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Antonio Giorgilli · Simone Paleari · Tiziano Penati
Extensive adiabatic invariants for nonlinear chains
Received: date / Accepted: date
Abstract We look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit.Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables wetransform it into a sum of two parts in involution. At variance with the usual method of introducingnormal modes, our constructive procedure allows us to exploit the complete resonance, while keepingthe extensive nature of the system. Next we construct a nonlinear approximation of an extensiveadiabatic invariant for a perturbation of the discrete nonlinear Schrodinger model. The fluctuationsof this quantity are controlled via Gibbs measure estimates independent of the system size, for alarge set of initial data at low specific energy. Finally, by numerical calculations we show that ouradiabatic invariant is well conserved for times much longer than predicted by our first order theory,with fluctuation much smaller than expected according to standard statistical estimates.
Keywords Adiabatic invariant · Thermodynamic limit · Extensive Hamiltonian lattice · Resonantnormal form · Ergodicity
1 Introduction
In the celebrated report of Fermi, Pasta and Ulam (FPU) [20] the fundamental question was reopenedwhether a small perturbation of an integrable system could act as a trigger for a relaxation to equi-librium in the sense of Statistical Mechanics. For the model investigated in that report, namely adiscretization of a non linear string, equipartition was expected among the normal modes of the chain.In contrast, the first numerical experiments showed that the energy may remain concentrated on a fewmodes for a long time, with no tendency to equipartition. Yet, more than 50 years of studies have notcompletely answered the question raised by the authors of that report.
In the present paper we follow an approach somehow different from the traditional one, i.e., wetry to exploit the resonances among the particles, disregarding the behaviour of the normal modes.Our main point is that looking at the system as a chain of identical oscillators, a good candidate fora conserved quantity over a long time may be identified by exploiting both the resonance and theextensive character of the Hamiltonian. By “conserved quantity” we mean a function independent of
A. GiorgilliE-mail: [email protected]
S. Paleari,Tel.: +39-02-50316131Fax: +39-02-50316090E-mail: [email protected]
T. Penati,E-mail: [email protected]
Depart. of Mathematics “F. Enriques”, via Saldini, 50 - Milan (Italy)
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the Hamiltonian which is an approximate first integral for the Hamiltonian in the sense of perturbationtheory, and could remain invariant for long times, possibly exponentially long in Nekhoroshev’s sense.The existence of such a function clearly means that a relaxation to a statistical equilibrium can notbe attained in a short time.
1.1 Overview of our results
We first consider a periodic linear chain as described by the Hamiltonian
H(x, y) = Ω
N∑j=1
y2j + x2j2
+ µxjxj+1 , xN = x0 , (1)
where Ω are the local frequencies and µ is a coupling parameter. This is the linear part of a Klein–Gordon (KG) model, and with an appropriate choice of the parameters it also coincides with the linearpart of the FPU model. The system is well known to be integrable. The usual procedure indeed isto introduce normal modes with a linear transformation, essentially using the theory developed byLagrange [28]. Thus N first integrals in involution are immediately found, and the motion is quasi-periodic with a given frequency spectrum (see sect. 2). However, we are particularly interested inexploiting the closeness of our model to a system of identical oscillators, which is completely resonantbecause all frequencies are equal. This aspect is lost when the normal modes are introduced. Thus wefollow a different path by considering µ as a perturbation parameter and looking for a normal formH = hΩ +Z0 where hΩ is a system of identical oscillators (the diagonal part of the Hamiltonian) andZ0 contains the coupling terms, but is in involution with hΩ , i.e., hΩ , Z0 = 0. With this approachthe integrability of the system is forgotten, since we exploit the existence of a single first integral Z0
in addition to the Hamiltonian. This is a more general property, definitely weaker than integrability,which may appear in a wide class of systems of interest in Statistical Mechanics. From a technical pointof view, the existence of a linear transformation which gives the system the wanted normal form followsfrom standard results in unfolding theory for linear systems (see [1]). An explicit application to a modelsimilar to ours can be found in [26], ch. 6. However the usual formulations, being essentially based onthe use of implicit function theorem, fail to give an explicit estimate of the range of applicability of thewanted transformations, and in particular give a very bad behaviour with respect to the number N ofparticles. On the other hand we need such a quantitative control, and in particular we are interestedin finding estimates independent of the number N of particles. Thus we use a constructive algorithmin order to determine the normalizing transformation as a convergent sequence of close to identitytransformations of increasing order in µ, also producing an explicit estimate of the allowed rangefor µ. We emphasize that the technique used here can be extended to a system including nonlinearcorrections. This is work in progress. It turns out that the new variables are exponentially localizedaround the old ones, with a decay rate proportional to µ.
Assuming now that the quadratic part of the Hamiltonian has the wanted normal form, we add anonlinear perturbation, that we choose to be
H2 =β
4
N∑j=1
x4j . (2)
This is a particularly simple choice, which however possesses the interesting property that its reso-nant normal form at fourth order coincides with the standard discrete Nonlinear Schrodinger (dNLS)Hamiltonian. A nonlinear fourth order correction Z2 for the adiabatic invariant Z0 is easily found (seesect. 3), and this is the quantity that we study in the present paper.
The spontaneous attempt would be to prove the adiabatic invariance of the function Ψ = Z0 +Z2,i.e., to find an inequality such as |Ψ(t) − Ψ(0)| . ε for t . 1/ε. However, proving such an inequalitywith the methods of perturbation theory turns out to be a difficult task, because the usual estimatesexhibit a bad dependence on the number N of degrees of freedom. This difficulty is partially overcomeusing the complete resonance, due to the local oscillators being identical, and the extensive character ofthe system. Indeed we are able to remove the dependence on N in the relevant parameters (see sect. 2).
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However, in a dynamical approach we can not exclude states with most of the energy concentrated ona few oscillators.
The latter difficulty can be tackled using a statistical approach. A remarkable result in this directionhas been given recently in [35], where it is proved that at any fixed time t < T the set of initial datafor which |Ψ(t)− Ψ(0)| remains small has large measure. Although the time threshold T can be takenexponentially long with the specific energy ε, this result is definitely weaker than the adiabatic one,since it does not exclude large oscillations of Ψ(τ) within the interval [0, t]. Moreover, since the setdepends on the time t, it is possible that to different times (shorter than T ) correspond different setof good initial data.
Using dynamical considerations, which combine the Hamiltonian formalism with classical Tcheby-chev and Markov inequalities, we prove that for a set of large Gibbs measure in R2N also the fluctuationsare bounded over the whole time interval of order 1/ε; thus we improve the previous result, in that itis closer to the usual adiabatic statement (see section 3, Proposition 3). In order to show that the fluc-tuations are much smaller than what allowed (see Proposition 4), we have performed estimates whichexploit the main technical results obtained in [35]. The length of the time interval could be substan-tially improved using both the present probabilistic approach and the classical methods of perturbationtheory, which may give t ∼ 1/εr, for some r > 1. Work in this direction, namely the construction of anonlinear approximation of a first integral at an arbitrary order, is in progress and we plan to publishit in the next future. The numerical work presented in this paper provides motivation and support tothe claim above.
We perform a wide numerical investigation by calculating the time average and the mean squaredeviation of the approximate first integral Ψ for different values of the parameters ε (specific energy),µ (coupling parameter) and N (the number of oscillators). Then we compare the values so foundwith the phase average and the mean square deviation evaluated according to the Maxwell-Boltzmanndistribution at specific energy . In most cases we observe a time average quite different from the phaseaverage, and with a much smaller fluctuation than expected. This shows that our approximate firstintegral keeps an almost constant value for a time interval longer than expected from adiabatic theory,so that ergodicity and relaxation to equilibrium in a short time is excluded.
1.2 A note on the relations with previous works
Before coming to the technical sections it may be useful to recall the connections of our work with theprevious ones and furnish some motivations for our choice.
The traditional approach in numerical studies usually exploits the complete integrability of thequadratic part of the Hamiltonian. One is typically interested in investigating how the energy, initiallygiven only to the first normal mode or to a small number of modes, flows towards the other modes.This was indeed the first purpose declared in the FPU report. The relaxation time to equilibrium hasbeen investigated using a variety parameters, and different results have been found depending not juston the initial distribution of energy, but also on the distribution of the initial phases of the oscillators[8,10–12,14,21–23,25,27,31–33]. Investigations of the same character have been performed also for 2dimensional models [3,7,9]. Only in a few cases initial conditions with energy shared among all modeshave been considered, looking at parameters of statistical interest [15,16,34]. Two recent reports onthe scenario emerging from this kind of studies may be found in [4,30] where the possible existence ofmetastability phenomena is widely discussed.
Some attempts of theoretical explanation of the phenomena observed in the previous works havebeen made in the framework of classical perturbation theory. The traditional approach, going back toWhittaker, Cherry and Birkhoff [13,17,18,36], consists in looking for non linear first integrals that arecorrections of the harmonic actions, i.e., functions Φj = 1
2 (x2j + y2j ) + . . . such that Φj , H = 0, where·, · is the Poisson bracket. Such functions are expressed by formal power series that are expected tobe divergent, but significant results may be found using truncated approximations of the first integralsso found. This is exploited in Nekhoroshev’s theory on exponential stability. In rough terms, a typicalresult (see, e.g., [24]) is stated as follows: Let the (specific) energy ε of the system be small enough.Then we have
|Φj(t)− Φj(0)| < εb for |t| < T ∗e(ε∗/ε)a
with some positive constants T ∗, ε∗, a and b.
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The problem is hidden in the bad dependence of the constants T ∗, ε∗ and a on the number N ofdegrees of freedom. Removing such a dependence usually requires very strong non resonance conditionson the frequencies ωj and very special initial conditions. The first result in this direction may be foundin [5], where the long time persistence of states with localised energy is proved for a disordered lattice.However we should remark that, notwithstanding the considerable effort spent by many researcher,nobody has succeeded in removing the unwanted dependence on N in all the constants appearing in theestimates above. In view of this remark it is worth to explore a different strategy, thus abandoning theapproach through normal modes. In this respect we put the accent on the fact that in order to provethat a system is not ergodic it is enough to find just one first integral independent of the Hamiltonian.
As a general fact, in a completely non resonant system (the case investigated by Birkhoff) onehas something like a ∼ 1/N , which makes the statement meaningless for large systems. Such a baddependence has been shown to disappear in case the system is completely resonant, e.g., when ωj = Ωfor all j (identical oscillators). This has been done in [6] where it is shown that, thanks to the completeresonance, we get a = 1. However for T ∗ and ε∗ analytical estimates give a dependence on some powerof N . A remarkable improvement has been found in [2], but still one has to impose the conditionthat the total energy should be small, while the specific energy is the quantity of interest in StatisticalMechanics. In crude but essentially correct words the critical fact is that a dynamical approach seems tobe unable to distinguish the case of energy uniformly distributed among all the oscillators from the casewith all the energy concentrated on a single one. This makes the analytical estimates very pessimistic,but it is quite natural to observe that bad situations such as the concentration of all the energy in asingle mode are statistically unlike. Statistical considerations have been introduced in the remarkablerecent paper [35], that we have already mentioned, in which an extensive quantity that behaves as agood adiabatic invariant has been found, thus getting a result which applies in statistical sense, validfor small specific energy. The price to be paid is that most informations concerning the dynamics arelost. The present work is connected with that one, since we perform a dynamical investigation on aquantity which is essentially the same as in the quoted paper.
We focus our attention on two main aspects. On the one hand, we exploit the result already foundin [6], namely that resonance may create stable states that persist for an exponentially long time. Onthe other hand, we exploit the symmetry of the system in order to remove the critical dependence ofthe results on the number N of degrees of freedom. How this is made is described in detail in sect. 2.
We close this introduction with a short comment on the relation of the present paper with someresults and common knowledge of Statistical Mechanics. It is known indeed (see, e.g., [29]) that thequantities of interest, roughly speaking extensive ones, are essentially constant on the energy surface.In particular it can be shown that their distribution, with respect to the microcanonical measure,approaches a delta function in the thermodynamic limit. Therefore it might seem not surprising thatthe quantity we propose as an adiabatic invariant appears to be almost constant during the evolution.However, we point out two facts. The first one is that the time average of our quantity does not coincidewith the phase average, even though this could be a consequence of the finite number of particles (butnot extremely small: 512) used in our calculations. The second one is that the observed fluctuationsare much smaller than the expected ones according to phase calculations. The latter fact is certainlysurprising, and unexpected within the framework of Lanford’s paper.
The problem now is to understand whether the existence of an additional almost conserved quantitybesides the Hamiltonian has some impact on the Statistical Mechanics for the system. We think theanswer should be positive insofar the invariant measure should be changed, at least for the time scalein which the adiabatic invariant remains well conserved. In our opinion these aspects deserve furtherinvestigation.
The paper is organised as follows. In sect. 2 we construct the resonant normal form for the quadraticHamiltonian (1), including also the proof of convergence. In sect 3 we introduce the quartic nonlinearity,calculate the non linear correction of the first integral. Then we state all the related Propositions whichprovide an adiabatic control. In sect. 4 we present the results of the numerical calculations. An appendixwith some technical details and the proofs of the Propositions of sect. 3 follows.
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2 Resonant normal form for the quadratic Hamiltonian
We consider the linear chain (1) presented in the introduction; we rewrite the Hamiltonian here
H(x, y) = Ω
N∑j=1
y2j + x2j2
+ µxjxj+1
with periodic boundary conditions xj = xj+N and yj = yj+N . Here (x, y) ∈ R2N are the canonicalcoordinates and momenta, respectively, Ω is the linear frequency of the local oscillator and µ is acoupling parameter.
The system is well known to be integrable, for a linear transformation introduces the normal modes,with a frequency spectrum
Ω2k = 1− 4µ
Ω + 2µsin2
(kπ
N
). (3)
However, as we stressed in the introduction, we disregard this aspect, since we want to restrict ourattention to extensive quantities, and in particular we want to exploit the complete resonance amongthe local oscillators.
It will be useful to introduce complex coordinates (ξ, η) ∈ Cn via the canonical transformation
xj =1√2
(ξj + iηj) , yj =i√2
(ξj − iηj) , j = 1, . . . , N . (4)
A relevant general property is that if f(x, y) =∑
j,k cj,kxjyk (in multiindex notation) is a real poly-
nomial, then the transformation (4) produces a polynomial g(ξ, η) =∑
j,k bj,kξjηk with complex
coefficients bj,k satisfyingbj,k = −b∗k,j . (5)
Conversely, this is the condition that the coefficients of g(ξ, η) must satisfy in order to assure thattransforming it back to real variables x, y we get a real polynomial.
In complex coordinates the Hamiltonian reads
H(ξ, η) = hΩ(ξ, η) + f(ξ, η) (6)
where
hΩ(ξ, η) =
N∑j=1
iΩξjηj ,
f(ξ, η) =µ
2
N∑l=1
(iξlηl+1 + iξl+1ηl + ξlξl+1 − ηlηl+1) .
(7)
Our aim is to prove the following
Proposition 1 For |µ| ≤ Ω the Hamiltonian (6) may be given the normal form H(ξ, η) = hΩ(ξ, η) +ζ(ξ, η) satisfying the condition LhΩζ = 0 .
For the details see Lemma 3, where λ = µ/Ω is actually used.
Remark 1 In the statement we give an explicit estimate of the range for µ, which can be evaluatedthanks to the constructive character of the proof. As a side remark, we mention that having explicitlyperformed some steps in the construction of the normal form with our algorithm, we realized, withour surprise, that the norms (as defined in the proof below) resulting from the explicit constructioncoincide at every perturbation step with the norms estimated with our method. This suggests thatthe result might be optimal. The explicit construction has been performed using MACSYMA, a freeComputer Algebra System.
The property mentioned in the introduction that the new coordinates associated with the normalform are exponentially localized around the old ones turns out to be a byproduct of the proof (seethe end of Section 2.2). This shows that the coordinates of the normal form are closer to the originalcartesian coordinates than to the normal modes ones.
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Table 1 Properties of Poisson brackets
·, · N (2)0 N (2) R(2)
N (2)0 0 0 R(2)
N (2) 0 N (2) R(2)
R(2) R(2) R(2) N (2)
2.1 Algebraic framework
Let us give a better characterisation of the normal form in complex variables. A main role is playedby the Lie derivative along the canonical vector field generated by hΩ , namely the linear operatorLhΩ · = ·, hΩ acting on homogeneous polynomials in ξ, η. The relevant property is that LhΩ isdiagonal on the basis of the monomials in ξ, η . For a straightforward calculation gives
LhΩξjηk = iΩ (|j| − |k|) ξjηk , (8)
where |j| = |j1|+ . . .+ |jN | and similarly for |k| .Let us denote by P(s) the linear space of the homogeneous polynomials of degree s in the 2n
canonical variables ξ1, . . . , ξn, η1, . . . , ηn . In view of LhΩ being in diagonal form it is natural to introducethe subspaces of P(s)
N (s) = L−1hΩ (0) , R(s) = LhΩ (P(s))
which satisfyN (s) ∩R(s) = 0 , N (s) ⊕R(s) = P(s) .
It will also be interesting to consider the commutant N (2)0 of the kernel N (s) (also called the centralizer
in unfolding theory), namely the set
N (2)0 =
f ∈ N (s) : f, g = 0 ∀g ∈ N (2)
.
This set is particularly interesting because it contains the first integrals of the Hamiltonian. Thus itgives the first approximation for possible canditates as conserved quantities when non linear terms areadded.
Let us now restrict our attention to the space P(2) of quadratic polynomials. The composition table1 states the properties of the Poisson bracket.
Let us check these properties. The cases with zeros are true by definition of kernel and of commutantof the kernel. Recall the Jacobi’s identity for Poisson brackets, according to which we have LhΩf, g =LhΩf, g+ f, LhΩg for any pair of functions f, g . If f, g ∈ N (s) then LhΩf, g = 0 , which provesthat f, g ∈ N (s) . If f ∈ N (s) and g ∈ R(s) then we may write g = LhΩL
−1hΩg , and so we have
f, LhΩL−1hΩg = LhΩf, L−1hΩg in view of LhΩf = 0 , which proves that f, g ∈ R(s) . If f, g ∈ R(s)
then in general we have f, g ∈ P(s−2) , which just means that we know nothing about the result.However, in the case of quadratic polynomials we can prove f, g ∈ N (2). Let us explain this point insome detail.
Let us characterise the subspacesN (2) andR(2). Recalling (8) we immediately check that if |j| = |k|then we have ξjηk ∈ N (s), else if |j| 6= |k| then we have ξjηk ∈ R(s), whatever is the degree s of themonomial. But if we restrict the monomial to be quadratic then we have
f ∈ N (2)0 =⇒ f =
∑l
iclξlηl , cl ∈ R
f ∈ N (2) =⇒ f =∑l≤m
(al,mξlηm − a∗l,mξmηl) ,
f ∈ R(2) =⇒ f =∑l≤m
(bl,mξlξm − b∗l,mηlηm) ,
(9)
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where the condition (5) has been taken into account. With this in mind it is straightforward to checkthat if f, g ∈ R(2) then we get f, g ∈ N (2), which gives the lower right case of the diagram (1). Westress once again that this is true only for quadratic polynomials.
2.2 Formal normalisation algorithm
Let us rearrange the Hamiltonian (6) as
H(ξ, η) = hΩ(ξ, η) + ζ(ξ, η) + g(ξ, η) , ζ ∈ N (2) , g ∈ R(2) . (10)
From (7) we easily get, initially,
hΩ(ξ, η) =
N∑j=1
iΩξjηj ,
ζ(ξ, η) =iµ
2
N∑l=1
(ξlηl+1 + ξl+1ηl) ,
g(ξ, η) =µ
2
N∑l=1
(ξlξl+1 − ηlηl+1) .
(11)
We now proceed to removing the unwanted function g(ξ, η) by following the usual procedure basedon Lie series. We emphasise that only the properties ζ ∈ N (2), g ∈ R(2) are used here, so that ourargument does not depend on the particular form (11) of the initial Hamiltonian. We look for thegenerating function χ of a canonical transformation producing a new Hamiltonian
exp(Lχ)H = hΩ + ζ +(g + LχhΩ
)+∑s≥2
1
s!LsχhΩ +
∑s≥1
1
s!Lsχζ +
∑s≥1
1
s!Lsχg . (12)
The generating function χ is determined by solving the homological equation
LhΩχ = g , (13)
which in view of g ∈ R(2) admits a unique solution g ∈ R(2) (we set to zero the arbitrary kernel termin the solution). Selecting in the expression above for the Hamiltonian all terms which are independentof ζ we get ∑
s≥2
1
s!LsχhΩ+
∑s≥1
1
s!Lsχg = −
∑s≥2
1
s!Ls−1χ LhΩχ+
∑s≥2
1
(s− 1)!Ls−1χ g
=∑s≥2
s− 1
s!Ls−1χ g
=∑m≥1
(2m− 1)1
(2m)!L2m−1χ g +
∑m≥1
2m
(2m+ 1)!L2mχ g .
(14)
Here the first line is obtained by translating the index s by 1 in the second sum. Moreover in the lastline the table (1) has been used in order to collect all monomials belonging to the kernel N (2) in thefirst sum and all monomials belonging to the range R(2) in the second sum. With a similar calculationwe split the part depending on ζ as∑
s≥1
1
s!Lsχζ =
∑m≥1
1
(2m)!L2mχ ζ +
∑m≥1
1
(2m− 1)!L2m−1χ ζ (15)
were terms belonging to N (2) and R(2) are collected in the first and the second sum, respectively.
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We conclude that the transformed Hamiltonian may be written as
H = hΩ + ζ ′ + g′ , ζ ′ ∈ N (2) , g′ ∈ R(2) (16)
where
ζ ′ = ζ +∑m≥1
2m− 1
(2m)!L2m−1χ g +
∑m≥1
1
(2m)!L2mχ ζ ,
g′ =∑m≥1
1
(2m− 1)!L2m−1χ ζ +
∑m≥1
2m
(2m+ 1)!L2mχ g .
(17)
The transformed Hamiltonian is similar to (10), but is not yet in normal form because g′ does notvanish. However, we may hope that the size of g′ has decreased in view of the following heuristicargument. In view of (11) we see that both ζ and g contain a factor µ, which should be considered asa perturbation parameter. The same factor is inherited by χ, because the homological equation (13)is a linear one. Therefore every term in the expansion (17) of ζ ′ − ζ and g′ contains at least a factorµ2. Thus, although the Hamitonian is still quadratic, we recover the usual perturbation scheme basedon expansions in powers of a small parameter µ, and we can proceed by iterating our procedure. Theproblem is to prove the convergencence of the infinite sequence of transformations. This is what we aregoing to state in a rigorous way, also producing an explicit estimate for the convergence radius in µ.
It is useful to write the explicit form of the solution of the homological equation (13). To this endrecall that by (9) we may write, generically,
g(ξ, η) =∑l<m
(bl,mξlξm − b∗l,mηlηm) , (18)
where the property (5) is taken into account. Recalling that LhΩξjηk = −iΩ
(|j| − |k|
)ξjηk we imme-
diately get the explicit form of the generating function as
χ(ξ, η) =i
2Ω
∑l<m
(bl,mξlξm + b∗l,mηlηm) . (19)
Let us add a last remark. According to the theory of Lie series we can work out the processof constructing a normal form by transforming the Hamiltonian. However, the composition of theflows of the generating functions may also be used in order to give an explicit form to the canonicaltransformation, ξ′ = ϕ(ξ, η) , η′ = ψ(ξ, η) say, that gives the Hamiltonian the normal form. Here,the functions ϕ(ξ, η) and ψ(ξ, η) are given as power series. A straightforward analysis shows that themonomials involving both the j–th and the k–th coordinates have µ|j−k| as a coefficient. Thus thechange of coordinates shows an exponential decay with the distance along the chain.
2.3 Cyclic symmetry
Let us denote generically the canonical variables as (q1, . . . , qn, p1, . . . , pn) . The cyclic permutationoperator τ is defined as
τ(q1, . . . , qN ) = (q2, . . . , qN , q1) , τ(p1, . . . , pN ) = (p2, . . . , pN , p1) .
We say that a function f(q, p) is cyclically symmetric in case f(τ(q), τ(p)
)= f(q, p).
Cyclically symmetric functions may be constructed as follows. Let a function F (q, p) be given. Anew function F⊕(q, p) is constructed as
F⊕(q, p) =
N∑l=1
F (τ lq, τ lp) . (20)
The upper index ⊕ should be considered as an operator defining the new function. We shall saythat F⊕(q, p) is generated by the seed F (q, p). Generally speaking the decomposition of a cyclicallysymmetric function in the form (20) needs not be unique.
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It is an easy matter to check that cyclic symmetry is preserved by sum and multiplication bya constant; thus cyclically symmetric homogeneous polynomials are subspaces of the homogeneouspolynomials. Moreover, cyclic symmetry is preserved also by product and Poisson bracket between twofunctions, by the transformation from real to complex variables and vice versa, and by the solution (19)of the homological equation (13).
Furthermore, the Hamiltonian (1) is cyclically symmetric, being generated by the seed Ω2 (y21 +x21)+
µx1x2. In view of the properties stated above the whole normalisation process of sect. (2.2) is restrictedto these subspaces (since we add no arbitrary term to the generating function χ). This remark playsa central role in proving that the normalisation procedure converges for any number N of particles inthe chain.
2.4 Quantitative estimates
Let f(ξ, η) be a homogeneous polynomial of degree s in ξ, η. We define its polynomial norm as
‖f‖ =∑j,k
|fj,k| . (21)
For the cyclically symmetric function f⊕ generated by the seed f we shall use the norm∥∥f⊕∥∥⊕ = ‖f‖ . (22)
An obvious remark is that the norm so defined depends on the choice of the seed, but this will beharmless in the following. The relevant fact is that the inequality
‖f⊕‖ ≤ N∥∥f⊕∥∥⊕ (23)
holds true for any choice of the seed. For example, the three cyclic terms (11), have respectively∥∥hΩ∥∥⊕ = Ω ,∥∥ζ∥∥⊕ = µ =
∥∥g∥∥⊕ .We come now to investigating how the norms change during the process of normalisation of
sect. (2.2). Although the technical lemmas stated below can be formulated for homogeneous polynomi-als of any degree s, we shall restrict our attention to the class of quadratic homogeneous polynomialsf(ξ, η) =
∑|j|+|k|=2 cj,kξ
jηk in complex variables with coefficients satisfying the property (5), so that
under the change of variables (4) they become real polynomials in x, y. This will somehow simplify thediscussion, also producing better estimates.
Lemma 1 Let f(ξ, η) and g(ξ, η) be homogeneous quadratic polynomials satisfying (5). Then f, g isa homogeneous quadratic polynomial satisfying the same property, and one has
‖f, g‖ ≤ 2‖f‖ ‖g‖ . (24)
Moreover, there exists a seed of f⊕, g⊕ such that one has∥∥f⊕, g⊕∥∥⊕ ≤ 2∥∥f⊕∥∥⊕ ∥∥g⊕∥∥⊕ . (25)
The proof is just a sequence of tedious calculations that we omit.
Lemma 2 Let g⊕ be a cyclically symmetric quadratic polynomial as in (18). Then the solution of thehomological equation LhΩχ
⊕ = g⊕ given by (19) satisfies
‖χ⊕‖⊕ ≤ 1
2Ω‖g⊕‖⊕ (26)
This is obvious in view of the form (19) of χ⊕.
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2.5 Convergence of the normalisation procedure
The following lemma gives quantitative estimates for the normalisation step of sect. 2.2
Lemma 3 LetH0(ξ, η) = hΩ(ξ, η) + ζ(ξ, η) + g(ξ, η) (27)
be a cyclically symmetric function, and assume that
‖ζ‖⊕ ≤ λΩ , ‖g‖⊕ ≤ νΩ (28)
for some positive λ and ν. Then there exists a canonical transformation that gives the Hamiltonian theform
H ′0(ξ, η) = hΩ(ξ, η) + ζ ′(ξ, η) + g′(ξ, η) (29)
with‖ζ ′‖⊕ ≤ λ′Ω , ‖g′‖⊕ ≤ ν′Ω (30)
whereλ′ = λCh ν + ν Sh ν − (Ch ν − 1) ,
ν′ = λ Sh ν + ν(Ch ν − 1)− (Sh ν − ν) .(31)
Proof. The new functions ζ ′ and g′ are given by (17). Using lemmas 1 and 2 we get
‖ζ ′ − ζ‖⊕ ≤ νΩ∑m≥1
2m− 1
(2m)!ν2m−1 + λΩ
∑m≥1
1
(2m)!ν2m ,
‖g′‖⊕ ≤ λΩ∑m≥1
1
(2m− 1)!ν2m−1 + νΩ
∑m≥1
2m
(2m+ 1)!ν2m .
(32)
In view of the well known formulae∑m≥1
1
(2m− 1)!α2m−1 = Shα ,
∑m≥0
1
(2m)!α2m = Chα . (33)
we have ∑m≥1
2m− 1
(2m)!ν2m−1 =
∑m≥1
1
(2m− 1)!ν2m−1 − 1
ν
∑m≥1
1
(2m)!ν2m
= Sh ν − 1
ν(Ch ν − 1) ,∑
m≥1
2m
(2m+ 1)!ν2m =
∑m≥1
1
(2m)!ν2m − 1
ν
∑m≥1
1
(2m+ 1)!ν2m+1
= (Ch ν − 1)− 1
ν(Sh ν − ν) .
(34)
Replacing these expressions in (32) we get (31), which concludes the proof.We finally come to the proof of proposition 1. Recall that the normal form is obtained by iterating
infinitely many times the step of sect. 2.4, and we should prove that the unwanted term g′ vanishes inthe limit. The simplest way is to consider the recursive formulae (31) as a map of the first quadrantof the λ, ν plane into itself.
Our aim is to prove the following: the closed triangle with vertexes (0, 0), (0, 1), (1, 0) is invariant,and belongs to the basin of attraction of the segment 0 ≤ λ ≤ 1 , ν = 0 which is a set of attractivefixed points for the map. Thus, by iterating the map we have ν′ → 0 while λ′ tends to some positivevalue λ ≤ 1. This proves that proposition 1 holds true for λ + ν ≤ 1. Let us give a few details. Allpoints on the line ν = 0 are fixed points for the map and are attractive for 0 ≤ λ < 1. Remarking that
λ′ + ν′ − 1 = (λ+ ν − 1)(Ch ν + Sh ν) = (λ+ ν − 1)eν (35)
we find that the line λ+ ν = 1 is invariant. Restricting attention to the segment with extremes (1, 0) e(0, 1), which belongs to the first quadrant, we remark that λ′ > λ for 0 < ν < 1. Thus, all points of thissegment tend to the fixed point λ = 1, ν = 0 . Moreover, considering λ+ν = α < 1 and remarking thatin the first quadrant we have 0 ≤ ν ≤ α we conclude that for positive ν we have 0 < λ′ + ν′ < λ+ ν.This is what we wanted to prove.
11
3 A non linear model
We investigate now a nonlinear model as described by the Hamiltonian
H = hΩ + Z0 +H2. (36)
where
hΩ =Ω
2
N∑j=1
(x2j + y2j ) , Z0 =µ
2
N∑j=1
(xj+1xj + yj+1yj) , H2 =γ
4
N∑j=1
x4j , (37)
still keeping the periodic boundary conditions
xj = xj+N , yj = yj+N . (38)
The quadratic part1 is readily obtained by transforming back to real variables the HamiltonianH(ξ, η) = hΩ(ξ, η) + Z0(ξ, η) with Z0(ξ, η) given by (11). The nonlinear potential makes our model aslight variation of the so called quartic Klein-Gordon model, with a nonlinear quartic potential actingon every particle.
We observe that the non linear model still has the cyclic symmetry introduced in Paragraph 2.3.Indeed, setting
hΩ =Ω
2(x21 + y21) , ζ0 = x1x2 + y1y2 , h2 =
γ
4x41 (39)
we havehΩ = h⊕Ω , Z0 = ζ⊕0 , H2 = h⊕2 . (40)
3.1 A nonlinear approximate first integral
Following a classical procedure, we look now for approximate first integrals: the procedure requiresan initial term and we could use either hΩ or Z0, i.e. we could look for a function Φ = hΩ + Φ2 orΨ = Z0 +Ψ2. Actually, as it will be clear in a while, the construction of one of the two quantities givesautomatically also the other one.
Let us consider Φ; we would like it to be in involution with H, so we determine Φ2 in order tominimise the corresponding Poisson bracket:
Φ,H = hΩ , H2+ Φ2, hΩ+ ρ ,
with ρ = Φ2, Z0+Φ2, H2 small because the first term contains µ and the second is of higher order.We thus solve the equation
LhΩΦ2 = LhΩH2. (41)
This is easily done by splitting H2 = HN2 + HR2 over the subspaces N (4) and R(4); recalling thatLhΩH2 = LhΩH
R2 , we immediately write the solution of (41) as Φ2 = HR2 . We could add an arbitrary
term belonging to N , but this will not produce any actual advantage, so we keep the simplest solutionabove. It is not difficult to check that the corresponding procedure for Ψ would bring to Ψ2 = HN2 . Weremark that this term would also appear as Z2 in a resonant normal form2 H = hΩ +Z0 +Z2 + h.o.t.;thus we denote Z2 = HN2 .
Summarising, we have:
Φ = hΩ + Φ2 , Φ2 : = HR2 ; (42)
Ψ = Z0 + Z2 , Z2 : = HN2 ; (43)
H = Φ+ Ψ . (44)
1 If we introduce the complex variables ψj = xj + iyj , then this quadratic part turns to be the quadraticpart of the dNLS model
H(ψ) =Ω − 4µ
2
∑j
|ψj |2 −µ
4
∑j
|ψj+1 − ψj |2.
2 We recall that hΩ + Z0 + Z2 is a dNLS model.
12
In order to split H2 as required we use again the complex coordinates introduced by (4). Wetransform
H2 =γ
16
∑j
(ξ4j + 4iξ3j ηj − 6ξ2j η
2j − 4iξjη
3j + η4j
).
Splitting the function is now a straightforward matter, and by transforming back to real variables weget
Z2 =3γ
32
∑j
(x2j + y2j )2, (45)
and since Φ2 = H2 − Z2 we get
Φ2 =γ
32
∑j
(5x4j − 6x2jy2j − 3y4j ). (46)
We thus have two candidates as approximate conserved quantities. Given their relation (44) withthe Hamiltonian, it is clear that they are equivalent, and indeed one has Ψ,H = −Φ,H = −ρ.Actually it is enough to work out our estimates and to show numerical data for only one of the two.Our preference is for Ψ because its first order approximation contains only the contributions due tothe resonance. Thus we shall exploit the equation
Ψ,H = −ρ .
The remainder ρ is easily computed, and we split it into two parts, namely ρ = ρ40 + µρ21 with
ρ40 = −3γ
8
∑j
(x5jyj + x3jy
3j
), (47)
ρ21 =γ
32
∑j
(10x3j (σy)j − 6xjy
2j (σy)j + 6x2jyj(σx)j + 6y3j (σx)j
), (48)
where we have set (σw)j := wj+1 + wj−1.
3.2 Estimates of Ψ along the dynamics on the energy surface
We come now to producing some analytical estimates for the time evolution of our quantity Ψ . Weshow that Ψ behaves as a good extensive adiabatic invariant for time scales of the order 1/ε for a largeset of initial data on the phase space M := R2N endowed with the Gibbs measure. We control indeedits time fluctuations, checking the correct dependence on N .
We recall that the Gibbs measure mG of a measurable subset A of the phase space M, defined as
mG(A) :=1
Z(β)
∫A
e−βH(x,y)dxdy , Z(β) =
∫Me−βH(x,y)dxdy ,
is invariant under the Hamiltonian flow generated by H. Here β depends only on the energy density ε.Considering the flow φt(x, y) with (x, y) ∈ R2N we denote ∆tΨ(x, y) := Ψ
(φt(x, y)
)− Ψ((x, y)). In
view of Ψ = Ψ,H = −ρ we have
∆tΨ(x, y) = −∫ t
0
ρ φs(x, y)ds. (49)
For a given measurable function f the time average f with its standard deviation σt[f ] and the phaseaverage 〈f〉 with its deviation σG [f ] are defined as
〈f〉 :=1
Z(β)
∫R2N
f(x, y)e−βH(x,y)dxdy, σ2G [f ] := 〈f2〉 − 〈f〉2,
and
f(t, x, y) :=1
t
∫ t
0
f(φs(x, y))ds, σ2t [f ] := (f2)−
(f)2.
13
The results in this section are based on Tchebychev and Markov inequalities. We give here therelevant statements, deferring the technical details to the Appendix. The aim is to show that the setof the “bad initial data”. i.e., initial points for which Ψ is not well conserved, has a small measure.We emphasize that here we produce estimates in the spirit of adiabatic theory, namely up to timesof order O(1/ε). This because we consider only the first order correction for our adiabatic invariant.We expect to find definitely better results by pushing the pertubation analysis to higher orders, thusfinding estimates for times of order O(1/εr) for some r, or even for an exponentially long time, asin [35]. The numerical results of the next section do indeed suggest that the time may actually bemuch longer than expected from our present estimates.
We first report a result analogous to Corollary 1 in [35], the difference being that our times are atmost of order O(1/ε).
Proposition 2 (Control of ∆Ψ and ∆Ψ) Let the time t be fixed. For any positive δ one has
mG((x, y) ∈ R2N s.t. |∆tΨ(x, y)| > δ
)≤ C t
2
δ2〈ρ2〉 , (50)
and
mG((x, y) ∈ R2N s.t. |∆tΨ(x, y)| > δ
)≤ C t
2
δ2〈ρ2〉 . (51)
We add some comments. The first estimate (50) controls the variation of Ψ at time t with respectto its initial value. Indeed, it is possible to estimate 〈ρ2〉 . ε2σ2
G [Ψ ] (see Lemma 6), so that theestimate (50) may be rewritten as
mG(z ∈ R2N s.t. |∆tΨ(x, y)| > δσG [Ψ ]
)≤ C
δ2
(t
t
)2
, t =1
ε.
This means that for most initial data the value of Ψ at time t is close to the initial one; however thisgives no information on the behaviour of Ψ for all the intermediate times, since the set of bad initialdata may well depend on t. Thus, in principle, large fluctuations are not excluded, even though possiblyconcentrated in short time intervals. It should be noted that the dependence on the parameter δ iscoming directly from the Tchebychev inequality that lies at the basis of the result.
The second estimate (51) refines the first one in that it controls the variation of the time average Ψat time t with respect to its initial value. Thus it adds the further constraint that if large fluctuationsof Ψ do actually occur then they must compensate each other, because the time average is neverthelesspreserved.
The improvement with respect to Proposition 2 is contained in the next two propositions. Here wegive a control of the increase of standard deviation in time, comparing it with the phase deviation.Such a comparison is meaningful for the following reason. In view of Lanford’s arguments (see [29])it is expected that an extensive quantity may be somewhat constant in time since for N → ∞ it isasymptotically constant on the energy surface with respect to the canonical measure. However, weshow here that the fluctuation induced by the dynamics is definitely smaller than the fluctuation thatis allowed if the orbit actually explores the whole energy surface.
Proposition 3 (Control of time fluctuations of Ψ) For any positive δ and for positive t one has
mG(z ∈ R2N s.t. σ2
t [Ψ ] ≥ δσ2G [Ψ ]
)≤ Ct2〈ρ2〉
δσ2G [Ψ ]
,
with C = O(1).
The second proposition refines the result of the first one, since it adds an explicit control over time.
Proposition 4 (Adiabatic invariance of Ψ) If the energy density ε is small enough, then thereexists a threshold µ∗(ε) such that for µ < µ∗ one has
mG(z ∈ R2N s.t. σ2
t [Ψ ] ≥ δσ2G [Ψ ]
)≤ C
δ
(t
t
)2
, t =1
ε,
with C = O(1).
14
The second statement exploits the smallness of 〈ρ2〉
σ2G [Ψ ]
. ε2. Actually, under suitable hypotheses on
ε and µ, we are able to show that with large probability the time variation is smaller than the phasevariation, at least on a time scale t . 1/ε. This result is much stronger than Proposition 2, since wecontrol a positive function: if the deviation at time t is small, then the total amount of fluctuationsalong the orbit must be small, since no compensations are allowed and only the “averaging” 1/t mayhelp to decrease the final value. This result is thus stronger than a purely statistical one, and very closeto an adiabatic estimate. The actual difference is that it applies only for a large set of initial data,while in adiabatic theory one looks for results that apply to all of them. Indeed, when we combine thestatements of both Propositions 2 and 4 we can conclude that for a large set of initial data and on atime scale t . 1/ε the fluctuation |Ψ(t, z) − Ψ(0, z)| is bounded by a small constant for most of thetimes, i.e., large fluctuation may occur only for very short time intervals, negligible with respect to1/ε.
We remark that an optimal result would be given by another Tchebychev type estimate, but thisrequires the calculation of the phase standard deviation of the time standard deviation, while we areonly able to evaluate its phase average, using a Markov estimate.
4 Numerical results
In this section we present the numerical results. In a first part we will describe the initial settings ofour simulations and the statistical quantities we have actually computed, in order to properly readthe forthcoming figures. Then we will focus on three different aspects of the simulations: the lack ofergodicity (and its difference with equipartition of the energies of the particles), existence of an almostexactly conserved integral of motion for long time scales and its dependence on the main parameters(N, ε, µ); in all calculations we set Ω = γ = 1.
4.1 Classes of initial data and phase averages
In our numerical experiments, we mainly considered five different classes of initial data. One is rem-iniscent of many previous works in the field of nonlinear lattices, like the FPU chain [11,12], and ischaracterised by the energy initially given to a packet of consecutive N
16 particles, whose width is thusproportional to the total length N of the chain: energy is equidistributed within the packet, but withrandom phases (for the effect of coherent phases see e.g. [8]). The position of the packet in the chainis irrelevant due to the periodic boundary conditions. A second class is that with initial data in a setthat mimic the canonical ensemble, like in [16]. The single initial data were sampled out according tothe Maxwell–Boltzmann distribution at specific energy ε. This was actually implemented as follows.Each initial datum was extracted from a Gibbs ensemble (with the quadratic part only of the totalHamiltonian) at temperature ε, and was then rescaled to let it fit the constraint H = Nε. The thirdand fourth classes are those we call “pseudo first mode” and “pseudo last mode”. Indeed, like in truefirst and last normal mode (for a fixed boundary condition case), we have all the particles in phaseor in anti-phase respectively; at variance with real normal modes, here amplitudes are all equal. Thechoice of the latter two classes is justified since they roughly provide respectively a maximisation andminimisation of Z0, and this will be useful in our analysis. In a few cases we add a fifth class by givinginitial equidistribution of energy within all particles of the chain, with random phases (see Fig. 3).
Recalling that we are interested in the canonical ensemble, we calculate the phase average and thecorresponding standard deviation with the Gibbs measure for the relevant quantities, in particular forZ0 and Ψ . We do it by numerical evaluation via a Montecarlo method. Then we compare the phaseaverage and deviation with the time average and time standard deviation along the single orbits. Thisis what we do in all the panels of Fig. 1, 2, 3, 4 and 5. In most cases we report the results for bothfunctions Z0 and Ψ = Z0 + Z2 in order to exhibit the improvement due to the first order correctionfor the adiabatic invariant. The results are represented as follows.
All the straight lines are related to the phase statistics. The phase average is given by the blackone; the dotted straight lines represent the phase average plus/minus one standard deviation σ, andthe thinner dot lines give the phase average plus/minus three standard deviations. The area between
15
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Fig. 1 Lack of ergodicity (I). Temporal evolution of Ψ (upper panels) and Z0 (lower panels) comparedto phase averages, for different types of initial data. In all the panels N = 512, ε = 10−2 and µ = 10−3. Leftpanels: Initial data on a packet. Right panels: Initial data at random.
±σ is coloured in yellow3 (light grey in a black and white print) and outside ±σ up to ±3σ in lightyellow (even lighter gray).
The curves represent the time statistics along the temporal evolution of the orbit. The dotted-dashedblack curve is the time average. The area between the average plus/minus one standard deviation iscoloured in violet (gray). The dotted curve in light red represents a sampling of instantaneous valuesof the quantity under investigation. All the time statistics is calculated from time zero.
4.2 Lack of ergodicity
According to ergodic theory, in the limit of t→∞ the time averages are expected to converge to thecorresponding phase averages for almost every initial condition. The first qualitative conclusion we candraw from our numerical calculation is that our nonlinear model does not exhibit an ergodic behaviourup to quite long times, in our case 109. Let us illustrate these results.
In Fig. 1 and 2, we report the results for each of the four classes of initial condition considered,keeping the same specific energy and coupling parameter. The lower two panels of each figure representthe behaviour of Z0, which changes with time and at least in one case (left bottom panel of fig. 2) itappears to converge to the phase average. But if we look at Ψ (upper panels of each figure), it appearsto remain almost constant for all the different initial conditions, so that starting in different regionsin the phase space one has well separated evolutions without any indication of convergence to thecommon phase average, within the (quite long) time considered in the numerical simulation.
Remarkably, the lack of ergodicity appears to coexist with energy equipartition among the particles.In Fig. 3 we follow two evolutions from the first and the fifth classes of initial data, i.e., in the left panel
3 color in electronic version
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Fig. 2 Lack of ergodicity (II). Temporal evolution of Ψ (upper panels) and Z0 (lower panels) comparedto phase averages, for different types of initial data. In all the panels N = 512, ε = 10−2 and µ = 10−3. Leftpanels: Initial data on a pseudo first mode. Right panels: Initial data on a pseudo last mode.
energies are taken initially in equipartition through all the chain, while in the right one, energies areon the usual packet, in both cases with random phases. Looking also at the time evolutions of all theparticle’s energies (pictures not shown), one observes what follows. For initial data on a packet, by thetime in which the time average converges to phase average, the energy spreads along the whole chain,and then visually appears to have almost reached the equipartition among particles. Conversely, in theother case, the time average remains well apart from the phase average, although one has equipartitionfrom the very beginning and no visual difference is observed with respect to the previous case.
By the way, the left panel provides the only example we found in which the time average of Ψactually shows a convergence towards the phase-average.
4.3 Ψ is a good adiabatic invariant
We come now to briefly report about the dependence on the specific energy ε and on the couplingconstant µ, paying particular attention to the fluctuation. This will better illustrate the good propertiesof Ψ as an almost conserved quantity during the evolution of the system.
We select two values of the specific energy, namely ε = 10−3 in Figure 4, and ε = 1 in Figure 5;in both cases we consider three values of the coupling parameter, namely µ = 10−4, 10−2, and 1. Wecompare again the behaviour of Z0 and Ψ . For both quantities we see that the time average behavesas in the previous figures 1 and 2. However, let us pay attention to the fluctuation. The time varianceis expected to be comparable with the phase variance. As a matter of fact, for small specific energies(Fig. 4) we see that for both Z0 and Ψ the variance is definitely smaller than expected. Only at quitehigh values of the specific energy the time variance of Z0 becomes comparable with the phase varianceand in some cases also the averages are very close; concerning Ψ , its time variance gets large too, butits time average still lacks to converge to the phase one.
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Fig. 3 Lack of ergodicity (III). Temporal evolution of Ψ compared to phase averages for two differenttypes of initial data. In all the panels N = 512, ε = 1 and µ = 10−1. Left panel: Initial data on a packet. Rightpanel: Equipartition on the whole chain, random phases.
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Fig. 4 Ψ is a good adiabatic invariant (I). Temporal evolution of Ψ (upper panels) and Z0 (lower panels)compared to phase averages for different values of specific energy and coupling. In all the panels N = 512,ε = 10−3 and initial data at random (see subsection 4.1). Left column: µ = 10−4. Central column: µ = 10−2.Right column: µ = 5× 10−1. Row 1: Ψ . Row 2: Z0.
The different panels are shown for a fixed value of N (512) and for orbits in a fixed class of initialdata (canonical ensemble), but similar conclusions can be drawn using different values for N anddifferent choices of initial data.
4.4 Dependence on N
We come now to investigate the dependence onN . Since all the analytical estimates predict the averagesto grow with N and the deviations with
√N , as expected for an extensive quantity, both for phase and
time statistics, we plot in Fig. 6 the temporal evolution of Ψ/N (left panel) and of σt[Ψ ]/√N (right
panel).Actually, every series of data is an average over 10 orbits all extracted from the canonical ensemble.
This has been done to partially compensate the finite size effects producing slightly scattered datafor low values of N ; this is indeed visible in left panels for values of N < 1024. The computational
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Fig. 5 Ψ is a good adiabatic invariant (II). Temporal evolution of Ψ (upper panels) and Z0 (lower panels)compared to phase averages for different values of specific energy and coupling. In all the panels N = 512,ε = 1 and initial data at random (see subsection 4.1). Left column: µ = 10−4. Central column: µ = 10−2. Rightcolumn: µ = 5× 10−1. Row 1: Ψ . Row 2: Z0.
10-10
10-8
10-6
10-4
10-2
100
100
101
102
103
104
105
106
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
100
101
102
103
104
105
106
Fig. 6 Dependence on N . Temporal evolution of σt[Ψ ]/√N (left panel) and Ψ/N (right panel) for different
values of specific energy and number of particles N ; in all the orbits µ = 10−4 and initial data sampled outfrom the canonical ensemble. N ranges from 32 to 8192 (thin lines for values less than 1024, and thick symbolsfrom 1024 to 8192); ε assumes the values 1, 10−1, 10−2 and 10−3 from top to bottom.
price of the multiple integration for every choice of parameters had to be compensated decreasing theintegration time to 106.
The plot of Ψ/N shows perfectly the linear dependence on N , with a very short transient (hardlyvisible in the picture) of the order of 10 time units. The four different bunches of (perfectly superim-posed) data series also show the regular dependence on ε, roughly like ε2, compatible with the analoguephase-average dependence on ε (indeed, the main contribute is given by Z2, whose phase average isapproximately Nε2, as follows from the results in the Appendix).
The plot of σt[Ψ ]/√N shows very well the asymptotically dependence on
√N for the deviation.
In this case the transient is longer, in particular for low specific energies. This is expected since, atleast for some particular parameter regimes, it can be shown analytically and numerically checked thatσt[Ψ ] has a component which depends linearly on N ; such terms vanish with time.
19
The most useful conclusion we could draw from these data is the following. Since both time andphase averages and deviations behave in the same way on N , the property shown in subsection 4.3 andin particular in Fig. 4 for a fixed value of N , i.e. that time fluctuations for Ψ are much smaller thanthe corresponding phase fluctuations, persists in the thermodynamic limit. The last claim is actuallythe numerical analogue of Propositions 3 and 4.
5 Appendix
5.1 Bounds for σ2G [Ψ ] and 〈ρ2〉
We provide here some details about the lower bound for σ2G [Ψ ] and the upper bound for 〈ρ2〉 (where ρ
is the remainder defined in Subsection 3.1), with respect to the Gibbs measure in R2N . Our theoreticalapproximations are in good agreement with the numerical computation of the same quantities via aMontecarlo method, still based on a Gibbs measure (as explained in Section 4).
We give here two technical Lemmas which will be useful in the forthcoming computations. Thestatements are given in a rather simple form, which is enough for our purposes in this paper. Theproofs may be found in [35]. Thus we omit the details of the proof, but add some comments.
The first lemma states that the average of a measurable function depending on a fixed (with respectto N) subset of indexes is independent on N . The statement refers to the simplest case of monomials.
Lemma 4 Consider the monomials fh := xshh and gk := xskk with h 6= k. If µ is small enough, thenthere exists chk(β, µ) independent of N such that
|〈fhgk〉| ≤ chk(β, µ), ∀N. (52)
In view of the cyclic symmetry of the Hamiltonian, the statement may be understood as a naturalconsequence of Lanford’s results about the average of extensive functions [29]. Indeed, by defining
P (x) := (fg)⊕,
and by the translational invariance, one has
N
∫R2N
fhgkdmG =
∫R2N
PdmG ∼ CN.
Moreover, the claim is obvious if one approximates the Hamiltonian with its uncoupled part by settingµ = 0, due to the factorization of the Gibbs measure. Indeed if H(x, y) = h⊕(x1), then one has
e−βH(x,y) =
N∏j=1
e−βh(xj) .
Taking into account the coupling terms of the Hamiltonian requires some extra consideration, (see [35]sect. 5, lemma 4).
The second lemma states that the correlation between observables depending of two different sitesdecreases geometrically with the distance between the sites.
Lemma 5 Consider the monomials fh := xshh and gk := xskk with h < k. If µ is small enough, then
|〈fhgk〉 − 〈fh〉〈gk〉| ≤ µk−hchk(β) (53)
with a constant chk(β) independent of N .
The claim follows from the remarkable result proved in [35], Section 6, where some ideas of [19]have been used. We emphasize that Lemma 5 allows us to approximate the average of fhgk with theproduct of the two different averages
〈fhgk〉 ≈ 〈fh〉〈gk〉+O(µk−h
). (54)
Moreover if one of the two functions has zero average then (53) becomes
|〈fhgk〉| ≤ µk−hchk(β). (55)
20
5.1.1 Upper bound for 〈ρ2〉
We recall that ρ = ρ40 + µρ21 as given by (47) and (48), and use the trivial inequality
ρ2 = ρ240 + µ2ρ221 + 2µρ40ρ21 ≤ 2(ρ240 + µ2ρ221
).
Let us focus first on ρ240. Writing
ρ240(x, y) =∑j
f2j + 2∑h<k
fhfk, fj(x, y) = x5jyj + x3jy3j
and using lemma 4 we have 〈f21 〉 ≤ c1(β, µ) with a constant independendent of N . Thus we have〈∑
j f2j 〉 = Nc1(β, µ).
The second sum involves N2 terms; however, by using Lemma 5 (see formula (55) and observe that〈fj〉 = 0), we get
|〈fhfk〉| ≤ cm(β)µk−h , m = |h− k|where cm(β) depends only on the distance m between the indices, in view of the cyclic symmetry.Hence, exploiting
∑m≥1 µ
m = O(µ) for small µ, we get
2∑h<k
|〈fhfk〉| ≤ µc2(β)N.
We conclude〈ρ240〉 ≤ C40(β, µ)N.
In a similar way we can estimate 〈ρ221〉, thus getting
〈ρ2〉 ≤ NCρ(β, µ). (56)
5.1.2 Lower bound for σG [Ψ ]
We pay particular attention to the dependence of the deviation σG [Ψ ] on N . Recalling (39) and settingζ2 = 3γ
32 (x21 + y21)2 we first write
Ψ = ψ⊕, ψ = ζ0 + ζ2.
The square deviation reads
σ2G [Ψ ] = 〈Ψ2〉 − 〈Ψ〉2 =
∑j
σ2G [ψj ] + 2
∑h<k
(〈ψhψk〉 − 〈ψh〉〈ψk〉) ,
where we have denoted ψj = ψ(τ j−1x, τ j−1y). The first sum is clearly proportional to N , namely∑j
σ2G [ψj ] = Nc1(β, µ)
with a constant c1(β, µ) independent of N . For the second sum we apply Lemma 5, and conclude
2∑h<k
|〈ψhψk〉 − 〈ψh〉〈ψk〉| ≤ µc2(β, µ)N
with a constant c2(β, µ). Thus the lower bound for the square deviation is proportional to N and isgiven by
σ2G [Ψ ] ≥
∑j
σ2G [ψj ]− 2
∑h<k
|〈ψhψk〉 − 〈ψh〉〈ψk〉| ≥ NCσ(β, µ) , (57)
with Cσ(β, µ) = c1(β, µ) + µc2(β, µ). In particular, for µ small enough, one has
σ2G [Ψ ] ≈
∑j
σ2G [ψj ] . (58)
21
5.2 Adiabatic estimates of Ψ(t, z)
We come now to the proofs of Propositions 2, 3 and 4.
5.2.1 Proof of Proposition 2
The proof consist in an estimate of the deviation
σ2G [∆tΨ ] = 〈(∆tΨ)2〉 − 〈∆tΨ〉2.
Due to the invariance of the measure under the Hamiltonian flow Φt(·), we first notice that 〈Ψ(t)〉 = 〈Ψ〉,so that we also have4 〈∆tΨ〉 = 0. We should evaluate 〈(∆tΨ)2(t)〉. The main idea is to swap integrationin time and space. We first use Fubini’s theorem∫
M
(∆sΨ(x, y)
)2dx dy =
∫Mdx dy
∫[0,s]×[0,s]
ρ(φτ1(x, y)
)ρ(φτ2(x, y)
)dτ1dτ2,
and then, after swapping, we apply the usual inequality∫M
∣∣ρ(φτ1(x, y))ρ(φτ2(x, y)
)∣∣dx dy ≤ ‖ρ(φτ1(x, y))‖L2M‖ρ(φτ2(x, y)
)‖L2M
= ‖ρ‖2L2M,
which gives the final estimate
〈(∆tΨ)2〉 ≤ 1
4t2〈ρ2〉. (59)
Concerning the time average, we denote
∆tΨ(x, y) :=1
t
∫ t
0
Ψ(φs(x, y)
)ds− Ψ(x, y)
and we look again for an upper bound for
σ2G [∆tΨ ] = σ2[∆tΨ ] = 〈(∆tΨ)2〉 ,
since 〈∆Ψ(t)〉 = 〈∆Ψ(t)〉 = 0. Once more, the idea is to swap integrations in time and space twice;first with the time average
1
Z(β)
∫M
(∆tΨ(x, y)
)2dx dy =
1
t2Z(β)
∫[0,t]2
ds1ds2
∫M∆s1Ψ(x, y)∆s2Ψ(x, y)dx dy ,
and then with the inner integration∫M∆s1Ψ(x, y)∆s2Ψ(x, y)dx dy =
∫[0,s1]×[0,s2]
dτ1dτ2
∫Mρ(φτ1(x, y)
)ρ(φτ2(x, y)
)dxdy;
hence, following the steps of the previous proof, one gets
〈(∆tΨ)2〉 ≤ 1
4t2〈ρ2〉.
4 This implies immediately that
0 =
∫MΨ Φsdx dy −
∫MΨdx dy =
∫M
∫ s
0
ρ Φτ (x, y)dτdx dy =
∫ s
0
dτ
∫Mρ(x, y)dx dy = s〈ρ〉,
which means that 〈ρ〉 = 0.
22
5.2.2 Proof of Proposition 3
By definition, one can rewrite
σ2t [Ψ ] = (∆tΨ)2(x, y)−∆tΨ
2(x, y) ≤ (∆tΨ)2(x, y) .
Using also (59) this implies
〈σ2t [Ψ ]〉 ≤ 〈∆tΨ2(x, y)〉 =
1
Z(β)
∫M
1
t
∫ t
0
(∆sΨ(x, y)
)2ds dx dy =
1
t
∫ t
0
〈(∆sΨ)2〉ds ≤ 1
12t2〈ρ2〉 ,
hence
〈σ2t [Ψ ]〉 ≤ 1
12t2〈ρ2〉 . (60)
For the positive random variable σ2t [Ψ ](z, s), Markov inequality claims that
mG((x, y) s.t. σ2
t [Ψ ] ≥ δσ2G [Ψ ]
)≤ 〈σ
2t [Ψ ]〉
δσ2G [Ψ ]
≤ Ct2
δ
〈ρ2〉σ2G [Ψ ]
.
5.2.3 Proof of Proposition 4
The proof of Proposition 4 is based on a quantitative estimate of the ratio 〈ρ2〉/σ2G [Ψ ], which provides
its smallness with respect to the energy density. This is obtained by first exploiting the previous upperand lower bounds (56) and (57) in order to remove the dependence on N
〈ρ2〉σ2G [Ψ ]
≤ CρCσ
(β, µ),
and then approximating the dependence on β of Cρ/Cσ in the limit of large values of β. The rigorousjustification of this approximation (in a suitable range of validity of the parameters µ and β) can befound once more in [35], Sect. 5. We rephrase it in the following
Lemma 6 For any ε small enough, there exists µ∗(ε) such that, for µ < µ∗ one has
〈ρ2〉σ2G [Ψ ]
. ε2. (61)
From an intuitive level instead, we can somehow expect that, at fixed N and β, the Gibbs measureis continuos with respect to the small coupling parameter µ. If so, as µ→ 0, the measure (still at fixedN and β) has to converge to the Gibbs measure
mun,G(A) :=1
Zun(β)
∫A
e−β(hΩ+H2)dx dy, Zun(β) =
∫R2N
e−β(hΩ+H2)dx dy ,
which can be factorized in the 2N directions. However we disregard the general and delicate problemof continuity of the measure, since our interest is strictly confined to averages of monomials dependingonly on few and fixed sites. In this case, for µ small enough it is possible to approximate the averagesuniformly with N with the measure mun,G , as proved in [35] (more precisely, one can provide both anupper and a lower bound of the average independent on N and then take µ small).
Thus, to show (61), we focus on the scaling with β of the averages of, for example, xs11 xshh with
s1 + sh = 2r (since they compose the remainder ρ and the adiabatic invariant Ψ) and taking mun,G asmeasure. In this special case, it reduces to
〈xs11 xshh 〉 =
∫R e−β(hΩ(x1)+h2(x1))xs11 dx1
∫R e−β(hΩ(xh)+h2(xh))xshh dxh∫
R e−β(hΩ(x1)+h2(x1))dx1
∫R e−β(hΩ(xh)+h2(xh))dxh
,
since all the others factors have been simplified. So we focus on the dependence on β of
Fs(β) :=
∫R e−β(x2+x4)xsdx∫
R e−β(x2+x4)dx
,
23
for large β. In the limit β → ∞ (namely ε → 0) the quadratic part is actually more relevant, soasymptotically we have Fs(β) ∼ 1
βs2
. The above argument gives the scaling
〈xs11 xshh 〉 ∼
1
βr≈ εr, β 1 (ε 1),
hence in the factorized Gibbs measure (uncoupled nonlinear oscillators) the bigger is the degree of themonomial, the smaller (in ε) is the average and, moreover, for small ε the Gibbs measure is essentiallygiven by the harmonic oscillators.
If we restore the small coupling µ, it is not possible to exploit any factorization of the measure;nevertheless we have already stressed that, for any large enough β, there exists µ∗(β), such that forµ < µ∗ we get
Cρ . c40〈(x21 + y21)6〉+ c21µ2〈(x41 + y41)
(y21(σx)21 + x21(σy)21
)〉 ∼ ε4
(ε2 + µ2
),
andCσ & σ2
G [ψ1] ∼ ε2(ε2 + µ2
),
which finally yields〈ρ2〉σ2G [Ψ ]
. ε2.
Acknowledgements
We thank D. Bambusi, G. Benettin, A. Carati, H. Christodoulidi, L. Galgani, A. Maiocchi and A. Ponnofor useful discussion. We also thank D.Morale for some comments on the probabilistic part. S.P. andT.P. are partially supported by Indam–GNFM grant (Progetto giovani ricercatori) “Metodi perturba-tivi e multiscala in reticoli nonlineari”.
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