Serge Aubry - Physics Department . Technion

Suppression of Energy Diffusion in
Random Nonlinear Systems
Haifa June 2008
Serge Aubry
LLB,CEA Saclay, France
Coll. G. Kopidakis, S. Komineas, S. Flach
Serge Aubry, Aubry,
LLB, FRANCE

What is the long time behavior of an initially localized wavepacket
1-Does it spread to zero amplitude (diffusion)?
2- does it remain localized (absence of diffusion)?
3- Does a part of the energy spread and another part remains localized?
The answer is known in linear systems
Diffusion if the linear spectrum is continuous
Absence of diffusion when the linear spectrum is purely discrete
(for example in disordered systems with Anderson localization)
Nonlinear systems?
Serge Aubry, Aubry,
LLB, FRANCE

I-Nonlinear spatially periodic systems:
(with no disorder)
Incomplete diffusion of a wavepacket
Possible formation of Discrete Breathers
(spatially localized time periodic solutions)
Sievers and Takeno 1988 (ILM)
Campbell and Peyrard 1990 (Discrete Breathers-DBs)
Generic Nonlinear Excitations (non restricted to
special integrable models)
in 1D, 2D, 3D… periodic arrays of nonlinear oscillators
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Examples: Klein-Gordon systems 1,2 or 3D
(anharmonic oscillators coupled by harmonic springs
FPU systems (with gapless accoustic phonons)
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nonlinear springs
The system at infinity behaves like linear and generate
an effective frequency dependant damping on the dynamics
Filtering the resonant components
(Consequence:no damping for Discrete Breathers)
Discrete Breathers may be dynamical ATTRACTORS
in INFINITE Hamiltonian systems

Initial wavepacket
Time periodic
solution
Initial
Spontaneous Formation of a
DISCRETE BREATHER
no radiation
Limit
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Sievers and Takeno (1988):
from a initial localized wavepacket
radiation
Chaotic or « quasiperiodic »
transient
Transient
radiation
A large part of the initial energy
remains localized as a DB, the rest
spreads to zero at infinity
The second moment diverges but
the participation number does not

Existence proofs od Discrete Breathers (spatially localized
Time periodic solutions)
Existence Proofs of DBs in many nonlinear models
1- by the Principle of Anticontinuity
(MacKay SA 1994 Klein-Gordon
2- Bambusi 1996
3- Livi Mackay Spicci 1997 dimer FPU
4- SA 1998
5- SA, Kopidakis and Kadelburg 2001 (Variational)
Arioli Gazola
6- G. James 2003
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II- Exact Time Periodic Solutions (Intraband Discrete Breathers)
in Random and Nonlinear Systems
(with linear Anderson lOCALIZATION)
Kopidakis Aubry 1999-2000
Random systems: When the linear spectrum is discrete.
(Anderson localization) no linear phonon radiation is possible
Existence Proofs of Intraband DBs (IDBs)
Theorem: Albanèse and Frohlich 1991
Anderson modes persist in nonlinear models as time periodic
solutions with frequencies in a fat Cantor set.
Numerical Calculations (at computer accuracy) of intraband
Discrete Breathers : KA 1999-2000
1- Quasi-continuation in L 2 : Each Anderson mode generates a family
of IDBs with frequency in a FAT CANTOR SET (with nonvanishing measure)
2- Continuation in L ∞ : Each Anderson mode becomes an extended multiDB
with infinite energy
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Fat Cantor Set
(finite Lebesgue measure)
(Pseudo)continuation of an Anderson mode
at nonvanishing amplitude (scheme)
Nonlinear modes disappears in a small frequency gap at each resonance
Nonlinear Serge Aubry, Aubry modes , LLB, avoid FRANCE
resonances
frequency

Numerical Calculation of Localized Time periodic
Solutions
Using the Nonlinear Response Manifold to a Localized Time
Periodic Driving Force
Calculate the time periodic solution u n(t) at frequency
versus the driving force F (using Newton method)
F small linear response.
The response is smooth for most
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Projection of
NLRM
KA 2000
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RDNLS: a straightforward proof of existence
for Time Periodic Extended States
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Periodic solutions
are extrema of
When the ground-state corresponds to a
(linearly stable) extended solution (but many other extrema)

III- Exact Quasiperiodic Solutions
in Random Nonlinear Systems
Some exact Results
Fröhlich Spencer Wayne (1986) for
random Hamiltonian Systems with pair interactions
Bourgain Wang (2008) for RDNLS
W. Craig yet unpublished (?) for RDNLS
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Anderson Representation of DNLS
norm current p —> p’
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New complex variables

Almost Periodic Solutions:
Fröhlich, Spencer, Wayne (1986)
Assume
C p,p’ depends only on µ p and µ p’ (pair interactions)
Then, with large probability, there are quasiperiodic lattice
vibrations of finite total energy which lie on some infinitedimensional,
compact invariant tori in phase space.
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Quasiperiodic Periodic Solutions:
Bourgain Wang (2008)
In random DNLS, there are quasiperiodic solution
corresponding to finite dimension invariant tori
Almost periodic Periodic Solutions:
Walter Craig (private communication)
In random DNLS, there are « many » almost periodic solutions
corresponding to infinite dimension invariant tori
(use of Nash-Moser method?)
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Assume p random in interval B
Probability of resonance at lowest order (leading order)
Assume

The probability to find an initial wave packet which
evolves almost periodically, is large and goes to one as
the norm of the wave packet goes to zero
What about the wavepacket which are initially resonant?
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quadratic
IV Rigorous Results for DNLS Hamiltonians
proving the absence of complete diffusion of
a large enough Wavepacket
with invariance by phase rotation
(implies norm l 2 conservation)
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Nonlinear higher order
H NL > 0 strictly positive
(or negative)

Assume the wavepacket spreads uniformy to zero:
(lim t=∞ ||Ψ|| ∞ =0 ),
then at infinite time the nonlinear contribution H NL to the
energy is zero since
and the norm ||Ψ|| 2 is time constant. Then
energy cannot be
conserved and consequently the wave packet cannot
spread uniformly to zero.
Since the higher order nonlinear energy grows faster
than the norm the wavepacket cannot spread uniformly
to zero when its amplitude is large enough
Serge Aubry, Aubry,
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A family of models without diffusion
(random or not random) :
DNLS Models with Dispersionless Linear Phonons
H L is time constant, then H NL is time constant
Any initial l 2 wavepacket cannot spread to zero
independantly whether there is disorder or not
Serge Aubry, Aubry,
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Example:
What is the behavior of a wavepacket if no spreading to zero
is possible.
(periodic case: independant of n:
generation of mobile Discrete Breathers (« Discrete
Compactons ») which radiates wavelets and stops
(numerically observed for periodic systems by S. Flach)
Consequence: The limit profile is quasiperiodic.
Random case: Same conjecture
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In the Anderson base same model, the random DNLS
Model belong to this family of
NONdiffusive Hamiltonian except that operator L
(though it is diagonal), is not proportional to unity
but is random.
Why extra randomness would generate diffusion?
Conjecture: for RDNLS, the limit profile of any
initially localized wavepacket should be or
become quasiperiodic
(as an infinite dimension invariant torus).
Serge Aubry, Aubry,
LLB, FRANCE

V Diffusion of a Wavepacket in nonlinear random
Arrays: Early numerical investigation
When linear modes are localized (Anderson localization)
no diffusion but nonlinearity couples the linear Anderson modes
Then, energy diffusion?
Many early works conclude subdiffusion!
Bourbonnay-Maynard (1990) (random FPU)
Shepelyansky (1994), Molina (1997)…. (DNLS)
Snyder-Kirpatrick(random FPU) (2006)
Pikovsky-Shepeliansky(2008)
Numerical investigation of the second moment versus time
Diffusion is subcritical t α
where α ~ 2/5 (Shepelyansky) … 0.27 (Molina) etc....
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Subdiffusion was claimed only on
the base of the second moment divergency
(not clear numerically poorly defined exponents)
Remark: Divergency of the second moment does not
implies diffusion at all!!!!!
Counter example: The second moment of a wavepacket
in a linear harmonic chain of atoms coupled by
random springs diverges as t 1/2
but there is no diffusion because of Anderson localization
Coll: Analytic estimations R. Schilling, S. Lepri and SA
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There are initial conditions with no diffusion at all
For example: Intraband discrete breathers
Almost no diffusion (?) for small amplitude initial
wavepacket or close to anticontinuous limit
Quasiperiodic solutions?
There are initial conditions with apparently initially
strong diffusion (with strong chaos Shepelyansky)
The most studied!!!
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LLB, FRANCE

Conjectures for Infinite Nonlinear Arrays with linear Anderson
Localization (purely discrete spectrum with no mobility threshold)
There are two kinds of Initial Conditions which both may be obtained
both with finite probability
1- Solutions which are purely quasiperiodic from the begining (infinite
dimension invariant tori). Their probability goes to 1 at small norm
or close to the anticontinuous limit or close linearly stable IDB
solutions.
2-Solutions which are initially Chaotic but slowly converge to a
quasiperiodic profile. Self-organization profile.
Initial resonances are spontaneously removed by energy radiation
through non excited Anderson modes which may generate a long tail.
No Diffusion at all for any initial l 2 wave packet.
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VI-New Numerical Investigations
G. Kopidakis, S. Komineas, S. Flach and SA (PRL (2008))
http://www.mpipks-dresden.mpg.de/pages/forschung/forschung_advanced.html
Random DNLS Hamiltonian(2 invariants)
Random Quartic Klein-Gordon Hamiltonian (1 invariant)
Serge Aubry, Aubry,
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Very similar behaviors

Energy density profile in the Anderson space
(Quartic KG model)
Serge Aubry, Aubry,
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Few Anderson modes are and remains strongly excited
V=0.25
β=1
W=4

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LLB, FRANCE
t=1.2. 10 8 red
Profile (log)
In Anderson space
(Quartic Klein Gordon model
W=4
In real space:
random DNLS
t=10 7 black
t=2. 10 8 red

Participation number of norm distribution for the random DNLS model
for different amplitudes of the single site initial wave packet
Serge Aubry, Aubry,
LLB, FRANCE

Participation number of norm distribution in Anderson space
for the random DNLS model and for different amplitudes of
the single Anderson mode initial wave packet
Serge Aubry, Aubry,
LLB, FRANCE

3 orders of magnitude
longer time of evolutions
Serge Aubry, Aubry,
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Second moment
Participation number
The second moment
versus time seems to diverge.
The participation number
does not!
e n = energy density in
Anderson space for KG
e n =|Ψ n | 2 for DNLS

Serge Aubry, Aubry,
LLB, FRANCE
Time Fourier transform
of the participation number
The limit profile tends
to be quasiperiodic!
Same result for the time
Fourier transform at a
single central site

Numerical arguments for KAM tori as limit profile
of an initially localized wavepacket
1-The participation ratio of the energy distribution does not
diverge.
2- The amplitude of the at any site does not decay to zero
though it oscillates indefinetely.
3-The largest Lyapounov exponent initially drops by 2 or 3
orders of magnitude while the wavepacket does not spread to
small amplitude. It continues to decay but slower and slower.
4- Simultaneously, the Fourier spectrum of the atomic motion
near center from broad band becomes narrow band with thin
line widths
Conclusion: The attractor of the system (limit profile)
might be a KAM tori ?
but very slow convergence (Arnol’d diffusion?)
What about the evolution for times beyond those numerically available?
Need a mathematical proof?
Serge Aubry, Aubry,
LLB, FRANCE

Scenario for the Absence of spreading
Limit profile at t= + ∞ of the energy density with long tail
is quasiperiodic (nevertheless the second moment may diverge??!!!))
Serge Aubry, Aubry,
LLB, FRANCE
Tail generation
by weak excitation
of Anderson modes

Finite size effects (numerically tested)
For small system, the wave packet cannot spread out.
No long tail can develop. Radiated energy from resonances
remains confined and interacts again with the core of the wavepacket.
Initial Chaos persists forever there is (roughly) energy equipartition
after long enough time (unless the initial solution generates
a KAM torus)
For an infinite system the energy of the initial wave packet is not extensive
and corresponds to zero temperature. It cannot thermalize.
For an FINITE system, initial chaos of the wavepacket (if
any) persist at any time while it spontaneously disappears
for an INFINITE system. Then, the wavepacket
slowly converges to a nonvanishing quasiperiodic limit
profile.
Serge Aubry, Aubry,
LLB, FRANCE

Warning about the problem of reliability of numerical simulations for Long
Time Evolution (Arnol’d diffusion regime)
1-Nonlinear terms are responsible for energy exchanges between the Anderson
modes. At small amplitudes, the system becomes nearly integrable because
nonlinearities are weak. Thus, numerical noise becomes relatively important (with
respect to the nonlinear terms) and could play the essential role for generating
(small) energy diffusion.
-2 A well-known similar problem of « Arnol’d diffusion » was encountered for the
numerical integration of solar planetary systems. The problem was solved by using
secular equations involving only slow variables (orbits parameters).
With such techniques Jacques Laskar could integrate with controlled accuracy
over 5. 10
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9 years solar system models using a step size of 200-500 years that is
about only 107 integration steps (see J. Laskar Icarus 196 1-15 (2008)) (while
ordinary techniques would require a step size of only 8 days!) and very short
(reliable) runs.

Similarity with the problem of Chaotic diffusion in the
Solar System.
It is generally believed that at the beginning, the planet
motions in the solar system were rather chaotic.
After a long time, collisions and planet escapes spontaneously
regularized the motions of the left planets to become
« almost » quasiperiodic as today.
However, weak chaos still persist today from the original one.
It has been proved numerically for the motion of
some planets like Mercury (e.g. see J. Laskar Icarus 196 1-
15 (2008) and refs. ) using sophisticated methods where the
numerical errors could be carefully controlled.
Serge Aubry, Aubry,
LLB, FRANCE

VII Summary and concluding remarks
The possible divergency of the second moment does not implies energy spreading
as believed in early publications.
Numerical observations (over times which are not too long
for the above reasons) shows
1- There are initial conditions with practically no diffusion from the beginning
(quasiperiodic?) and others which looks (initially) chaotic.
2- the participation number does not (clearly) diverge in any case. Moreover,
it is rigorously proven it cannot diverge in DNLS models with norm conservation
and large enough amplitude wave packet.
3- The same qualitative behavior for the spreading of an initially localized
wavepacket in DNLS models with norm conservation and quartic KG models
without norm conservation
4-Conjecture: Unless the wavepacket corresponds initialy to a quasiperiodic
solution, it converges (slowly) to a limit profile which is a quasiperiodic and
spatially localized solution (KAM torus).
This profile may not have second moment or may have a second moment?
(open question)
Serge Aubry, Aubry,
LLB, FRANCE

Conjectures:: There should be absence of diffusion for any initially
localized wave packet in any nonlinear model at any dimension
with purely discrete linear spectrum (Random, quasiperiodic or else)
There is diffusion (at least partial in case of time periodic Discrete Breather
formation) in all models for which the linear spectrum has an absolutely
continous component.
There is diffusion as soon the model is a at non vanishing temperature
or if it submitted to a random noise (with broad spectrum)
Numerical problem with random FPU model: The linear localization
length diverges at low frequency. The long tail already exists in the linear
case (diverging second moment) and cannot be considered as an
effect of nonlinearity
Serge Aubry, Aubry,
LLB, FRANCE
Final Comments