Approach to equilibrium in a chain of nonlinear oscillators

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Approach to equilibrium in a chain of nonlinear oscillators

F. Fucito, F. Marchesoni, E. Marinari, G. Parisi, L. Peliti, S. Ruffo, A.

Vulpiani

To cite this version:

F. Fucito, F. Marchesoni, E. Marinari, G. Parisi, L. Peliti, et al.. Approach to equilib- rium in a chain of nonlinear oscillators. Journal de Physique, 1982, 43 (5), pp.707-713.

�10.1051/jphys:01982004305070700�. �jpa-00209442�

707

LE JOURNAL DE PHYSIQUE

Approach ^to equilibrium ^{in a chain of} nonlinear oscillators

F. Fucito (*) (+), ^F. Marchesoni (**), ^{E. Marinari} (*) (+), ^{G. Parisi} (***), ^{L. Peliti} (*) (+ +), ^{S. Ruffo} (**)

and A. Vulpiani (*)

(*) Istituto di Fisica « G. Marconi », Università di Roma, Italy.

(**) Istituto di Fisica, Università di Pisa, Italy, ^and INFN, ^Pisa.

(***) ^{Istituto di} Fisica, Facoltà di Ingegneria, Università di Roma, Italy ^and INFN, ^Frascati.

(+) INFN, ^Roma.

(++) GNSM-CNR, Unità di Roma.

(Recu ^{le 6 aout} 1981, révisé le 30 décembre, accepté ^{le 18} janvier 1982)

Résumé.

Nous considérons l’approche ^à l’équilibre thermodynamique d’une chaîne d’oscillateurs non linéaires

couplés ^où l’énergie ^est initialement fournie aux modes de plus grande longueur d’onde. La considération des effets des singularités ^{dans le} plan complexe de la solution des équations ^{du mouvement} permet ^de prédire qu’à

des temps relativement courts le spectre d’énergie ^{a une} dépendance exponentielle ^en fonction du nombre d’onde k lorsque ^{celui-ci est} grand. ^Elle permet également ^de distinguer ^{entre deux} régions temporelles : ^{une à} temps ^courts où la pente ^de l’exponentielle dépend linéairement de ln t, ^{et une à des} temps intermédiaires où elle est proportion-

nelle à (In t)-1/2. ^Ces prédictions ^{sont en accord avec les} simulations numériques que nous avons faites sur ce

système.

Abstract.

We consider the approach ^to thermodynamical equilibrium ^{in a chain of} coupled ^nonlinear oscillators when energy is initially ^fed only ^{to the} longest wavelength ^modes. Consideration of the effects of singularities ⁱⁿ

the complex plane ^{of the} solutions of the equations ^of motion allows us to predict ^that at not too long ^{times the}

energy spectrum ^{has an} exponential tail in the wavenumber k and to distinguish ^{between two time} regions : ^{a short}

time region ^{where the} slope ^{of the} exponential ^{shows a linear} dependence ^on In t, and ^an intermediate region ^where

it is proportional ^to (In t)-1/2. ^These predictions ^are successfully compared ^{with the numerical} simulations we have

performed ^{on the} system.

J. Physique ⁴³ (1982) ^{707-713 MAi} 1982, ¹

Classification

Physics ^Abstracts

05.20

1. Introduction.

The approach ^to equilibrium ^of

a classical hamiltonian system ^{is still an} open and

interesting problem. ^{The main} rigorous ^{results are}

the KAM [1] ^{and Sinai} [2] ^theorems. They ^describe

the behaviour of two opposite ^{extreme cases of a} nonintegrable hamiltonian system. ^The Sinai theorem shows the validity ^{of the} ergodic hypothesis ^{for a}

gas of hard spheres. ^{The main} consequence of the KAM theorem is the permanence of invariant tori in _presence of a small perturbation ^{of an} integrable system. ^{In this case the} long time behaviour of the system ^{will not} be described by ^a Boltzmann distri- bution. The _range of validity of the KAM theorem for a system ^{with a} large ^{number of} degrees ^{of freedom}

is not yet understood.

Numerical simulations have been extensively

examined for a linear chain of coupled ^nonlinear oscillators, starting from the famous work of Fermi,

Pasta and Ulam [3]. ^{More recent works} [4-6] ^have

shown that the regions ^of phase space where ordered motion takes place ^{are much more} extended than

expected ^{on the basis} of the KAM theorem. Some authors [6] ^stress the existence of a threshold in energy which separates ordered from disordered behaviour. All these works study long times in order to

distinguish between ordered and stochastic asymptotic

behaviour. The present ^work analyses short times in order to understand the _way chaotic behaviour

sets in. Our system ^{is a} discrete version of a classical one-dimensional field obeying ^a nonlinear Klein-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004305070700

708

Gordon equation. ^This system ^has already ^been

considered in reference [6]. ^We consider initial condi- tions such that energy is concentrated in the longest wavelength ^{modes and we} study ^{how it} propagates ^to shorter wavelengths. ^In the times we consider the bulk of the _energy is still contained in the initially

excited modes, ^and only ^{a small} quantity ^{is fed to}

the shorter wavelengths. ^{In order to} allow for an

approximate analytic ^{treatment we focus our atten-}

tion on the process by which this small amount is distributed. We are then able to compare ^our pre- dictions with the results of the numerical simulations.

The paper is organized ^as follows : the model is introduced in section 2, ^{where the} analytic ^treatment

at short and intermediate times is given. The results of the numerical simulations as well as their compa- rison with our theoretical predictions ^are given ⁱⁿ

section 3. In section 4 we draw some conclusions and

point ^{out some} possible directions for further work.

2. Analytical treatment.

We study ^{the one-di-}

mensional nonlinear Klein-Gordon equation :

where the real, one-component ^field p(x, t) ^{is defined}

on the interval - L/2 ^x L/2 ^with periodic ^boun- dary conditions. The corresponding hamiltonian den-

sity ^is given by :

where the field 7r is canonically conjugate ^to po The system may be either considered as a one-dimensional elastic string ^{in an} external anharmonic potential,

or as a one-dimensional classical field theory.

We study the time evolution of the field _~ as a

function of the initial condition p(x, 0),

It is known that if H is finite at t ⁼ 0, the solution of equation (2.1) exists and is unique [7]. ^{It is useful}

to define the space Fourier transform of the field p

as follows :

We shall consider as a typical initial condition :

We are interested in the behaviour, ^{in the} region I k I > kI, ^{of the} spectrum W(k, t), ^defined by :

We expect the field _p to reach asymptotically ^a

thermal equilibrium distribution, given by ^{a Boltz-}

mann factor exp( - f3H) ^{for some} value of the inverse

temperature f3 determined by the initial conditions.

In this _{case, at} values of the wavenumber k so large

that the mass and nonlinear terms of H are negligible,

one would have :

This behaviour of W corresponds ^{to functions} p(x, t)

which are not differentiable with respect ^{to x. Now} it is known that since T(x, 0) ^is analytical ^{as a function}

of _x, the solution T(x, t) will remain analytical ^at any finite time t [7]. Equation (2.6) ^can only be valid for infinite time. This means that, ^{as time} goes on, singu-

larities of p(x, t) appear in the complex ^x plane ^which

creep toward the real axis and accumulate onto it at infinite times. We show below that these singu-

larities ^are simple poles.

If we assume this to be the _case, then we may relate these singularities ^{to the} large k ^behaviour

of W by ^means of the theorem of residues. Let us

consider the integral defining 0 (eq. (2.3)) ^{in the} complex ^x plane. ^At positive (negative) k ^{we close}

the contour in the lower (upper) complex ^half plane.

We then obtain the following expression ^for 0(k, t) :

where the sum runs over all poles located in the relevant half plane, Rj ^being their residue and xj + iy; ^their

location. We therefore obtain the following asymptotic

behaviour of W(k, t ) ^at large k :

where yS(t) ^{is the} imaginary part of the location of the pole which lies nearest to the real axis.

Our strategy ^{is then to} evaluate the most likely

value of ys(t) by extending ^the approach ^{of Frisch}

and Morf [8] ^{to a} deterministic partial differential

equation.

- We should in principle evaluate the analytic

continuation to complex ^x of the solution p(x, t)

of equation (2.1). ^We prefer ^{instead to} consider the solution 9(x ⁺ iy, t ) ^{of the} complex partial ^diffe-

rential equation :

where _~ is an analytical function of z ⁼ x + iy. ^Such

an equation ^{is the} complexification ^of equation (2.1)

with the hypothesis ^that T(x, t ) ^{can be} analytically

continued. In other words we follow the commutative

diagram given ^{below :}

We first consider the case of an ordinary differential

equation obtained from equation (2.9) by neglecting

the laplacian >> ^term a2paz2. ^{Let us} consider the solution 4>«fJo, t) ^{of this} equation ^at real times t,

as a function of its (complex) ^initial condition (po :

If we write 4> = 4>R + i4>I ^{we have the} following

differential equations ^for 4>R’ 4>1 :

The force field associated to this system ^{is drawn} in figure ¹ (only ^{the first} quadrant ^is shown). Although

the origin ^{is a stable} point, ^{we find a} pair of unstable

points ^{on the} imaginary axis, from which a trajectory

escapes ^to infinity. ^The point ^at infinity appears ^{as a} saddle point, ^{and may} be reached in a finite times from

points ^{on the} imaginary axis. This is the origin ^{of the} singularities of 4>(cpo, t). It is then clear that these

singularities ^are simple poles, ^{located at} that value of _To from which infinity ^may be reached in time t.

We may estimate the location of these poles by computing the time needed to reach infinity starting

from a given point ^{on the} imaginary 45 axis. Since the motion is in this case one-dimensional (all along ^the

Fig. ^1.

Force field of equation (2.11). Only the upper

right quadrant ^is shown. The position of the instable point

on the imaginary ^axis (1m cp

m/gl/2) ^{is marked.}

imaginary axisx ^we may introduce a potential ^accord- ing ^{to :}

’

The time needed to reach infinity starting ^{from the} point (po ⁼ i(p" 0 may be then easily computed ^{from the}

conservation of « energy » in the one-dimensional motion :

At large ~o ^{the time T behaves as :}

On the contrary, ^no singularity will appear ^at any time if

for the time T diverges (although slowly) as Im (po I approaches this value from above. From _equa- tion (2.14) ^we therefore obtain the location of the

pole ^of 4>( CPo, ^{t) at} large CPo’s, ^{therefore at} short t’s :

If the effect of the laplacian ^{term in} equation (2.9)

may be neglected ^{at short} enough times, ^{the above} analysis may be also applied ^to the one-dimensional chain ^as follows : the solution T(z, t) ^of equation (2 . 9)

will be given by 4>(cpo(z), t), ^where CPo(z) ^{is the} analytical

continuation of the initial condition T(x, 0) ^{to the} complex ^x plane. ^In particular, ^a typical ^initial

condition

yields ^{in the} complex ^x plane

We have therefore at large y I ^:

According ^{to our} hypothesis, ^the singularity ^of

CPo(z, ^{t) will be located where Im} CPo, ^as estimated

710

from equation (2. 19), will be of the order of that

given by equation (2.16). ^Therefore

We now consider the effect of the previously neglected laplacian ^{term. We} claim that the analysis

we just made retains its validity ^at very short times.

The laplacian ^{term will not} in fact be able to prevent

a 9(z, t) ^from escaping ^to infinity if its initial ima-

ginary part ^is large enough. Localized o bursts » will therefore appear in (p(x, t), ^whose spatial ^cohe-

rence is only ^{due to} correlations present in the initial conditions.

The laplacian ^{term will on the} contrary ^{be essential} in introducing singularities ^{in the} part ^{of the} complex ^x plane ^which corresponds ^{to small initial values of}

Im _cpo If the term were not present, ^Im T would remain bounded at all times. If however ! Im cp [

were to become larger ^than m/gI/2, the mechanism

we just described would drive it to infinity ^{in a} very short time, ^{and a} singularity ^{will be} produced. ^We

must now estimate the time needed for Im _T to become larger ^than m/g 1/2. ^The analysis ^becomes simpler ^{if we consider a} harmonic chain ( g ⁼ 0) ⁱⁿ

the limit of infinite length (L - oo). ^We expect ^this analysis ^to be also valid for our case if g ^{is not too} large. ^It is known that the one-point probability

distribution function of a classical harmonic field in one-dimensional is gaussian [9]. ^{The variance Q2}

of the gaussian will be then given ^{at small} y’s by :

The probability p, for I 1m cp to ^be larger ^than m/g 1/2

will be therefore given by :

The time needed for this to happen ^{will therefore}

be of order 1/pc. ^This yields ^the following ^estimate of YS :

The main effect of the nonlinear terms in this regime

will be to change the value of (12. If a2 were time

independent, equation (2.21) ^{will be} essentially ^correct.

Let us distinguish between the role of short and long wavelength modes. At the times we are interested in,

most of the _energy is contained in the long ^wave- length modes, ^which may be assumed to be in a kind of thermal equilibrium among themselves. Their contribution to (J2 _may be then considered as essential-

ly ^constant in time. The short wavelength ^{modes will}

however also contribute to (J2. As long ^as W(k, t)

is small in the large k region, ^their contribution ^is

negligible. ^{As time goes on, however,} W(k, t ) ^will

start increasing, ^{what will} increase the value of Q2 and fasten therefore the transfer of _energy to short

wavelength modes. This triggers ^a catastrophic ^process

which our analytical ^{tools are unable to handle. We}

cannot therefore draw conclusions about the behaviour at very long times before thermal equilibrium ^is

reached.

We summarize the results for W(k, t) ^{at short and}

intermediate times :

3. Numerical results.

We present in this section the results of numerical simulations of equation (2 .1 )

for different initial conditions, ^{in order to} verify ^the validity ^{of the} analytical ^{results we} obtained in the

previous ^section.

We consider a chain of N points ^with periodic boundary conditions which represents ^a discretization of equation (2 .1 ). ^{We set L} (the length ^{of the} chain) equal ^to 1 ; ^{and we} consider the field cp(x) ^{to be repre-}

sented by _gj ⁼⁼ cpU ^{Ax -} 1/2), ^{where Ax =} 1/N. ^The periodic boundary conditions then imply _9jlN ⁼ _9j’

The Fourier transform of the discrete field Pj ^is defined as follows :

Fig. ^2.

^A typical spectrum, ^In Wn(t) ^{vs. n for A}

5,

g = 5, N=64, t=2, ^{T = 1.}

The quantity corresponding ^{to the} W(k, t ) ^{of the} pre- vious section will be :

We have performed simulations for N ⁼ 64 and N

128. The initial conditions were chosen to be :

all other Fourier coefficient having ^{been set to zero.}

The integration ^of equation (2 .1 ) ^was performed by

means of the central difference algorithm [10] :

where the discretized force Fj({ ^{T }) is given by :}

In the limit At - 0 equation (3. 3) ^{reduces to}

The initial condition cPj(O) ^{= 0} corresponds ^to (pj(- ^{et) =} w;(0).

We have done simulations keeping ^{m2 = 0.01 and} varying A and g ^{in the range} 0.5 : 10. At has been taken in the _range 10-3 + 0.5 ^x 10- 3. We have checked that the _energy is conserved within 0.1 %; ^no

significant differences have been seen in runs with different At. The results we report have been obtained via computations ^{with 15 or 16} significant digits.

Since part ^{of the} arguments ^{in the} previous ^section

are probabilistic, ^{it is} convenient to introduce the average of Wn(t) centred around the time t :

In order to eliminate oscillations, ^{T must be chosen}

to be of the order of magnitude ^{of the} period of oscilla- tion of a mode of wavenumber kn ^{= 2} nn/ L ^{in a har-}

monic chain. Since m is quite ^{small in our} case, this

yields comparatively large values of T, ^{of order} 0.1 : 0.2 in the short time region and 2 : 10 in the intermediate time region. The deviations of W,,(t)

from its average may be understood ^on the basis of the statistical properties of the field (p(x, t) ^at large k,

which is analysed in detail in a forthcoming paper [11].

We now present the results for short times. We have verified that the spectrum Wn(t ) ^{can be} very well described by ^an exponential ^{in the} region ^{of not too}

small n. A typical example is shown in figure ^{2. It is}

easy ^to obtain the value of the slope S (t ) ^{of this} expo- nential from the data :

This slope is related to ys(t) ^{of the} previous ^section by :

We show in figure ^{3 the} dependence ^of S (t ) ^on

In (tAgI/2) with different values of t, _g and A. We see

Fig. ^3.

Slope S (arbitrary units) ^{vs. In} (Ag1/2 t), ^N

^{64, T}

^{0.1 : 0.2.}

712

Fig. ^4.

Slope ^S (arbitrary units) ^vs. (A2 g ^In t)- 1/2 , ^N

64, ^T

^5.

that the data gather ^{on an universal curve, which is} essentially ^a straight line, giving ^{therefore a confirma-}

tion to the analysis of short times performed ⁱⁿ

section 2. This plot may be therefore considered ^{as a} check of equation (2.20). ^{All error bars} represent essentially uncertainties in the estimation of S(t).

We have verified that the value of Wn(t ) ^coincides

with its time _average at short times. This is due to the dominance of a single complex singularity. As the time increases and we enter the intermediate time region,

we expect ^{that more than one} complex singularity ^at roughly ^{the same} distance from the real x axis will

Fig. ^5.

Slope ^S (arbitrary units) ^vs. (In t) - 112 , ^N

128, A ⁼ 3, g

5, ^T

¹⁰ except ^{for t = 8 where T =} 2t

appear. Oscillations will therefore be present ⁱⁿ Wn(t)

as a consequence of the interference _among such

singularities.

In the intermediate time region ^we expect ^that

S (t) oc (A2 g In t)- 112. ^{We have} plotted ^{S vs. this}

variable in figure ^4, varying ^{either t at fixed A} and g,

or g ^at fixed t and A. We see again that the data

gather ^{on a} single straight ^line. Therefore the func- tional dependence of S (t ) ^{is as} expected ^{on the basis of} equation (2.23) ^{for not too} long times. The behaviour

at longer ^{times of a} solution with a given ^{value of A} and g ^{is shown in} figure ^{5. One sees both the} (In t)- li2

behaviour and a break at longer times. This is connect- ed with the observation done at the end of the previous

section.

4. Conclusions.

We have seen that it is possible

to give ^definite predictions ^about the behaviour of a

nonintegrable hamiltonian system ^{with an infinite} number of degrees of freedom in the nonasymptotic

time region. This has been achieved by ^{the use of the} analytical continuation of the equations of motion in the complex domain, ^a method introduced in a similar

context by Frisch and Morf [8]. The role of the infinite volume limit has also been clarified. The free evolution of the system is indeed such that 9 may only ^become arbitrarily large in the infinite volume limit. In this

case no linear perturbation ^may be considered small if

one follows the system ^at long enough times. It is

possible that intermittance concepts ^are relevant in this context [8, 11]. Further work aims at clarifying

this connection [11]. ^{One of our main} results is that the system approaches equilibrium ^{with a} logarithmic dependence ^{on t, so that the} nonequilibrium spectrum

may persist ^for extremely long ^{times, and} may be mistaken for a stationary ^{state if the} observation time is not sufficiently long. ^{It is} amusing ^to remark that this quasi-equilibrium distribution is similar to Wien’s law for black body radiation, ^{with a} slowly varying

« Planck’s constant ».

Acknowledgments.

We thank U. Frisch for hav-

ing ^{made available to us the} paper in ref. [8] ^before publication. ^We also thank L. Galgani, ^A. Giorgilli

and M. Vitaletti for helpful conversations and cons-

tructive criticism.

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