Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit

We consider the Fermi-Pasta-Ulam-Tsingou (FPUT) chain composed by $N \gg 1$ particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature 1/beta. Given a fixed $1<m \ll N$ , we prove that the first $m$ integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $\beta^{1-2\varepsilon}$, $\forall \varepsilon>0$, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.


Introduction and main results
The FPUT chain with N particles is the system with Hamiltonian which we consider with periodic boundary conditions q N " q 0 , p N " p 0 and b ą 0. We observe that any generic nearest neighborhood quartic potential can be set in the form of V F pxq through a canonical change of coordinates. Over the last 60 years the FPUT system has been the object of intense numerical and analytical research. Nowadays it is well understood that the system displays, on a relatively short time scale, an integrable-like behavior, first uncovered by Fermi, Pasta, Ulam and Tsingou [13] and later interpreted in terms of closeness to a nonlinear integrable system by some authors, e.g. the Korteweg-de Vries (KdV) equation by Zabusky and Kruskal [40], the Boussinesq equation by Zakharov [41], and the Toda chain by Manakov first [30], and then by Ferguson, Flaschka and McLauglin [12]. On larger time scales the system displays instead an ergodic behavior and approaches its micro-canonical equilibrium state (i.e. measure), unless the energy is so low to enter a KAM-like regime [24,25,35].
In the present work we give a quantitative result of the integrable behavior of the FPUT system that hold in the thermodynamic limit. Namely we show that a family of first integrals of the Toda system are adiabatic invariants (namely almost constant quantities) for the FPUT system. We bound their variation for times of order β 1´2ε , ε ą 0, where β is the inverse of the temperature of the chain. Such estimates hold for a large set of initial data with respect to the Gibbs measure of the chain and they are uniform in the number of particles, thus they persist in the thermodynamic limit. In this way we show that the FPUT chain has, in measure, an integrable-like behaviour on time scales of order β 1´2ε , thus we give an insight of the so-called FPUT paradox.
In the last few years, there has been a lot of activity in the problem of constructing adiabatic invariants of nonlinear chain systems in the thermodynamic limit, see [8,9,17,18,28,29]. In particular adiabatic invariants in measure for the FPUT chain have been recently introduced by Maiocchi, Bambusi, Carati [28] by considering the FPUT chain a perturbation of the linear harmonic chain. Our approach is based on the remark [12,30] that the FPUT chain (1.1) can be regarded as a perturbation of the (nonlinear) Toda chain [38] H T pp, qq :" which we consider again with periodic boundary conditions q N " q 0 , p N " p 0 . The equations of motion of (1.1) and (1.2) take the form 9 q j " BH Bp j " p j , 9 p j "´B H Bq j " V 1 pq j`1´qj q´V 1 pq j´qj´1 q, j " 0, . . . , N´1, (1.3) where H stands for H F or H T and V for V F and V T respectively. According to the values of b in (1.1), the Toda chain is either an approximation of the FPUT chain of third order (for b ‰ 1), or fourth order (for b " 1). We remark that the Toda chain is the only nonlinear integrable FPUT-like chain [11,36]. The Toda chain admits several families of N integrals of motion in involution (e.g. [15,23,39]). Among the various families of integrals of motion, the ones constructed by Henon [21] and Flaschka [14] are explicit and easy to compute, being the trace of the powers of the Lax matrix associated to the Toda chain. In the following we refer to them simply as Toda integrals and denote them by J pkq , 1 ď k ď N (see (2.12)).
As the J pkq 's are conserved along the Toda flow, and the FPUT chain is a perturbation of the Toda one, the Toda integrals are good candidates to be adiabatic invariants when computed along the FPUT flow. This intuition is supported by several numerical simulations, the first by Ferguson-Flaschka-McLaughlin [12] and more recently by other authors [4,6,10,19,34]. Such simulations show that the variation of the Toda integrals along the FPUT flow is very small on long times for initial data of small specific energy. In particular, the numerical results in [4,6,19] suggest that such phenomenon should persists in the thermodynamic limit and for "generic" initial conditions.
Our first result is a quantitative, analytical proof of this phenomenon. More precisely, we fix an arbitrary m P N and provided N and β sufficiently large, we bound the variations of the first m Toda integrals computed along the flow of FPUT, for times of order β 1´2ε ppb´1q 2`C 1 β´1q 1 2 , (1.4) where ε ą 0 is arbitrary small and C 1 is a positive constant, independent of β, N . Such a bound holds for initial data in a set of large Gibbs measure. Note that the bound (1.4) improves to β 3 2´2 ε when b " 1, namely when the Toda chain becomes a fourth order approximation of the FPUT chain. Such analytical time-scales are compatible with (namely smaller than) the numerical ones determined in [4][5][6].
An interesting question is whether the Toda integrals J pkq 's control the normal modes of FPUT, namely the action of the linearized chain. It turns out that this is indeed the case: we prove that the quadratic parts J p2kq 2 (namely the Taylor polynomials of order 2) of the integral of motions J p2kq , are linear combinations of the normal modes. Namely one has where E j is the j th normal mode (see (2.19) for its formula), pp p, p qq are the discrete Hartley transform of pp, qq (see definition below in (2.16)) and p c pkq are real coefficients.
So we consider linear combinations of the normal modes of the form where pp g j q j is the discrete Hartley transform of a vector g P R N which has only 2t m 2 u`2 nonzero entries with m independent from N . Our second result shows that linear combinations of the form (1.6), when computed along the FPUT flow, are adiabatic invariants for the same time scale as in (1.4). Actually, exploiting the fact that the Toda integrals are invariant for the Toda dynamics, we deduce also that the linear combinations in (1.6), when computed along the flow of Toda chain, are adiabatic invariants for all times. This is our third result. Examples of linear combinations (1.6) that we control are . (1.7) These linear combinations weight in different ways low and high energy modes.
Our results are mainly based on two ingredients. The first one is a detailed study of the algebraic properties of the Toda integrals. The second ingredient comes from adapting to our case, methods of statistical mechanics developed by Carati [8] and Carati-Maiocchi [9], and also in [17,18,28,29].

Toda integrals as adiabatic invariants for FPUT
We come to a precise statements of the main results of the present paper. We consider the FPUT chain (1.1) and the Toda chain (1.2) in the subspace which is invariant for the dynamics. Here L is a positive constant.
Since both H F and H T depend just on the relative distance between q j`1 and q j , it is natural to introduce on M the variables r j 's as due to the periodic boundary condition q N " q 0 . We observe that the change of coordinates (2.2) together with the condition (2.3) is well defined on the phase space M, but not on the whole phase space R NˆRN . In these variables the phase space M reads We endow M by the Gibbs measure of H F at temperature β´1, namely we put where as usual Z F pβq is the partition function which normalize the measure, namely Given a function f : M Ñ C, we will use the probability (2.5) to compute its average xf y, its L 2 norm }f }, its variance σ 2 f defined as In order to state our first theorem we must introduce the Toda integrals of motion. It is well known that the Toda chain is an integrable system [21,38]. The standard way to prove its integrability is to put it in a Lax-pair form. The Lax form was introduced by Flaschka in [14] and Manakov [30] and it is obtained through the change of coordinates b j :"´p j , a j :" e 1 2 pqj´qj`1q " e´1 2 rj , 0 ď j ď N´1 . (2.10) By the geometric constraint (2.3) and the momentum conservation ř N´1 j"0 p j " 0 (see (2.1)), such variables are constrained by the conditions The Lax operator for the Toda chain is the periodic Jacobi matrix [39] Lpb, aq :"¨b (2.11) We introduce the matrix A " L`´L´where for a square matrix X we call X`the upper triangular part of X and in a similar way by X´the lower triangular part of X otherwise.
A straightforward calculation shows that the Toda equations of motions (1.3) are equivalent to dL dt " rA, Ls.
It then follows that the eigenvalues of L are integrals of motion in involutions.
In particular, the trace of powers of L, 2.12) are N independent, commuting, integrals of motions in involution. Such integrals were first introduced by Henon [21] (with a different method), and we refer to them as Toda integrals. We give the first few of them explicitly, written in the variables pp, rq: (2.13) Note that J p2q coincides with the Toda Hamiltonian H T .
Our first result shows that the Toda integral J pmq , computed along the Hamiltonian flow φ t HF of the FPUT chain, is an adiabatic invariant for long times and for a set of initial data in a set of large Gibbs measure. Here the precise statement: for every time t fulfilling |t| ď β 1´2έ In (2.14) P stands for the probability with respect to the Gibbs measure (2.5).
We observe that the time scale (2.15) increases to β 3 2´ε for b " 1, namely if the Toda chain is a fifth order approximation of the FPUT chain.
Remark 2.2. We observe that our estimates in (2.14) and (2.15) are independent from the number of particles N . Therefore we can claim that the result of theorem 2.1 holds true in the thermodynamic limit, i.e. when lim N Ñ8 xHF y N " e ą 0 where xH F y is the average over the Gibbs measure (2.5) of the FPUT Hamiltonian H F . The same observation applies to theorem 2.4 and theorem 2.5 below.
Our Theorem 2.1 gives a quantitative, analytical proof of the adiabatic invariance of the Toda integrals, at least for a set of initial data of large measure. It is an interesting question whether other integrals of motion of the Toda chain are adiabatic invariants for the FPUT chain. Natural candidates are the actions and spectral gaps.
Action-angle coordinates and the related Birkhoff coordinates (a cartesian version of actionangle variables) were constructed analytically by Henrici and Kappeler [22,23] for any finite N , and by Bambusi and one of the author [1] uniformly in N , but in a regime of specific energy going to 0 when N goes to infinity (thus not the thermodynamic limit). The difficulty in dealing with these other sets of integrals is that they are not explicit in the physical variables pp, rq. As a consequence, it appears very difficult to compute their averages with respect to the Gibbs measure of the system. Despite these analytical challenges, recent numerical simulations by Goldfriend and Kurchan [19] suggest that the spectral gaps of the Toda chain are adiabatic invariants for the FPUT chain for long times also in the thermodynamic limit.

Packets of normal modes
Our second result concerns adiabatic invariance of some special linear combination of normal modes. To state the result, we first introduce the normal modes through the discrete Hartley transform. Such transformation, which we denote by H, is defined as and one easily verifies that it fulfills The Hartley transform is closely related to the classical Fourier transform F , whose matrix elements are F j,k :" 1 ? N e´i 2πjk{N , as one has H " ℜF´ℑF . The advantage of the Hartley transform is that it maps real variables into real variables, a fact which will be useful when calculating averages of quadratic Hamiltonians (see Section 5.2).
A consequence of (2.16) is that the change of coordinates is a canonical one. Due to ř j p j " 0, ř j q j " L, one has also p p 0 " 0, p q 0 " L{ ? N . In these variables the quadratic part H 2 of the Toda Hamiltonian (1.1), i.e. its Taylor expansion of order two nearby the origin, takes the form We observe that (2.18) is exactly the Hamiltonian of the Harmonic Oscillator chain. We define the j th normal mode.
To state our second result we need the following definition: is the integer part of m 2 . For any N ą m, a vector x P R N is said to be m-admissible if there exits a non zero vector y " py 0 , y 1 , . . . , y Ă m q P R Ă m`1 with K´1 ď ř j |y j | ď K, K independent from N , such that x k " x N´k " y k , for 0 ď k ď r m and x k " 0 otherwise.
We are ready to state our second result, which shows that special linear combinations of normal modes are adiabatic invariants for the FPUT dynamics for long times. Here the precise statement: Theorem 2.4. Fix m P N and let g " pg 0 , . . . , g N´1 q P R N be a m-admissible vector (according to Definition 2.3). Define
Again when b " 1 the time scale improves by a factor β 1 2 . Finally we consider the Toda dynamics generated by the Hamiltonian H T in (1.2). In this case we endow M in (2.4) by the Gibbs measure of H T at temperature β´1, namely we put where as usual Z T pβq is the partition function which normalize the measure, namely We prove that the quantity (2.20), computed along the Hamiltonian flow φ t HT of the Toda chain, is an adiabatic invariant for all times and for a large set of initial data: Theorem 2.5. Fix m P N; let g P R N be an m-admissible vector and define Φ as in (2.20). Then there exist N 0 , β 0 , C ą 0 such that for any N ą N 0 , β ą β 0 , 0 ă ε ă 1 2 , one has for all times.
Remark 2.6. It is easy to verify that the functions Φ in (2.20) are linear combinations of (2.25) (choose g ℓ " g N´ℓ " 1, g j " 0 otherwise). Then, using the multi-angle trigonometric formula cosp2nxq " p´1q n T 2n psin xq, cosp2nxq " T 2n pcos xq, where the T n 's are the Chebyshev polynomial of the first kind, it follows that we can control (1.7).
Let us comment about the significance of Theorem 2.4 and Theorem 2.5. The study of the dynamics of the normal modes of FPUT goes back to the pioneering numerical simulations of Fermi, Pasta, Ulam and Tsingou [13]. They observed that, corresponding to initial data with only the first normal mode excited, namely initial data with E 1 ‰ 0 and E j " 0 @j ‰ 1, the dynamics of the normal modes develops a recurrent behavior, whereas their time averages 1 t ş t 0 E j˝φ τ HF dτ quickly relaxed to a sequence exponentially localized in j. This is what is known under the name of FPUT packet of modes.
Subsequent numerical simulations have investigated the persistence of the phenomenon for large N and in different regimes of specific energies [4,6,7,16,26,31] (see also [2] for a survey of results about the FPUT dynamics).
Analytical results controlling packets of normal modes along the FPUT system are proven in [1,3]. All these results deal with specific energies going to zero as the number of particles go to infinity, thus they do not hold in the thermodynamic limit. Our result controls linear combination of normal modes and holds in the thermodynamic limit.

Ideas of the proof
The starting point of our analysis is to estimate the probability that the time evolution of an observable Φptq, computed along the Hamiltonian flow of H, slightly deviates from its initial value. In our application Φ is either the Toda integral of motion or a special linear combination of the harmonic energies and H is either the FPUT or Toda Hamiltonian. Quantitatively, Chebyshev inequality gives , @λ ą 0. (2.26) So our first task is to give an upper bound on the variance σ Φptq´Φp0q and a lower bound on the variance σ Φp0q . Regarding the former bound we exploit the Carati-Maiocchi inequality [9] σ 2 Φptq´Φp0q ď @ tΦ, Hu 2 D t 2 , @t P R, (2.27) where tΦ, Hu, denotes the canonical Poisson bracket Next we fix m P N, consider the m-th Toda integral J pmq , and prove that the quotient scales appropriately in β (as β Ñ 8) and it is bounded uniformly in N (provided N is large enough). It is quite delicate to prove that the quotient in (2.29) is bounded uniformly in N and for the purpose we exploit the rich structure of the Toda integral of motions. This manuscript is organized as follows. In section 2 we study the structure of the Toda integrals. In particular we prove that for any m P N fixed, and N sufficiently large, the m-th have disjoint supports if the distance between j and k is larger than m. Then we make the crucial observation that the quadratic part of the Toda integrals J pmq are quadratic forms in p and q generated by symmetric circulant matrices. In section 3 we approximate the Gibbs measure with the measure were all the variable are independent random variables. and we calculate the error of our approximation. In section 4 we obtain a bound on the variance of J pmq ptq´J pmq p0q with respect to the FPUT flow and a bound of linear combination of harmonic energies with respect to the FPUT flow and the Toda flow. Finally in section 5 we prove our main results, namely Theorem 2.1, Theorem 2.4 and Theorem 2.5. We describe in the Appendices the more technical results.

Structure of the Toda integrals of motion
In this section we study the algebraic and the analytic properties of the Toda integrals defined in (2.12). First we write them explicitly: :" rL m s jj is given explicitly by where it is understood r j " r j mod N , p j " p j mod N and A pmq is the set The quantity r m :" tm{2u, N 0 " N Y t0u and ρ pmq pn, mq P N is given by We give the proof of this theorem in Appendix D.
Remark 3.2. The structure of J pN q is slightly different, but we will not use it here.
We now describe some properties of the Toda integrals which we will use several times. The Hamiltonian density h pmq j pp, rq depends on set A pmq and the coefficient ρ pmq pn, kq which are independent from the index j. This implies that h pmq j is obtained by h pmq 1 just by shifting 1 Ñ j; we formalize this property below with the notion of cyclic functions.
A second immediate property, as one sees inspecting the formulas (3.3) and (3.4), is that there exists C pmq ą 0 (depending only on m) such that We first define each of these properties rigorously, and then we show that the Toda integrals enjoy them.
Cyclicity. Cyclic functions are characterized by being invariant under left and right cyclic shift. For any ℓ P Z, and x " px 1 , x 2 , . . . , x N q P R N we define the cyclic shift of order ℓ as the map For example S 1 and S´1 are the left respectively right shifts: It is immediate to check that for any ℓ, ℓ 1 P Z, cyclic shifts fulfills: Consider now a a function H : Clearly S ℓ is a linear operator. We can now define cyclic functions: It is clear from the definition that a cyclic function fulfills S ℓ H " H @ℓ P Z. It is easy to construct cyclic functions as follows: given a function h : R NˆRN Ñ C we define the new function H by H is clearly cyclic and we say that H is generated by h.
Support. Given a differentiable function F : R NˆRN Ñ C, we define its support as the set and its diameter as where d is the periodic distance dpi, jq :" min p|i´j|, N´|i´j|q . (3.13) Note that 0 ď dpi, jq ď tN {2u. We often use the following property: if f is a function with diameter K P N, and K ! N , where S j is the shift operator (3.7). With the above notation and definition we arrive to the following elementary result. Circulant symmetric matrices. We begin recalling the definition of circulant matrices (see e.g. [20,Chap. 3]).
We will say that A is represented by the vector a.
In particular circulant matrices have all the form where each row is the right shift of the row above. Moreover, A is circulant symmetric if and only if its representing vector a is even, i.e. one has a k " a N´k , @k. (3.16) One of the most remarkable property of circulant matrices is that they are all diagonalized by the discrete Fourier transform (see e.g. [20,Chap. 3]). We show now that circulant symmetric matrices are diagonalized by the Hartley transform: Lemma 3.6. Let A be a circulant symmetric matrix represented by the vector a P R N . Then where p a " Ha.
Proof. First remark that a circulant matrix acts on a vector x P R N as a periodic discrete convolution, where it is understood a ℓ " a ℓ mod N . As the Hartley transform of a discrete convolution is given by we obtain (3.17), using that the Hartley transform maps even vectors (see (3.16)) in even vectors.
Our interest in circulant matrices comes from the following fact: quadratic cyclic functions are represented by circulant matrices. More precisely consider a quadratic function of the form where A, B, C are NˆN matrices. Then one has This result, which is well known (see e.g. [20]), follows from the fact that Q cyclic is equivalent to A, B, C commuting with the left cyclic shift S 1 , and that the set of matrices which commute with S 1 coincides with the set of circulant matrices. We conclude this section collecting some properties of Toda integrals. Denote by J pmq 2 the Taylor polynomial of order 2 of J pmq at zero; being a quadratic, symmetric, cyclic function, it is represented by circulant symmetric matrices. We have the following lemma.
Lemma 3.7. Let us consider the Toda integral Then h pmq 1 pp, qq has the following Taylor expansion at p " r " 0: where each ϕ pmq k pp, rq is a homogeneous polynomial of degree k " 0, 1, 2 in p and r of diameter m and coefficients independent from N . The reminder ϕ pmq ě3 pp, rq takes the form ϕ pmq ě3 pp, rq :" ÿ pk,nqPA pmq |k|ě3 pp, rq is a cyclic function of the form with A pmq , B pmq circulant, symmetric NˆN matrices; their representing vectors a pmq , b pmq are m-admissible (according to Definition 2.3) and . m . The proof is postpone to Appendix A. We conclude this section giving the definition of m-admissible functions and we prove a lemma that characterize them in terms of tJ plq 2 u N l"1 : Definition 3.8. G 1 , G 2 : R NˆRN Ñ R N are called m-admissible functions of the first and second kind respectively if there exists a m-admissible vector g P R N such that Remark 3.9. From definition 3.8 and (3.20) one can deduce that both G 1 and G 2 can be represented with circulant and symmetric matrices. Indeed we have that G 1 " p ⊺ G 1 r where pG 1 q jk " g pj´kq mod N and similarly for G 2 .
An immediate, but very useful, corollary of Lemma 3.7 , is the fact that the quadratic parts of Toda integrals are a basis of the vector space of m-admissible functions. Then there are two unique sequences tc j u Ă m j"0 , td j u Ă m j"0 , with max j |c j |, max j |d j | independent from N , such that: is the quadratic part (3.24) of the Toda integrals J pmq in (3.1).
Proof. We will prove the statement just for functions of the first kind. The proof for functions of the second kind can be obtained in a similar way. Let J From Lemma 3.7 the matrix B " rb p2l`1q k s Ă m k,l"0 is upper triangular and the diagonal elements are always different from 0 (see in particular formula (3.25)). This implies that the above linear system is uniquely solvable for pc 0 , . . . , c Ă m q.

Averaging and covariance
In this section we collect some properties of the Gibbs measure dµ F in (2.5). The first property if the invariance with respect to the shift operator. Namely for a function f : which follows from the fact that pS j q˚dµ F " dµ F . It is in general not possible to compute exactly the average of a function with respect to the Gibbs measure dµ F in (2.5). This is mostly due to the fact that the variables p 0 , . . . , p N´1 and r 0 , . . . , r N´1 are not independent with respect to the measure dµ F , being constrained by the conditions ř i r i " ř i p i " 0. We will therefore proceed as in [28], by considering a new measure dµ F,θ on the extended phase space according to which all variables are independent. We will be able to compute averages and correlations with respect to this measure, and estimate the error derived by this approximation.
For any θ P R, we define the measure dµ F,θ on the extended space R NˆRN by j"0 rj dp dr, where we define Z F,θ pβq as the normalizing constant of dµ F,θ . We denote the expectation of a function f with respect to dµ F,θ by xf y θ . We also denote by If }f } θ ă 8 we say that f P L 2 pdµ F,θ q.
The measure dµ F,θ depends on the parameter θ P R and we fix it in such a way that ż R r e´θ r´βVF prq dr " 0. Following [28], it is not difficult to prove that there exists β 0 ą 0 and a compact set I Ă R such that for any β ą β 0 , there exists θ " θpβq P I for which (4.3) holds true. We remark that (4.3) is equivalent to require that xr j y θ " 0 for j " 0, . . . , N´1 and as a consequence We observe that A ř N´1 j"0 r j E " 0 with respect to the measure dµ F . The main reason for introducing the measure dµ F,θ is that it approximates averages with respect to dµ F as the following result shows. The above lemma is an extension to the periodic case of a result from [28], and we shall prove it in Appendix C. As an example of applications of Lemma 4.1, we give a bound to correlations functions.
Lemma 4.2. Fix K P N. Let f, g : R NˆRN Ñ C such that : 1. f, g and f g P L 2 pdµ F,θ q, 2. the supports of f and g have size at most K P N.
Moreover, if f and g have disjoint supports, then Proof. We substitute the measure dµ F with dµ F,θ and then we control the error by using Lemma 4.1. With this idea, we write xf gy´xf y xgy " xf gy´xf gy θ (4.7) xf gy θ´x f y θ xgy θ (4.8) xf y θ xgy θ´x f y xgy , (4.9) and estimate the different terms. We will often use the inequality |xf y θ | ď }f } θ , (4.10) valid for any function f P L 2 pdµ F,θ q.
Combining the three bounds above and redefining C " maxtC, C 1 u one obtains (4.5). To prove (4.6) it is sufficient to observe that if f and g have disjoint supports, then xf gy θ " xf y θ xgy θ and consequently (4.8) is equal to zero.
In order to make Lemma 4.2 effective we need to show how to compute averages according to the measure (4.2). Lemma 4.3. There exists β 0 ą 0 such that for any β ą β 0 , the following holds true. For any fixed multi-index k, l, n, s P N N 0 and d, d 1 P t0, 1, 2u, there are two constants C p1q k,l P R and C p2q k,l ą 0 such that where p k " ś N j"1 p kj j and r l "  j"0 rj dp dr; (4.12) here θ is selected in such a way that ż R r e´θ r´βVT prq dr " 0. We show in Appendix B that it is always possible to choose θ to fulfill (4.13) (see Lemma B.1) and we also prove Lemma 4.3 for Toda. In Appendix C we prove Lemma 4.1 for the Toda chain.

Bounds on the variance
In this section we prove upper and lower bounds on the variance of the quantities relevant to prove our main theorems.

Upper bounds on the variance of J pmq along the flow of FPUT
In this subsection we only consider the case M endowed by the FPUT Gibbs measure. We denote by J pmq ptq :" J pmq˝φt HF the Toda integral computed along the Hamiltonian flow φ t HF of the FPUT Hamiltonian. The aim is to prove the following result: Proof. As explained in the introduction, applying formula (2.27) we get The above expression enables us to exploit the fact that the FPUT system is a fourth order perturbation of the Toda chain. To proceed with the proof we need the following technical result. @j, moreover the diameter of the support of H j is at most m; (ii) there exist N 0 , β 0 , C, C 1 ą 0 such that for any N ą N 0 , β ą β 0 , any i, j " 1, . . . , N , the following estimates hold true: The proof of the lemma is postponed at the end of the subsection. We are now ready to finish the proof of Proposition 5.1. Substituting (5.6) in (5.5) we obtain Eı .
We now apply estimates (5.9), (5.10) to get for some positive constants C 1 and C 2 .

Proof of Lemma 5.2
We start by writing the Poisson bracket tJ pmq , H F´HT u in an explicit form. First we observe that for any 1 ď m ă N one has from (2.12) for all j " 1, . . . , N . In the above relation h pm´1q j is the generating function of the m´1 Toda integral defined in (3.2).
Next we observe that This implies also that p0, 0qq`R 1 pr j´2 q´R 1 pr j´1 q˘(

5.14)
where, to obtain the second identity, we are using that h pm´1q j p0, 0q is by (3.15) and (3.21) a constant independent from j and the second term in the last relation is a telescopic sum. Define . To prove item piiq we start by expanding R 1 pr j´1 q´R 1 pr j q in Taylor series with integral remainder. Since we get that where explicitly ψ 3 prq :" r 3 where ϕ pmq j , j " 0, 1, 2, are defined in (3.21). Thus the squared L 2 norm of H j is given by (we suppress the superscript to simplify the notation)

Lower bounds on the variance of m-admissible functions
From now on we consider M endowed with either the FPUT or the Toda Gibbs measure; the following result holds in both cases.
Proposition 5.3. Fix m P N, let G be an m-admissible function of the first or second kind (see Definition 3.8). There exist N 0 , β 0 , C ą 0 such that for any N ą N 0 , β ą β 0 , one has Proof. We first prove (5.26) when G " G 1 " p ⊺ G 1 r where G 1 is a circulant, symmetric matrix represented by the m-admissible vector a P R N . We now make the change of coordinates pp, rq " pHp p, Hp rq which diagonalizes the matrix G 1 (see (3.17)), getting So we have just to compute where we used that p p k , p r j are random variables independent from each other. We notice that p p 1 , p p 2 , . . . , p p N´1 are i.i.d. Gaussian random variable with variance β´1, p p 0 " 0 (see (2.1)), so that we have xp p j y " 0 and xp p j p p i y " δi,j β i, j " 1, . . . , N´1 (remark that this holds true both for the FPUT and Toda's potentials as the p-variables have the same distributions). As a consequence, (5.27) becomes: Since G 1 is circulant symmetric matrix so is G 2 1 and its representing vector is d :" g ‹ g. Next we remark that the identity Applying this property to (5.28) we get For the case of admissible functions of the second kind, one has G 2 " p ⊺ G 2 p`r ⊺ G 2 r with G 2 circulant, symmetric and represented by an m-admissible vector. Since p and r are independent random variables one gets Then arguing as in the previous case one gets (5.26).
By applying Proposition 5.3 to the quantity J pmq 2 that is an m-admissible function of the first or second kind depending on the parity of m, we obtain the following result. of the Taylor expansion of the Toda integral J pmq near pp, rq " p0, 0q satisfies for some constant C ą 0.
In a similar way we obtain a lower bound on the reminder J pmq ě3 of the Taylor expansion of the Toda integral J pmq near p " 0 and r " 0.
Lemma 5.5. Fix m P N. There exist N 0 , β 0 , C ą 0 such that for any N ą N 0 , β ą β 0 , one has Proof. Recall from Lemma 3.7 that J pmq ě3 is a cyclic function generated by r h and its variance is given by We can bound the correlations in (5.33) exploiting Lemma 4.2, provide we estimate first the L 2 pdµ F,θ q and L 2 pdµ T,θ q norms of r h . Proceeding with the same arguments as in Lemma 5.2, one proves that there existsC ą 0 such that for any N ą N 0 , β ą β 0 , Combining Corollary 5.4 and Lemma 5.5 we arrive to the following crucial proposition.
Proposition 5.6. Fix m P N. There exist N 0 , β 0 , C ą 0 such that for any N ą N 0 , β ą β 0 , one has constant. By Corollary 5.4 and Lemma 5.5 we deduce that for N and β large enough, which leads immediately to the claimed estimate (5.37).

Proof of the main results
In this section we give the proofs of the main theorems of our paper.

Proof of Theorem 2.1
The proof is a straightforward application of Proposition 5.1 and 5.6. Having fixed m P N, we apply (2.26) with Φ " J pmq and λ " β´ε to get from which one deduces the the statement of Theorem 2.1.

Proof of Theorem 2.4 and Theorem 2.5
The proofs of Theorem 2.4 and Theorem 2.5 are quite similar and we develop them at the same time. As in the proof of Theorem 2.1, the first step is to use Chebyshev inequality to bound where the time evolution is intended with respect to the FPUT flow φ t F or the Toda flow φ t T . Accordingly, the probability is calculated with respect to the FPUT Gibbs measure (2.5) or the Toda Gibbs measure (2.22).
Next we observe that the quantity Φ :" ř N´1 j"1 p g j E j defined in (2.20) can be written in the form where g P R N is a m-admissible vector and G 2 pp, rq is a m-admissible function of the second kind, as in Definition 3.8. As the inequality (2.26) is scaling invariant, prove (6.2) is equivalent to obtain that Applying Proposition 5.3 we can estimate σ 2 G2 . We are then left to give an upper bound to σ 2 G2ptq´G2 . By Lemma 3.10, there exists a unique sequence tc j u Ă m´1 j"0 , with max j |c j | independent from N , such that G 2 pp, rq " are defined in (3.24). Hence we bound Next we interpolate J p2lq 2 with the integrals J p2lq and exploit the fact that they are adiabatic invariants for the FPUT flow and integrals of motion for the Toda flow. More precisely By the invariance of the two measures with respect to their corresponding flow and Lemma 5.5, we get both for FPUT and Toda the estimate for some constantC 1 ą 0 and for β ą β 0 and N ą N 0 . As (6.6) is zero for the Toda flow (being J p2lq ptq constant along the flow), we get for some constant C 1 ą 0 and for β ą β 0 and N ą N 0 . Combing Proposition 5.3 with (6.8) we conclude that and by choosing λ " β´ε with 0 ă ε ă 1 2 we arrive to the expression (2.24), namely we have concluded the proof of Theorem 2.5.
We are left to estimate (6.6) for FPUT, but this is exactly the quantity bounded in Proposition 5.1. We conclude that for some constant C j ą 0, j " 1, 2, 3 and for β ą β 0 and N ą N 0 . Combing Proposition 5.3 with (6.10) we obtain Choosing λ " β´ε with 0 ă ε ă 1 4 , (6.11) is equivalent to for some redefine constant C 1 ą 0 and for every time t fulfilling |t| ď β 1´2έ We have thus concluded the proof of Theorem 2.4.

A Proof of Lemma 3.7
In order to prove Lemma 3.7 we describe more specifically the Toda integrals and characterize their quadratic parts. Equation ÿ pk,nqPA pmq p´1q |k| ρ pmq pn, kq p k e´n ⊺ r , with supp k, supp n Ď B d Ă m p0q :" tj : dp0, jq ď r mu, |k|`2|n| " m.
These, together with the explicit formula of ρ pmq pn, kq, prove (3.21). It is easy to see that defining we immediately get that it is a constant that is zero for m odd; moreover thanks to the boundary condition (2.4) and the linearity of J Case m even. As before there exist two matrices A pmq , D pmq represented by m-admissible vectors such that: We have just to prove that the two matrices are equal; to do this we exploit the involution property of the Toda integrals. Indeed we know that J pjq , J pkq ( " 0, for any j, k. It follows easily that also their quadratic parts must commute: To compute explicitly the Poisson bracket we change coordinates via the Hartley transform (2.16) getting that: where ω j " 2 sin`π j N˘. As the Hartley transform is a symplectic map, by (A.1) we get which implies that p a j " p d j for all j ‰ 0. To prove that also p a 0 " p d 0 we come back to the original variables getting that: This means that a which proves the statement.

B Proof of Lemma 4.3
We prove the lemma for both the FPUT and Toda measure. First of all we observe that for d, v " 2, 3: This means that we have actually to prove that for any fixed multi-index k, l, n P N N 0 there exist two constants C p1q k,l P R and C p2q k,l ą 0 such that: Moreover since for the two measures dµ F,θ , dµ T,θ all p and r are independent random variables and moreover the p j are independent and normally distributed according to N p0, β´1q, it follows C Here k!! denotes the double factorial. Instead the distribution of the r j is different for the two measures, so we need to calculate it separately for the FPUT and Toda chain.
FPUT chain. Let's start considering @ r l min pe´n r , 1q D θ : @ r l min`e´n r , 1˘D θ " Since for β large enough θpβq is uniformly bounded, it follows that there is a positive constant C l such that: @ r l min`e´n r , 1˘D θ ě p´1q l C l β l 2 .

(B.8)
We notice that if l is even then the right end side of (B.8) is positive. The proof for @ r l max pe´n r , 1q D θ follows in the same way so we get the claim for the FPUT chain.
Toda chain. For the Toda chain the computation is a little bit more involved, so we prefer to split it in different parts.
Lemma B.1. Consider the measure 4.2, then there exists a β 0 ą 0 such that for all β ą β 0 there exists θ " θpβq P r1{3, 2s such that Proof. First we prove that, for any β large enough, we can chose θpβq in a compact interval I such that xr j y θ " 0. We notice that: where Γpzq is the usual Gamma function and we used the following equality: In the case k " 1 one obtains Introducing the digamma function ψpzq " Γ 1 pzq Γpzq [27] and using the inequality it is easy to show that there exists β 0 ą 0 such that @β ą β 0 one has ψˆ1 3`β˙ď logˆ1 3`β˙´1 2p1{3`βq ď log β and ψp2`βq ě logp2`βq´1 2`β ě log β.
Since x Þ Ñ ψpxq is continuous on p1,`8q, by the intermediate value theorem there exists θpβq P r1{3, 2s fulfilling ψpθ`βq " log β which implies by (B.11) that xr j y θ " logpβq´Γ 1 pθ`βq Γpθ`βq " 0. (B.12) We will prove the remaining part of the claim by induction; (B.10) leads in the case k " 2 to: Γpθ`βq pψpθ`βq´lnpβqq" xr j y θ pψpθ`βq´lnpβqq`ψ p1q pθ`βq " ψ p1q pθ`βq, where ψ psq is the s th polygamma function defined as ψ psq pzq :" B s ψpzq Bz s . For x P R it has the following expansion as x Ñ`8 : where B k are the Bernoulli number of the second kind. Therefore So the first inductive step is proved. Next suppose the statement true for k and let us prove it for k`1.
We are now ready to prove the last part of Lemma 4.3 for the Toda chain: The last integral can be estimated in the same way as in the previous lemma, moreover the lower bound follows in the same way, so we get the claim also for the Toda chain.

C Measure approximation
In this section we show how to approximate the measure dµ, in which the variables are constrained, with the measure dµ θ , where all variables are independent. The proof follows the construction of [28] (where it is done for Dirichlet boundary conditions) which applies both to the Gibbs measure of FPUT (2.5) and Toda (2.22). To simplify the construction we consider a general potential V : R Ñ R and make the following assumptions: (V1) There exist β 0 ą 0 and a compact interval I Ă R such that for any β ą β 0 , there exists θ " θpβq P I such that ż R r e´θ r´βV prq dr " 0.
Both the FPUT potential V F pxq and the Toda potential V T pxq satisfy the assumptions (V1)-(V3) by the results of Appendix B.
We define the constraint measure dµ V on the restricted phase space M as and the unconstrained measure dµ V θ on the extended phase space R NˆRN as j"1 rj dp dr; (C.5) as usual Z V pβq and Z V,θ pβq are the normalizing constants of dµ V , dµ V θ respectively . We denote the expectation of f with respect to the measure dµ V as xf y V , and with respect to the measure dµ V θ as xf y V,θ . We also denote by }f } V,θ :" @ f 2 D 1{2 V,θ the L 2 norm of f with respect to the measure dµ V θ . The main result is the following one: Theorem C.1. Assume that (V1)-(V3) hold true. Fix K P N and assume that f : R NˆRN Ñ R have support of size K (according to definition 3.11) and finite second order moment with respect to dµ V θ . Then there exist C, N 0 and β 0 such that for all N ą N 0 , β ą β 0 one hašˇˇx C.1 Proof of Theorem C.1 Introduce the structure function The important remark is that Ω N pxq is N -times the convolution of the function e´β V pxq with itself thus it is the density function of the sum of N iid random variables distributed as e´β V pxq . Next, for θ P R, we define the conjugate distribution As before, we remark that U pθq N pxq it is N -times the convolution of the function e´β V pxq´θx with itself thus it is the density function of the sum of N iid random variables tY pθq n pβqu 1ďnďN distribute as Y pθq n pβq " Y pθq :" moreover thanks to (C.1) we know that @ Y pθq D " 0.
The central limit theorem says that the rescaled random variable Y pθq n pβq converges in distribution to a normal N p0, 1q. We want to apply a more refined version of this result, called local central limit theorem, which describes the asymptotic of this convergence.
In particular we will use a local central theorem whose proof can be found in [33,Theorem VII.15]; to state it, we first define the functions where H j is the j-th Hermite polynomial, γ d is the d-th cumulant 1 of Y pθq n pβq, and Bpνq is the set of all non-negative integer solutions k 1 , . . . , k ν of the equalities k 1`2 k 2`¨¨¨`ν k ν " ν, and s " k 1`k2`¨¨¨`kν .
Theorem C.2 (Local central limit). Let tX n u be a sequence of iid variables such that (i) For any 1 ď n ď N , one has E rX n s " 0.
(iii) The random variable 1 σ ? N ř N n"1 X n has a bounded density p N pxq. Then there exists C ą 0 such that where the q ν 's are defined in (C.10). 1 We recall that γ d " ř Cpdq d!p´1q m 1`. ..`m d´1 pm 1`. . .`m d´1 q! ś d l"1 α m l l m l !pl!q m l where α l is the l th moment of the random variable and Cpdq is the set of all non-negative integer solution of ř l lm l " d.
Proof. We verify that the assumptions of Theorem C.2 are met in case X n " Y pθq n pβq. Item piq and piiq hold true thanks to assumptions (V1) and (V2), in particular piiq is true with k " 4. To verify piiiq, we note that Y pθq n pβq has density given by σ ?
This last function is N -times the convolution of g θ prq :" e´θ r´βV prq . By assumption (V3), g θ P L 8 pRq and by (V2) it belongs also to L 1 pRq. So Young's convolution inequality implies that σ ? N U pθq N pσ ? N xq is bounded uniformly in x, hence piiiq of Theorem C.2 is verified. We apply Theorem C.2 with p N pxq " σ ? N U pθq N pσ ? N xq, then rescale the variable x to get (C.12).
We study also the structure function and the normalized distribution We have the following result: Lemma C.4. For any N ě 1, any β ą 0, one has (C.14) Proof. The function r U N is the N -times convolution of Gaussian functions of the form gpξq :" b β 2π e´β 2 ξ 2 . Since convolution of Gaussians is a Gaussian whose variance is the sum of the variances, (C.13) follows.
We can finally prove Theorem C.1: Proof of Theorem C.1. The proof follows closely [28]. We assume that f is supported on 1, . . . , K, the other cases being analogous. Using that and denoting r p :" pp 1 , . . . , p K q and r r :" pr 1 , . . . , r K q, we write xf pr p, r rqy V " where dr µ :" exp´´β ř K j"1 p 2 j 2´β ř K j"1 V pr j q¯dr pdr r. As, by (C.8) and (C. 13), where U pθq pr p, r rq :" . Now we use that ż R KˆRK e´θ ř K j"1 rj pz θ pβqq K pr z θ pβqq K U pθq pr p, r rq dr µ " x1y V´x 1y V,θ " 0 so that we can write the difference xf y V´x f y V,θ as Using Cauchy-Schwartz we obtain thaťˇˇx so in order to prove (C.6) we are left to show that uniformly in N and β one has Using (C.12) and (C.14), we have thaťˇˇˇˇU N´K˙`K N´KṄ ext we use that |e´a 2´b2´1 | ď a 2`b2 , the explicit expression the estimate γ 3 6 σ 3 ď C for some C independent of β (which follows by (C.2) as in our case γ 3 ď Cβ´3 {2 ), to obtain that there exists C ą 0 such that @N ě N 0 , @β ě β 0 ,ˇˇˇˇU Substituting x "´ř K j"1 r j , ξ "´ř K j"1 p j , and computing the L 2 norm (with respect to dµ V θ ) of the terms in the r.h.s. of the last formula give the claimed estimate (C.15).

D Proof of Theorem 3.1
In this appendix we prove Theorem 3.1. From the structure of the matrix Lax matrix L in (2.11), we immediately get rL m s jj pa, bq " S j´1 prL m s 11 pa, bqq , where S j is the shift defined in (3.7), thus we have to prove formula (3.2) just for the case j " 1.
To accomplish this result we need to introduce the notion of super Motzkin path and super Motzkin polynomial, that generalize the notion of Motzkin path and Motzkin polynomial [32,37].
Definition D.1. A super Motzkin path p of size m is a path in the integer plane N 0ˆZ from p0, 0q to pm, 0q where the permitted steps from p0, 0q are: the step up p1, 1q, the step down p1,´1q and the horizontal step p1, 0q. A similar definition applies to all other vertices of the path.
The set of all super Motzkin paths of size m will be denoted by sM m . In order to introduce the super Motzkin polynomial associated to these paths we have to define their weight. This is done in the following way: to each up step that occurs at height k, i.e. it joins the points pl, kq and pl`1, k`1q, we associate the weight a k , to a down step that joins the points pl, kq and pl`1, k´1q we associate the weight a k´1 , to each horizontal step from pl, kq to pl`1, kq we associate the weight b k . Since k P Z, the index of a k and b k are understood modulus N .
At this point we can define the total weight wppq of a super Motzkin path p to be the product of weights of its individual steps. So it is a monomial in the commuting variables pb, aq " pb´Ă m , . . . , b Ă m , a´Ă m , . . . , a Ă m q, where r m " tm{2u. We remark that the total weight do not characterize uniquely the path. We are now ready to give the definition of Motzkin polynomial: where the super Motzkin polynomial sP m pa, bq is defined in (D.1) and a j " a j mod N , b j " b j mod N .
Proof. In general we have that: To every element of the sum we associate the path with vertices p0, 0q, p1, r j 1´1 , q, p2, r j 2´1 q, . . . , pℓ, r j ℓ´1 q, . . . , pm´1, r j m´1´1 q, pm, 0q This is a super Motzkin path p j and we can associate the weight wpp j q as in the description above therefore we have L 1,j1 L j1,j2 . . . L jm´1,1 " wpp j q This is clearly a bijection. The sum of the weights of all possible super Motzkin paths, is defined to be the super Motzkin polynomial sP m pa, bq and thus we get the claim.
Proceeding as in [32, Proposition 1], we are able to prove the following result, which together with Proposition D. Proof. For a give super Motzkin path p starting at p0, 0q and finishing at p0, mq let k i be the number of horizontal steps at height i and let n i be the number of step up from height i to i`1.
We remark the number n i of step up from height i to i`1 is equal to the number of step down from i`1 to i. We define the vectors k " pk´Ă m , k´Ă m`1 , . . . , k Ă m q and n " pn´Ă m , n´Ă m`1 , . . . , n Ă m q and we associate the product Ă m ź i"´Ă m a 2ni i b ki i .
Next we need to sum over all possible super Motzkin path p of length m connecting p0, 0q to p0, mq. Since the number of steps up is equal to the number of steps down, one necessarily have Ă m ÿ i"´Ă m p2n i`ki q " m .
Furthermore since the path is connected it follows that it is not possible to have a vertex at height i`1 without have a vertex at height i ą 0 and the other way round if i ă 0. Therefore one has @i ě 0, n i " 0 ñ n i`1 " k i`1 " 0, @i ă 0, n i`1 " 0 ñ n i " k i " 0 .
This proves the definition of the set A pmq in (D.5). The final step of the proof is to count the number of paths associated to the vectors k " pk´Ă m , k´Ă m`1 , . . . , k Ă m q and n " pn´Ă m , n´Ă m`1 , . . . , n Ă m q. We want to show that this number is equal to ρ pmq pn, kq. A horizontal step at height i can occur just after a step up to height i, another horizontal step at height i, or a step down to height i. This leaves a total of n i`ni`1 different positions at which a horizontal step at height i can occur. Since we have k i of horizontal steps, the number of different configurations with these step counts is the number of ways to choose k i elements from a set of cardinality n i`ni`1 with repetitions allowed, i.e.`n i`ni`1`ki´1 ki˘.
The number of different configurations with n i steps at height i and n i`1 at height i`1 is given by the number of multi-sets of cardinality n i`1 taken from a set of cardinality n i and this number is equal to`n i`ni`1´1 ni`1˘.
For the horizontal steps at height 0, they can also occur at the beginning of the path, this increase the number of possible positions by 1, so the number of these configurations with these steps counts is`n 0`n´1`k0 k0˘. In this way we have obtained the coefficient ρ pmq pn, kq.