THE ANDERSON LOCALIZATION PROBLEM, THE
FERMI - PASTA - ULAM PARADOX AND THE
GENERALIZED DIFFUSION APPROACH
V.N. Kuzovkov
ERAF project Nr. 2010/0272/2DP/2.1.1.1.0/10/APIA/VIAA/088
Institute of Solid State Physics, University of Latvia, 8 Kengaraga, LV-1063 Riga, Latvia
Abstract
About a half a century ago two celebrated papers were published temporally close to each other which gave birth to two fundamental directions of theoretical physics. In 1955 the Fermi-Pasta-Ulam
(FPU) paradox [1] was formulated which suggested the nonequipartition of energy among normal modes of an anharmonic atomic chain. This phenomenon is closely connected with the problems of
ergodicity, integrability, chaos and the stability of motion [2]. A few years later, in 1958, Anderson suggested the possibility of electron localization (AL) in a random system, provided that the disorder
is sufficiently large [3]. This idea is one of the foundations for the understanding the electronic properties of disordered systems [4].
Our purpose is to establish a close connection} between the two fundamental problems [5]. That is, the AL is nothing else but the FPU effect in a modified dynamical system with interacting normal
modes. Besides, the delocalized states in the AL problem are analogous to the stable quasi-periodic motion (recurrence in the FPU problem) and, respectively, the localized states are similar to the
thermalization (motion instability) in the FPU. Our FPU modification includes: (i) non-trivial change of the mode ensemble. In the FPU without interactions all normal modes are stable. In contrast, in
the AL both stable and unstable modes also exist. (ii) a modified mode interaction: the stochasticity in the AL is introduced directly, through random forces linear in coordinates and rather indirectly,
through nonlinear terms in the dynamical equations.
Introduction
AL and Generalized diffusion
n 1  ( E   n )n  n 1
Stochasticity:
FPU: a non-trivial effect in the dynamics of nonlinear systems.
AL: explicitly through random potentials.
Multiplicative noise
determines the exponential
character of the divergence
(generalized diffusion) [7]:
Diffusion:
FPU: the equations for nonlinear dynamical systems can under certain conditions describe
pseudo-random walks which leads to the diffusion behaviour and diffusion increase in mean
energy (i.e.divergence).
AL: the random forces impose random walk amplitudes (the dynamics is bounded in the phase
space for unperturbed system) which can lead to the diffusion divergence.
2
 exp( 2n)
1


General (exact) solution for D dimensional case:
p  D 1
L(k )  E  2 j 1 cos( k j )
1
1
2
 1
H ( z)
(2 ) p
Classical Hamiltonian map
The concept of the localization operator
H(z) is a general and abstract description
of the problem of localization.
Instead of analyzing wave functions, it is
sufficient to analyse properties of H(z) by
means of the theory for functions of
complex variables:
n 1  ( E   n )n  n 1
( n , n1 )  ( qn , pn )
Two-dimensional Hamiltonian map:
n
Lyapunov exponent γ
and localization length ξ
The diffusion concept permits to connect the AL and FPU problems. The Schrödinger equation
for AL (quantum tight-binding model) can be interpreted in terms of the classical Hamiltonian
map [6]. As a result, the AL problem can be reformulated in terms of interacting mode dynamics
which opens the opportunity for the detailed comparison of AL with FPU.
Tight-binding equation as Newton
equation for discrete time n:
 n n
pn 1  pn   qn   n qn
2
p

1
 dk ( z  1)2 / z  L2 (k )
The localization operator H(z) is a nonanalytic function of the complex variable z.
The unit circle |z|=1 divides the complex
plane into two analytic domains: the interior
and exterior of the unit circle. Multiple
solutions can result in the formal analysis of
the problem.
 phase diagram of the system (regions of
localized and delocalized solutions),
 the localization length.
qn 1  qn  pn 1   2  E
2
Kicked-oscillator Hamiltonian
p  q  (t ) q
H


2
2
2
2
2 2
2
 (t  nt )
Unstable (inverted) modes: |E|>2
Stable (normal) modes: |E|<2 (old band)
Stability and thermalization
It defines the system with unperturped
Hamiltonian of oscillator which is affected by a
periodic sequence of kicks (δ-pulses).
Oscillator with random walk!
Random walks and Normal diffusion
n 1  n   n
 n  0,  n
n
2
2
2
Causality principle:
 ψ1 is a function of ε0,
 ψ2 is a function of ε0, ε1, etc,
 ψn and εn are statistically
independent.

2
n 1
 n  
2
2
The FPU problem: the paradox in chaos theory that many complicated enough physical
systems exhibited almost exactly periodic behavior instead of ergodic behavior.
Numerical simulations of a chain of harmonic oscillators coupled with a quadratic or cubic
nonlinearity show that energy, initially placed in a low-frequency normal mode of the linear
problem stay almost completely locked within a few neighbor modes, instead of being
distributed among all modes of the system (thermalization). Recurrence of energy to the
originally excited mode is also observed (FPU effect).
The existence of a stochasticity threshold in the FPU problem: If the nonlinearity is below a
stochasticity threshold, the dynamics of the system remains similar to the one of the
unperturbed system for large time scales. For a strong nonlinearity the overlap of nonlinear
resonances leads to a strong dynamical chaos, destroying the FPU effect.
 0   2n
2
To detect the diffusion, it is sufficient to
demonstrate the divergence of the second
moment of the amplitude and to establish its
law of time-dependence. For a normal
diffusion the mean square displacement is
linear in time, while for an annormal diffusion
it is nonlinear.
Results
The Anderson localization and Fermi-Pasta-Ulam problems are very complementary:
counter-intuitive result in one problem has its analog in another problem.
A stable quasi-periodic motion (the recurrent behavior) in the FPU corresponds to the
existence of the delocalized states in the AL problem.
The thermalization effect in FPU has an analog in the localized states in the AL.
This work has been supported by ERAF 010/0272/2DP/2.1.1.1.0/10/APIA/VIAA088
References
[1] E. Fermi et al , Los Alamos, Report. No. LA-1940 (1955).
[2] G.P. Berman et al , Chaos 15, 015104 (2005).
[3] P.W. Anderson, Phys. Rev. 109, 1492 (1958).
[4] E. Abrahams et al , Rev. Mod. Phys. 73, 251 (2001).
[5] V.N. Kuzovkov, Phys. Scr. 84, 065002 (2011).
[6] F.M. Izrailev et al , Phys. Rev. Lett. 82, 4062 (1999).
[7] V.N. Kuzovkov et al , Physica A 369, 251 (2006).