 0.05, 0.1); (c) D vs. q (curves 1, 2, 3 correspond to e ¼ 0:3; 0:6; 0:99). A domain of ordered state is situated below curves in plate a; in plate c the ordered state is bounded by corresponding curves.
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is crossed. Solutions of the phase diagram (32) are shown in Fig. 6. It is seen that despite the form of curves eðDÞ being topologically identical to the case of kðpÞ ¼ p, the influence of the nonlinearity parameter q changes the situation in principle (Fig. 6a). Here the bifurcation magnitude of the noise intensity D is shifted toward the
0.8
4
0.6 ηq0
3 2 0.4 1 0.2
ε0
0.0 0.0
0.2
0.4
(a)
0.6
ε
0.8
1.0
0.4
Sq0
0.3
4
0.2
3 2 4 2
1
0.1
3
1 ε0
0.0 0.0
(b)
0.2
0.4
ε
0.6
0.8
1.0
Fig. 7. Stationary values of order parameter (a) and autocorrelator (b) vs. control parameter e (curves 1, 2 correspond to q ¼ 0:2, D ¼ 0:01; 0:05; curves 3, 4 correspond to q ¼ 1:2, D ¼ 0:05; 0:07).
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large magnitudes if q increases. A more complicated picture of the ordering is shown in Fig. 6b. It is seen that at moderate values of q the bifurcation realizes at high temperature whereas at small and large q the ordered state appears at low temperature. Fig. 6c shows that a transition to the ordered state is possible at large D if q is taken from moderate values, whereas at large and small q a magnitude of the critical noise intensity is small. The stationary behavior of the order parameter and the autocorrelator is shown in Fig. 7, where solid and dotted lines correspond to the stable point NðZþ 0 ; S0 Þ þ and saddle CðZ0 ; S 0 Þ in the plane. Coordinates of these points are defined as follows: 1=2 , Z 0 ¼ ½e 3S
S 0
1=2 ! ð3 qÞe 6ð5 qÞ 1 2 1 1 ¼ ðbq D þ e ð1 qÞ=4Þ . 3ð5 qÞ ð3 qÞ2 e2
ð33Þ
It is seen that in the case under consideration the behavior of the order parameter vs. e is the same as in the previous case, but the form of dependencies is changed. Here with e-growth magnitudes of Z 0 and S 0 increase. Finally, the topology of phase portraits and corresponding time dependencies of main modes are the same as at kðpÞ ¼ p. At high temperature the system is in disordered state and characterized by the single node (Fig. 3a). The appearance of the ordered state at low temperature is accompanied by the appearance of two stationary points C and N (see Fig. 3b). Performing the stability analysis in the vicinity of these points, one can find that the Lyapunov exponent lq ¼ lðeÞ=q, where lðeÞ is defined for the system evolving in time t=q. Here at small q the Lyapunov exponent takes large magnitudes in the vicinity of the node N and, therefore, the system attains the stable ordered state faster than at large q. The above analysis shows that for the systems with a potential constructed on the basis of catastrophes theory the x2 -term yields the exponential behavior of the statistical moments (see for example Refs. [12,4]). One can conclude that the algebraic form of time dependencies can be observed when the system is in the potential V ðxÞ / x1þb , b40. In other words, the scaling laws appear if the system has fractal properties or its potential is a homogeneous function with a fractional exponent b at D ¼ const. It can be seen from the form of the NFPE that the drift term f ðxÞ defines the exponential behavior of the time dependence cðtÞ of the probability density function pðx; tÞ ¼ cðtÞfðxÞ. On the other hand, as was shown above, the scaling laws are realized in the limit f ðxÞ ¼ 0 and D ¼ const. It is easy to see that such a regime is observed in the case DðxÞ / xd with d40. In other words, the algebraic form of time dependencies is realized if the system has a self-affine structure of the phase space. 3.3. System with self-affine phase space Let us pass to the limit of homogeneous functions considering the variable y ¼ xðtÞ=aðtÞ which allows one to write down p ¼ a1 jðyÞ. We suppose that the diffusion
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coefficient scales as D ¼ ad DðyÞ; for the drift term being the first term in the NFPE we assume f qD=qx ¼ ab F ðyÞ. It allows one to find the law of motion and scaling properties of the diffusion process. Making use of the automodel assumption, at kðpÞ ¼ p the NFPE reads as _ ¼ ab F j þ a2þqþd Dj1q ayj
d j. dy
(34)
For the time dependence we obtain _ b ¼ const ¼ n; aa
b ¼ 2 þ q þ d .
(35)
Hence, in the representation aðtÞ / ðt t0 Þm for the exponent m, we have m¼
1 . 3qd
(36)
In the case kðpÞ ¼ p1=q the equation _ ¼ qab F j þ ad1=q Dj1=q1 qayj
d j dy
(37)
is used to find the scaling properties of the law of motion. Here, for the exponent m one has m¼
q ; 1 þ qð1 dÞ
b ¼ d 1=q .
(38)
1.0 2′ 0.8
superdiffusion
1′ 2
0.6 µ
1
0.4
subdiffusion
0.2
0.0 0.0
0.5
1.0
1.5
2.0
q
Fig. 8. Domains of anomalous diffusion in the m2q plot for the system with multiplicative noise (solid and dotted lines are related to Eqs. (36) and (37) respectively): curves 1, 2 correspond to d ¼ 0:0; curves 10 ; 20 correspond to d ¼ 0:5.
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For the spatial dependence we obtain following expressions, 8 Ry > ny F ðyÞ 1=ð1qÞ > 1q > ð1 qÞ dy at kðpÞ ¼ p ; > < fðy0 Þ DðyÞ jðyÞ ¼ > Ry ny F ðyÞ q=ð1qÞ > ð1þqÞ=q > fðy Þ ð1 qÞ dy at kðpÞ ¼ p1=q ; > 0 : DðyÞ
279
(39)
where terms with n define a contribution related to the time derivative in the Fokker–Planck equation; at n ¼ 0 we pass to the stationary equilibrium distributions. Comparing expressions for the exponent m in the linear and nonlinear cases, one can conclude that the ordinary diffusion process with m ¼ H ¼ 1=2 can be reached if d ¼ 1 q at k ¼ p or d ¼ 1=q 1 at k ¼ p1=q . In other words, the nonextensive system with the multiplicative noise can behave itself as a system of ordinary diffusion. By putting q ¼ 1 in both cases we will receive the expression m ¼ ð2 dÞ1 [18] for the system with multiplicative noise. The regime of superdiffusion is characterized by m41=2. As it is seen from Eqs. (36) and (38) an increase in d leads to a transition to the superdiffusion regime (see Fig. 8). In other words, the process generated by the p-dependent Langevin equation scales in time as an anomalous diffusion process. The condition mo1 allows one to set the possible interval for q as follows: 1oqo2 at kðpÞ ¼ p and q40 at kðpÞ ¼ p1=q .
4. Conclusions We have investigated the system behavior within the nonlinear approach. Systems with nonextensive statistics are considered under the supposition of the independent and dependent diffusion coefficient on the stochastic quantity x. We set scaling laws of motion for a free particle system and discussed the picture of transitions in the system. It was found that the system behaves itself as a system with multiplicative fluctuations, where the bifurcation values are dependent on the noise intensity. Considering Tsallis statistics, we elucidate that the index q of nonlinearity (nonextensivity) of the system shifts the bifurcation magnitudes at which the ordered state appeared. The scaling form of the motion law is defined for both cases of Tsallis statistics. We have shown that the system with nonextensive statistics can be transformed from the regime of anomalous to ordinary diffusion by means of the variation of the multiplicative noise exponent.
Acknowledgement The authors are thankful to Prof. A.I. Olemskoi for inspiring discussions and helpful comments.
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References [1] C. Tsallis, J. Stat. Phys. 52 (1988) 479. [2] C. Tsallis, in: S. Abe, Y. Okamoto (Eds.), Nonextensive Statistical Mechanics and its Applications, Lecture Notes in Physics, Springer, Berlin, 2000. [3] P. Castiglione, A. Mazzino, P. Muratore-Ginanneschi, A. Vulpiani, Physica D 134 (1999) 75. [4] E.W. Montroll, B.J. West, in: E.W. Montroll, J.L. Lebowitz (Eds.), Studies in Statistical Mechanics, VII, North-Holland, Amsterdam, 1979, p. 62; C. Tsallis, D.J. Bukman, Phys. Rev. E 54 (1996) R2197. [5] P.T. Landsberg, J. Stat. Phys. 35 (1984) 159; L.G. Taff, Celestial Mechanics, Wiley, New York, 1985. [6] D. Kahneman, A. Tversky, Econometrica 47 (1979) 263; A. Tversky, D. Kahneman, J. Risk and Uncertainty 5 (1992) 297. [7] H. Risken, The Fokker–Planck Equation, Springer, Berlin, 1984. [8] Y.-H. Taguchi, H. Takayasu, Europhys. Lett. 30 (1995) 499. [9] D.C. Clayton, Nature 246 (1974) 131. [10] C. Tsallis, R.S. Mendes, A.R. Plastino, Physica A 261 (1998) 534. [11] S. Abe, S. Martinez, F. Pennini, A. Plastino, Phys. Lett. A 281 (2001) 126. [12] A.R. Plastino, A. Plastino, Phys. Lett. A 222 (1995) 347. [13] L. Borland, Phys. Rev. E 57 (1998) 6634. [14] O.I. Olemskoi, A.V. Khomenko, D.O. Kharchenko, Physica A 323 (2003) 263. [15] G. Kaniadakis, P. Quarati, Phys. Rev. E 49 (1994) 5103. [16] G. Kaniadakis, Phys. Lett. A 288 (2001) 283. [17] A.I. Olemskoi, D.O. Kharchenko, Physica A 293 (2001) 178; D.O. Kharchenko, S.V. Kohan, EPJ B 29 (2002) 97. [18] D.O. Kharchenko, Fluct. Noise Lett. 2 (2002) L287. [19] A.I. Olemskoi, JETP Lett. 71 (2000) 412. [20] G.W. Ford, M. Kac, P. Mazur, J. Math. Phys. 6 (1965) 504. [21] R. Zwanzig, J. Stat. Phys. 9 (1973) 215.
Physica A 354 (2005) 262–280 www.elsevier.com/locate/physa
Evolution of a stochastic system within the framework of Tsallis statistics Dmitrii O. Kharchenko, Vasilii O. Kharchenko Sumy State University, Rimskii-Korsakov St. 2, 40007 Sumy, Ukraine Received 29 June 2004; received in revised form 21 December 2004 Available online 19 April 2005
Abstract The general approach of a nonlinear Fokker–Planck equation is applied to investigate the behavior of main statistical moments of a stochastic system. It was shown that the system described by Tsallis statistics can undergo transitions inherent to multiplicative noise-induced transitions. The scaling laws of motion are defined to identify an anomalous time behavior of the system with multiplicative noise. r 2005 Elsevier B.V. All rights reserved. PACS: 02.50.r; 05.40.a; 72.70.+m Keywords: Tsallis statistics; Noise; Statistical moments; Anomalous diffusion
1. Introduction The study of processes which obey nonextensive statistics [1] remains a cornerstone of modern statistical mechanics. There is a wide spectrum of complex systems for which the standard and powerful Boltzmann–Gibbs (BG) approach cannot give correct results [2]. Typical and well-known examples of processes where standard BG statistics arrives at difficulties to interpret the experimental data are Le´vy-type anomalous diffusion (see, for example, [3] and references therein), Corresponding author.
E-mail address: [email protected] (D.O. Kharchenko). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.01.057
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correlated-type anomalous diffusion [4], self-gravitating processes [5], and financial risks and decisions [6]. To add to this list we may mention systems with long-range microscopic memory [7], granular systems [8], solar neutrinos [9], and so on. A satisfactory description of such processes is achieved within the framework of generalized (nonextensive) statistical mechanics [2], where power-law distributions, algebraic correlations and violation of the habitual additivity (extensivity) rule for entropy functional can be introduced. The nonextensive statistics is based on a generalized entropic form which keeps all thermodynamic relations, from which BG relations can be derived in a special limit [10,11]. An evolution of systems obeying the nonextensive statistics can be described with the help of a nonlinear Fokker–Planck equation (NFPE) or fractional Fokker– Planck equation (associated with fractional order evolution equation for a physical quantity). A formalism based on the NFPE has been widely discussed since 1995, after work by Plastino and Plastino [12]. Usually, the form of NFPE is constructed in the framework of special assumptions about correlation between micro- and macro-levels in a hierarchical system. An application of the NFPE to a free particle system was discussed in Ref. [13], and a study of the self-organized criticality with the help of the NFPE of a fractional order is proposed in Ref. [14]. Recently, the generalized scheme for nonlinear kinetics was considered and the NFPE was obtained on the basis of a master equation [15,16]. As shown in Ref. [16], in the framework of such an approach one can pass on to the nonextensive statistics normally. In works that concern a study of dynamics of systems with nonextensive statistics, main attention is paid to investigate the form of the NFPE or to define main laws of nonextensive thermodynamics. However, the dynamics of the system obeying a generalized approach is an open question. Usually, one deals with a free particle system to investigate the Le´vy-type processes and to find relations between the parameter of nonextensivity q and exponent inherent to Le´vy-type diffusion. The statistical picture of the system in quadratic potential of the stochastic quantity x is proposed in Refs. [12,4], where the explicit form of the distribution was calculated. The present effort is an attempt to show a picture of the system evolution in terms of first and second moments of stochastic quantity within the framework of the generalized approach. We will investigate the behavior of main statistical moments to show in what way the passage to the generalized approach transforms the picture of the ordering of the system described by x4 -potential. We will set scaling properties of corresponding time dependencies and define the influence of multiplicative noise on fractal properties of the phase space of the system. The work is organized as follows. We represent main ideas to derive the NFPE (Section 2). Based on the form of NFPE, evolution equations for first and second statistical moments are obtained in Section 3. Here we will show the form of the corresponding Langevin equation containing terms defining the hierarchical coupling between micro- and macro-levels of the system. Two special cases inherent to Tsallis statistics are considered in Sections 3.1 and 3.2. Section 3.3 is devoted to investigating a multiplicative noise influence where the main scaling laws of motion are defined. The main results are collected in Section 4.
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2. Nonlinear Fokker–Planck equation A keypoint to the nonextensive statistics is a master equation for the probability density function p ¼ pðx; tÞ in the form Z qpðx; tÞ ¼ ½wðx0 ; x x0 Þgðp0 ; pÞ wðx; x0 xÞgðp; p0 Þ dx0 , (1) qt where wðx; x0 Þ is the transition rate which depends on the nature of interactions between the particle and the bath and is a function of starting x and arrival x0 sites. The factor gðp; p0 Þ is an arbitrary function of the particle populations of the starting and the arrival sites. This function satisfies the following conditions: gð0; p0 Þ ¼ 0 if the starting site is empty, the transition probability is equal to zero; gðp; 0Þa0 requires that in the case the arrival site is empty the transition probability depends on the population of the starting site. In the case gðp0 ; pÞ ¼ p0 and gðp; p0 Þ ¼ p we have the standard liner kinetics and the master equation corresponds to the Chapman– Kolmogorov equation. The master equation (1) yields the NFPE in the form [16] qp q qDðxÞ q q ln kðpÞ qp ¼ f ðxÞ DðxÞgðpÞ , (2) gðpÞ þ qt qx qx qx qp qx where gðpÞ ¼ gðp; pÞ and the function kðpÞ40 is defined through the condition q ln kðpÞ q gðp; p0 Þ ¼ ln 0 . qp qp gðp ; pÞ p¼p0
(3)
The first and second moments Rover wðx; x0 Þ correspond R to the drift and diffusion terms, respectively, i.e., f ðxÞ dx0 x0 wðx; x0 Þ, 2DðxÞ dx0 x0 2 wðx; x0 Þ. In the stationary equilibrium state, a solution ps ¼ pðx; 1Þ of Eq. (2) takes the form Z x f ðx0 Þ ln kðps Þ ¼ bðU ef ðxÞ mÞ; U ef ðxÞ ¼ dx0 þ ln DðxÞ , (4) Dðx0 Þ where gðpÞ remains an arbitrary function, b is the constant related to the noise R intensity and m takes care of the normalization condition p dx ¼ 1. A nonlinearity of the Fokker–Planck equation is defined through the form of the function kðpÞ: at ln kðpÞ ¼ ln p we pass to the BG statistics; a q-deformation of the logarithm ln k ! lnq k promotes Tsallis statistics, where lnq k ¼ ðk1q 1Þ=ð1 qÞ.
3. Equations for averages The aim of this paper is to consider the evolution of a system described by Eq. (2). To this end, we explore the behavior of the first moment hxi being an order parameter Z, a variance hðdxÞ2 i (dx ¼ hxi x) which plays the role of a one-time correlation function S and a two-time correlator (Green function) Gðt; t0 Þ ¼ hxðtÞxðt0 Þi.
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To derive corresponding evolution equations we use the standard technique. Let us multiply Eq. (2) by xn (n ¼ 1; 2) and integrate over x. As a result, we arrive at an evolution equation for the first and second moments in the form d dDðxÞ q q ln kðpÞ hxi ¼ f ðxÞ þ DðxÞp , dt dx qx qp d 2 dDðxÞ q q ln kðpÞ hx i ¼ 2 x f ðxÞ þ DðxÞp dt dx qx qp q ln kðpÞ þ 2 DðxÞp . ð5Þ qp The obtained equations and Eq. (2) allow us to write down an effective Langevin equation as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx dDðxÞ q q ln kðpÞ q ln kðpÞ ¼ f ðxÞ þ DðxÞp þ DðxÞp xðtÞ , (6) dt dx qx qp qp where the generalized Langevin source xðtÞ is defined as a white noise with standard properties: hxðtÞi ¼ 0, hxðtÞxðt0 Þi ¼ 2dðt t0 Þ. In the case ln kðpÞ ¼ ln p we pass to the linear Fokker–Planck equation and arrive at the standard Langevin equation in Itoˆ interpretation with a multiplicative noise of pffiffiffiffiffiffiffiffiffiffi amplitude DðxÞ. In such a case, the multiplicative noise appearance is induced by the influence of environment. At ln kðpÞa ln p the multiplicative character of the noise in Eq. (6) is related to the appearance of terms with p-dependence. Such dependencies of drift and diffusion terms at D ¼ const can be explained as a twosided feedback between different (micro- and macro-) hierarchical levels of the system [13]. Therefore, the function kðpÞ governs a procedure of the hierarchical coupling of the system states. A solution of the problem related to a system with multiplicative noise within the framework of linear kinetics (ln kðpÞ ¼ ln p) was proposed in Ref. [17], where an arbitrary multiplicative noise (in a power-law form) is considered. It was shown that the multiplicative noise governs the system dynamics (it changes the bifurcation values of the system parameters): the power-law dependence of the amplitude of multiplicative noise leads to an anomalous behavior of the system (its phase space acquires fractal properties) [18]. Systems with the nonlinear kinetics ln kðpÞa ln p are formed hierarchically and, therefore, should have fractal properties. A typical and well-known example is a motion of the ‘‘particle’’ on a hierarchical tree in an ultrametric space [19]). Unfortunately, a description of such objects is carried out on the basis of the macroscopic approach, where the NFPE is postulated initially. Using a theory of diffusion processes, one can derive the corresponding Fokker–Planck equation and write down the effective Langevin equation containing terms that depend on the probability density function p. Initially, such a kind of problem was discussed in Refs. [20,21]. Next, fixing ln½kðpÞ through a q-deformed logarithm, we consider the system with the Tsallis statistics for both cases of kðpÞ a linear form of kðpÞ ¼ p (Section 3.1) and a power-law form of kðpÞ ¼ p1=q (Section 3.2). In the framework of this approach we
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focus on properties of the ordering process. Considering the model of the system with D ¼ const and x4 -potential V ðxÞ ¼
e 2 x4 x þ , 2 4
(7)
where e ¼ ðT c TÞ=T c is a control parameter acting as a temperature counted from a critical value T c , we discuss the dynamics of the main statistical moments. 3.1. Linear form: kðpÞ ¼ p Assuming ln½kðpÞ ¼ lnq p , we arrive at the NFPE in the form qp q qDðxÞ qDðxÞp1q q2 ¼ f ðxÞ þ p þ 2 DðxÞp2q . qt qx qx qx qx
(8)
(9)
Next, we use the Fokker–Planck equation (9) to find the long-time scaling behavior of processes. To this end, we focus on a free particle system supposing f ðxÞ ¼ 0 and D ¼ const. In such a case we arrive at equivalent equations for the stochastic process xðtÞ and probability density function as follows: sffiffiffiffiffiffiffiffiffiffiffi dx 2D ð1qÞ=2 ¼ p xðtÞ , (10) dt 2q qp D q2 2q ¼ p . qt 2 q qx2
(11)
The nonlinear diffusion equation (11) can be solved in an automodel regime: xðtÞ ¼ yaðtÞ, pðx; tÞ ¼ ðaðtÞÞ1 jðx=aðtÞÞ. Such an assumption allows one to identify a width of the diffusion burst which increases as tm . In general, the exponent m defines anomalous scaling in time which is reduced to the Hurst exponent H ¼ 1=2 of the diffusion process in BG statistics. Therefore, instead of Eq. (11) we get da d d2 j y j ¼ aq2 D 2 j2q . (12) dt dy dy _ 2q ¼ const ¼ n, Eq. (12) reads Assuming aa nyj ¼
D d 2q j . 2 q dy
The solution of Eq. (13) has the form of Tsallis distribution 1=ð1qÞ ð1 qÞn ðy y0 Þ2 , jðyÞ ¼ jðy0 Þ1q 2D
(13)
(14)
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where conditions jðy ! 1Þ ¼ 0 and ðdjðyÞ=dyÞy!1 ¼ 0 are used. The time dependence aðtÞ is in the form aðtÞ ¼ ½aðt0 Þ3q þ nð3 qÞðt t0 Þ1=ð3qÞ .
(15)
R Making use of such a scaling procedure, we define hxi ¼ xpðxÞ dx and set hx2 ðctÞi ¼ c2m hx2 ðtÞi. Hence, we obtain an expression for the exponent m in the form m ¼ ð3 qÞ1 [13]. The scaling law can be obtained from the solution of Eq. (10), where for the terms up to the first order in dt, we have 0
Z tZ
t0
pð1qÞ=2 ðxðtÞ; tÞpð1qÞ=2 ðxðt0 Þ; t0 ÞhdW ðtÞ dW ðt0 Þi
hxðtÞxðt Þi ¼ D 0
0
¼ hx2 ðtÞit2=ð3qÞ ,
ð16Þ
where t ¼ minðt; t0 Þ and the definition of the white noise xðtÞ ¼ dW ðtÞ=dt is used; W ðtÞ is the Wiener process. Considering the dynamics of the statistical modes, we use the deterministic force f ðxÞ ¼ qV =qx and construct a close loop system for the order parameter Z ¼ hxi and one-time autocorrelation function S ¼ hðZ xÞ2 i. Deriving these equations, we express averages with p-dependencies through the form of the stationary distribution function pðxÞ. A stationary solution of Eq. (9) is the Tsallis distribution: pðxÞ ¼ Z1 ½1 bð1 qÞV ðxÞ1=ð1qÞ ,
(17)
where Z is the normalization constant, b ¼ Z1q =D. Taking into account Eq. (7), evolution equations for the order parameter and autocorrelation function read as dZ ¼ Zðe Z2 3SÞ , dt dS 1q 1 4 2b1 D 2 2 2 2 ¼ 2Sðe 3ðS þ Z ÞÞ þ eðS þ Z Þ ½Z þ 6SZ þ 3S þ , dt 2q 2 2q ð18Þ where t ¼ ð2 qÞt. An equation for the Green function Gðt; t0 Þ ¼ hxðtÞxðt0 Þi is in the form q G ¼ ðe 3ðS þ Z2 ÞÞG . (19) qt Consider a stationary picture of the system behavior. Assuming dZ=dt ¼ dS=dt ¼ 0, we find that the first solution of the order parameter equation defines the disordered state being the attractive point C 0 ðZ0 ; S 0 Þ with coordinates 1=2 ! ð5 3qÞe 12ð9 5qÞb1 D 1þ 1þ Z0 ¼ 0; S 0 ¼ (20) 3ð9 5qÞ ð5 3qÞ2 e2
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on a phase plane ðZ; SÞ. The ordered state appears if we take D and e inside a domain bounded by a solution of the phase diagram equation 24b1 D ð9 5qÞ ¼ 4ð5 3qÞ2 6ð9 5qÞð1 qÞ . e2
(21)
Solutions of Eq. (21) are shown in Fig. 1. It is seen that a decrease in the nonlinearity index q shifts the bifurcation point toward large noise intensity D (Fig. 1a). Formally, such a dependence shows that in the case qa1 the system behaves as a system with a multiplicative noise [17]. Indeed, here a decrease in the index q promotes the dependence of the bifurcation point through the noise intensity D. Fig. 1b shows the influence of the noise intensity on the bifurcation point
1 1.6
0.8
1 2
disorder 0.6
1.2
D 0.4
5 0.4
5 6 7
0.0 0.0
0.2
0.4
ε
0.6
0.8
4
0.8
3 4
0.2
(a)
3
q
2
order
0.0 0.0
1.0
0.2
(b)
0.4
ε
0.6
0.8
1.0
1.6
q
1.2
0.8
0.4
1
3
2
0.0 0.0 (c)
0.2
0.4
0.6
0.8
D
Fig. 1. Phase diagrams for the system with kðpÞ ¼ p: (a) e vs. D (curves 1–7 correspond to q ¼ 0:01; 0:3; 0:6; 0:8; 0:99; 1:3; 1:6); (b) e vs. q (curves 1–5 correspond to D ¼ 0:02; 0:05; 0:1; 0:2; 0:4); (c) D vs. q (curves 1–3 correspond to e ¼ 0:3; 0:6; 0:99. A domain of ordered state is situated below the corresponding curves in a and c plates.
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with variation of the index q. Here the ordered state with Za0 is realized at large e if we increase D. The critical noise intensity increases with an increase in e (Fig. 1c). According to the obtained phase diagrams, we find that the ordered state is characterized by dependencies shown in Fig. 2. It is seen that the ordered state is realized in a discontinuous manner inherent to the first-order phase transition. Such a picture of the order parameter (variance) dependence can be explained introducing the correlation effects, i.e. Sa0. It is well known that on the macroscopic level we can neglect correlations and arrive at the standard Landau picture of phase transitions of the second order, where Z_ ¼ hf ðxÞi ¼ f ðZÞ, Z0 / e1=2 . The ordered state is characterized by two stationary points on the phase plane ðZ; SÞ with coordinates: 1=2 Z , 0 ¼ ½e 3S
S 0
1=2 ! ð5 3qÞe 6ð9 5qÞ 1 2 1 1 ¼ ðb D þ e ð1 qÞ=4Þ . 3ð9 5qÞ ð5 3qÞ2 e2
ð22Þ
þ þ Here upper and lower signs define the saddle CðZ 0 ; S 0 Þ and node NðZ0 ; S 0 Þ on the phase plane (see Fig. 3.). Dotted lines in Fig. 2 display the position of saddle S and solid lines represent the attractive point N position. As seen from Fig. 2, with an increase in q the ordered state is characterized by small values of the order parameter. The system behavior does not change essentially if we take qo1 or q41. To consider dynamics of the system we present solutions of Eq. (18) in Fig. 3. Here at small temperatures the phase portrait is characterized by the single point C 0 defining the disordered state (Fig. 3a). When e increases two additional points þ þ CðZ 0 ; S 0 Þ and NðZ0 ; S 0 Þ appear (Fig. 3b). In the vicinity of the point C the slowing down of the order parameter ZðtÞ and one-time correlator SðtÞ occurs (Fig. 4). Analysis of the time dependencies is performed with the help of the Lyapunov’s exponents method. Here for the system evolving in time t we find lq ¼ ð2 qÞlðeÞ, where lðeÞ is defined for the system evolving in time tð2 qÞ. It is seen that points N and C are node and saddle if qo2. The system slowly attains these points if q takes large magnitudes. Let us take a look at the correlation processes in the vicinity of points C 0 and N (see Fig. 5). Solutions of Eq. (19) are shown in Fig. 5 at different values of e and q at identical initial conditions. It is seen that at eoe0 the monotonic decrease appears at large initial magnitude Zð0Þ (Fig. 5a). In case e4e0 , a maximum of dependence disappears at condition Zð0ÞoZc corresponding to the domain of the ordered state (Fig. 5b).
3.2. Power-law form: kðpÞ ¼ p1=q Assuming ln½kðpÞ ¼ lnq ½p1=q ,
(23)
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0.8 4
2 3
5
1
η0
0.6
0.4
0.2
ε0 0.0 0.5
0.6
0.7 ε
(a)
0.8
0.9
1
0.5
S0
0.4
0.3 2 4 1 3
0.2
2 3 5
5
0.1
ε0
0.0 0.0
(b)
4
1
0.2
0.4
ε
0.6
0.8
1.0
Fig. 2. Stationary values of order parameter (a) and autocorrelator (b) vs. control parameter e (curves 1, 2 are plotted at D ¼ 0:2 and q ¼ 0:4; 0:7; curve 3 corresponds to D ¼ 0:05, q ¼ 1; curves 4, 5 are plotted at D ¼ 0:05, q ¼ 1:2; 0:7). The bifurcation temperature is denoted as e0 .
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S
0.6
0.4 C0 0.2
0.0 0.0
0.2
0.4 η
(a)
S
0.3
0.6
C0
0.2
C
0.1 N
0.0 0.0 (b)
0.2
0.4
η
0.6
0.8
Fig. 3. Phase portraits at D ¼ 0:2: (a) e ¼ 0:4, q ¼ 0:6; (b) e ¼ 0:7, q ¼ 0:3.
271
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0.8 3
0.6 η
2
0.4 1
0.2
0.0 0.0
5
10
(a)
15 τ
20
25
0.3
1
S
0.2
2
0.1 3
0.0 0.0
(b)
5
10 τ
15
Fig. 4. Time dependencies of the order parameter (a) and the autocorrelator (b) corresponding to different trajectories on the phase portrait in Fig. 3 at q ¼ 0:3, e ¼ 0:7, D ¼ 0:2 (curves 1, 2, 3 correspond to Sð0Þ ¼ 0, Zð0Þ ¼ 0:215; 0:225; 0:99).
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273
0.10
G(t,0)
0.08
0.06 1 0.04 2 0.02 3 0.00 2
0.0
4
(a)
6 τ
8
10
0.12
G(τ ,0)
1 2 0.08
0.04 3
0.00 0.0
(b)
2
τ
4
6
Fig. 5. Correlation functions in the vicinity of points C 0 , N, at D ¼ 0:2: (a) e ¼ 0:4, q ¼ 0:6 (curves 1, 2, 3 correspond to (Sð0Þ ¼ 0, Zð0Þ ¼ 0:2), (Sð0Þ ¼ 0:6, Z ¼ 0:4), (Sð0Þ ¼ 0:1, Z ¼ 0:8); (b) e ¼ 0:4, q ¼ 0:6 (curves 1, 2, 3 correspond to (Sð0Þ ¼ 0, Zð0Þ ¼ 0:215), (Sð0Þ ¼ 0, Z ¼ 0:225), (Sð0Þ ¼ 0:1, Z ¼ 0:99).
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the NFPE can be written as follows: qp q qDðxÞ 1 qDðxÞpð1qÞ=q 1 q2 ¼ f ðxÞ þ DðxÞp1=q . pþ qt qx qx q q qx2 qx
(24)
As in the previous subsection, we investigate the free particle system at first. In such a case, the diffusion equation reads qp q2 ¼ D 2 p1=q . qt qx
(25)
Using the results of the previous subsection, we can find solutions in the automodel regime for both the function aðtÞ and the spatial component jðyÞ: aðtÞ ¼ ½aðt0 Þ1þ1=q þ nð1 þ 1=qÞðt t0 Þq=ð1qÞ ,
(26)
q=ð1qÞ nð1 qÞ 1=qþ1 2 ðy y0 Þ jðyÞ ¼ jðy0 Þ . 2D
(27)
It is easy to see that the correlation function hxðtÞxðt0 Þiq is the second-order moment defined through the probability density function1 Z t hxðtÞxðt0 Þiq ¼ p1=q1 ðxðtÞ; tÞ dt ¼ hx2 ðtÞit2q=ð1þqÞ . (28) 0
Hence, for the exponent m we have m ¼ q=ð1 þ qÞ .
(29)
It differs from the expression obtained in the linear case of kðpÞ. Here one has m ¼ 0 at q ¼ 0, whereas m ¼ H ¼ 1=2 at q ¼ 1. Considering the evolution of the system in x4 -potential, we write down equations for the order parameter Z ¼ hxiq and autocorrelation function S ¼ hðdxÞ2 iq in the form dZ ¼ Zðe Z2 3SÞ , dt dS ¼ 2Sðe 3ðS þ Z2 ÞÞ þ ð1 qÞ eðS þ Z2 Þ 12½Z4 þ 6SZ2 þ 3S 2 þ 2b1 q D. dt ð30Þ An evolution equation for the Green function coincides with Eq. (19). In these equations we use the renormalized time t ¼ t=q, and the constant bq is defined from the normalization condition. R Here the averages are defined through the standard scheme hxn i ¼ xn p dx. Due to the construction k ¼ p1=q , the probability density p should be considered as an escort probability density. Indeed, R introducing f ðxÞ ¼ p1=q ðxÞ, Raverages should be meant as q-averages, i.e., hxn iq ¼ xn f q ðxÞ dx whereas the normalization condition is f q ðxÞ dx ¼ 1 (see Ref. [16]). 1
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Here the disordered state is characterized by a single stable node C 0 with coordinates: 1=2 ! ð3 qÞe 12ð5 qÞb1 D 1þ 1þ . (31) Z0 ¼ 0; S 0 ¼ 3ð5 qÞ ð3 qÞ2 e2 The ordered state appears when the bifurcation line obtained as a solution of the equation 24b1 D ð5 qÞ ¼ ðq 3Þ2 6ð5 qÞð1 qÞ e2
(32)
1
2.0
1 0.20 2 0.15
disorder
3
q
D
4
0.10
5
1.0
6
0.5
3
0.05
0.00
order
0.0 0.0
(a)
2
1.5
0.2
0.4
ε
0.6
0.8
1.0
0.0
0.2
(b)
0.4
ε
0.6
0.8
1.0
1.6
q
1.2 1
2
3
0.8
0.4
0.0 0.00
(c)
0.05
0.10
0.15
0.20
D
Fig. 6. Phase diagrams for the system with kðpÞ ¼ p1=q : (a) e vs. D (curves 1–6 correspond to q ¼ 0:01, 0.1, 0.6, 0.99, 1.3, 1.5); (b) e vs. q (curves 1–3 correspond to D ¼0.01, 0.05, 0.1); (c) D vs. q (curves 1, 2, 3 correspond to e ¼ 0:3; 0:6; 0:99). A domain of ordered state is situated below curves in plate a; in plate c the ordered state is bounded by corresponding curves.
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is crossed. Solutions of the phase diagram (32) are shown in Fig. 6. It is seen that despite the form of curves eðDÞ being topologically identical to the case of kðpÞ ¼ p, the influence of the nonlinearity parameter q changes the situation in principle (Fig. 6a). Here the bifurcation magnitude of the noise intensity D is shifted toward the
0.8
4
0.6 ηq0
3 2 0.4 1 0.2
ε0
0.0 0.0
0.2
0.4
(a)
0.6
ε
0.8
1.0
0.4
Sq0
0.3
4
0.2
3 2 4 2
1
0.1
3
1 ε0
0.0 0.0
(b)
0.2
0.4
ε
0.6
0.8
1.0
Fig. 7. Stationary values of order parameter (a) and autocorrelator (b) vs. control parameter e (curves 1, 2 correspond to q ¼ 0:2, D ¼ 0:01; 0:05; curves 3, 4 correspond to q ¼ 1:2, D ¼ 0:05; 0:07).
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large magnitudes if q increases. A more complicated picture of the ordering is shown in Fig. 6b. It is seen that at moderate values of q the bifurcation realizes at high temperature whereas at small and large q the ordered state appears at low temperature. Fig. 6c shows that a transition to the ordered state is possible at large D if q is taken from moderate values, whereas at large and small q a magnitude of the critical noise intensity is small. The stationary behavior of the order parameter and the autocorrelator is shown in Fig. 7, where solid and dotted lines correspond to the stable point NðZþ 0 ; S0 Þ þ and saddle CðZ0 ; S 0 Þ in the plane. Coordinates of these points are defined as follows: 1=2 , Z 0 ¼ ½e 3S
S 0
1=2 ! ð3 qÞe 6ð5 qÞ 1 2 1 1 ¼ ðbq D þ e ð1 qÞ=4Þ . 3ð5 qÞ ð3 qÞ2 e2
ð33Þ
It is seen that in the case under consideration the behavior of the order parameter vs. e is the same as in the previous case, but the form of dependencies is changed. Here with e-growth magnitudes of Z 0 and S 0 increase. Finally, the topology of phase portraits and corresponding time dependencies of main modes are the same as at kðpÞ ¼ p. At high temperature the system is in disordered state and characterized by the single node (Fig. 3a). The appearance of the ordered state at low temperature is accompanied by the appearance of two stationary points C and N (see Fig. 3b). Performing the stability analysis in the vicinity of these points, one can find that the Lyapunov exponent lq ¼ lðeÞ=q, where lðeÞ is defined for the system evolving in time t=q. Here at small q the Lyapunov exponent takes large magnitudes in the vicinity of the node N and, therefore, the system attains the stable ordered state faster than at large q. The above analysis shows that for the systems with a potential constructed on the basis of catastrophes theory the x2 -term yields the exponential behavior of the statistical moments (see for example Refs. [12,4]). One can conclude that the algebraic form of time dependencies can be observed when the system is in the potential V ðxÞ / x1þb , b40. In other words, the scaling laws appear if the system has fractal properties or its potential is a homogeneous function with a fractional exponent b at D ¼ const. It can be seen from the form of the NFPE that the drift term f ðxÞ defines the exponential behavior of the time dependence cðtÞ of the probability density function pðx; tÞ ¼ cðtÞfðxÞ. On the other hand, as was shown above, the scaling laws are realized in the limit f ðxÞ ¼ 0 and D ¼ const. It is easy to see that such a regime is observed in the case DðxÞ / xd with d40. In other words, the algebraic form of time dependencies is realized if the system has a self-affine structure of the phase space. 3.3. System with self-affine phase space Let us pass to the limit of homogeneous functions considering the variable y ¼ xðtÞ=aðtÞ which allows one to write down p ¼ a1 jðyÞ. We suppose that the diffusion
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coefficient scales as D ¼ ad DðyÞ; for the drift term being the first term in the NFPE we assume f qD=qx ¼ ab F ðyÞ. It allows one to find the law of motion and scaling properties of the diffusion process. Making use of the automodel assumption, at kðpÞ ¼ p the NFPE reads as _ ¼ ab F j þ a2þqþd Dj1q ayj
d j. dy
(34)
For the time dependence we obtain _ b ¼ const ¼ n; aa
b ¼ 2 þ q þ d .
(35)
Hence, in the representation aðtÞ / ðt t0 Þm for the exponent m, we have m¼
1 . 3qd
(36)
In the case kðpÞ ¼ p1=q the equation _ ¼ qab F j þ ad1=q Dj1=q1 qayj
d j dy
(37)
is used to find the scaling properties of the law of motion. Here, for the exponent m one has m¼
q ; 1 þ qð1 dÞ
b ¼ d 1=q .
(38)
1.0 2′ 0.8
superdiffusion
1′ 2
0.6 µ
1
0.4
subdiffusion
0.2
0.0 0.0
0.5
1.0
1.5
2.0
q
Fig. 8. Domains of anomalous diffusion in the m2q plot for the system with multiplicative noise (solid and dotted lines are related to Eqs. (36) and (37) respectively): curves 1, 2 correspond to d ¼ 0:0; curves 10 ; 20 correspond to d ¼ 0:5.
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For the spatial dependence we obtain following expressions, 8 Ry > ny F ðyÞ 1=ð1qÞ > 1q > ð1 qÞ dy at kðpÞ ¼ p ; > < fðy0 Þ DðyÞ jðyÞ ¼ > Ry ny F ðyÞ q=ð1qÞ > ð1þqÞ=q > fðy Þ ð1 qÞ dy at kðpÞ ¼ p1=q ; > 0 : DðyÞ
279
(39)
where terms with n define a contribution related to the time derivative in the Fokker–Planck equation; at n ¼ 0 we pass to the stationary equilibrium distributions. Comparing expressions for the exponent m in the linear and nonlinear cases, one can conclude that the ordinary diffusion process with m ¼ H ¼ 1=2 can be reached if d ¼ 1 q at k ¼ p or d ¼ 1=q 1 at k ¼ p1=q . In other words, the nonextensive system with the multiplicative noise can behave itself as a system of ordinary diffusion. By putting q ¼ 1 in both cases we will receive the expression m ¼ ð2 dÞ1 [18] for the system with multiplicative noise. The regime of superdiffusion is characterized by m41=2. As it is seen from Eqs. (36) and (38) an increase in d leads to a transition to the superdiffusion regime (see Fig. 8). In other words, the process generated by the p-dependent Langevin equation scales in time as an anomalous diffusion process. The condition mo1 allows one to set the possible interval for q as follows: 1oqo2 at kðpÞ ¼ p and q40 at kðpÞ ¼ p1=q .
4. Conclusions We have investigated the system behavior within the nonlinear approach. Systems with nonextensive statistics are considered under the supposition of the independent and dependent diffusion coefficient on the stochastic quantity x. We set scaling laws of motion for a free particle system and discussed the picture of transitions in the system. It was found that the system behaves itself as a system with multiplicative fluctuations, where the bifurcation values are dependent on the noise intensity. Considering Tsallis statistics, we elucidate that the index q of nonlinearity (nonextensivity) of the system shifts the bifurcation magnitudes at which the ordered state appeared. The scaling form of the motion law is defined for both cases of Tsallis statistics. We have shown that the system with nonextensive statistics can be transformed from the regime of anomalous to ordinary diffusion by means of the variation of the multiplicative noise exponent.
Acknowledgement The authors are thankful to Prof. A.I. Olemskoi for inspiring discussions and helpful comments.
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