Deterministic diffusion in one-dimensional maps - Calculation of diffusion constants by harmonic inversion of periodic orbit sums

Abstract

A method is proposed for the calculation of diffusion constants for one-dimensional maps exhibiting deterministic diffusion. The procedure is based on harmonic inversion and uses a known relation between the diffusion constant and the periodic orbits of a map. The method is tested on an example map for which results calculated by different other techniques are available for comparison. Comment: 10 pages, 2 figures, accepted for publication in Phys. Lett. A

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arXiv:nlin/0111057v1 [nlin.CD] 26 Nov 2001
Deterministic diffusion in one-dimensional
maps – Calculation of diffusion constants by
harmonic inversion of periodic orbit sums
Kirsten Weibert, J¨ org Main, and G¨ unter Wunner
Institut f¨ ur Theoretische Physik 1, Universit¨ at Stuttgart, D-70550 Stuttgart,
Germany
Abstract
A method is proposed for the calculation of diffusion constants for one-dimensional
maps exhibiting deterministic diffusion. The procedure is based on harmonic inver-
sion and uses a known relation between the diffusion constant and the periodic orbits
of a map. The method is tested on an example map for which results calculated by
different other techniques are available for comparison.
1Introduction
Deterministic diffusion plays an important rˆ ole in a variety of physical phe-
nomena and applications, e.g. particle confinement in magnetic fields [1], con-
ductivity in metals [2], etc. A general theory of diffusion for systems with
continuous time is presented in [3]. In systems with periodic phase-space struc-
ture, diffusion may be induced by chaotic dynamics inside the elementary cells,
causing an irregular jumping between the cells. The investigation of determin-
istic diffusion is an interesting task both from the theoretical point of view,
concerning the question of how the typical stochastic properties arise from a
purely deterministic dynamics, as well as with regard to its various applica-
tions in different physical contexts.
It has been observed that even the simplest model systems – one-dimensional
maps – which exhibit deterministic diffusion are capable of explaining the be-
haviour of relevant physical systems [4,5]. For such maps, it has been demon-
strated that the diffusive properties are related to the periodic orbits of the
map restricted to an elementary cell [4–7]. In particular, the diffusion constant
characterizing the diffusive spreading can be obtained from the leading zero
of a zeta function consisting of a product over contributions from all peri-
odic orbits of the reduced map. Similar to the dynamical zeta functions which
Preprint submitted to Physics Letters A8 February 2008

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appear in semiclassical quantization problems, this product usually does not
converge. In former work [4–6], the convergence problem was handled by the
application of cycle expansion techniques. However, cycle expansion strongly
depends on the existence of a symbolic dynamics and the shadowing of long
orbits by combinations of short ones.
On the other hand, it has been demonstrated [8,9] that the convergence prob-
lems of the zeta functions and corresponding response functions arising in
semiclassical quantization problems can be circumvented by the application
of harmonic inversion, which is a universal method in the sense that it does
not rely on a symbolic code or other special properties of the system. The
method has recently been extended from Hamiltonian systems to maps [10].
In this paper we demonstrate that harmonic inversion can also be used for the
calculation of diffusion constants of one-dimensional maps. In analogy with
the procedure for semiclassical quantization, we rewrite the zeta function as a
response function, from which we construct a periodic orbit signal including a
finite set of periodic orbits. The analysis of the signal by harmonic inversion
yields the zeros of the underlying zeta function, from which the diffusion con-
stant is calculated. We apply this technique to an example map for which the
periodic orbits can easily be calculated, and which has already been examined
with the help of cycle expansion techniques [11], rendering a comparison with
previous results possible.
We start by describing the general procedure. We consider one-dimensional
mapsˆf : R → R defined by an anti-symmetric functionˆf,
ˆ xn+1=ˆf(ˆ xn) ,
ˆf(−ˆ x) = −ˆf(ˆ x)(1)
for points ˆ x inside the elementary cell [−1/2,1/2), and a discrete translational
symmetry outside the elementary cell,
ˆf(ˆ x + n) =ˆf(ˆ x) + n ,n ∈ Z . (2)
From this map, a reduced map f : [−1/2,1/2) → [−1/2,1/2) can be con-
structed by restricting the dynamics to the elementary cell via the definition
x= ˆ x − [ˆ x + 1/2]
f(x)=ˆf(x) − [ˆf(x) + 1/2] ,
(3)
(4)
where [·] denotes the integer part of the quantity in the brackets. Each pe-
riodic orbit of the reduced map f(x), consisting of a sequence of fixed points
{x1,x2,...,xn}, corresponds to an orbit of the full mapˆf(ˆ x) which returns to
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an equivalent point,
ˆfn(xj) = xj+ σp,σp∈ Z .(5)
The diffusion constant associated with the map is determined by the lattice
translation σ after n applications of the map, averaged over initial conditions
in the elementary cell
D = lim
n→∞
1
2n?σ2?n.(6)
As was shown in Ref. [6], for hyperbolic or almost hyperbolic maps, the diffu-
sion constant can be determined from the periodic orbits of the reduced map
(4) by considering the zeta function
Z(z,β) =
∞
?
m=0
?
p
?
1 −znpeσpβ
|Λp| Λm
p
?
,(7)
where the second product runs over all primitive periodic orbits of the reduced
map; npand Λp=?n
the orbit, respectively, σpdenotes the lattice translation of the corresponding
orbit of the full map, and β is a free parameter. According to Ref. [6], the
diffusion constant D can be expressed in terms of the leading zero z0(β) of the
zeta function as a function of β,
i=1f′(xi) are the topological length and the stability of
D ≈ −1
2
?∂2
∂β2z0(β)
?
β=0
.(8)
Additionally, from the symmetry of the map, it follows that [∂z0/∂β]β=0= 0.
Using these relations, an expansion of lnz0(β) for small β yields
lnz0(β) ≈ −β2D , (9)
which can directly be used to calculate the diffusion constant from the leading
zero for small β.
The problem that remains is how to actually calculate the leading zero, since
the expression (7) for the zeta function usually diverges. In past works, this
problem has been tackled by application of cycle expansion techniques. As
an alternative, we demonstrate in the following how the zeta function can be
expressed in terms of a response function, and how the leading zero can then
be obtained by harmonic inversion of a periodic orbit signal constructed from
this response function.
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2Harmonic inversion for maps
We start by defining the quantity w = −i lnz and rewriting the zeta function
(7) as a function of w,
Z(w,β) =
∞
?
m=0
?
p
?
1 −eiwnpeσpβ
|Λp| Λm
p
?
.(10)
The density of zeros of the zeta function (10) can be written in the form
ρ(w,β) = −1
πIm g(w,β) ,(11)
where the response function g(w,β) is given by
g(w,β) = g0(w,β) +
∂
∂wlnZ(w,β)(12)
with g0(w,β) a smooth, slowly varying function of w. Using the definition
t(m)
p
=eiwnpeσpβ
|Λp| Λm
p
,(13)
we rewrite the second term of the response function as
∂
∂wlnZ(w,β)=
∞
?
∞
?
∞
?
m=0
?
p
(−inp)
t(m)
p
1 − t(m)
p
(14)
=
m=0
?
p
∞
?
∞
?
r=1
(−inp)
?
t(m)
p
?r
(15)
=
m=0
?
(−in0)eiwnpoeσpoβ
p
r=1
(−inp)eiwrnperσpβ
|Λr
p| Λrm
p
(16)
=
?
po
|Λpo|
∞
?
m=0
Λ−m
po, (17)
where the first sum in (17) now runs over all periodic orbits (p.o.) including
multiple traversals of primitive orbits, and n0denotes the topological length
of the underlying primitive orbit. After carrying out the sum over m, the
response function finally reads
g(w,β) = g0(w,β) +
?
po
(−in0)
eσpoβ
|1 − Λpo|eiwnpo.(18)
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On the other hand, the response function can be written in terms of the zeros
wkof the zeta function (10) and their multiplicities dk,
g(w,β) =
?
k
dk
w − wk(β).(19)
Like the zeta function (10), the expression (18) for the response function will
usually not converge in those regions of the complex variable w, where the
poles of g(w,β) are located, and the zeros of the zeta function cannot directly
be calculated from (18). Instead, we apply harmonic inversion to obtain an
analytical continuation [8,9]. The poles wk(β) of the analytical continuation of
g(w,β) in (18) are the zeros of the analytical continuation of the zeta function
Z(w,β) in (10). To apply harmonic inversion, we now construct a periodic
orbit signal by Fourier transformation of the response function (18),
C(s,β) =
1
2π
∞
?
−∞
g(w,β) e−iswdw .(20)
Considering only the second, oscillating part of g(w,β), this results in
C(s,β) =
?
po
−in0eσpoβ
|1 − Λpo|
δ(s − npo) .(21)
The corresponding signal resulting from Fourier transformation of Eq. (19)
reads
C(s,β) = −i
?
k
dke−iswk(β),(22)
which contains the zeros wk(β) of the zeta function (10) as frequencies. The
zeros wk(β) can now be determined by adjusting the periodic orbit signal (21)
– including all orbits up to a maximum topological length nmax– to the form
of the corresponding signal (22) by harmonic inversion. According to Eq. (9)
and with the relation w = −i lnz, the diffusion constant can be obtained
from the leading frequency w0using the relation
D ≈−i
β2w0(β) ,(23)
which for small β should be independent of β.
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