Memory Effects in the Fermi–Pasta–Ulam Model

Abstract

We study the intermediate scattering function (ISF) of the strongly-nonlinear Fermi–Pasta–Ulam Model at thermal equilibrium, using both numerical and analytical methods. From the molecular dynamics simulations we distinguish two limit regimes, as the system behaves as an ideal gas at high temperature and as a harmonic chain for low excitations. At intermediate temperatures the ISF relaxes to equilibrium in a nontrivial fashion. We then calculate analytically the Taylor coefficients of the ISF to arbitrarily high orders (the specific, simple shape of the two-body interaction allows us to derive an iterative scheme for these). The results of the recursion are in good agreement with the numerical ones. Via an estimate of the complete series expansion of the scattering function, we can reconstruct within a certain temperature range its coarse-grained dynamics. This is governed by a memory-dependent Generalized Langevin Equation (GLE), which can be derived via projection operator techniques. Moreover, by analyzing the first series coefficients of the ISF, we can extract a parameter associated to the strength of the memory effects in the dynamics.

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Acknowledgements

We thank T. Voigtmann, T. Franosch and A. Zippelius for useful discussions. Computer simulations presented in this paper were carried out using the bwForCluster NEMO high-performance computing facility.

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Correspondence to Graziano Amati.

Appendix

Appendix

Two Body Potential

In this appendix we derive the expression for the two-well interaction potential between nearest neighbors used (Eq. 1). We would like to fix a parametrization in terms of a global energy scale \(\alpha \), a length scale \(\sigma \) and a factor tuning the unbalance between two wells. Therefore we can define

$$\begin{aligned} V(r) = \alpha \left[ \left( \frac{r}{\sigma }\right) ^2 + A\left( \frac{r}{\sigma }\right) ^3 + B \left( \frac{r}{\sigma }\right) ^4\right] \end{aligned}$$

We then require that the distance \(r=\sigma \) corresponds to an equilibrium position, and we impose \(V(\sigma )\equiv -\epsilon \), \(\epsilon \in {\mathbb {R}}\):

$$\begin{aligned} \frac{{\mathrm {d}} V(r)}{{\mathrm {d}} r}\Bigr |_{r=\sigma }&= 0 = \frac{\alpha }{\sigma }\left[ 2+3A+4B\right] \\ V(\sigma )&= -\,\epsilon = \alpha [1 +A+B] \\ \end{aligned}$$

The relations are solved by

$$\begin{aligned} A&= A(\eta ) = -\,2-4\eta \\ B&= B(\eta ) = 1+3\eta \end{aligned}$$

With the dimensionless parameter \(\eta = \frac{\epsilon }{\alpha }\) assuming positive values in case the minimum in \(r=\sigma \) is negative.

Analytical Expression for \(\omega _n\)

In this appendix we present the analytical expressions of the series coefficients \(\omega _n\); while an exact solution is easily determined at any order in the ideal gas regime, a general expression is not available in case a nonlinear interaction is added to the dynamics. For the non-integrable regime, we can still derive the analytical solution of the lowest orders; in particular in A.2.2 we compute exactly \(\omega _4\), which is the first coefficient that depends on the dynamics.

Ideal Gas Regime

By expanding the kinetic part of the Liouvillian (Eq. 11) in the general expression in Eq. 5 we get:

$$\begin{aligned} \omega _n^{{{\,\mathrm{id}\,}}}&=\left\langle \left[ \left( \frac{1}{m}\sum _{\gamma =1}^{N-2}(p_\gamma -p_{\gamma -1})\frac{\partial }{\partial r_\gamma }\right) ^ne^{iKr_j}\right] e^{-iKr_j} \right\rangle _\beta = \left( \frac{iK}{m}\right) ^n\left\langle (p_j-p_{j-1})^n\right\rangle _\beta \nonumber \\&=\left( \frac{iK}{m}\right) ^n\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \left\langle p_j^k \right\rangle _\beta \left\langle (-p_{j-1})^{n-k}\right\rangle _\beta \nonumber \\&\simeq \frac{1}{\pi } \left( \frac{\beta }{2m}\right) \left( \frac{iK}{m}\right) ^n \sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) (-1)^{n-k} \left[ \int _{-\infty }^{+\infty } {\mathrm {d}} p_j\; e^{-\beta \frac{p_j^2}{2m}} p_j^k \right] \nonumber \\&\quad \quad \left[ \int _{-\infty }^{+\infty } {\mathrm {d}} p_{j-1}\; e^{-\beta \frac{p_{j-1}^2}{2m}} p_{j-1}^{n-k} \right] \nonumber \\&= \frac{1}{\pi }\left( \frac{\beta }{2m}\right) \left( \frac{iK}{m}\right) ^n\frac{(-1)^n+1}{2}\sum _{k=0}^{\frac{n}{2}}\left( {\begin{array}{c}n\\ 2k\end{array}}\right) \left( \frac{2m}{\beta }\right) ^{ k+\frac{1}{2}} \left[ \int _{-\infty }^{+\infty } {\mathrm {d}} x\; e^{-x^2} x^{2k} \right] \nonumber \\&\quad \times \left( \frac{2m}{\beta }\right) ^{\frac{n+1}{2}-k}\left[ \int _{-\infty }^{+\infty } {\mathrm {d}} x\; e^{-x^2} x^{n-2k} \right] \nonumber \\&=\frac{1}{\pi } \left( \frac{\beta }{2m}\right) \left( \frac{iK}{m}\right) ^n \frac{(-1)^n+1}{2} \left( \frac{2m}{\beta }\right) ^{\frac{n}{2}+1}\nonumber \\&\quad \times \left[ \sum _{k=0}^{\frac{n}{2}}\left( {\begin{array}{c}n\\ 2k\end{array}}\right) \Gamma \left( k+\frac{1}{2}\right) \Gamma \left( \frac{n}{2}-k+\frac{1}{2}\right) \right] \nonumber \\&=\frac{1}{\pi }(-1)^{\frac{n}{2}}\left( \frac{2K^2}{m\beta }\right) ^{\frac{n}{2}} \frac{(-1)^n+1}{2}\sqrt{\pi }2^{\frac{n}{2}}\Gamma \left( \frac{n+1}{2}\right) \nonumber \\&=\frac{1}{\sqrt{\pi }}(-1)^{\frac{n}{2}}\left( \frac{4K^2}{m\beta }\right) ^{\frac{n}{2}} \frac{(-1)^n+1}{2}\Gamma \left( \frac{n+1}{2}\right) \end{aligned}$$
(26)

where we used

$$\begin{aligned} \left\langle p_j^m \right\rangle _\beta = \left( \frac{2m}{\beta }\right) ^{\frac{m}{2}}\frac{1}{\sqrt{\pi }}\frac{(-1)^m+1}{2}\Gamma \left( \frac{m+1}{2}\right) \end{aligned}$$

and

$$\begin{aligned}&\sum _{k=0}^{\frac{n}{2}} \left( {\begin{array}{c}n\\ 2k\end{array}}\right) \Gamma \left( k+\frac{1}{2}\right) \Gamma \left( \frac{n}{2}-k+\frac{1}{2}\right) \\&\quad =\sum _{k=0}^{\frac{n}{2}} \frac{n!}{(2k)!(n-2k)!}\cdot \sqrt{\pi }\,\frac{(2k)!}{4^k k!}\cdot \sqrt{\pi }\, \frac{(n-2k)!}{4^{\frac{n}{2}-k} \left( \frac{n}{2}-k\right) !}\\&\quad =\frac{\pi n!}{4^{\frac{n}{2}}\left( \frac{n}{2}\right) !}\sum _{k=0}^{\frac{n}{2}} \frac{\left( \frac{n}{2}\right) !}{k!\left( \frac{n}{2} -k\right) !} = \frac{\pi n!}{4^{\frac{n}{2}}\left( \frac{n}{2}\right) !} \sum _{k=0}^{\frac{n}{2}} \left( {\begin{array}{c}\frac{n}{2}\\ k\end{array}}\right) =\frac{\pi n!}{4^{\frac{n}{2}}\left( \frac{n}{2}\right) !} 2^{\frac{n}{2}} \\&\quad =\sqrt{\pi }2^{\frac{n}{2}} \frac{\sqrt{\pi }n!}{4^{n/2}\left( \frac{n}{2}\right) !} = \sqrt{\pi }2^{\frac{n}{2}}\Gamma \left( \frac{n+1}{2}\right) \end{aligned}$$

\(\omega _4\) in the Anharmonic Regime

Let us decompose the Liouville operator into kinetic and potential part

$$\begin{aligned} {\mathcal {L}}= & {} {\mathcal {L}}_r + {\mathcal {L}}_p \\ {\mathcal {L}}_r\equiv & {} \sum _{j=1}^{N-1}\frac{1}{m}\left( p_j- p_{j-1}\right) \frac{\partial }{\partial r_j}\quad \, {\mathcal {L}}_p\equiv \sum _{j=1}^{N-1}-\frac{\partial V(r_j)}{\partial r_j}\left( \frac{\partial }{\partial p_j}-\frac{\partial }{\partial p_{j-1}}\right) \end{aligned}$$

The fourth dynamical correlator is then given by:

where we defined the force acting on a single d.o.f. with

$$\begin{aligned} F_{\eta }(r) \equiv -\,\frac{{\mathrm {d}} V_{\eta }(r)}{\mathrm {dr}} \end{aligned}$$

and the equilibrium variance \(\sigma _\beta ^2(A, B) = \left\langle AB\right\rangle _\beta -\left\langle A \right\rangle _\beta \left\langle B \right\rangle _\beta \). In the last two lines we reduced the phase averages to one-dimensional integrals, in the assumption of weak statistical correlation between positions and momenta, as discussed in the text. Finally in the last line we noticed that \(\sigma ^2(F_\eta , F_\eta ) = \left\langle F_\eta ^2 \right\rangle _\beta \) because \(\left\langle F_\eta \right\rangle _\beta =0\).

Harmonic Limit of the ISF

In the limit \(\beta \gg 1\) the dynamics of the system approaches the one of a harmonic chain, because the interaction can be effectively linearized at the bottom of the lowest potential well. In this section we will derive an approximated expression for the ISF in this limit; we will mainly follow the arguments in [25,26,27].

The two body potential in this regime can be effectively approximated by its series expansion at order two:

$$\begin{aligned} V^h(r_j)= & {} V(\sigma ) + \frac{m}{2} \Omega ^2(q_j-q_{j-1}+\sigma )^2 \\ \Omega ^2\equiv & {} \frac{V''(\sigma )}{m} \end{aligned}$$

The linearized dynamics is diagonalized by the normal modes, a set of \(N-2\) fictitious oscillators parametrized by the Lagrangian coordinates\(\left\{ \eta _j, \dot{\eta }_j\right\} _{j=1}^{N-2}\); their explicit expression is determined as the discrete Fourier transform of the configurations and momenta; in particular

$$\begin{aligned} q_j(t)&=\sqrt{\frac{2}{N-1}}\sum _{k=1}^{N-2} \eta _k(t)\sin \left( \frac{\pi jk}{N-1}\right) +j\sigma \end{aligned}$$
(27)
$$\begin{aligned} p_j(t)&=m\sqrt{\frac{2}{N-1}}\sum _{k=1}^{N-2} \dot{\eta }_k(t)\sin \left( \frac{\pi jk}{N-1}\right) \end{aligned}$$
(28)

The frequency of oscillation of each mode is given by

$$\begin{aligned} \omega _j= 2\Omega \sin \left( \frac{\pi j}{2(N-1)}\right) \end{aligned}$$

We can insert the modes’ decomposition of the configurations in the ISF in the quasi-integrable regime \(\beta \gg 1\); in this case the phase averages \(\left\langle \cdot \right\rangle _{h, \beta }\) are computed w.r.t. the harmonic Hamiltonian

$$\begin{aligned} H^h(\mathbf{q, p}) \equiv \sum _{j=1}^{N-2}\frac{p_j^2}{2m} + (N-1)V(\sigma ) + \frac{V''(\sigma )}{2}\sum _{j=1}^{N-1}(q_j-q_{j-1}+\sigma )^2 \end{aligned}$$

The ISF is then expanded in the following chain of identities:

$$\begin{aligned} C^h(t)&\equiv \left\langle e^{iK\left( r_j(t)-r_j(0)\right) }\right\rangle _\beta = \left\langle e^{iK\left( q_j(t)-q_{j-1}(t)-q_j(0)+q_{j-1}(0)\right) } \right\rangle _\beta \nonumber \\&=\left\langle \exp \left( iK\sqrt{\frac{2}{N-1}}\sum _{l=1}^{N-2}\left( \eta _l(t)-\eta _l(0)\right) \left[ \sin \left( \frac{\pi j l}{N-1}\right) -\sin \left( \frac{\pi (j-1)l}{N-1}\right) \right] \right) \right\rangle _\beta \nonumber \\&\quad =\frac{1}{Z_\beta }\int _{{\mathbb {R}}^{2(N-2)}}\prod _{l=1}^{N-2}{\mathrm {d}} \eta _l{\mathrm {d}} \dot{\eta }_l\;\exp \Bigg (\sum _{l=1}^{N-2}\Bigg [-\frac{m\beta }{2}\dot{\eta }_l^2-m\beta \frac{\omega _l^2}{2}\eta _l^2\nonumber \\&\qquad +\,iK\sqrt{\frac{2}{N-1}}\left( (\cos (\omega _l t)-1)\eta _l(0)+\frac{\dot{\eta }_l(0)}{\omega _l}\sin \left( \omega _l t\right) \right) \nonumber \\&\qquad \times \left( \sin \left( \frac{\pi j l}{N-1}\right) -\sin \left( \frac{\pi (j-1)l}{N-1}\right) \right) \Bigg ]\Bigg )= \end{aligned}$$
(29)
$$\begin{aligned}&=\left( \frac{ m\beta }{2\pi }\right) ^{N-2}\left( \prod _{m=1}^{N-2}\omega _m\right) \left\{ \int _{{\mathbb {R}}^{N-2}} \prod _{l=1}^{N-2}{\mathrm {d}} \dot{\eta }_l\; \exp \left( -\,\frac{m\beta }{2}\dot{\eta }_l^2\right. \right. \nonumber \\&\qquad \left. \left. +\,iK\sqrt{\frac{2}{N-1}}\frac{\dot{\eta _l}}{\omega _l}\sin \left( \omega _l t\right) 2\cos \left( \frac{\pi (2j-1)l}{2(N-1)}\right) \sin \left( \frac{\pi l}{2(N-1)}\right) \right) \right\} \nonumber \\&\qquad \times \left\{ \int _{{\mathbb {R}}^{N-2}}\prod _{l'=1}^{N-2}{\mathrm {d}} \eta _{l'}\;\exp \left( -\frac{m\beta \omega _{l'}^2}{2}\eta _{l'}^2\right. \right. \nonumber \\&\quad \left. \left. +iK\sqrt{\frac{2}{N-1}}\eta _{l'}\left( \cos \left( \omega _{l'} t\right) -1\right) 2\cos \left( \frac{\pi (2j-1)l'}{2(N-1)}\right) \sin \left( \frac{\pi l'}{2(N-1)}\right) \right) \right\} \end{aligned}$$
(30)
$$\begin{aligned}&=\left( \frac{ m\beta }{2\pi }\right) ^{N-2}\left( \prod _{m=1}^{N-2}\omega _m\right) \Bigg \{\int _{{\mathbb {R}}^{N-2}}\prod _{l=1}^{N-2}{\mathrm {d}}\dot{\eta }_l\;\exp \left( -\left[ \sqrt{\frac{m\beta }{2}}\dot{\eta }_l\right. \right. \nonumber \\&\quad \left. \left. -\,\frac{iK}{\sqrt{m\beta (N-1)}}\frac{\sin \left( \omega _l t\right) }{\omega _l}2\cos \left( \frac{\pi (2j-1)l}{2(N-1)}\right) \sin \left( \frac{\pi l}{2(N-1)}\right) \right] ^2\right) \Bigg \} \nonumber \\&\quad \times \exp \left( -\frac{4 K^2}{m\beta (N-1)}\frac{\sin ^2\left( \omega _l t\right) }{\omega _l^2}\cos ^2\left( \frac{\pi (2j-1)l}{2(N-1)}\right) \sin ^2\left( \frac{\pi l}{2(N-1)}\right) \right) \Bigg \} \nonumber \\&\quad \times \left\{ \int _{{\mathbb {R}}^{N-2}}\prod _{l'=1}^{N-2}{\mathrm {d}} \eta _{l'}\;\exp \left( -\left[ \sqrt{\frac{m\beta }{2}}\omega _{l'}\eta _{l'}\right. \right. \right. \nonumber \\&\quad \left. \left. \left. -\,\frac{iK}{\omega _{l'}\sqrt{m\beta (N-1)}}\left( \cos \left( \omega _{l'} t\right) -1\right) 2\cos \left( \frac{\pi (2j-1)l'}{2(N-1)}\right) \sin \left( \frac{\pi l'}{2(N-1)}\right) \right] ^2\right) \right\} \nonumber \\&\qquad \times \exp \left( -\frac{4 K^2}{m\omega _{l'}^2\beta (N-1)}\left( \cos \left( \omega _{l'} t\right) -1\right) ^2\cos ^2\left( \frac{\pi (2j-1)l'}{2(N-1)}\right) \sin ^2\left( \frac{\pi l'}{2(N-1)}\right) \right) \nonumber \\&\quad =\left( \frac{m\beta }{2\pi }\right) ^{N-2}\left( \prod _{m=1}^{N-2}\omega _m\right) \left( \frac{2\pi }{ m\beta }\right) ^{\frac{N}{2}-1}\nonumber \\&\qquad \exp \left( -\,\frac{4 K^2}{m\beta (N-1)}\sum _{l=1}^{N-2}\frac{\sin ^2\left( \omega _l t\right) }{\omega _l^2}\cos ^2\left( \frac{\pi (2j-1)l}{2(N-1)}\right) \sin ^2\left( \frac{\pi l}{2(N-1)}\right) \right) \nonumber \\&\qquad \times \left( \frac{2\pi }{ \beta }\right) ^{\frac{N}{2}-1}\left( \prod _{m'=1}^{N-2}\omega _{m'}\right) ^{-1} \nonumber \\&\quad \quad \exp \left( -\frac{4 K^2}{m\beta (N-1)}\sum _{l'=1}^{N-2}\frac{\left( \cos \left( \omega _{l'} t\right) -1\right) ^2}{\omega _{l'}^2}\cos ^2\left( \frac{\pi (2j-1)l'}{2(N-1)}\right) \sin ^2\left( \frac{\pi l'}{2(N-1)}\right) \right) \nonumber \\&\quad =\exp \left( -\frac{8 K^2}{m\beta (N-1)}\sum _{l=1}^{N-2}\frac{1-\cos \left( \omega _l t\right) }{\omega _l^2} \cos ^2\left( \frac{\pi (2j-1)l}{2(N-1)}\right) \sin ^2\left( \frac{\pi l}{2(N-1)}\right) \right) \end{aligned}$$
(31)

In Eq. 29 we inserted the solution for the Lagrangian configurations \(\eta _j(t)=\eta _j(0)\cos (\omega _j t)+\frac{\dot{\eta }_j(0)}{\omega _j}\sin (\omega _j t)\), in Eq. 30 we used the trigonometric identity \(\sin \alpha -\sin \beta =2\cos \left( \frac{\alpha +\beta }{2}\right) \sin \left( \frac{\alpha -\beta }{2}\right) \).

We can simplify Eq. 31 by approximating the sum in the exponent with its continuum counterpart, according to:

$$\begin{aligned} \frac{\pi l}{N-1}&\rightarrow x \\ \frac{\pi }{N-1}&\rightarrow {\mathrm {d}} x \\ \frac{\pi }{N-1}\sum _{l=1}^{N-2}&\rightarrow \int _0^\pi {\mathrm {d}} x \end{aligned}$$

We then get:

$$\begin{aligned} C^h(t)\simeq&\exp \left\{ -\frac{8K^2}{\pi m\beta }\int _0^\pi {\mathrm {d}} x\frac{1-\cos (\omega (x) t)}{\omega (x)^2}\cos ^2\left( \frac{2j-1}{2}x\right) \sin ^2\left( \frac{x}{2}\right) \right\} \end{aligned}$$
(32)
$$\begin{aligned} =&\exp \left\{ -\frac{2K^2}{\pi m\beta \Omega ^2}\int _0^\pi {\mathrm {d}} x\left[ 1-\cos (\omega (x) t)\right] \cos ^2\left( \frac{2j-1}{2}x\right) \right\} \nonumber \\ =&\exp \left\{ -\frac{2K^2}{\pi m\beta \Omega ^2}\left[ \frac{\pi }{2}-\int _0^\pi {\mathrm {d}} x\cos (\omega (x) t)\cos ^2\left( \frac{2j-1}{2}x\right) \right] \right\} \end{aligned}$$
(33)

where we defined

$$\begin{aligned} \omega (x) = 2\Omega \sin \left( \frac{x}{2}\right) \end{aligned}$$
(34)

The integral term in Eq. 33 can be rewritten in a more compact fashion via a few additional manipulations; the final expression will also allow us to determine the long-time limit of the harmonic ISF. Via a first integration by part we get:

$$\begin{aligned} \int _0^\pi {\mathrm {d}} x \cos \left( \omega (x)t\right) \cos ^2\left( \frac{2j-1}{2} x\right)&= \int _0^\pi {\mathrm {d}} x \cos \left( 2\Omega \sin \left( \frac{x}{2}\right) t\right) \cos ^2\left( \frac{2j-1}{2} x\right) \nonumber \\&=\left[ \cos \left( 2\Omega \sin \left( \frac{x}{2}\right) t\right) \left( \frac{x}{2}+\frac{\sin \left( (2j-1)x\right) }{2(2j-1)}\right) \right] _{x=0}^{x=\pi }\nonumber \\&\quad +\int _{0}^\pi {\mathrm {d}} x2\Omega t \sin \left( 2\Omega t \sin \left( \frac{x}{2}\right) \right) \frac{1}{2} \cos \left( \frac{x}{2}\right) \nonumber \\&\quad \left[ \frac{x}{2}+\frac{\sin \left( (2j-1)x\right) }{2(2j-1)}\right] \nonumber \\&\simeq \frac{\pi }{2}\cos \left( 2\Omega t\right) +\frac{\Omega t}{2}\int _0^\pi {\mathrm {d}} x \sin \left( 2\Omega t \sin \left( \frac{x}{2}\right) \right) \cos \left( \frac{x}{2}\right) x \end{aligned}$$
(35)

where we noticed that the contribution of the last term is negligible for \(j=N/2\gg 1\). Via an additional integration by parts in the second term of Eq. 35 we get:

$$\begin{aligned} \int _0^\pi {\mathrm {d}} x \sin \left( 2\Omega t \sin \left( \frac{x}{2}\right) \right) \cos \left( \frac{x}{2}\right) x&= -\left[ x \cos \left( 2\Omega t\sin \left( \frac{x}{2}\right) \right) \frac{1}{\Omega t}\right] _{x=0}^{x=\pi }\nonumber \\&\quad +\,\int _{0}^{\pi } {\mathrm {d}} x \cos \left( 2\Omega t\sin \left( \frac{x}{2}\right) \right) \frac{1}{\Omega t} \nonumber \\&=\frac{\pi }{\Omega t}\left[ J_0(2\Omega t)-\cos \left( 2\Omega t\right) \right] \end{aligned}$$
(36)

We finally obtain the following analytical approximation for the ISF in the harmonic regime:

$$\begin{aligned} C^h(t) = \exp \left[ -\frac{K^2}{m \beta \Omega ^2}\left( 1 -J_0(2\Omega t)\right) \right] \end{aligned}$$
(37)

which is normalized such that \(C^h(0)=1\). The oscillating pattern of the function is encapsuled in the periodic behavior of the Bessel function \(J_0(\Omega t)\). Within this approximation it is straightforward to determine

$$\begin{aligned} \lim _{t\rightarrow +\infty } C^h(t) = e^{-\frac{K^2}{m\beta \Omega ^2}} \end{aligned}$$
(38)

\(\omega _n\) in the Harmonic Regime

In this appendix we present an exact method for the numerical calculation of the Taylor coefficients of the ISF in the harmonic limit. The series expansion of the correlation function in this regime can be directly computed from the time derivatives of Eq. 37:

$$\begin{aligned} \omega _{2n}^h=\left. \frac{{\mathrm {d}}^{2n} }{{\mathrm {d}} t^{2n}} \exp \left( -\frac{K^2}{m\beta \Omega ^2}(1-J_0(2\Omega t))\right) \right| _{t=0} \end{aligned}$$
(39)

In order to compute these derivatives, we can make use of Faa’ di Bruno’s formula for the 2n-th derivative of a composite function:

$$\begin{aligned} \frac{{\mathrm {d}}^{2n}}{{\mathrm {d}} x^{2n}}f\left( g(x)\right)&= \sum _{{\mathbf {m}}\in {\mathcal {K}}_{2n,2n}} \frac{(2n)!}{m_1!m_2!\cdots m_{2n}!}f^{(m_1+\cdots +m_{2n})}\left( g(x)\right) \prod _{j=1}^{2n}\left( \frac{g^{(j)}(x)}{j!}\right) ^{m_j} \end{aligned}$$
(40)
$$\begin{aligned} {\mathcal {K}}_{n,s}&\equiv \left\{ {\mathbf {m}}\in {\mathbb {N}}_0^{n}: \; \;\sum _{l=0}^{n} l\cdot m_l = s \right\} \end{aligned}$$
(41)

by establishing the correspondences

$$\begin{aligned} x&\leftrightarrow t \\ f(x)&\leftrightarrow \exp (t) \\ g(x)&\leftrightarrow -\frac{K^2}{m\beta \Omega ^2}(1-J_0(2\Omega t)) \end{aligned}$$

We now need to express the different contributions appearing in eqn. 40 in terms of the functions we are interested in. We note

$$\begin{aligned} f^{(m_1+\cdots +m_{2n})}\left( g(x)\right) \leftrightarrow \exp \left( -\frac{K^2}{m\beta \Omega ^2}(1-J_0(2\Omega t))\right) \end{aligned}$$
(42)

and

$$\begin{aligned} g^{(2j)}(x) \leftrightarrow \frac{{\mathrm {d}}^{2j}}{{\mathrm {d}} t^{2j}} \left[ -\frac{K^2}{m\beta \Omega ^2}(1-J_0(2\Omega t))\right] = \frac{K^2}{m\beta \Omega ^2}\frac{{\mathrm {d}}^{2j}}{{\mathrm {d}} t^{2j}}J_0(2\Omega t) \end{aligned}$$
(43)

where we omitted the constant term as in eqn. 40 we are only interested in strictly positive derivatives (\(j\ge 1\)). We can get a closed expression for the derivatives of \(J_0(2\Omega t)\) as follows:

$$\begin{aligned} J_0(2\Omega t')=&\left. J_0\left( 2\Omega (t+t')\right) \right| _{t=0}=\exp \left. \left( 2\Omega t'\frac{{\mathrm {d}}}{{\mathrm {d}} (2\Omega t)}\right) J_0(2\Omega t)\right| _{t=0}\nonumber \\ =&\sum _{j=0}^{+\infty }\frac{{t'}^j}{j!}\left. \frac{{\mathrm {d}}^j}{{\mathrm {d}} t^j}J_0(2\Omega t)\right| _{t=0}=\sum _{j=0}^{+\infty }\frac{(-1/4)^j}{j!^2}(2\Omega t')^{2j} \end{aligned}$$
(44)

where the last identity stems from the Taylor expansion of \(J_0(2\Omega t')\). We can then equate term by term the same powers of \(t'\) in the last identity of Eq. 44 to get \( \forall { j \ge 0}\):

$$\begin{aligned} \left. \frac{{\mathrm {d}}^{2j}}{{\mathrm {d}} t^{2j}}J_0(2\Omega t)\right| _{t=0}&= (-1)^j\frac{(2j)!}{j!^2}\Omega ^{2j}=(-1)^j\left( {\begin{array}{c}2j\\ j\end{array}}\right) \Omega ^{2j} \end{aligned}$$

while the odd derivatives are identically null, as \(J_0\) is an even function. This in particular implies \(m_{2j+1}\equiv 0\)\(\forall j\ge 0\) in Eq. 40. Due to this symmetry, the set in eqn. 41 can be rewritten as

$$\begin{aligned}&\tilde{{\mathcal {K}}}_{2n,s} \equiv \left\{ {\mathbf {m}}\in {\mathbb {N}}_0^{2n}: \; \;\sum _{l=0}^{n}2 l\cdot m_{2l} = s, \;\;m_{2l+1}\equiv 0 \;\;\forall \;\;0\le l\le n-1\right\} \end{aligned}$$

such that

$$\begin{aligned} \omega _{2n}^h=&\left. \frac{{\mathrm {d}}^{2n} }{{\mathrm {d}} t^{2n}} \exp \left( -\frac{K^2}{m\beta \Omega ^2}(1-J_0(2\Omega t))\right) \right| _{t=0} \nonumber \\ =&\sum _{{\mathbf {m}}\in \tilde{{\mathcal {K}}}_{2n,2n}} \frac{(2n)!}{m_2!m_4!\ldots m_{2n}!}\prod _{j=1}^{n}\left( \frac{(-1)^jK^2}{m\beta \Omega ^2(2j)!}\left( {\begin{array}{c}2j\\ j\end{array}}\right) \Omega ^{2j}\right) ^{m_{2j}} \nonumber \\ =&\sum _{{\mathbf {m}}\in \tilde{{\mathcal {K}}}_{2n,2n}} \frac{(2n)!}{m_2!m_4!\ldots m_{2n}!}\left( -\Omega ^2\right) ^{\sum _{j=1}^n j m_{2j}}\left( \frac{K^2}{m\beta \Omega ^2}\right) ^{\sum _{j=1}^n m_{2j}}\prod _{j=1}^n\left( \frac{1}{j!}\right) ^{2m_{2j}}\nonumber \\ =&\left( -\Omega ^2\right) ^{n}\sum _{{\mathbf {m}}\in \tilde{{\mathcal {K}}}_{2n,2n}} \frac{(2n)!}{f({\mathbf {m}})}\left( \frac{K^2}{m\beta \Omega ^2}\right) ^{g({\mathbf {m}})}h({\mathbf {m}}) \end{aligned}$$
(45)

where we defined

$$\begin{aligned} f({\mathbf {m}})\equiv \prod _{j=1}^n m_{2j}! \quad g({\mathbf {m}})\equiv \sum _{j=1}^n m_{2j} \quad h({\mathbf {m}})\equiv \prod _{j=1}^n\left( \frac{1}{j!}\right) ^{2m_{2j}} \end{aligned}$$

The elements of \({\tilde{K}}_{2n, 2n}\) can be determined in a recursive routine as described in [14]. The values in Eq. 45 can then be computed at arbitrarily high orders.

Liouvillian in Non Canonical Coordinates

We derive here a symmetric expression for the Liouvillian in the set of non canonical coordinates \(({\mathbf {r}},{\mathbf {p}})\) used in the work.

$$\begin{aligned} i{\mathcal {L}}&=\sum _{i=1}^{N-2}\left[ -\frac{\partial V({\mathbf {q}})}{\partial q_i}\frac{\partial }{\partial p_i}+\frac{p_j}{m}\frac{\partial }{\partial q_i}\right] = \sum _{i, j=1}^{N-2}\left[ -\frac{\partial r_j}{\partial q_i}\frac{\partial V({\mathbf {q}})}{\partial r_j}\frac{\partial }{\partial p_i}+\frac{p_j}{m}\frac{\partial r_j}{\partial q_i}\frac{\partial }{\partial r_j}\right] \\&= \sum _{i, j =1}^{N-2} \left[ -\frac{\partial (q_j-q_{j-1})}{\partial q_i}\frac{\partial V({\mathbf {q}})}{\partial r_j}\frac{\partial }{\partial p_i}+\frac{p_i}{m}\frac{\partial (q_j-q_{j-1})}{\partial q_i}\frac{\partial }{\partial r_j} \right] \\&= \sum _{i, j =1}^{N-2} \left[ -(\delta _{i,j}-\delta _{i,j-1})\frac{\partial V({\mathbf {q}})}{\partial r_j}\frac{\partial }{\partial p_i}+\frac{p_i}{m}(\delta _{i, j}-\delta _{i,j-1})\frac{\partial }{\partial r_j} \right] \\&= \sum _{i=1}^{N-2} \left[ -\left( \frac{\partial V({\mathbf {q}})}{\partial r_i}-\frac{\partial V({\mathbf {q}})}{\partial r_{i+1}}\right) \frac{\partial }{\partial p_i}+\frac{p_i}{m}\left( \frac{\partial }{\partial r_i}-\frac{\partial }{\partial r_{i+1}}\right) \right] \\&= -\, \left( \frac{\partial V({\mathbf {q}})}{\partial r_1}-\frac{\partial V({\mathbf {q}})}{\partial r_2}\right) \frac{\partial }{\partial p_1}+\frac{p_1}{m} \left( \frac{\partial }{\partial r_1}-\frac{\partial }{\partial r_2}\right) \\&\quad -\left( \frac{\partial V({\mathbf {q}})}{\partial r_2}-\frac{\partial V({\mathbf {q}})}{\partial r_3}\right) \frac{\partial }{\partial p_2}+\frac{p_2}{m} \left( \frac{\partial }{\partial r_2}-\frac{\partial }{\partial r_3}\right) +\cdots +\\&\quad - \left( \frac{\partial V({\mathbf {q}})}{\partial r_{N-2}}-\frac{\partial V({\mathbf {q}})}{\partial r_{N-1}}\right) \frac{\partial }{\partial p_{N-2}}+\frac{p_{N-2}}{m} \left( \frac{\partial }{\partial r_{N-2}}-\frac{\partial }{\partial r_{N-1}}\right) \\&=-\,\frac{\partial V({\mathbf {q}})}{\partial r_1}\left( \frac{\partial }{\partial p_1} - \boxed {\frac{\partial }{\partial p_0}} \right) + \frac{1}{m}(p_1-\boxed {p_0})\frac{\partial }{\partial r_1}+\cdots \\&\quad -\frac{\partial V({\mathbf {q}})}{\partial r_{N-1}}\left( \boxed {\frac{\partial }{\partial p_{N-1}}} - \frac{\partial }{\partial p_{N-2}} \right) + \frac{1}{m}(\boxed {p_{N-1}}-p_{N-2})\frac{\partial }{\partial r_{N-1}} \\&=\sum _{i=1}^{N-1} \left[ -\frac{\partial V(r_i)}{\partial r_i}\left( \frac{\partial }{\partial p_i}-\frac{\partial }{\partial p_{i-1}}\right) + \frac{1}{m}\left( p_i- p_{i-1}\right) \frac{\partial }{\partial r_i} \right] \end{aligned}$$

In the second last line we added to the sum the terms in boxes, as they are identically null due to the boundary conditions considered.

Polynomial Expansion of the Dynamical Correlators

In this appendix we expand the discussion in Sect. 4, in order to gain a deeper insight in the spreading operators. The dynamics of the ISF at finite times can be reconstructed via a recursive calculation of the coefficients of the polynomial expansion in Eq. 12. From the expression of \(i{\mathcal {L}}\) in Eq. 11 we note that only the d.o.f. j and \(j-1\) are effectively involved in \(i{\mathcal {L}}e^{iKr_j}\). At second order (i.e. in \((i{\mathcal {L}})^2 e^{iKr_j}\)) we have to add an additional layer of neighbors around j and so on. The whole chain is finally expected to participate in the dynamics for , while for higher orders additional effects will enter the evolution, due to reflections at the ends of the chain and interference of different wavefronts. In the following we will omit the analysis of these finite size effects; as \(N\gg 1\) the dynamics at short times is not affected by the boundaries.

A graphical representation of the spreading process is given in Fig. 13, where we plot for different particles l on the left and right of j the leading power m in the momentum at order n, such that \((i{\mathcal {L}})^n\sim p_l^m e^{iKr_j}\). We observe a monotonic extension of the non-locality for increasing values of n.

Fig. 13
figure13

Leading power m for momentum of the l-th particle in \((i{\mathcal {L}})^n e^{iKr_j}\)

Once the maximum order of the propagation \(n=n_{max}\) has been fixed, it becomes convenient to quantify boundaries of the domain at this given order. This way it is possible to allocate the minimum amount of CPU memory which is needed for the storage of the coefficients of the tensor \({\mathcal {I}}\).

We present in the following a detailed calculation of the contributions in the Liouville operator controlling the features described above. We leave the remaining part of the Appendix to the readers interested in the most technical aspects of the analysis.

By following the same arguments leading to the definition of Eq. 13, we can define the contribution in \((i{\mathcal {L}})^n e^{iKr_j}\) which connects the central particle of the chain with the farthest on its left:

(46)
(47)
(48)

From Eqs. 47 and 48 we can extract the momenta entering the dynamics at order n:

(49)

and we notice a slight asymmetry w.r.t. j as a consequence of the dynamics propagating slightly faster on the left side of j, as Eq. 11 suggests.

Another relevant factor tuning the numerical complexity of the problem is the maximum power of a general displacement \(r_k\) and momentum \(p_l\) in Eq. 12 at a certain order. This piece of information fixes the dimensions of the indexes \({\mathbf {m}}\) and \(\mathbf {s}\) of \({\mathcal {I}}\). The calculation of this leading order is easily performed for the d.o.f. adjacent to j, as we do not need to account for finite spreading of the dynamics along the chain. Let us focus in particular to the leading power of \(r_k\) for \(|k-j|\le 1\); the relevant contribution in \((i{\mathcal {L}})^n e^{iK r_j}\) is

(50)

with . It is analogously straightforward to identify that the contribution in \((i{\mathcal {L}})^n e^{iKr_j}\) associated to leading power of \(p_l^{s_l}\) for \(l\in \{j, j-1\}\) is

$$\begin{aligned} \left[ \left( p_j-p_{j-1}\right) \frac{\partial }{\partial r_j}\right] ^ne^{iK r_j} \sim p_l^{S_p} e^{i K r_j} \end{aligned}$$
(51)

with \(S_p=n\). For another d.o.f. arbitrarily far from j we could proceed as in Eq. 47-48 by defining suitable chains of differential operators in \((i{\mathcal {L}})^n e^{iKr_j}\). A subset of \({\mathcal {N}}_{r/p}^{(R/L)}<n\) terms must be devoted to the spreading of the interaction till the d.o.f. in scope is reached. Via direct counting in Eqs. 13, 46, 47, 48 respectively we can determine

Once the desired coordinate has been reached, the remaining \(n-{\mathcal {N}}_{r/p}^{(R/L)}\) factors must be combined in specific sequences in order to provide the differentials associated to the leading powers in \(r_k^{M_r}\) or \(p_l^{S_p}\). Such operators are respectively defined as

(52)
(53)

with

Note that Eqs. 52 and 53 are symmetric w.r.t. the index j, in contrast to the expression of the spreading operators themselves. We are now almost able to write the explicit expression of the functions \({\mathcal {M}}_r(n,j,|k-j|)\) and \({\mathcal {S}}_p(n,j,|k-j|)\), returning respectively the maximum power of \(r_k\) and \(p_l\) at order n. This can be accomplished via direct counting in Eqs. 52 and 53. We have to distinguish three different cases; for non null contributes we get:

and therefore

Analogously we can write for the momenta:

The different cases above can be expressed together to get:

Recursive Construction of the Dynamical Tensor

We complete here the derivation of the recursion relations for the construction of the tensor \({\mathcal {I}}^{(n)}\) defined in Eq. 12. The iterative scheme for \(i{\mathcal {L}}_p^\gamma \) has been derived in Eq. 17. We can proceed in the same fashion for the other differential operators in \(i{\mathcal {L}}\). Let us now consider the action

$$\begin{aligned} i{\mathcal {L}}_r^\gamma \equiv \frac{1}{m} (p_\gamma -p_{\gamma -1})\frac{\partial }{\partial r_\gamma }, \quad k_{\min } \le \gamma \le k_{\max } \end{aligned}$$

on a generic monomial of the expansion at order n. We have:

$$\begin{aligned}&\left\{ i{\mathcal {L}}_r^\gamma \left[ r_{k_{\min }}^{m_{k_{\min }}} \cdots r_\gamma ^{m_\gamma } \cdots r_{k_{\max }}^{m_{k_{\max }}} \right] \left[ p_{l_{\min }}^{s_{l_{\min }}} \cdots p_{\gamma -1}^{s_{\gamma -1}} p_\gamma ^{s_\gamma } \cdots p_{l_{\max }}^{s_{l_{\max }}} \right] \right\} e^{iKr_j} \\&\quad = \frac{m_\gamma }{m}\left\{ \left[ r_{k_{\min }}^{m_{k_{\min }}} \cdots r_\gamma ^{m_\gamma -1} \cdots r_{k_{\max }}^{m_{k_{\max }}} \right] \left[ p_{l_{\min }}^{s_{l_{\min }}} \cdots \left( p_{\gamma -1}^{s_{\gamma -1}} p_\gamma ^{s_\gamma +1}-p_{\gamma -1}^{s_{\gamma -1}+1} p_\gamma ^{s_\gamma }\right) \cdots p_{l_{\max }}^{s_{l_{\max }}} \right] \right\} \\&\qquad e^{i K r_j} \end{aligned}$$

We can then infer the following recursion relation:

$$\begin{aligned} {{\mathcal {I}}}^{(n+1)}_{\mathbf {m,s}}\Big \vert _{r,\gamma } \equiv \frac{m_\gamma }{m} \sum _{k=0,1}(-1)^k {\mathcal {I}}^{(n)}_{\mathbf {m}-{\hat{\mathbf {e}}}_\gamma ,\mathbf {s}+{\hat{\mathbf {e}}}_{\gamma -k}} \end{aligned}$$
(54)

We finally have to consider separately the action on \(i{\mathcal {L}}_r^j\) on \(e^{iKr_j}\):

$$\begin{aligned}&\left\{ \left[ r_{k_{\min }}^{m_{k_{\min }}} \cdots r_{k_{\max }}^{m_{k_{\max }}} \right] \left[ p_{l_{\min }}^{s_{l_{\min }}} \cdots p_{j-1}^{s_{j-1}} p_j^{s_j} \cdots p_{l_{\max }}^{s_{l_{\max }}} \right] \right\} i{\mathcal {L}}_r^j e^{iKr_j} \\&=\frac{iK}{m} \left\{ \left[ r_{k_{\min }}^{m_{k_{\min }}}\cdots r_{k_{\max }}^{m_{k_{\max }}} \right] \left[ p_{l_{\min }}^{s_{l_{\min }}} \cdots \left( p_{j-1}^{s_{j-1}} p_j^{s_j+1}-p_{j-1}^{s_{j-1}+1} p_j^{s_j}\right) \cdots p_{l_{\max }}^{s_{l_{\max }}} \right] \right\} e^{iKr_j} \end{aligned}$$

and therefore

$$\begin{aligned} {{\mathcal {I}}}^{(n+1)}_{\mathbf {m,s}}\Big \vert _{r,j, K} \equiv \frac{iK}{m} \sum _{k=0,1}(-1)^k {\mathcal {I}}^{(n)}_{\mathbf {m},\mathbf {s}+{\hat{\mathbf {e}}}_{j-k}} \end{aligned}$$
(55)

The final expression for the dynamical tensor at order \(n+1\) is obtained by summing the contributions of Eqs. 17, 54 and 55 over the subset \(\tilde{\Gamma }\) of the coordinates entering the interaction at order n:

$$\begin{aligned} {{\mathcal {I}}}^{(n+1)}_{\mathbf {m,s}} = \sum _{\gamma \in {\varvec{\tilde{\Gamma }}}}\left( {{\mathcal {I}}}^{(n+1)}_{\mathbf {m,s}}\Big \vert _{p,\gamma }+{{\mathcal {I}}}^{(n+1)}_{\mathbf {m,s}}\Big \vert _{r,\gamma }\right) +{{\mathcal {I}}}^{(n+1)}_{\mathbf {m,s}}\Big \vert _{r,j, K} \end{aligned}$$

with the initial condition

$$\begin{aligned} {\mathcal {I}}^{(0)}_{{\mathbf {m}},\mathbf {s}}= {\left\{ \begin{array}{ll} 1, &{}\quad \text {for}\;\;{\mathbf {m}}= \mathbf {e}_j, \mathbf {s}=\mathbf {0} \\ 0, &{}\quad \text {elsewhere} \end{array}\right. } \end{aligned}$$

Let us call \(n_{\max }\) the maximum order of \({\mathcal {I}}^{(n)}\) we aim to compute. The set \(\varvec{\tilde{\Gamma }}\) can be fixed once for all for any \(n\in \{0,\cdots , n_{\max }\}\) :

Analysis of the Radius of Convergence

The series in Eq. 4 can be formally written for any \(t\in {\mathbb {C}}\). However, the domain ensuring a finite convergence of the expansion is in general bounded to compact disks centered in the origin with an extension bounded by a finite radius of convergence r. In this appendix we briefly discuss the calculation of r and how it connects to the relaxation of the correlation function. Let us rewrite

$$\begin{aligned} C_j(t)&=\sum _{m=0}^{+\infty }c_mt^m\\ c_m&={\left\{ \begin{array}{ll} \frac{\omega _{m}}{m!}&{} m\text { even} \\ 0 &{} m\text { odd} \end{array}\right. } \end{aligned}$$

The Cauchy–Hadamard Theorem relates the radius of convergence r to the series’ coefficients according to

$$\begin{aligned} r=\frac{1}{\limsup _{n\rightarrow +\infty }\root n \of {\left|c_n\right|}}=\lim _{n\rightarrow +\infty }\root n \of {\frac{(2n)!}{\left|\omega _{2n}\right|}} \end{aligned}$$

The calculation of r is therefore possible from the estimate of the limit of the sequence \(\alpha _n\equiv 1 /\root n \of {\left|c_n\right|}\). The fist values of \(\alpha _n\) are shown in the Fig. 14. We can compute exactly the sequence \(\alpha _n\) in the case of an ideal gas, in order to determine the related radius of convergence \(r^{{{\,\mathrm{id}\,}}}\):

$$\begin{aligned} \alpha _n^{{{\,\mathrm{id}\,}}}&= \frac{1}{\root n \of {\left|c_n\right|}}= \root n \of {\frac{(2n)!}{\left|\omega _{2n}^{{{\,\mathrm{id}\,}}}\right|}}= \root n \of {\frac{(2n)!}{\left( \frac{4K^2}{m\beta }\right) ^n\frac{(2n)!}{4^n n!}}}=\frac{m\beta }{K^2}\root n \of {n!}=\simeq \frac{m\beta }{K^2}\root n \of {\sqrt{2\pi n}\left( \frac{n}{e}\right) ^n}\simeq \frac{m\beta }{eK^2}n \end{aligned}$$
Fig. 14
figure14

First terms of the sequence \(\alpha _n\), for four values of the inverse temperatures \(\beta \); inset: estimate of the radius of convergence as a function \(\beta \) via \(\alpha _{16}\)

where we approximated the factorial via the Stirling’s approximation. From the last identity we get that \(r^{{{\,\mathrm{id}\,}}}=+\,\infty \). It then follows that a sufficient condition for a correlation function not to decay as a Gaussian is that its radius of convergence is finite. We can compare the two regimes of low and high \(\beta \) the sequence of the \(\alpha _n\) for an ideal gas, for the nonlinear interaction and for the linearized potential (harmonic limit). The results are shown in the Fig. 15.

Fig. 15
figure15

Sequence of \(\alpha _n\) for the ideal gas limit (blue stars), for the nonlinear dynamics from the MD simulations (red dots) and for the harmonic regime (green diamonds) (Color figure online)

We can see that for both the temperatures the sequence of the \(\alpha _n\) for the nonlinear and harmonic dynamics tends to a finite plateau; the ideal gas case shows instead a linear increase as expected. As discussed above, the finite asymptotics for the dynamic regimes is sufficient to predict a non-Gaussian relaxation.

Estimate of the High-order Kernel Coefficients

In this appendix we derive the estimate Eq. 24. We can expand Eq. 19 as

$$\begin{aligned} \kappa _{2n-2}^I =&\, \omega ^I_{2n} + \sum _{m=1}^{n-1} \chi ^{(1)}_{m} \omega ^I_{2(n-m)}\omega ^I_{2m} + \sum _{m=1}^{n-1}\sum _{m'=1}^{m-1} \chi ^{(2)}_{m,m'} \omega ^I_{2(n-m)}\omega ^I_{2(m-m')}\omega ^I_{2m'} + \cdots \nonumber \\ =&\,\omega ^I_{2n} \left( 1 + \sum _{m=1}^{n-1} \chi ^{(1)}_{m} \frac{\omega ^I_{2(n-m)}\omega ^I_{2m}}{\omega ^I_{2n}}+ \sum _{m=1}^{n-1}\sum _{m'=1}^{m-1} \chi ^{(2)}_{m,m'} \frac{\omega ^I_{2(n-m)}\omega ^I_{2(m-m')}\omega ^I_{2m'}}{\omega ^I_{2n}} + \cdots \right) \end{aligned}$$
(56)

with suitable coefficients \(\chi ^{(j)}_{m_1,\ldots , m_j}\). We can explicitly show that the non-constant contributions in Eq. 56 vanish via Eq. 23; starting from the first terms we get:

$$\begin{aligned} \frac{\omega ^I_{2(n-m)}\omega ^I_{2m}}{\omega ^I_{2n}}&=(1-C_{\beta })\frac{(2(n-m))!}{(n-m)!}\frac{(2m)!}{m!}\frac{n!}{(2n)!} \le 2(1-C_{\beta })\frac{(2(n-1))!}{(n-1)!}\frac{n!}{(2n)!} \nonumber \\&\le 2(1-C_{\beta })\frac{n}{2n(2n-1)} \xrightarrow { n \rightarrow \infty } 0 \end{aligned}$$
(57)

where in the second line we noticed that \(m=1\) corresponds to a maximum of the function

$$\begin{aligned} r_{n,m} \equiv \frac{(2n-2m)!}{(n-m)!}\frac{(2m)!}{m!} \end{aligned}$$
(58)

This is shown by evaluating the increment

$$\begin{aligned} r_{n,m+1} =&\frac{(2n-2m-2)!}{(n-m-1)!}\frac{(2m+2)!}{(m+1)!}\\ =&\frac{(2n-2m)!}{(2n-2m)(2n-2m-1)}\frac{n-m}{(n-m)!}\frac{1}{m!(m+1)} (2m)!(2m+1)(2m+2) \\ =\,&r_{n,m} \frac{n-m}{(2n-2m)(2n-2m-1)} \frac{(2m+1)(2m+2)}{(m+1)} \\ =\,&r_{n,m}\frac{2m+1}{2n-2m-1} = r_{n,m}f_{n,m} \end{aligned}$$

where we defined

$$\begin{aligned} f_{n,m}\equiv \frac{2m+1}{2n-2m-1} \end{aligned}$$
(59)

We can see that the discrete derivative of \(f_{n,m}\) w.r.t. m is always positive:

$$\begin{aligned} f_{n, m+1}-f_{n,m} = \frac{4n}{(1+2m-2n)(3+2m-2n)} \ge \frac{4n}{(3+2m-2n)^2} \ge 0 \end{aligned}$$

The denominator of the last identity is always defined for \(m,n\in {\mathbb {N}}\). It then follows that \(f_{n,m}\) is increasing with m and

$$\begin{aligned} f_{n,m} = 1 \iff m = \frac{n-1}{2} \end{aligned}$$
(60)

Thus, \(r_{n,m+1} \le r_{n,m}\) if \(m\le (n-1)/2\) and vice-versa. Hence, the maximum values of \(r_{n,m}\) are obtained at the boundaries of the admitted values of m, i.e. \(m=1\) and \(m=n-1\), as \(r_{n,1}=r_{n, m-1}\).

Long-time Decay of the Memory Kernel

In this section we estimate the long-time tail of \(K_j^I(t)\) in Fig. 11, in the high temperature regime. From Fig. 10 and Appendix A.9 we know that \(\kappa _{2n}^I\simeq \omega _{2n+2}^I\) for \(n\gtrsim 40\). We can extend this estimate for any value of n and sum the resulting function, by assuming that for \(t\gg 1\) the higher series coefficients matter the most. The first issue that needs to be checked is whether the resulting series is convergent. From the ratio test we get:

$$\begin{aligned} \lim _{n\rightarrow +\infty } \left|\frac{t^{2n+2}\kappa _{2n+2}^I}{(2n+2)!}\frac{(2n)!}{t^{2n}\kappa _{2n}^I}\right|=\,&t^2\lim _{n\rightarrow +\infty } \left|\frac{a^{n+2}H_{2n+4}}{(2n+2)!}\frac{(2n)!}{a^{n+1}H_{2n+2}}\right|\\ =\,&at^2\lim _{n\rightarrow +\infty } \left|\frac{(-2)^{n+2}(2n+3)!!(2n)!}{(2n+2)!(-2)^{n+1}(2n+1)!!} \right|\\ =\,&2at^2 \lim _{n\rightarrow +\infty }\left|\frac{\frac{(2(n+2))!}{2^{n+2}(n+2)!}(2n)!}{(2n+2)!\frac{(2(n+1))!}{2^{n+1}(n+1)!}} \right|\\ =\,&2at^2 \lim _{n\rightarrow +\infty } \frac{(2n+4)(2n+3)}{2(2n+2)(2n+1)(n+2)} at^2\lim _{n\rightarrow +\infty }\frac{1}{n} =0 \end{aligned}$$

This shows that the radius of convergence of the series obtained with the approximation \(\kappa _{2n}\simeq \kappa _{2n}^I\) is infinite; we can therefore sum the approximation of the tail for any value of n:

$$\begin{aligned} K_j(t) \simeq _{t\gg 1} \sum _{n=0}^{+\infty }\frac{t^{2n}}{(2n)!} \omega ^I_{2n+2}&= \sum _{n=0}^{+\infty }\frac{t^{2n}}{(2n)!} \left[ (1-C_\beta )a^{n+1}H_{2n+2}+\delta _{n+1,0}C_\beta \right] \nonumber \\&=\sum _{n=0}^{+\infty }\frac{t^{2n}}{(2n)!}\left[ (1-C_\beta )a^{n+1}(-2)^{n+1}(2n+1)!!\right] \nonumber \\&= (1-C_\beta )\sum _{n=0}^{+\infty }\frac{t^{2n}}{(2n)!}a^{n+1}(-2)^{n+1}\frac{2^{n+1}\Gamma (n+\frac{3}{2})}{\sqrt{\pi }}\nonumber \\&= (1-C_\beta )\sum _{n=0}^{+\infty }(-1)^{n+1}\frac{t^{2n}}{(2n)!}a^{n+1}\frac{4^{n+1}}{\sqrt{\pi }}\frac{\sqrt{\pi }(2n+2)!}{4^{n+1}(n+1)!}\nonumber \\&= (1-C_\beta )a\sum _{n=0}^{+\infty }(-1)^{n+1}\left( at^2\right) ^n\frac{(2n+2)(2n+1)}{(n+1)!} \nonumber \\&= (1-C_\beta )a\sum _{n=0}^{+\infty }(-1)^{n+1}\left( at^2\right) ^n\frac{(2n+2)(2n+1)}{(n+1)!}\nonumber \\&= 2(1-C_\beta )a e^{-at^2}(2at^2-1) \end{aligned}$$
(61)

which means

$$\begin{aligned} \frac{K_j(t)}{K_j(0)} \simeq _{t\gg 1} e^{-at^2}(1-2at^2) \end{aligned}$$
(62)

From the result above we can choose the following ansatz for the long time limit:

$$\begin{aligned} K_j(t)\simeq _{t\gg 1} A t^2 e^{-Bt^2} \end{aligned}$$

The coefficients A and B can be fixed for each temperature by extending the sum in Eq. 25 continuously with continuous derivative from a time \(t^*=t^*(\beta )\) before its divergence. The parameters A and B are then fixed by

$$\begin{aligned} A&= \frac{K^*}{{t^*}^2 \exp \left( \frac{{K'}^*t^* - 2 K^* }{2K^* }\right) } \\ B&= \frac{2K^*-{K'}^* t^*}{2K^* {t^*}^2} \end{aligned}$$

where

$$\begin{aligned} K^*&\equiv K_{N_{max}}(t^*) \\ {K'}^*&\equiv \sum _{n=0}^{N_{max}} \frac{\kappa _{2n}}{(2n-1)!}{t^*}^{2n-1} \end{aligned}$$

We can additionally check that the series obtained by summing the first neglected terms in the expansion of the kernel Eq. 56 is convergent. Although this is far from proving the boundedness of the infinitely many orders neglected, it represents a necessary condition. The time-series of the first order correction is given by

$$\begin{aligned} K_1^I(t)\equiv&\sum _{n=0}^{+\infty } \frac{t^{2n}}{(2n)!}\sum _{m=1}^n\chi _m^{(1)}\omega _{2(n-m+1)}^I\omega _{2m}^I\nonumber \\ =&\sum _{n=0}^{+\infty }\frac{t^{2n}}{(2n)!}\sum _{m=1}^n\chi _m^{(1)}\left[ (1-C_\beta )a^{n-m+1}H_{2(n-m+1)}\right] \left[ (1-C_\beta )a^m H_{2m}\right] \nonumber \\ =&-\, (1-C_\beta )^2\sum _{n=0}^{+\infty } a^{n+1}\frac{t^{2n}}{(2n)!}\sum _{m=1}^n (-2)^{n-m+1}\left( 2(n-m)+1\right) !!(-2)^m(2m-1)!!\nonumber \\ =&-\,(1-C_\beta )^2\sum _{n=0}^{+\infty }(-2a)^{n+1}\frac{t^{2n}}{(2n)!}\sum _{m=1}^n\frac{\left( 2(n-m+1)\right) !}{2^{n-m+1}(n-m+1)!}\frac{(2m)!}{2^m m!} \nonumber \\ =&-\, (1-C_\beta )^2a\sum _{n=0}^{+\infty }\frac{\left( -at^2\right) ^n}{(2n)!}\sum _{m=1}^n\frac{\left( 2(n-m+1)\right) !}{(n-m+1)!}\frac{(2m)!}{m!}\equiv \sum _{n=0}^{+\infty }a_n \end{aligned}$$
(63)

where in the second line we used the property \(\chi ^{(1)}_m=-1\;\;\forall m\), as it can be explicitly checked from the recursive reconstruction of the relation Eq. 19 till high orders. Following the same argument presented in Appendix A.9, we can determine

$$\begin{aligned} \max _{m\in \left\{ 1,n\right\} }\frac{\left( 2(n-m+1)\right) !}{(n-m+1)!}\frac{(2m)!}{m!} = \left. \frac{\left( 2(n-m+1)\right) !}{(n-m+1)!}\frac{(2m)!}{m!}\right| _{\begin{array}{c} m=1\\ m=n \end{array}} = 2\frac{(2n)!}{n!} \end{aligned}$$

We can then check the convergence of the series in Eq. 63 via the direct comparison test:

$$\begin{aligned} \left|a_n \right|\le&\left|-(1-C_\beta )^22a\frac{ (-at^2)^n}{(2n)!}n\frac{(2n)!}{n!}\right|\equiv \left|b_n \right|\\ =&\left|-(1-C_\beta )^22a\frac{ (-at^2)^n}{(n-1)!}\right|, \quad n\ge 1 \end{aligned}$$

Via the ratio test we can notice that the series \(\sum _{n=0}^{+\infty } b_n\) is convergent:

$$\begin{aligned} \lim _{n\rightarrow +\infty }\left|\frac{b_{n+1}}{b_n}\right|= \lim _{n\rightarrow +\infty } \left|-a t^2\right|\frac{(n-1)!}{n!} = \left|-a t^2\right|\lim _{n\rightarrow +\infty }\frac{1}{n} =0 \end{aligned}$$

which then proves the convergence of Eq. 63.

Management of High-dimensional Tensors

The components of \({\mathcal {I}}_{\mathbf {ms}}^{(n)}\) can be stored in a one-dimensional pointer in row-major order. For example the indeces of a \(3\times 3\times 3\) tensor would be sorted as

$$\begin{aligned} \left\{ 000,\quad 001,\quad 002,\quad 010,\quad 011,\quad 012,\quad 021,\ldots ,222\right\} \end{aligned}$$

and by mapping the sequence of the indexes into integer numbers according to

$$\begin{aligned} (i, j, k)\rightarrow i \cdot 10^2 + j\cdot 10 +k \end{aligned}$$

we get a monotonically increasing sequence. In this case, let us define a general label

$$\begin{aligned} \mathbf {v} = \{v_0, \cdots , v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}\} \end{aligned}$$

of a tensor \({\mathcal {A}}\) of dimension \(\mathbf {D} =\left\{ D_0, \cdots , D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}\right\} \in {\mathbb {N}}^{n_{ind}}\) and complex images; \(D_j \equiv {{\,\mathrm{car}\,}}\left( \left\{ v_j\right\} \right) \) is the cardinality of the set of the allowed values of the \(v_j\). In our case \(v_j\in {\mathbb {N}}\cup \{0\}\) is a power of a certain coordinate of the system; it follows that \(v_j\in \{0,1,\cdots , D_j-1\}\). Any entry of the tensor \({\mathcal {A}}_{\mathbf {v}}\in {\mathbb {C}}\) can be mapped in a one dimensional array via the function

$$\begin{aligned} {{\,\mathrm{map}\,}}(\mathbf {v}, \mathbf {D}, {{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}})=v_0\prod _{\beta _0=1}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1} D_{\beta _0} + v_1\prod _{\beta _1=2}^{n_{ind}-1} D_{\beta _1} + \cdots v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1} = \sum _{\alpha =0}^{n_{ind}-1} v_\alpha \prod _{\beta _\alpha =\alpha +1}^{n_{ind}-1} D_{\beta _\alpha }\nonumber \\ \end{aligned}$$
(64)

It is straightforward to determine the map of last index of the tensor \(\mathbf {v}_{last} \equiv \{D_0-1, D_1-1, \cdots , D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}-1\}\):

$$\begin{aligned} {{\,\mathrm{map}\,}}(\mathbf {v}_{last}, \mathbf {D}, {{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}})=&\sum _{\alpha =0}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1} (D_\alpha -1)\prod _{\beta _\alpha =\alpha +1}^{n_{ind}-1} D_{\beta _\alpha } =\left[ (D_0-1)D_1 D_2\cdots D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}\right] \\&\quad +\,\left[ (D_1-1)D_2\cdots D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}\right] +\cdots + D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}-1\\ =&\prod _{\gamma =0}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1} D_\gamma -1 \equiv M_{\mathbf {D}} \end{aligned}$$

The identity follows as the sum is telescopic.

In the implementation of the recursion relations in Sect. 4 controlling \((i{\mathcal {L}})^n e^{iKr_j}\rightarrow (i{\mathcal {L}})^{n+1}e^{iK r_j}\) we are interested in integer increments (let us say by \(l\in {\mathbb {N}}\)) of a generic i-th ’column’ of the coefficients tensor: \({\mathcal {I}}^{(n)}_{\mathbf {ms}}\rightarrow I^{(n)}_{\mathbf {ms}+l{\hat{\mathbf {e}}_i}}\). By imposing \(v_i\equiv l \in \{0, \cdots , D_i-1\}\), it directly follows from Eq. 64:

$$\begin{aligned} {{\,\mathrm{map}\,}}(\{v_0,\cdots , v_{i-1},l,v_{i+1},\cdots , v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1} \}, \mathbf {D}, {{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}) = \sum _{\begin{array}{c} \alpha =0\\ \alpha \ne i \end{array}}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1} v_\alpha \prod _{\beta _\alpha =\alpha +1}^{n_{ind}-1} D_{\beta _\alpha } + l \prod _{\beta _i=i+1}^{n_{ind}-1} D_{\beta _{i}} \end{aligned}$$

It is moreover possible to determine the inverse map that, given an entry of a one-dimensional mapping and the list of the related dimensions, returns the multi-index associated to that component. For this we can proceed recursively, from the extraction of the last index \(v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}\) backwards. From Eq. 64 we get:

$$\begin{aligned} {{\,\mathrm{n_{{{\,\mathrm{map}\,}}}}\,}}\equiv&{{\,\mathrm{map}\,}}(\mathbf {v}, \mathbf {D}, {{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}) = \left( \sum _{\alpha =0}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-2}v_{\alpha }\prod _{\beta _\alpha =\alpha +1}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}D_{\beta _\alpha }\right) +v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}\nonumber \\ =&\left( \sum _{\alpha =0}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-2}v_{\alpha }\prod _{\beta _\alpha =\alpha +1}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-2}D_{\beta _\alpha }\right) D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}+v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1} \end{aligned}$$
(65)
$$\begin{aligned} v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}=&{{\,\mathrm{n_{{{\,\mathrm{map}\,}}}}\,}}\mod D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1} \end{aligned}$$
(66)

Eq. 66 follows from Eq. 65 being \(v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}<D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}\). We can proceed with the extraction of the second last component \(v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-2}\) via the knowledge of \(v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}\):

$$\begin{aligned} A_{n_{ind}-1}\equiv \frac{{{\,\mathrm{n_{{{\,\mathrm{map}\,}}}}\,}}-v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}}{D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1}}&= \left( \sum _{\alpha =0}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-3} v_\alpha \prod _{\beta _\alpha =\alpha +1}^{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-3} D_{\beta _\alpha }\right) D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-2} + v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-2} \\ v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-2}&= A_{n_{ind}-1} \mod D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-2} \end{aligned}$$

A recursion relation can then be established \(\forall \; \;i\in \{0,\cdots , {{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-1\}\):

$$\begin{aligned} v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-i-1}&=A_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-i}\mod D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-i-1} \\ A_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-i-1}&= \frac{A_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-i} - v_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-i-1}}{D_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}-i-1}} \end{aligned}$$

with initial condition \(A_{{{\,\mathrm{n_{{{\,\mathrm{ind}\,}}}}\,}}}={{\,\mathrm{n_{{{\,\mathrm{map}\,}}}}\,}}\).

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Amati, G., Meyer, H. & Schilling, T. Memory Effects in the Fermi–Pasta–Ulam Model. J Stat Phys 174, 219–257 (2019). https://doi.org/10.1007/s10955-018-2207-6

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Keywords

  • FPU model
  • FPU system
  • Fermi–Pasta–Ulam model
  • Fermi–Pasta–Ulam system
  • Coarse-graining
  • Generalized Langevin equation
  • Intermediate scattering function
  • Memory kernel
  • Harmonic chain