Weak chaos and anomalous transport: a deterministic approach

Communications in Nonlinear Science and Numerical Simulation 8 (2003) 137–148 www.elsevier.com/locate/cnsns

Weak chaos and anomalous transport: a deterministic approach Roberto Artuso *, Giampaolo Cristadoro Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Istituto Nazionale di Fisica della Materia, Unita
di Como, Universit a dellÕInsubria and Center for the Study of Dynamical Systems, Via Valleggio 11, I-22100 Como, Italy

Abstract We review how transport properties for chaotic dynamical systems may be studied through cycle expansions, and show how anomalies can be quantitatively described by hierarchical sequences of periodic orbits. Ó 2003 Elsevier B.V. All rights reserved. PACS: 05.45.+b Keywords: Anomalous diffusion; Zeta functions; Weak chaos

1. Introduction In this paper we consider deterministic dynamical systems that exhibit nontrivial transport properties (which may be associated to either normal or anomalous diffusion). We start by dealing with the case of hyperbolic systems where typically normal diffusion is observed (even though actual calculation of transport coefficients may be exceedingly difficult [1,2]), while the in the second part we consider weakly chaotic systems, where long trappings near regular phase-space regions may induce anomalies in diffusive properties. 2. Normal diffusion A prototype model of diffusing system is realized via a chain of one-dimensional maps obtained by lifting a torus map on the whole real line (see Fig. 1). *

Corresponding author. Address: Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Universita dellÕInsubria and Center for the Study of Dynamical Systems, Via Valleggio 11, I-22100 Como, Italy. E-mail address: artuso@fis.unico.it (R. Artuso). 1007-5704/03/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1007-5704(03)00025-X

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Fig. 1. An example of a lift map f .

We denote by f^ the torus map and by f the corresponding lift, which will satisfy the following symmetry requirements: f ðx þ nÞ ¼ n þ f ðxÞ

ð1Þ

so that the system has a spatial periodicity, and f ðxÞ ¼ f ðxÞ

ð2Þ

so that no net drift will be present in the system. The torus map, on S ¼ ½0; 1Þ is just the previous map on a single cell, taken mod 1. To study transport properties, like diffusion, we start by a bunch of trajectories, and follow their spreading in time: in particular normal diffusion is associated to linear growth of the variance (angular brackets denote averages with respect to a distribution of initial conditions): E D 2 ð3Þ ðf n ðx0 Þ  x0 Þ  2Dn for large times n: D is called the diffusion constant: if the process is gaussian D determines completely the behavior of all moments of the distribution. Our aim is to provide a statistical mechanics strategy to compute D [3]: it is actually convenient to start by introducing the generating function associated to the transport process: Gn ðuÞ ¼ heuðf

n ðx Þx Þ 0 0

i

ð4Þ

The key step consists now in linking the asymptotic behavior of the generating function to the trace of high powers of a transfer operator: in order to get a consistent picture the transfer operator has to be defined by using the dynamics associated to the torus map [3,4], with weights that reflect the full dynamics on the line: we thus define Z 1 ð1Þ ^ dygðyÞeuðf ðyÞxþry Þ dðf^ðyÞ  xÞ ð5Þ ðLu gÞðxÞ ¼ 0

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139

where ^ rð1Þ x ¼ f ðxÞ  f ðxÞ

ð6Þ

is an integer number which we call the jumping number (it actually says how many boxes are jumped to the right or to the left by acting on x with the lift map f . Since the exponential weighting factor is multiplicative, the transfer operator enjoys the semigroup property, and thus Z 1   dy Lnu q0 ðyÞ ð7Þ Gn ðuÞ ¼ 0

where q0 ðxÞ is the distribution of initial conditions. This works exactly as the calculation of the asymptotic behavior of the canonical partition function for a system admitting a transfer matrix: if we introduce the integral kernel of the operator Z 1 dygðyÞLu ðyjxÞ ð8Þ ðLu gÞðxÞ ¼ 0

we get the asymptotic behavior Z 1 dyLnu ðyjyÞ Gn ðuÞ 

ð9Þ

0

Thus the long time behavior of the generating function is dominated by the leading eigenvalue of the transfer operator, which we denote by ku;0 , Gn ðuÞ  knu;0

ð10Þ

Now, by expanding the generating function in powers of u, (10) leads to a formula linking the diffusion constant to the leading eigenvalue of the transfer operator  1 d2 ku;0  ð11Þ D¼ 2 du2 u¼0 Now we want to obtain ku;0 by cycle expansion techniques [5–7]: to this end we preliminary observe that ku;0 ¼ z1 where z is the smallest solution of detð1  zLu Þ ¼ 0

ð12Þ

We can now easily rephrase the condition (12) to explicitly show the role of periodic orbits: we employ the identity ! 1 X   zm ð13Þ detð1  zLu Þ ¼ exp ln detð1  zLu Þ ¼ exp  TrLmu m m¼1 Periodic orbits come into play once we use elementary properties of DiracÕs distribution, to evaluate traces of the transfer operator Z 0

1

dyLnu ðyjyÞ

¼

X yjf^n ðyÞ¼y

ðnÞ

eury jKðnÞ y  1j

ð14Þ

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where the jumping numbers of the orbit are n n ^n rðnÞ y ¼ f ðyÞ  f ðyÞ ¼ f ðyÞ  y

ð15Þ

and K are the cycle instabilities KðnÞ y

¼

n1 Y

f 0 ðf j ðyÞÞ

ð16Þ

j¼0

We may now proceed to derive the standard expressions for spectral determinants and zeta functions: the first step consists in using a geometric series expansion for the denominator in (14): TrLnu ¼

1 X X k¼0 yjf^n ðyÞ¼y

ðnÞ

eury

ðnÞ k jKðnÞ y jðKy Þ

ð17Þ

We remark that the geometric series expansion makes sense only if we deal with a hyperbolic system, for which jKðkÞ x j > 1 for any point and iteration order. Next we perform a subtle step: the sum in (13) and (14) is over periodic points of any possible order m: so for instance a fixed point contribution appears at any order, and every p orbit contribution appears whenever p divides m: moreover all points of a given periodic orbit contributes to the trace evaluation: but they all yield the same weight, as instabilities and jumping numbers are equal for different points of the same periodic orbit. Thus we may convert the sum over periodic points to a sum over orbits p, each taken once: we further denote by Kp and rp the instability and jumping number of the orbit p, each computed with respect to its prime period np . In this way we get 1 X 1 XX znp r eurrp detð1  zLu Þ ¼ exp  ð18Þ r jKp jr Kkr p fpg r¼1 k¼0 where fpg denotes the set of periodic orbits, each taken once, np being the prime period of the orbit. The sum over r reproduces each repetition of a single orbit p in the original summation over periodic points: we now realize that this sum may be performed exactly, leading to ! 1 XX znp eurp ð19Þ ln 1  detð1  zLu Þ ¼  exp jKp jKkp fpg k¼0 Thus the spectral determinant may be expressed by an infinite product over the so-called zeta functions: 1 Y detð1  zLu Þ ¼ Fu ðzÞ ¼ f1 ð20Þ k;ðuÞ ðzÞ k¼0

where each zeta function is a product over the whole set of prime cycles f1 k;ðuÞ ðzÞ ¼

Y fpg

znp eurp 1 jKp jKkp

! ð21Þ

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We now denote by zðuÞ the smallest zero of Fu ðzÞ (or equivalently of f1 0;ðuÞ ðzÞ) we may rewrite (11) as  1 d2 zðuÞ  ð22Þ D¼ 2 du2  u¼0

where we have taken into account that, from the standard theory of Perron–Frobenius operators, zð0Þ ¼ 1. The result (22) acquires a special significance as (generalized) zeta functions have been thoroughly considered both in the mathematics and physics literature (see for instance [7,8]: in particular they are particularly suited (provided the topology of the system may be controlled) to act as the starting point of sophisticated perturbation techniques to evaluate chaotic averages. In particular such techniques lead to exact results for one-dimensional maps with constant slope and for sawtooth and cat maps [3,9]: it has to be remarked that exactly solvable models correspond to cases where symbolic dynamics is simple (as when admissible symbol sequences are generated by a finite automaton), the generic case is typically quite complex, as clearly shown for one-dimensional maps and finite horizon Lorentz gas [1]. The symbolic dynamics complexity is just one source of possible troubles when formulas like (22) are to be applied: an even worse situation appears when dealing with weakly chaotic systems, where marginally stable orbits appear: it is quite apparent from our formal developments that we cannot cope directly with indifferent equilibria, as factors like Kp  1 vanish, leading to divergences in the former expressions. 3. Intermittency and transport Before dealing with transport properties we briefly introduce a typical weakly chaotic map [10]: xnþ1 ¼ f ðxn Þ ¼ xn þ cxzn ðmod 1Þ

ð23Þ

where c > 0 and z > 1 (the intermittency exponent). Such a map lacks full hyperbolicity, since the fixed point at x ¼ 0 is an indifferent equilibrium point. By taking c ¼ 1 the map consists of two full branches, with support on I0 ¼ ½0; pÞ and I1 ¼ ½p; 1, where p is determined by p þ pz ¼ 1. The inverse of f jIi will be denoted by /i . Though we have a binary partition ½0; 1 ¼ I0 [ I1 (such that f ðIi Þ ¼ ½0; 1), the marginal fixed point at x ¼ 0 cannot be included directly in cycle expansions, as K0  1 ¼ 0, and we have to exclude it from the infinite product (21): the symbolic dynamics can be now reproduced by a (countable) new alphabet: f1; 0k 1, k ¼ 1; 2; . . .g. The unit interval is accordingly partitioned into a sequence of subsets I1 , I0k 1 , where I1 ¼ ½p; 1 and I0k 1 ¼ /k0 ðI1 Þ. We call ‘ the widths of these intervals. Now the map is linearized in each interval [11]: the corresponding slopes will be sðI1 Þ ¼ ‘1 1 ¼ p=ð1  pÞ ¼ K, while sðI0k 1 Þ ¼ ‘0k1 1 =‘0k 1 : the corresponding cycles stabilies will be . The asymptotic behavior of ‘0k 1 is determined by the intermittency exponent K1 ¼ K, K0k 1 ¼ ‘1 0k 1 z, in fact if we put y0 ¼ 1, y1 ¼ p, yl ¼ /0 ðyl1 Þ for l ¼ 2; 3; . . . we have that ym1 ¼ ym þ ymz and d 1=ymðz1Þ  dm so that q ym  a a ¼ 1=ðz  1Þ k

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Fig. 2. A piecewise linear approximation to an intermittent map.

This implies that the widths will scale as ‘0k 1  q=k aþ1 . We now assume the former as an exact expression (and thus convert the original map to a piecewise linear approximation [11], see Fig. 2), so that ‘1 ¼ K1 and ‘0k 1 ¼ q=k aþ1 and choose q in such a way that probability conservation is respected: ‘1 þ

1 X

‘0n 1 ¼ 1 ¼ K1 þ q  fð1 þ aÞ

n¼1

so q ¼ ðK  1Þ=ðK  fð1 þ aÞÞ and the stabilities are finally written as K1 ¼ K, while K0k 1 ¼ K  fð1 þ aÞ  k aþ1 =ðK  1Þ.The dynamical zeta function, within the piecewise linear approximation, picks up only contributions coming from elementary cycles [5,7] (corresponding to the alphabetÕs letters) f1 0 ðzÞ ¼ 1 

z ðK  1Þ  z  gaþ1 ðzÞ K K  fða þ 1Þ

ð24Þ

where gs ðzÞ ¼

1 X zm ms m¼1

(which is the function appearing in the theory of Bose perfect gas), while

l z ðK  1Þ 1 z  glðaþ1Þ ðzÞ fl ðzÞ ¼ 1  l  K  fða þ 1Þ K

ð25Þ

From (24) we observe that, while by construction probability conservation is guaranteed 1  ðf1 0 ð1Þ ¼ 0Þ, the behavior of f0 ðzÞ as z  1 is not necessarily that of a simple zero. For instance g1 ðzÞ ¼  logð1  zÞ, which diverges as z 7! 1 . Now if 0 < s < 1 again gs ðzÞ diverges as z 7! 1 , with a behavior like gs ðzÞ  Cð1  sÞ  ð1  zÞs1 (notice that however such cases are not directly connected to our zeta function).

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When s > 1 (which is the case relevant to dynamical zeta functions (24)) the limit ðz 7! 1 Þ yields a finite result (fðsÞ): if s > 2 we have fðsÞ  gs ðzÞ  fðs  1Þ  ð1  zÞ

ð26Þ

while if s 2 ð1; 2 we have fðsÞ  gs ðzÞ  ð1  zÞgs1 ðzÞ

ð27Þ

from which we get the dominant contribution as z 7! 1 fð2Þ  g2 ðzÞ  ð1  zÞ logð1  zÞ

ð28Þ

fðsÞ  gs ðzÞ  ð1  zÞs1

ð29Þ

and

for s 2 ð1; 2Þ. The most important remark here is that the sequence of longer and longer cycles shadowing the marginal fixed point changes dramatically the analytic features of the dynamical zeta function: singularities develop in the form of branch points: when dealing with anomalous diffusion we will show how this leads to physically relevant consequences.

4. Thermodynamic formalism We have seen that for the intermittent map peculiar features are induced by the fact that there are orbits whose instability grows exponentially with the period together with cycles (those accumulating to the indifferent fixed point) whose instability grows polynomially. A way in which the overall scaling properties may be analyzed is through the thermodynamic analysis of the partition of the unit interval induced by periodic points of longer and longer period. At each period n we consider the set (xi;ðnÞ i ¼ 1; . . . ; 2n ) of periodic points and partition the unit interval with the intervals wiðnÞ ¼ ðxi;ðnÞ ; xiþ1;ðnÞ Þ; i ¼ 1; . . . ; 2n  1: then at each step we build the following partition function: Zn ðsÞ ¼

n 1 2X

ws iðnÞ

ð30Þ

i¼1

where the weighting parameter s (formally analogous to an inverse temperature) is allowed to vary over the whole real axis. The role of s is to probe different sizes of the partitioning intervals: as s ! 1 the sum (which will diverge) is dominated by the thinnest intervals, while at the opposite limit the behaviour is dominated by the fattest wiðnÞ . The relevance of (30) in accounting for scaling properties is easily appreciated if one recalls that the critical value of s, sc such that Zn ðsÞ ! 0ðn ! 1Þ s < sc Zn ðsÞ ! 1ðn ! 1Þ s > sc determines the Hausdorff dimension [12] of the set ðDH ¼ sc Þ (in the present example, as the underlying set is the unit interval DH ¼ 1). The scaling properties will be encoded in the free energy gðsÞ, defined in the thermodynamic limit

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gðsÞ ¼ lim gn ðsÞ n!1

where finite size approximations are determined by ðNn ¼ 2n  1Þ Zn ðsÞ ¼ Nngn ðsÞ

ð31Þ

gðsÞ is a monotonically increasing function of s, and its zero determines the Hausdorff dimension gðDH Þ ¼ 0, while it is easy to check that gð0Þ ¼ 1. Thus for a homogeneous fractal set (like the usual middle–third Cantor set) gðsÞ is a straight line with slope D1 H . In general the first derivative of gðsÞ will span the whole set of scales characterizing the set. If we define scaling exponents fliðnÞ g by liðnÞ ¼ 

log wiðnÞ log Nn

ð32Þ

and rewrite Zn ðsÞ ¼

Nn X

sliðnÞ

Nn

i¼1

it is easy to see that PðnÞ 0

g ðsÞ ¼ lðsÞ ¼ lim

n!1

i

PðnÞ i

liðnÞ s

liðnÞ Nn

liðnÞ s

Nn

so that lðsÞ gives the scaling exponent dominating the partition sum for weighting factor s. In our example we notice that finite scaling exponents are obtained only for intervals shrinking exponentially with the period, while if interval contract according to power-laws we get scalings that vanish like lnðnÞ=n. A useful function accounting for the distribution of scaling exponents is given by the scaling spectrum sðlÞ, which is defined once the sum (30) is reordered by increasing length size Z lmax dlNnsn ðlÞþls ð33Þ sðlÞ ¼ lim sn ðlÞ Zn ðsÞ ¼ n!1

lmin

Nnsn ðlÞ dl

gives the number of intervals (at the nth step of the hierarchical procedure) whose where scaling exponent is within l and l þ dl. The scaling spectrum sðlÞ is a highly irregular function of l for finite n: in practise one evaluates gn ðsÞ and replaces sðlÞ by its convex envelope SðlÞ, evaluated in the thermodynamic limit saddle point evaluation of (33).  1=2 2p log Nn NnðSn ðlÞþslÞ Nngn ðsÞ ¼ CðnÞ  Sn00 ðsÞ where lðsÞ is the solution of the extremum condition  dSn ðsÞ  ¼ s dl  l

So in our case we expect that the presence of marginality is signalled by a tail in the small scaling region of SðlÞ, which slowly advances towards zero as the hierarchical order increases. This is

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145

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.5

1

1.5

Fig. 3. SðlÞ for the partition induced by the map (23): full line N ¼ 10, dashed line N ¼ 15.

exactly what happens, see Fig. 3: the right branch of entropy is stable, and it is representative of exponential scalings: while the leftmost branch of entropy slowly moves towards zero, being ruled by intervals shrinking with power laws.

5. Anomalous diffusion Our model example is the map shown in Fig. 4, whose corresponding map on the torus is shown in Fig. 5. Branches whose support is in Ii , i ¼ 1; 2; 3; 4 have uniform slope (absolute value K), while f jI0 is of intermittent form (and we will consider as a model the piecewise linear approximation formerly dealt with). Once the  0 fixed point is pruned away the symbolic dynamics is determined by the countable alphabet f1; 2; 3; 4; 0i 1; 0j 2; 0k 3; 0l 4g, i; j; k; l ¼ 1; 2; . . . The partitioning of the subinterval I0 is induced by I0k ðrightÞ ¼ /kðrightÞ ðI3 [ I4 Þ (where /ðrightÞ denotes the inverse of the right branch of f^jI0 ) and the same reasoning applies to the leftmost branch. These are regions over which the slope of f^jI0 is constant. Thus we have the following stabilities and weights associated to letters: 1þa

0k 3; 0k 4

Kp ¼ kq=2

0l 1; 0l 2 3; 4 2; 1

l1þa

rp ¼ 1

Kp ¼ q=2 rp ¼ 1 Kp ¼ K rp ¼ 1 Kp ¼ K rp ¼ 1

where q is to be determined by probability conservation for f : 4 þ 2qfða þ 1Þ ¼ 1 K

ð34Þ

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Fig. 4. An intermittent diffusing map.

Fig. 5. The torus map corresponding to Fig. 4.

so that q ¼ ðK  4Þ=2Kfða þ 1Þ. The dynamical zeta function picks up only contributions from the alphabetÕs letters, as we have imposed piecewise linearity, and is written as f1 0;ðuÞ ðzÞ ¼ 1 

4 K4 z cosh u  z cosh u  gaþ1 ðzÞ K K  fð1 þ aÞ

and its first zero zðuÞ is determined by 4 K4 1 zþ z  gaþ1 ðzÞ ¼ K K  fð1 þ aÞ cosh u

ð35Þ

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147

By using implicit function derivation we see that D vanishes (i.e. z00 ðuÞju¼0 ¼ 0) when a 6 1. This is easily interpreted from a physical point of view, as marginal stability implies that a typical orbit will be sticked up for long times near the  0 indifferent fixed point, and the Ôtrapping timeÕ will be larger for higher values of the intermittency parameter z (recall a ¼ ðz  1Þ1 ). In order to get deviations from linear behavior we remember that  uðxn x Þ  X n 0 ku;j ð36Þ  e j¼0

and in terms of the spectral determinant (20) we have that X ku;j es d ln Fu ðes Þ ¼ ds 1  ku;j es j¼0

ð37Þ

so that [7]   1 Gn ðuÞ ¼ euðxn x0 Þ  2pi

Z

aþi1

ai1

ds esn

d ln Fu ðes Þ ds

and, in the case of one-dimensional diffusion we thus have ! Z aþi1 0 s F ðe Þ 1 d2 1 u lim ds est D¼ Fu ðes Þ 2n t!1 du2 2pi ai1

ð38Þ

ð39Þ

u¼0

As we are interested in the leading singularity we may use zeta functions instead of spectral determinants [7], thus getting ! Z aþi1 1 s 2 o f ðe Þ d 1 s 0;ðuÞ 2 ds est 1 ð40Þ hðxn  x0 Þ i  2 du 2pi ai1 f0;ðuÞ ðes Þ u¼0 The evaluation of inverse Laplace transforms for high values of the argument is most conveniently performed (if applicable) by using Tauberian theorems [13]. The asymptotic behavior of the integrand in (40) may then be evaluated from our estimate the behavior of Bose functions functions near z ¼ 1. Omitting prefactors (which can however be calculated by the same procedure) we have 8  for a > 1 < s2 2 o f1 ðes Þ  s 0;ðuÞ d  ðaþ1Þ  for a 2 ð0; 1Þ s  s  : du2 f1 0;ðuÞ ðe Þ u¼0 1=ðs2 log sÞ for a ¼ 1 from which we get the estimates (see [14,15]) 8 for a > 1