Introduction

One of the very first uses of electronic computing machines was Fermi, Pasta, and Ulam's simulation of wave propagation in a weakly nonlinear lattice model [#!ef:snp!#]. They were expecting to observe that the weak coupling of the normal modes of the system would induce a redistribution of energy from an initial large-scale excitation to an equal sharing (equipartition) of energy among all normal modes after some time. As is well known, they were instead surprised to see the system display regular behavior characteristic of integrable systems, with the initial state recurring on a rather short time scale. This discovery shifted attention to its explanation and ramifications [#!ei:nwsc!#,#!acn:smp!#] for several decades. In the last two decades, however, the Fermi-Pasta-Ulam (FPU) model has once again been utilized as a test model for numerically illustrating and exploring standard concepts in statistical mechanics [#!lc:fpupr!#]. The peculiar near-integrable behavior observed by Fermi, Pasta, and Ulam is characteristic of their model only for systems which are sufficiently small in size and energy. There exist well-defined regimes for which the FPU model is weakly nonlinear but stochastic [#!mcs:psgst!#,#!rl:etnlh!#,#!pp:esfpu!#], and it is in these regimes that one can hope to connect the outcome of direct numerical simulations with statistical mechanical concepts [#!lc:fpupr!#], such as relaxation to equipartition [#!lc:fpupr!#,#!jdl:ftetl!#,#!vvm:cbfce!#,#!pp:swtsr!#], entropy production [#!lc:fpupr!#,#!jdl:ueeoc!#,#!jdl:ftetl!#,#!cyl:wodls!#,#!vvm:cbfce!#,#!pp:swtsr!#], chaos as manifested by positive Lyapunov exponents [#!ca:qhpsf!#,#!lc:fpupr!#,#!td:memle!#,#!pp:esfpu!#,#!dls:lecfp!#], universal behavior of statistical functions [#!jdl:ueeoc!#,#!jdl:ftetl!#,#!cyl:wodls!#,#!pp:swtsr!#,#!dls:lecfp!#] and virial relations [#!ca:nmaps!#,#!ca:nsbfp!#]. The reason for using the FPU model for this purpose is that it is one of the simplest and most natural one-dimensional nonlinear models for statistical mechanics which can be conceived. An interesting alternative of comparable simplicity which has been the subject of recent research is the truncated Burgers model [#!rva:hssrc!#,#!am:smtbh!#,#!ajm:rsbtb!#].

Our intent is to use the FPU model to scrutinize a ``weak turbulence'' (WT) theory, a nonequilibrium statistical mechanical theory which attempts to describe the dynamical energy transfer among normal modes in a weakly nonlinear, dispersive, extended system [#!djb:nlrwd!#,#!jb:rwc!#,#!kh:onlet1!#,#!bbk:pt!#,#!acn:wtaai!#,#!acn:wti!#,#!vez:kst1!#]. The theory has been developed over the last four decades to describe the energy transfer in wave dynamics primarily in fluids and plasmas [#!acn:wtaai!#,#!acn:wti!#,#!vez:kst1!#], among other novel applications such as semiconductors [#!yvl:qwtas!#,#!yvl:ffsqb!#,#!yvl:slks!#].

These systems are generally too complex for effective comparison of the weak turbulence theory with direct numerical simulations. Only in recent years has a simple one-dimensional model with features representative of such fluid systems been explored by Cai, Majda, McLaughlin, and Tabak [#!dc:sbdwt!#,#!dc:dwtod!#,#!ajm:odmdw!#] to examine the assumptions underlying WT theory. We propose to apply the FPU model for a similar purpose, though the issues on which we focus are distinct. The implications of our studies for the framework of WT theory will be taken up in other works [#!prk:awttf!#].

Here we will present the content of our findings as they inform the relaxation process in the FPU model in the stochastic but weakly nonlinear regime. We will restrict attention to the $\alpha$ version of FPU model, which has purely quadratic nonlinearity in the equations of motion (Section 2). The $\beta$ version (with cubic nonlinearity) seems to be the subject of more work [#!ca:nmaps!#,#!jdl:ftetl!#,#!xl:dame!#,#!vvm:cbfce!#,#!pp:esfpu!#,#!pp:swtsr!#], but the $\alpha$ version has attributes which make it more suitable for a first test case for WT theory. Most of the previous statistical mechanical work concerning the FPU models of which we are aware focuses primarily on computing particular statistical measures of the process, such as the time until equipartition is reached [#!jdl:ftetl!#,#!pp:swtsr!#], the Lyapunov exponents characterizing the degree of chaos [#!lc:fpupr!#,#!td:memle!#,#!pp:esfpu!#,#!dls:lecfp!#], or more exotic quantities characterizing the geometry of the trajectories [#!ca:qhpsf!#,#!mcs:psgst!#]. Another recent line of research has been tracing the path of energy transfer starting from a small set of excited modes [#!gc:retip!#,#!ky:mioda!#,#!ky:preei!#]. Because the WT theory has the potential to describe the process of energy transfer in the system from beginning to end, we have instead sought to characterize the entire evolution of the energy spectrum from large-scale excitation to eventual equipartition. We will consider the energy transfer in spectral terms, in contrast to the physical space viewpoint developed for the $\beta$-FPU model by Lichtenberg and coworkers [#!vvm:cbfce!#].

The energy spectrum in the $\alpha$-FPU model approaches equilibrium through a series of qualitatively distinct phases which we illustrate in Section 3. At the initial time, the energy is concentrated in a small set of low-wavenumber modes. This energy then proceeds to higher wavenumbers first through a standard superharmonic cascade, and then shifts to a nonlocal transfer of energy from low wavenumbers to a band of intermediate wavenumber modes. The energy in this intermediate wavenumber band then rolls back through an inverse cascade to lower wavenumbers again. This process then creates approximate equipartition only over a set of modes extending up to a cutoff wavenumber, beyond which the energy content falls off exponentially rapidly [#!ac:tdsrb!#,#!lg:peels!#]. We refer to the location of the transition between the flat and rapidly decaying parts of the energy spectrum as the ``knee.''

After presenting this pictorial ``life history'' of a large-scale excitation in the $\alpha$-FPU system, we present in Section 4 some specific quantitative predictions of WT theory and compare them with the numerical results. At the coarsest level, WT theory suggests the presence of two nonlinear time scales. Over the first nonlinear time scale, energy is exchanged through triads of modes which remain resonant over this time scale. A consideration of the resonances in the dispersion relation indicates that only modes of sufficiently small wavenumber can participate in nearly resonant triads [#!dls:lecfp!#]. This in turn suggests that this triad interaction phase should correspond to the formation of the partial equipartition up to the knee. The subsequent relaxation of the energy spectrum to global equipartition requires the slower energy exchange among resonant quartets of normal modes, which are much more abundant. Our present focus is on the triad interaction phase.

An adaptation of the WT theory allows predictions of both the order of magnitude of the time scale and the location of the knee which agree excellently with direct numerical simulation.