Boris Gershgorin, Yuri V. Lvov and David Cai
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180
The Fermi-Pasta-Ulam (FPU) lattice was introduced in their classical work  to address fundamental issues of statistical physics such as equipartition of energy, ergodicity. The attempt to resolve the mystery of the FPU recurrence (the system did not thermalize as was expected but rather kept returning to the initial state ) has spurred many great mathematical and physical discoveries, such as the celebrated KAM theorem and soliton physics . Despite this remarkable progress, there are still fundamental open questions that are under vigorous debate , such as what is the route to thermalization and how to fully characterize thermalized -FPU system. Furthermore, in the last decade discrete breathers (DB) as spatially localized, time periodic lattice excitations were discovered . Arising from energy localization in nonlinear lattices, they play important roles in many dynamics in fiber optics, condensed matter physics and molecular biology . The existence of DBs has been addressed rigorously . Important conceptual issues naturally arise, such as what is the role of DBs on the route to equilibrium  and how do they manifest in thermalization of the FPU system? Resolution of these issues will certainly provide deep insight into the fundamental understanding of route to thermalization for general nonlinear physical systems. Most of the results regarding DBs in -FPU chains have so far only addressed their behavior in the transient state of weakly nonlinear regimes before thermalization occurs [8,9].
In this Letter, we investigate the FPU dynamics in the strongly nonlinear limit. We demonstrate that, quite surprisingly, even for strong nonlinearity, the -FPU system in thermal equilibrium behaves like weakly nonlinear waves in properly chosen variables. Such behavior results from the collective effect of strongly nonlinear interactions effectively renormalizing linear dispersion relation. This observation enables us to use a well-developed weak turbulence (WT) formalism  for the description of the -FPU chains even in a strongly nonlinear regime. Furthermore, in addition to the nonlinear waves, we observe the DB excitations in the thermalized state of -FPU chains. Previously such DBs were observed only during transient stages towards thermalization . Here we show via numerical simulation that DBs actually persist and coexist with renormalized waves in the thermalized state. Thus, in the thermalized -FPU, there are two kinds of quasi-particle excitations, one localized in -space as renormalized nonlinear waves/phonons, and the other, localized in -space as DBs.
The -FPU chain is described by the Hamiltonian,
First, for the FPU system in thermal equilibrium ,
with moderate and strong nonlinearities and ,
we measured the power spectrum
function of , where
denotes the time averaging. Although it was
expected that, for weak
, where is an
effective temperature (Rayleigh-Jeans distribution for
waves ), it is surprising to find that the same scaling
holds even for strong nonlinearities as shown in Fig. 1.
To understand why
scales as even in the strongly
nonlinear limit, we
computed the - spectrum of as shown in
Fig. 2(a). In the weakly nonlinear limit the
spectrum would have resonant peaks along a curve given by the linear
. We observe that, when the
nonlinearity is no longer small, the
resonances move to higher frequencies, as the dispersion relation is
renormalized by the nonlinear part . The renormalized dispersion
indicated by the sine-like structure of the resonance peaks (the
solid line in Fig. 2(a)). We verified
numerically that the main contribution of the nonlinear potential
energy ( %) comes from the terms
constrained on . We note that when and or
and , these terms can be combined with the quadratic part of
(2) to effectively renormalize the frequency .
More specifically, Hamiltonian (2) can be rewritten as
the sum of a new renormalized quadratic part and a remaining
with the renormalized dispersion relation,
To further study the renormalization of interactions we measured the ratio of the quartic to the quadratic parts of the energy before and after renormalization procedure (i.e., vs ) for different values of with energy fixed (Fig. 2(c)). This figure also shows that the effective renormalized linear part becomes more dominant. Therefore even for strongly nonlinear regimes, with as large as 128, the renormalized waves (or purely nonlinear phonons) have weakened interactions. Note that the resonance of has a finite width (shown with the dashed lines in Fig. 2(a) which are the level of for each ). We note that for (the highest mode in the system) the resonances become the broadest. According to WT , the energy exchange among waves occurs on the resonance manifold given by and . Although one can show that there are no exact resonances in -FPU chains on the discrete lattice, the nonlinearity induced near resonance interactions (i.e. where is a resonance width) can occur. This allows us to use the WT theory if the interaction is considered to be weak. In order to characterize the system as weakly nonlinear waves, the renormalized waves are described by the new normal variables The relationship between bare and renormalized is Under the random phases approximation  for we have Therefore the power spectrum has scaling for both and . (Note that is shown in Fig. 1). As a consequence, the effective temperature, , for the renormalized waves, is related to the bare temperature by .
The renormalization picture is further corroborated by the time evolution of each individual wave . For the bare waves , there are large temporal modulations in both modulus and phase as a result of strong nonlinear interaction among these modes on the linear dispersion time scale (as shown in Fig. 3(a) and (b)). In weakly nonlinear systems, wave amplitudes and corresponding phases are expected to evolve slowly on the linear dispersion time scale. For renormalized waves (Fig. 3(c) and (d))these modes indeed have characteristics of weakly interacting waves, with a small modulation in and phase.
Now we turn to the numerical evidence for the persistence of DBs in the thermalized -FPU chain. As observed before , there are DBs present in the transient state. As nonlinearity increases, the duration of transient becomes shorter. After the energy redistributes among all the modes to achieve thermal equilibration, our simulations show that the spatially localized, high frequency excitations still exist. These DBs can interact with each other and may be destroyed by collision processes with other DBs or with the renormalized waves. The spatial structure of these excitations very much resembles the idealized breather oscillations in the absence of spatially extended waves: they ``live'' above the high frequency edge of the dispersion band and their lifetime is sufficiently long (on the order of 10-100 DB oscillations) to behave like a quasiparticle. Note that, under certain conditions, supersonic solitons may arise from the -FPU system as another kind of localized excitations [15,16]. However, they were not observed in our thermalized system. Fig. 4 is the energy density plot which shows the time evolution of energy of each particle for the transient (recording starting time ) and thermalized ( ) states, respectively. Fig. 4(c) and (d) display the energy as a function of site at corresponding to Fig. 4(a) and (b) respectively. In the transient case (Fig. 4(a)) the spatially localized objects (dark stripes) that carry sufficiently large amount of energy are clearly observed. Fig. 4(c) is a snapshot of the energy density plot (4(a)) at . Here the DBs are seen as localized peaks . After thermalization the spatial structure looks different (Fig. 4(b)). The system now consists of the renormalized waves (straight cross-hatch traces in Fig. 4(b)). On the top of these waves, the localized structures similar to DBs manifest themselves as the wavy dark trajectories (in Fig. 4(b)). Although the snapshot (Fig. 4(d)) of the energy density plot (Fig. 4(b)) indicates that in thermal equilibrium the energy is more evenly distributed among particles, spatially localized structures are clearly observed.
Since there are renormalized waves in the system, which also carry energy, we need to find a way to distinguish between these waves and DBs. We use a frequency filter that cuts out the lower side of the Fourier spectrum and leaves the high frequency part unmodified, i.e., , where denotes the real part, is a time Fourier transform, eliminates all frequencies below and is a dynamical variable that is being filtered. By applying this filter to the displacement to obtain , we can show the existence of DBs even for strong nonlinearities, for example, . Fig. 5(a) shows a clear example of a DB excitation reconstructed using the filtered with . Fig. 5(b) shows a typical spatial profile of the DB taken from Fig. 5(a), which strongly resembles the idealized DB . Finally, in Fig. 6, we present the evidence that there is a turbulence of DBs, which chaotically ride on renormalized waves. The corresponding energy density distribution along with the distribution of filtered displacement in a zoomed region is displayed in Fig. 6(a) and (b), respectively. After the lower modes from the displacement are filtered, one can clearly observe that the remaining high frequency oscillations are spatially highly localized, with the same characteristics as an idealized breather. The detailed time dynamics of the DB shows the main characteristics of breathers: the values of change signs periodically (as indicated by the alternating white and black spots along the trajectory) as the DB moves in space, with a spatial span of 2 or 3 sites only, as seen in Fig. 6(b).
In conclusions, we have presented an interesting dynamical scenario of the -FPU chains in thermal equilibrium: (i) For strong nonlinearity the linear dispersion relation is effectively renormalized, which allows one to treat even strongly nonlinear systems as if they were weakly nonlinear; (ii) On top of renormalized waves, the strongly nonlinear system is also characterized by the turbulence of discrete breathers.
Acknowledgments we thank Sergei Nazarenko for discussions.