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? 


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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 


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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 


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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). 


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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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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) 


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Very similar behaviors

Energy density profile in the Anderson space 

(Quartic KG model) 


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Few Anderson modes are and remains strongly excited 

V=0.25 

β=1 

W=4


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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 


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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 


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3 orders of magnitude 

longer time of evolutions 


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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


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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? 


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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??!!!)) 


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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. 


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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. 


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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) 


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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 


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Final Comments

Serge Aubry - Physics Department . Technion

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