Anomalous deterministic transport

1

1Deterministic (anomalous) transportRoberto Artuso1, and Giampaolo Cristadoro

1.1Introduction

A fascinating upshot of dynamical systems theory concerns the possibility

that chaos, at a microscopic level, may induce, once averages are taken,

stochastic behavior, for instance by generating normal, random-walk like,

transport properties. As a matter of fact diffusive properties of determinis-

tic chaotic systems (most remarkable examples being the standard map [1–4],

one-dimensional maps [5–7] and Lorentz gas (with finite horizon) [8, 9]) have

been actively studied since the early stage of chaos theory. An important com-

mon feature of the afore-mentioned examples is space-periodicity, a property

we will extensively use in the next sections.

Systematic investigations of such systems soon revealed the possibility of

generating anomalous transport, typically signaled by non-gaussian scaling

of the second moment of the diffusing variable (definitions will be provided

in the next section): the origin of such anomalies may be qualitatively tracked

down to a weakening of chaotic properties, namely intermittency in one-

dimensional maps [10, 11], regular islands punctuating the chaotic sea in the

standard map [3, 12, 13] and opening up an infinite horizon for the Lorentz

gas [14–16]. Indeed anomalous transport properties represent a relevant phe-

nomenon in statistical mechanics [17]: our interest here is to provide quan-

titative techniques that may shed some light on simple examples, and that

furthermore may give a clue to universal features of such phenomena.

While it is apparent from many contributions to this volume that such a

behavior may be fruitfully scrutinized in terms of random processes (see also

[18, 19]), our main impetus is to show that crisply deterministic techniques

may be applied here, and they are both capable of getting sharp estimates

and providing insight on universal behavior (we include a brief discussion

of probabilistic approaches in section (1.6)). We point out that the theoretical

framework that we here apply to deterministic transport has been employed

1) Corresponding author.

2 1 Deterministic (anomalous) transport

in a huge number of contexts, both in a classical and in a quantum framework

[20–23].

In the next two sections we will give an account of the theoretical approach:

further sections will be devoted to the discussion of examples both for normal

and anomalous transport.

1.2Transport and thermodynamic formalism

Though the technique we will introduce may be applied to generic dynamical

systems, in order to simplify the notation it is convenient to illustrate it within

the simplest possible example, so we consider a one-dimensional (discrete-

time) mapping f on the real line, enjoying the following symmetry properties:

f (−x) = − f (x) (1.1)

and

f (x+ n) = f (x) + n n ∈ N. (1.2)

The requirement (1.1) is not strictly necessary, it guarantees that no net drift

is present in the system (once a uniform initial distribution on a unit cell is

taken), so that the discussion of the second moment is easier (we notice that

the evaluation of the firstmoment maybe of interest too, as in the discussion of

deterministic ratchets [24]: see [25] for an example discussed along the present

lines). The condition (1.2) is much more relevant: it encodes space translation

symmetry and it implies that the map on the real line may be viewed as the

lift of a circle map f on the unit interval:

f (x) = f (x)|mod 1 (1.3)

This property is also shared by higher dimensional examples to which our

technique can be applied: for instance the toral map associated to a Lorentz

gas with a square lattice of scatterers is a Sinai billiard with periodic boundary

conditions (and this holds in any dimension)2 . Indeed violation of such a

property, for instance by introducing a weak quenched disorder, may lead to

quite a different physics, see for instance [28] and references therein, where

transport properties are considered for such a case, a deterministic analogue

of random walks in random environments [29, 30].

Given our dynamical system f , we are interested in moments of the diffus-

ing variable:

Mq(n) = 〈|xn − x0|q〉0 (1.4)

2)We note, however, that in higher dimensions taking properly intoaccount symmetry properties may require considerable effort [26, 27]

1.2 Transport and thermodynamic formalism 3

where 〈· · · 〉0 denotes an average over initial conditions (typically uniformlydistributed on some compact set, for instance an elementary cell). The evalua-

tion of even integer moments may be performed from the generating function

Gn(β) = 〈eβ( f n(x0)−x0)〉0 (1.5)

(due to symmetry property (1.1) odd ordermoments computed from (1.5) van-

ish). The idea is to deal with the generating function in the same way the

partition function is computed in lattice models admitting a transfer matrix,

i.e. by expressing (1.5) as the trace of the n-th power of a transfer operator.

The delicate point is reduction to torus map: as a matter of fact if we just look

at the first iterate (and take initial conditions uniformly spread over the unit

interval), we get

G1(β) =∫ 1

0dx∫ +∞

−∞dy eβ( f (x)−x)δ(y− f (x)) (1.6)

so that the “indices" of the kernel are mismatched, coming from two different

sets ([0, 1] and f ([0, 1])). But ∀y ∈ R such that y = f (x) there exists one andonly one z ∈ [0, 1] such that z = f (x) and y = z+ nx, nx ∈ Z, so that (1.6) maybe rewritten as

G1(β) =∫ 1

0dx∫ 1

0dz eβ( f (x)+nx−x)δ(z− f (x)) (1.7)

Now the starting and arriving domain coincide with the fundamental cell

(over which the torus map is defined) and we may introduce a generalized

transfer operator [31], acting on smooth functions, Lβ as

(

Lβh)

(x) =∫ 1

0dz h(z) eβ( f(z)+nz−z)δ(x− f (z)) (1.8)

whose (singular) integral kernel is

Lβ(y, x) = eβ( f (y)+ny−y)δ(x− f (y)) (1.9)

The transfer operator (1.8) is a modification of the usual Perron Frobenius op-

erator, describing measure evolution for dynamical systems: due to the expo-

nential form of the weight it maintains the semigroup property, and we may

write the generating function as

Gn(β) =∫ 10 dx

∫ 10 dyL

nβ(x, y)

=∫ 10 dx

∫ 10 dzn−1 · · ·

∫ 10 dz1

∫ 10 dxLβ(x, z1) · · · Lβ(zn−1, y)

(1.10)


This is, as we mentioned at the beginning of the section, the analogue of ex-

pressing the canonical partition function of a lattice system in terms of a trans-

fer matrix

QN(T,H) = ∑{σi}

e−βH({σi}) = tr T N (1.11)

In equilibrium statistical mechanics the next step is to express the trace as the

sum of N-th powers of the eigenvalues of T : the leading eigenvalue deter-mines in such a way the Gibbs free energy per particle in the thermodynamic

limit. Here we adopt the same strategy, ignoring the fact that the integral ker-

nel is singular (so that Lβ is not compact, like in ordinary Fredholm theory; the

reader interested in a rigorous approach should consult, for instance, [32, 33])

and we label eigenvalues of (1.8) in decreasing order (with respect to their ab-

solute value) λ0(β), λ1(β), . . . . In the large n limit (here the large time limitreplaces the thermodynamic limit, N → ∞ in (1.11)) we may thus write

Gn(β) =∫ 1

0dx∫ 1

0dyLnβ(x, y) ∼ ∑

j=0

λj(β)n (1.12)

In particular if the leading eigenvalue is isolated and positive (i.e. some gener-

alized Perron theorem holds) the generating function is asymptotically dom-

inated by powers of λ0(β); on the other hand, by power expanding in β we

obtain

Gn(β) = 1 + β〈(xn − x0)〉0 +β2

2!〈(xn − x0)

2〉0 + O(β3) ∼ λ0(β)n (1.13)

and we may thus relate the diffusion constant to the leading eigenvalue of the

generalized transfer operator in the following way

D =1

2

d2λ0(β)

dβ2

∣

∣

∣

∣

β=0

(1.14)

where

M2(n) ∼ 2dD · n (1.15)

(normal diffusion). As we will see, it may well be the case that D either vanish

or diverge (anomalous diffusion).

Such an approach, involving a generalized transfer operator to get the

asymptotic behavior of the generating function, has been proposed in [26,34].

Of course, since the asymptotic behavior of the whole generating function is

dominated by the leading eigenvalue of the transfer operator, we may gener-

alize (1.14) to yield expressions of higher moments (or cumulants) of the dif-

fusing variable. We will return to this point, but firstly we turn our attention

to the way in which the leading eigenvalue λ0(β)may be actually computed.

1.3 The periodic orbits approach 5

1.3The periodic orbits approach

The leading eigenvalue λ0(β) is the inverse of the smallest z(β) solving thesecular equation

det(

1− z(β)Lβ

)

= 0, (1.16)

which, by using the identity ln det(1−A) = −∑ m−1trAm, may be rewritten

as (dependence upon β of z will be implicit in the following)

exp−∞

∑k=1

zk

ktrLkβ = 0 (1.17)

Periodic orbits come into play as soon as we evaluate traces, and use elemen-

tary properties of Dirac’s δ: as a matter of fact

trLkβ =∫ 10 dxL

kβ(x, x)

= ∑y| f k(y)=ye

βn(k),y

|1−Λ(k),y|

(1.18)

that is a sum over periodic points: the weights that are picked up for each of

them are the instability

Λ(k),z =d f k

dz(z) =

k−1

∏m=0

f ′( fm(z)) (1.19)

and the jumping number of the orbit once unfolded on the real line

n(k),z : fk(z) = z+ n(k),z (1.20)

Notice that both n and Λ will be the same for each point of a given periodic

orbit. At each order k (1.18) picks up contributions both from orbits of “prime

period" k, and as well from orbits of smaller periods s such that s divides k (in

particular fixed points contribute to all orders). Now suppose z is a point of a

periodic orbit of prime period s and k = s ·m, then we have n(k),z = m · n(s),zand Λ(k),z = Λm

(s),zthus the only independent quantities entering the former

expressions are stabilities and jumping numbers of prime cycles. We now

impose the requirement that f is a chaotic map, in the form of a hyperbolicity

assumption, that in one dimension simply reads

| f ′(x)| > 1 ∀x ∈ [0, 1] (1.21)

Typically this property will generate normal transport properties: in section

(1.5) we will see how violation of such an inequality even at a single point will


dramatically modify transport features 3. If (1.21) holds than the denominator

in (1.18) may always be expanded as a geometric series and we may write

∞

∑k=1

zk

k ∑z| f k(z)=z

eβn(k),z

|1− Λ(k),z|= ∑

{p}

∞

∑j=0

∞

∑r=1

npznp·r

np · r

eβrσp

|Λp|rΛj·rp

(1.22)

where σp = n(np),zp (zp being any point of the periodic orbit labeled by p) and,

analogously, Λp = Λ(np),zp . In eq. (1.22) {p} indicates a sum over all “primeperiodic orbits" (of prime period np), j is the geometric series index coming

from expanding the denominators, while r counts repetitions of prime cycles

(as in the original sum each p cycle appears at every r · np order). Now wemay sum up the r (logarithmic) series, thus getting

det(1− zLβ) = exp∑{p}

∞

∑j=0

ln

(

1− znpeβ·σp

|Λp|Λjp

)

=∞

∏j=0

ζ−1β,(j)

(z) (1.23)

where dynamical zeta functions are thus defined as

ζ−1β,(j)

(z) = ∏{p}

(

1− znpeβ·σp

|Λp|Λjp

)

, (1.24)

while their infinite product is usually called the spectral determinant

Fβ(z) =∞

∏j=0

ζ−1β,(j)

(z) = det(1− zLβ) (1.25)

When j increases, the coefficients of znp in (1.23) become smaller and smaller:

this suggests that z(β) is a zero of the lowest order zeta function (1.24)

ζ−1β,(0)

(z) = ∏{p}

(

1− znpeβ·σp

|Λp|

)

. (1.26)

This is checked in exactly solvable models [23], and, under suitable hypothe-

ses, may be proved for particular classes of dynamical systems.

Cycle expansions [21–23] consist in expanding (1.26) into a power series

ζ−1β,(0)

(z) = 1−∞

∑m=1

γm(β)zm (1.27)

Finite l-order estimates (that require information coming from periodic or-

bits whose prime period does not exceed l) come from polynomial trunca-

tion of (1.27): this leads to a genuine perturbation scheme if we are able to

3)We remark that while chaos is an efficient randomizing mechanismto induce stochasticity, highly nontrivial transport properties mayalso appear in systems that lack exponential sensitivity upon initialconditions [35–39]

1.3 The periodic orbits approach 7

control how finite order estimates converge to the asymptotic limit. From a

mathematical point of view this amounts to investigate analytic properties

of dynamical zeta functions, typically by finding a domain in which they are

meromorphic: specific examples and heuristic arguments are provided in [23],

while a guide to the relevant mathematical literature may be found in [40,41].

In practice detailed knowledge on the topology of the system allows one to

write the dynamical zeta function (1.26) is such a way that the role of funda-

mental cycles is highlined :

ζ−1β,(0)

(z) = 1− ∑f

t f + ∑n

cn (1.28)

where we have incorporated z in the definition of cycle weights t, and we fac-

tored away the contribution of cycles which are not shadowed by combination

of lower order orbits: in the case of a unrestricted grammar the fundamental

cycles are just the contributions from the alphabet’s letters [21, 23]. The sim-

plest example comes a complete binary grammar, with alphabet {0, 1}4: theexpansion of the dynamical zeta function is

ζ−1β,(0)

(z) = (1− t0) · (1− t1) · (1− t01) · · ·

= 1− t0 − t1 − (t01 − t0 · t1) · · ·(1.29)

and the fundamental cycles are just the two fixed points, labeled by the al-

phabet’s letter 0 and 1. The fundamental cycles thus provide the lowest order

approximation in the perturbative scheme: a general chaotic systems is ap-

proximated at the lowest level with its simplest poligonalization, non unifor-

mity is incorporated perturbatively by considering curvature corrections (cn)

of higher and higher order. The whole scheme relies on a symbolic encod-

ing of the dynamics, and while we remark that finding a proper code for a

given system is a highly nontrivial task in general, we have to emphasize that

this cannot be considered as a shortcut on the theory proposed here, as the

topological complexity cannot be eluded in any sensible treatment of general

properties of chaotic systems.

To investigate the behavior of higher moments, or to scrutinize throughly

deviations from full hyperbolicity we need to extend the former considera-

tions in the following way: first we reorder the eigenvalues of the transfer

operator so that the dominant ones come first and write in general:

Fβ(e−s) = ∏

i

(1− λie−s) (1.30)

so that

d

dsln Fβ(e

−s) = ∑i

λie−s

(1− λie−s)(1.31)

4) Such is the case for instance for the Bernoulli shift or the quadraticfamily at Ulam point.


and if we now take the inverse Laplace transform, we get

1

2πi

∫ a+i∞

a−i∞ds esn

d

dsln[

Fβ(e−s)]

= ∑i

λni ∼ Gn(β) (1.32)

As the asymptotic behavior is dominated by the leading eigenvalue wemay

use the dynamical zeta function instead of the spectral determinant, thus

λn0 (β) ∼1

2πi

∫ a+i∞

a−i∞ds esn

d

dsln[

ζ−1β,(0)

(e−s)]

(1.33)

When β = 0 the transfer operator (1.8) coincides with the Perron Frobeniusoperator [42] and so λ0(0) = 1 (the corresponding eigenfunction being thedensity of the invariant measure). Even moments are given by Taylor expan-

sion of Gn(β) around β = 0 (when (1.1) is satisfied odd moments vanish):

Mk(n) = 〈(xn − x0)k〉0 =

∂k

∂βkGn(β)

∣

∣

∣

∣

∣

β=0

∼∂k

∂βk1

2πi

∫ a+i∞

a−i∞ds esn

d

dsln[

ζ−1β,(0)

(e−s)]

∣

∣

∣

∣

∣

β=0

(1.34)

and in particular we may rewrite (1.14) as

D = limn→∞

d2

dβ2

1

2πi

∫ a+i∞

a−i∞ds est

∂sζ−1β,(0)

(e−s)

ζ−1β,(0)

(e−s)

β=0

(1.35)

High order derivatives in the argument of the inverse Laplace transform are

then evaluated by making use of Faà di Bruno formula:

dn

dtnH(L(t)) =

n

∑k=1

∑k1,...,kn

n!

k1! · · · kn!

dkH

dLk(L(t)) · B~k(L(t)) (1.36)

B~k(L(t)) =

(

1

1!

dL

dt

)k1

· · ·

(

1

n!

dnL

dtn

)kn

(1.37)

~k = {k1, . . . kn}with ∑ ki = k, ∑ i · ki = n (1.38)

In view of formulas like (1.35) of (1.34) transport properties are deduced

from dynamical zeta functions, that organize knowledge about the system as

encoded in the set of periodic orbits. There relationships provide information

both on prefactors (1.35) and on asymptotic growth (1.34): the latter feature

is particularly relevant if we are facing the problem of anomalous diffusion,

as in this case the important piece of information concerns the spectrum ν(q),determining the asymptotic behavior of q-order moments

Mq(n) ∼ nν(q) (1.39)

1.4 One dimensional transport: kneading determinant 9

Gaussian (normal) transport is described by a single scale spectrum ν(q) =q/2, while anomalous transport typically shows a non trivial behavior, which

cannot be encoded by a single exponent, but rather typically exhibits a phase

transition [43–47].

We notice a remarkable feature of the formula (1.35): it leads to a normal

diffusing behavior from a balance between localized orbits (those with σp = 0),and ballistic orbits (with σp 6= 0): such a balance was also considered in earlyefforts to explain diffusive properties of sawtooth and cat maps via periodic

orbits [48] (see [49] for a treatment of the problem according to the present

technique).

1.4One dimensional transport: kneading determinant

While in the case of anomalous transport the interest is typically concentrated

in understanding the time asymptotic behavior, in normal transport the atten-

tion is focused on the prefactor of the second moment (the diffusion coeffi-

cient), that carries the crucial information of the process.

If we are interested in using periodic orbit theory for a precision calculation

of the diffusion coefficient we will eventually have to face the problem of the

full understanding of cycle organization. A good control of the periodic or-

bit organization is thus necessary if we want to proceed via cycle expansion

techniques. In fact, the success of the zeta function approach strongly relies

on the ability of the user to control the (typically exponential) proliferation of

periodic orbits. In particular, the cycle expansion technique guarantees nice

convergence properties of the zeta functions once we have identified the set

of periodic orbits that build the fundamental and curvature terms (1.28) [21, 23].

This in turns implies a complete understanding of the underlying symbolic

dynamic. Unfortunately such a control is rather exceptional and many inter-

esting examples present extreme complexity in orbit-coding and control over

finite order estimates becomes problematic. As a paradigmatic example we

will consider a simple one-dimensional map of the real line that determinis-

tically generates normal diffusion with a diffusion coefficient that sensitively

depends on the control parameter that define the map [50, 51]. In particular

the cycles organization change discontinuously under parameters variation,

thus invalidating any meaningful attempt to directly use periodic orbit the-

ory. This example is used to show how it is possible to build (at least in the

one-dimensional case) the dynamical zeta functionwithout any need of periodic

orbits, using the kneading trajectories of the systems. The kneading trajecto-

ries naturally order the admissible symbolic sequences of the systems and thus

incorporate all the information needed to generate grammar rules. It seems


Tab. 1.1 The map gΛ(x) with Λ = 3.23.

thus natural to try to directly use this trajectories into an expression closely

related to dynamical zeta functions. In fact Milnor and Thurston [52] were

able to relate the topological entropy to the determinant of a finite matrix (the

kneading determinant) where the entries are formal power series (with coef-

ficients determined by the kneading trajectories). Later Baladi and Ruelle [53]

have generalized the result, incorporating a constant weight (see also [54, 55]

and references therein for more general results). We will use this extension to

derive an explicit expression of the diffusion coefficient in the full parameter

range [56].

Let’s start by introducing the reference dynamical system [50]: for Λ ≥ 2let’s define the map gΛ : R → R:

gΛ(x) =1

2+ Λ(x−

1

2) x ∈ [0, 1] (1.40)

and extending it on the real line by the symmetry property (1.2). Let gΛ(x) :[0, 1] → [0, 1] be circle map corresponding to gΛ:

gΛ(x) = gΛ(x)|mod1 (1.41)

For integer values of the slope Λ dynamical zeta functions may be written

down explicitly and the diffusion coefficient D is easily computed from the

1.4 One dimensional transport: kneading determinant 11

smallest zero [23, 34, 57–59]:

D =(Λ − 1)(2Λ − 1− 3 (−1)Λ))

48(1.42)

For a generic value of the control parameter the situation is more involved:

typically there is an infinite set of pruning rules that do not allow to write

down the exact dynamical zeta function. Following the notation of Baladi and

Ruelle, we define ǫ(x) = ±1, whether f (x) is increasing or decreasing andt(x) as a constant weight for x ∈ [ai−1, ai], where a0 < a1 < ... < aN are the

ordered sequence of end points of each branch (see Fig.(1.1)). The restriction

of these functions on the i-interval will take the constant values:

ǫi = 1 (1.43)

ti =z eβσi

Λ(1.44)

where σi is the jumping number associated to the i-th branch.

We associate to each point x the address vector in ZN−1

~α(x) = [sgn(x− a1), ..., sgn(x− aN−1)] (1.45)

and we define the invariant coordinate of x by the formal series:

~θ(x) =∞

∑n=0

[

n−1

∏k=0

(ǫt)(gkΛ(x))

]

~α(gnΛ(x)) (1.46)

with the convention that the product is equal to one if n = 0 (the invariantcoordinate θ is single valued once we put ǫ(ai) = 0).Defining φ(a±) = limx→a± φ(x), we compute the discontinuity vector at thecritical points ai for i = 1, ..,N− 1:

~Ki(z, β) =1

2

[

~θ(a+i ) −~θ(a−i )]

(1.47)

The kneading matrix K(z, β) is defined as the (N − 1) × (N − 1) matrix with~Ki; i = 1, ..,N− 1 as rows. Let’s call ∆(z, β) = detK(z, β) the kneading deter-minant. It is possible to show that in our case the dynamical zeta function is

equal to the kneading determinant up to a rational function (see [53] for more

general results and a proof of relation (1.48)). In particular, if we denote { p}the set of prime periodic orbits that include a critical point (i.e. g

n pΛ

(ai) = ai),we have:

∆(z, β) = R(z, β) ζ−1β,(0)

(z) (1.48)

R(z, β) =

[

1−1

2(ǫ1t1 + ǫNtN)

]

∏{ p}

[1− t p(z, β)]−1 (1.49)


Tab. 1.2 Fractal diffusion coefficient as a function of the slope of the map gΛ(x) computedfrom the smallest zero of the dynamical zeta function. The insert is a blow up of a part of themain figure

Equation(1.48) explicitly relates a quantity build upon periodic orbit of sys-

tem (the dynamical zeta function) with a quantity build from the iterate of

kneading trajectories (the kneading determinant). It is simple to show that

in our case the kneading determinant can be explicitly written in term of the

trajectory of the pair of critical points a0 and aN :

∆(z, β) = 1+z

2Λ

N−1

∑i=1

eβ(σi+1/2)[

eβ2~θ(a+0 ) − e−

β2~θ(a−N)

]

i(1.50)

Moreoverwe can see from formula (1.48) that the smallest zero of the knead-

ing determinant coincides with the smallest zero of the dynamical zeta func-

tion for β → 0 and then we can derive, via the implicit function theorem, thediffusion coefficient as:

D = −1

2

(

∂2∆(z, β)

∂β2/

∂∆(z, β)

∂z

)∣

∣

∣

∣

z=1,β=0

(1.51)

By using (1.50) it is in principle possible to explicitly write a (lengthy) ex-

pression for the diffusion coefficient in term of the kneading trajectory (see [56]

for details). The evaluation of (1.51) for a parameter choice Λ ∈ [2, 3] is shownin Fig. (1.2) and can be compared, for example, with Fig.10 in [50] .

Though elegant, such a technique is essentially confined to the 1d setting:

the difficulties of computing D via (1.35) in higher dimensions is made clear

by attempts to get accurate estimates for the case of the Lorentz gas with finite

horizon [60–62].

1.5 An anomalous example 13

1.5An anomalous example

In many situations the dynamics of a system is far from being completely hy-

perbolic. Quite often accessible regions of phase space behave quasi-regularly

and strongly influence the overall properties of the system: trajectories tend

to stick close to regular regions, slowly moving away from it. Chaotic wan-

dering is then interrupted by long segments of quasi-regular motion (see Fig.

(1.3) for an example in one dimension). This peculiar aspect of the dynamic

canmodify important quantities like decay of correlation or return time statis-

tics, that typically show a power-law decay. In particular stickiness of trajec-

tories can induce anomalous diffusion in open systems. In the last few years it

has been realized [16, 63–69] that power-law separation of nearby trajectories

in weakly chaotic systems (in particular one dimensional intermittent maps,

or infinite horizon Lorentz gas models) deeply modify the analytic properties

of zeta functions, which typically exhibit branch points. The modified ana-

lytic structure of the zeta function may induce anomalous behavior (nonlinear

diffusion [23, 70]): in the thermodynamic language this would correspond to

critical behavior, with a gapless transfer operator slowing down correlations

decay. While the detailed understanding of the mechanism for stickiness in

generic Hamiltonian systems is still an open problem [71–74], it is fruitful to

investigate a simpler case, where the analysis can be performed in a detailed

way: a one dimensional map with a marginal fixed point.

0 50 100 150 200 250 300n

0

0.2

0.4

0.6

0.8

1

x(n)

0 0.2 0.4 0.6 0.8 1x(n)

0

0.2

0.4

0.6

0.8

1

x(n+

1)

0 0.2 0.4 0.6 0.8 1x(n)

0

0.2

0.4

0.6

0.8

1

x(n+

1)

0 50 100 150 200 250 300n

0

0.2

0.4

0.6

0.8

1

x(n)

Tab. 1.3 A segment of trajectory (300 iterations) with the same initial condition is showedfor two torus maps: a completely hyperbolic map (left) and one with intermittent fixed points(right).


The map on the fundamental cell is implicitly defined on [−1, 1] in the fol-lowing way [45]:

x =

12γ

(

1+ f (x))γ

0 < x < 1/(2γ)

f (x) + 12γ

(

1− f (x))γ

1/(2γ) < x < 1(1.52)

for negative values of x the map is defined as f (−x) = − f (x) (thus satisfyingthe symmetry requirement (1.1)): the map is plotted in fig. (1.4). The map

lacks full hyperbolicity due to the presence of two marginal fixed points (at

x = ±1), where the slope is exactly one. The lift of the map on the real lineis defined once we assign jumping numbers σL = −1 to the left branch andσR = +1 to the right branch: whenever a particle remains trapped near amarginal point the corresponding unfolded trajectory on the real line consists

of successive jumps to the neighboring cell. Symmetry requirement (1.2) is

assumed, so the map is thus extended on the full real line.

The presence of complete branches leads to an unrestricted grammar in the

symbolic code {L, R}, where the corresponding partition obviously consist ofL = [−1, 0] and R = [0, 1]. However the presence of the marginal fixed points(L, R) causes problems in using zeta function techniques: these fixed points

have |Λ| = 1, and cannot be included in any trace formula like (1.18), as theywould lead to divergences. In this case we are forced to prune away the L (R)

fixed points thus moving to a new countable infinite alphabet with unrestricted

grammar:

{LjR, RkL} j, k ∈ N+ (1.53)

In this new alphabet all LnR (RnL) cycles lack curvature counterterms (they

are not shadowed by any combination of shorter cycles (1.28,1.29), due to the

lack of the symbols L and R alone) and the fundamental cycle part of the zeta

function thus becomes:

ζ−1fund(z) = 1− tR − tLR − tL2R − · · · − tLnR − · · ·+ (R→ L) (1.54)

where again t indicates the appropriate weight. It is clear that, in contrast with

the normal example, we do not have a finite order polynomial and the analytic

structure of (1.54) will be strongly dependent on the intermittency exponent

γ. In order to proceed we need to compute the instabilities and jumping num-

ber of the cycles LnR(RnL) that come closer and closer to the marginal points.Firstly we note that the slope of the map in the chaotic region is not bounded

from above. This feature is intimately connected to the peculiar ergodic prop-

erties of the map: it has a constant invariant measure [45] for any values of γ

as is easily seen from the summation property

∑y= f−1(x)

1

f ′(y)= 1 (1.55)


that follows directly from the expressions (1.52). In this sense the behavior

of (1.52) is similar to what happens in an area preserving map example [73],

where a parabolic fixed point coexists with a Lebesgue invariant measure, and

is quite different from the usual Pomeau-Manneville [75, 76] case where the

torus map has non trivial ergodic properties: an absolutely continuous in-

variant measure only exists for α = 1/(γ − 1) > 1 (see for instance [77] and

references therein) while the ergodic behavior is much more complex when

α < 1; in any case sticking induces peaking of the measure around marginal

fixed points (see [78, 79]).

In order to use the peculiarities of the map into the derivation of the insta-

bilities we can start by identifying for each branch a laminar and an injection

regions (for example we call B = [0, 1/(2γ)] and B = [1/(2γ), 1] respectivelythe injection and laminar regions of the right branch (see figure (1.4)): A and

A will denote the symmetric regions in the left subinterval.

Tab. 1.4 Linearization of the map (1.52). Near marginal points, laminar regions, a Gaspard-Wang type partition is used while a finer one is used in the injection zone.

Nowwe refine partitions in the following way: if fR and fL denote the right

and left branches of the map we set Bi = f−iR (B) and Ai = f−iL (A): we thenrefine the turbulent sets by Ai = f−1L (Bi) and Bj = f−1R (Aj). Now in eachlaminar set the map is linearized according to Gaspard-Wang approximation

[78, 79]: linearization in the turbulent sets {Ai, Bj} is then constructed in such


a way to preserve the summation property (1.55):

Λ−1Bi

= 1− Λ−1Ai

(1.56)

The width of the sets Aj is easily seen to shrink with a power law in j:

Ak ∼C

kα+1(1.57)

where again α = 1/(γ − 1). By using (1.57) and (1.56) we get an estimate ofperiodic orbits instabilities:

ΛAkB =[A0A1

A1A2. . .Ak−1Ak

][

1−AkAk−1

]−1[1−A1A0

]−1(1.58)

∼ kα+2 (1.59)

This agrees with numerical simulations for the non-linearized map (see figure

(1.5)). Note that with this partition it is possible to estimate the behavior of

0 1 2 3 4 5α

0

1

2

3

4

5

6

7

χ(α)

0 1 2 3 4 50

1

2

3

4

5

6

7

Tab. 1.5 The exponent χ(α) of power law decay of instabilities ΛAkB ∼ kχ(α) as a function ofthe parameter α (open circles) and the best fit χ(α) = (0.88α + 1.9) (full line).

the instabilities of all orbits, not only the one accumulating to the marginal

point [80]. Once we notice that, for the cycles dominating the zeta function

expression, the jumping number is given by the length of the periodic orbit

(with the appropriate sign), the dynamical zeta function is (for the linearized


map, where curvature corrections vanish):

ζ−1β,(0)

(z) = 1− ζ(α + 2)z∞

∑k=1

zk

kα+2cosh(βk) (1.60)

To estimate the asymptotic behavior of the generating function (1.34) we

need to single out the leading singularity in the logarithmic derivative of the

zeta function. The divergences for z → 1 in derivatives expansions (1.36)

depend on the appearance of the Bose function :

gµ(z) =∞

∑l=1

zl

lµ(1.61)

that indeed may alter the analytic features. This function appears as a con-

sequence of the particular sequences of orbits whose stability increases only

polynomially with the period (1.58), a signature of non-hyperbolic behav-

ior [23, 78, 79]. We recall how Bose functions behave as z → 1− : the result

depends upon µ (related to the intermittency exponent)

gµ(z) ∼

(1− z)µ−1 µ < 1

ln(1− z) µ = 1ζ(µ) + Cµ(1− z)µ−1+ Dµ(1− z) µ ∈ (1, 2)ζ(2) + C2(1− z) ln(1− z) µ = 2ζ(µ) + Cµ(1− z) µ > 2

(1.62)

Now look at a generic term in Faà di Bruno expansion of the factor

∂n

∂βnln ζ−1

β,(0)(z)

∣

∣

∣

∣

β=0

(1.63)

in (1.34) and denote it by Dk1...kn . Taking into account that

∂i

∂βiζ−1

β,(0)(z)

∣

∣

∣

∣

β=0

∼

{

0 i odd

zgα+2−i(z) i even(1.64)

we have that

Dk1...kn ∼1

(ζ−10,(0)

(z))k∏j

(gα+2−j(z))k j =

D+k1 ...kn

D−k1 ...kn

(1.65)

where the D+ pick up the contributions from the product of Bose functions,

and all jmust be even, due to (1.64).

Since the dynamical zeta function has a simple zero we get

D−k1...kn

∼ (1− z)k, (1.66)


while the terms appearing in D+ modify the singular behavior near z = 1only for sufficiently high j

gα−j(z) ∼

{

(1− z)α+1−j j > α

ζ(α + 2− j) j < α(1.67)

If all {j} are less than α then the singularity is determined by D−: keeping in

mind that the highest k value is achieved by choosing j = 2 and k2 = n/2, weget

ν(q) =q

2q < α (1.68)

When q exceeds α we have to take into account possible additional singulari-

ties in D+, and thus we get

Dn ∼1

(1− z)ρ (1.69)

where ρ is determined by

ρ = sup{k1...kn}

(k+ ∑j>α

(j− α)kj) =

{

n/2 n < 2α

n+ 1− α n > 2α(1.70)

which, once we take (1.68) into account, yields the full spectrum of transport

moments

ν(q) =

{

q/2 q < 2α

q+ 1− α q > 2α(1.71)

which may also be checked numerically for the full map (see figure (1.6)).

The non trivial structure of the spectrum , that presents a sort of phase tran-

sition for q = 2α, is a common feature of many systems with anomalous trans-port properties [43, 46, 47]. In this example we showed how [44] the essential

ingredient in the analysis of weakly chaotic systems seems to be a proper char-

acterization of a particular sequence of periodic orbits: those probing closer

and closer dynamical features of the marginal structures: the parameter rul-

ing the presence of a phase transition (and the explicit form of the spectrum)

is α, that is the exponent describing the polynomial instability growth of the

family of periodic orbits coming closer and closer to the marginal fixed point,

and thus describing the sticking to the regular part of the phase space. In this

way, differently from prefactors, that depend critically on the fine structure of

the map, the spectrum ν(q) is fully determined by a local analysis, near themarginal fixed points, and this corroborates the idea that a single exponent

determines the universality class of the system, as regards the transport mo-

ments. Again if we try to evaluate prefactors in this context, fine details of the

full dynamics are needed, as shown in [81].

1.6 Probabilistic approximations 19

Tab. 1.6 Spectrum of the transport moments for the map (1.52) with γ = 1.5: the best fit onnumerical data is ν(q) = 0.50q + 0.04 for the dotted line, and ν(q) = 0.98q − 1.82 for the fullline.

1.6Probabilistic approximations

While the approach we described is purely deterministic (and in particular

cases can be rigorously justified, even for intermittent systems [68, 69]) we

briefly describe now how probabilistic methods may also be employed.

The main point in this class of probabilistic approaches is to suppose that

any orbit might be partitioned according to a sequence of times t1 < t2 <

· · · < tn < · · · such that the time laps ∆j = tj − tj−1 form a sequence ofrandom variables with common distribution ψ(∆) d∆ and the orbit propertiesbefore and after tn are independent of n. A typical choice for {tj} in the caseof intermittent maps consists in collecting the reinjection times in the laminar

region: the second property mentioned above is thus related to the “random-

ization” operated by the chaotic phase. From the stochastic processes point of

view this amounts to employ renewal theory [82] to describe the dynamics of

the system.

We now illustrate how an approximate form of the distribution ψ(T) maybe obtained in the case of Pomeau-Manneville map [11]

xn+1 = xn + xzn|mod 1 (1.72)

The map consists of two full branches, with support on [0, p) (I0 the laminarregion) and [p, 1] (I1 the chaotic region), where p+ pz = 1 . In a continuous


time approximation [83, 84], (1.72) is turned into the differential equation

xt = xzt (1.73)

whose solution we write as

xt =

[

1

xz−10− (z− 1)t

]− 1z−1

(1.74)

From (1.74) we can obtain the exit time T(x0) for each x0 ∈ I0, as xt will exit I0as soon as xt ≥ p:

T(x0) =1

z− 1

[

1

xz−10−1

pz−1

]

.

Now

ψ(T) =∫ p

0dx P(x, T− T(x)) , (1.75)

where P(x, t) is the probability of being injected at time t from I1 to x0 ∈ I0 (wepartition the time sequence so that tj is the entrance time in the chaotic region).

In the limit of large T, the dominant contribution to (1.75) comes from

ψ(T) ∝

∣

∣

∣

∣

dx0(T)

dT

∣

∣

∣

∣

if we suppose that the reinjection probability is smooth and chaotic residence

times vanish sufficiently fast. Within this approximation we thus get

ψ(T) ∝

(

(z− 1)T+1

pz−1

)− zz−1

T >> 0

so that

ψ(T) ∼1

Tzz−1

for large values of T.

From the distribution ψ(T) we can get information about correlation func-tions: suppose we consider an observable A that may change only during

transitions between neighboring time lapses: its autocorrelation function may

be written as the time average

CAA(t) = 〈A(t0 + t)A(t0)〉t0 − 〈A〉2 (1.76)

Our probabilistic assumption amounts to disregard correlations if t0 and t+ t0belong to different intervals (yielding a contribution 〈A〉2 to CAA(t), while

1.7 Conclusions 21

if t0 and t + t0 belong to the same interval correlation in complete (and thecorresponding contribution is 〈A2〉). So, if we denote by Φ(t) the probabilitythat no lapse transition has occurred between t0 and t+ t0, we easily rewritethe correlation function as

CAA(t) =(

〈A2〉 − 〈A〉2)

Φ(t) (1.77)

But now Φmay be written in terms of the distribution function as

Φ(t) =1

〈∆〉

∫ ∞

tdu

∫ ∞

ud∆ ψ(∆) (1.78)

Once we control correlations, the behavior of the second moment, via Green-

Kubo formulas [11], is easily found, and the results agree with our determin-

istic approach (see also [43]) 5. This approach has been reformulated in a clean

way in [85,86], and its virtues and shortcuts have been scrutinized, in the case

of the Lorentz gas with infinite horizon, in [16].

1.7Conclusions

We have presented the essential features of a theory of deterministic trans-

port (for systems enjoying space periodicity) based upon periodic orbit the-

ory. This technique, besides being crisply deterministic, presents a number of

appealing features: it is invariant under smooth conjugacies of the dynamical

system and it offers a way to present in a hierarchical way the problem, in

cases where typically a perturbative parameter does not exist. It also presents

subtle points: for instance the evaluation of the diffusion constant generically

requires a considerable amount of control over fine details of the dynamics.

The theory allows also to deal with anomalous transport: in particular mo-

ments of the diffusing variable may be investigated, and the corresponding

spectrum ν(q) computed. The shape of such a spectrum is found to be de-termined by cycles probing closer and closer sticking regions: thus only local

quantities enter the final results and this suggests interesting universal fea-

tures of deterministic anomalous transport.

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5) Of course the original Pomeau-Manneville map is defined on theunit torus, so no transport properties can be established for it:the actual model one considers if a lift on the real line, where themarginal fixed point actually unfolds to a ballistic point (i.e the lami-nar branch has a non-zero jumping number).

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25

Index

anomalous diffusion, 9, 13

Bose function, 17

curvature terms, 7cycle expansions, 6

diffusion constant, 4, 8, 12dynamical zeta functions, 6

Faà di Bruno, 8, 17fundamental term, 7

generating function, 3, 4

instabilities, 5intermittency, 13, 14

jumping numbers, 5

kneading determinant, 9, 12kneading matrix, 11

leading eigenvalue, 4

Lorentz gas, 2

marginal fixed point, 14

normal diffusion, 5, 9

partition function, 3periodic orbit theory, 5Perron-Frobenius operator, 3perturbative scheme, 7phase transition, 18prime periodic orbits, 5pruning rulees, 11

space translation symmetry, 2spectral determinant, 6spectrum of transport moments, 8, 18

thermodynamic formalism, 2torus map, 2transfer matrix, 3transfer operator, 3transport moments, 2, 8

Documents

Anomalous deterministic transport