Introduction

The study of discrete one-dimensional chains of particles with the nearest-neighbor interactions provides insight to the dynamics of various physical and biological systems, such as crystals, wave systems, and biopolymers [1,2,3]. In the thermal equilibrium state, such nonlinear chains can be described by the canonical Gibbs measure [4] with the Hamiltonian

$$H=\sum_j \frac{p_j^2}{2}+\frac{(q_j-q_{j+1})^2}{2}+V(q_j-q_{j+1}),$$ (1)

where $p_j$ and $q_j$ are the momentum and the displacement from the equilibrium position of the $j$-th particle, respectively, $V(q_j-q_{j+1})$ is the anharmonic part of the potential, and the mass of each particle and the linear spring constant are scaled to unity. In this article, we only consider the potentials of the restoring type, i.e., the potentials for which the Gibbs measure exists. In order to study interactions of waves in such systems, one usually introduces the canonical complex normal variables $a_k$ via

$$a_k=\frac{P_k-\imath \omega _kQ_k}{\sqrt{2\omega _k}},$$ (2)

where $P_k$ and $Q_k$ are the Fourier transforms of $p_j$ and $q_j$, respectively, and $\omega _k=2\sin(\pi k/N)$ is the linear dispersion relation of the waves represented by $a_k$. In terms of the $a_k$, the Hamiltonian (1) becomes

$$H=\sum\omega _k\vert a_k\vert^2+V(a),$$ (3)

where $V(a)$ is the combination of various products of $a_k$ and $a_k^*$ corresponding to various wave-wave interactions. If the potential in Eq. (1) is harmonic, i.e., $V\equiv 0$, then $a_k$ correspond to ideal, free waves, which have no energy exchanges among different $k$ modes. In thermal equilibrium, the Boltzmann distribution $\exp(-\theta^{-1}\sum\omega _k\vert a_k\vert^2)$ with temperature $\theta$, gives rise to the following properties of free waves

$$\langle a_k^*a_l\rangle = n_k\delta^k_l,$$ (4)

$$\langle a_ka_l\rangle = 0,$$ (5)

for any $k$ and $l$, where $n_k\equiv\langle \vert a_k\vert^2\rangle =\theta/\omega _k$ is the power spectrum. If the anharmonic part of the potential is sufficiently weak, then corresponding waves $a_k$ remain almost free, and Eqs. (4) and (5) would be approximately satisfied in the weakly nonlinear regime. However, when the nonlinearity becomes stronger, waves $a_k$ become strongly correlated, and, in general, the correlations between waves [Eq. (5)] no longer vanish. In particular, $\langle a_ka_{N-k}\rangle \neq 0$, as will be shown below. Naturally, the question arises: can the strongly nonlinear system in thermal equilibrium still be viewed as a system of almost free waves in some statistical sense? In this article, we address this question with an affirmative answer: it turns out that the system (1) can be described by a complete set of renormalized canonical variables $\tilde{a}_k$, which still possess the wave properties given by Eqs. (4) and (5) with a renormalized linear dispersion. The waves that correspond to these new variables $\tilde{a}_k$ will be referred to as renormalized waves. Since these renormalized waves possess the equilibrium Rayleigh-Jeans distribution [5] and vanishing correlations between waves, they resemble free, non-interacting waves, and can be viewed as statistical normal modes. Furthermore, it will be demonstrated that the renormalized linear dispersion for these renormalized waves has the form $\tilde{\omega}_k=\eta(k)\omega _k$, where $\eta (k)$ is the linear frequency renormalization factor, and is independent of $k$ as a consequence of the Gibbs measure.

In our method, the construction of the renormalized variables $\tilde{a}_k$ does not depend on a particular form or strength of the anharmonic potential, as long as it is of the restoring type with only the nearest neighbor interactions, as in Eq. (1). Therefore, our approach is non-perturbative and can be applied to a large class of systems with strong nonlinearity. However, in this article, we will focus on the $\beta$-FPU chain to illustrate the theoretical framework of the renormalized waves. We will verify that $\tilde{a}_k$ effectively constitute normal modes for the $\beta$-FPU chain in thermal equilibrium by showing that (i) the theoretically obtained renormalized linear dispersion relationship is in excellent agreement with its dynamical manifestation in our numerical simulation, and (ii) the equilibrium distribution of $\tilde{a}_k$ is still a Rayleigh-Jeans distribution and $\tilde{a}_k$'s are uncorrelated. Note that similar expressions for the renormalization factor $\eta$ have been previously discussed in the framework of an approximate virial theorem [6] or effective long wave dynamics via the Zwanzig-Mori projection [7]. However, in our theory, the exact formula for the renormalization factor is derived from a precise mathematical construction of statistical normal modes, and is valid for all wave modes $k$ -- no longer restricted to long waves.

Next, we address how renormalization arises from the dynamical wave interaction in the $\beta$-FPU chain. We will show that the $\beta$-FPU chain can be effectively described as a four-wave interacting Hamiltonian system of the renormalized resonant waves $\tilde{a}_k$. We will study the resonance structure of the $\beta$-FPU chain and find that most of the exact resonant interactions are trivial, i.e., the interactions with no momentum exchange among different wave modes. In what follows, the renormalization of the linear dispersion will be explained as a collective effect of these trivial resonant interactions of the renormalized waves $\tilde{a}_k$. We will use a self-consistency argument to find an approximation, $\eta _{sc}$, of the renormalization factor $\eta$. As will be seen below, the self-consistency argument essentially is of a mean-field type, i.e., the renormalization arises from the scattering of a wave by a mean-background of waves in thermal equilibrium via trivial resonant interactions. We note that our self-consistency, mean-field argument is not limited to the weak nonlinearity. Very good agreement of the renormalization factor $\eta$ and its dynamical approximation $\eta _{sc}$ -- for weakly as well as strongly nonlinear waves -- confirms that the renormalization is, indeed, a direct consequence of the trivial resonances.

We will further study the properties of these renormalized waves by investigating how long these waves are coherent, i.e., what their frequency widths are. Therefore, we consider near-resonant interactions of the renormalized waves $\tilde{a}_k$, i.e., interactions that occur in the vicinity of the resonance manifold, since most of the exact resonant interactions are trivial, i.e., with no momentum exchanges, and they, cannot effectively redistribute energy among the wave modes.

We will demonstrate that near-resonant interactions of the renormalized waves $\tilde{a}_k$ provide a mechanism for effective energy exchanges among different wave modes. Taking into account the near-resonant interactions, we will study analytically the frequency peak broadening of the renormalized waves $\tilde{a}_k$ by employing a multiple time-scale, statistical averaging method. Here, we will arrive at a theoretical prediction of the spatiotemporal spectrum $\vert\hat{a}_k(\omega )\vert^2$, where $\hat{a}_k(\omega )$ is the Fourier transform of the normal variable $\tilde{a}_k(t)$, and $\omega$ is the frequency. The predicted width of frequency peaks is found to be in good agreement with its numerically measured values.

In addition, for a finite $\beta$-FPU chain, we will mention the consequence, to the correlation times of waves, of the momentum exchanges that cross over the first Brillouin zone. This process is known as the umklapp scattering in the setting of phonon scattering [8]. Note that, in the previous studies [9] of the FPU chain from the wave turbulence point of view, the effects arising from the finite nature of the chain were not taken into account, i.e., only the limiting case of $N\rightarrow\infty$, where $N$ is the system size, was considered.

The article is organized as follows. In Section II, we discuss a chain of particles with the nearest-neighbor nonlinear interactions. We demonstrate how to describe a strongly nonlinear system as a system of waves that resemble free waves in terms of the power spectrum and vanishing correlations between waves. We show how to construct the corresponding renormalized variables with the renormalized linear dispersion. In Section III, we rewrite the $\beta$-FPU chain as an interacting four-wave Hamiltonian system. We study the dynamics of the chain numerically and find excellent agreement between the renormalized dispersion, obtained analytically and numerically. In Section IV, we describe the resonance manifold analytically and illustrate its controlling role in long-time averaged dynamics using numerical simulation. In Section V, we derive an approximation for the renormalization factor for the linear dispersion using a self-consistency condition. In Section VI, we study the broadening effect of frequency peaks and predict analytically the form of the spatiotemporal spectrum for the $\beta$-FPU chain. We provide the comparison of our prediction with the numerical experiment. We present the conclusions in Section VII.