On the role of the Integrable Toda model in one-dimensional molecular dynamics

We prove that the common Mie-Lennard-Jones (MLJ) molecular potentials, appropriately normalized via an affine transformation, converge, in the limit of hard-core repulsion, to the Toda exponential potential. Correspondingly, any Fermi-Pasta-Ulam (FPU)-like Hamiltonian, with MLJ-type interparticle potential, turns out to be $1/n$-close to the Toda integrable Hamiltonian, $n$ being the exponent ruling repulsion in the MLJ potential. This means that the dynamics of chains of particles interacting through typical molecular potentials, is close to integrable in an unexpected sense. Theoretical results are accompanied by a numerical illustration; numerics shows, in particular, that even the very standard 12--6 MLJ potential is closer to integrability than the FPU potentials which are more commonly used in the literature.Comment: 22 pages, 14 figures, Submitted in Journal of Statistical Physic

Burgers Turbulence in the Fermi-Pasta-Ulam-Tsingou Chain

We prove analytically and show numerically that the dynamics of the Fermi-Pasta-Ulam-Tsingou chain is characterized by a transient Burgers turbulence regime on a wide range of time and energy scales. This regime is present at long wavelengths and energy per particle small enough that equipartition is not reached on a fast timescale. In this range, we prove that the driving mechanism to thermalization is the formation of a shock that can be predicted using a pair of generalized Burgers equations. We perform a perturbative calculation at small energy per particle, proving that the energy spectrum of the chain Ek decays as a power law, Ek & SIM; k-zeta ot thorn , on an extensive range of wave numbers k. We predict that zeta ot thorn takes first the value 8=3 at the Burgers shock time, and then reaches a value close to 2 within two shock times. The value of the exponent zeta 1/4 2 persists for several shock times before the system eventually relaxes to equipartition. During this wide time window, an exponential cutoff in the spectrum is observed at large k, in agreement with previous results. Such a scenario turns out to be universal, i.e., independent of the parameters characterizing the system and of the initial condition, once time is measured in units of the shock time

Energy Localization in the Peyrard-Bishop DNA model

We study energy localization on the oscillator-chain proposed by Peyrard and Bishop to model the DNA. We search numerically for conditions with initial energy in a small subgroup of consecutive oscillators of a finite chain and such that the oscillation amplitude is small outside this subgroup for a long timescale. We use a localization criterion based on the information entropy and we verify numerically that such localized excitations exist when the nonlinear dynamics of the subgroup oscillates with a frequency inside the reactive band of the linear chain. We predict a mimium value for the Morse parameter $(\mu >2.25)$ (the only parameter of our normalized model), in agreement with the numerical calculations (an estimate for the biological value is $\mu =6.3$). For supercritical masses, we use canonical perturbation theory to expand the frequencies of the subgroup and we calculate an energy threshold in agreement with the numerical calculations

The Fermi-Pasta-Ulam problem in the thermodynamic limit: scaling laws of the energy cascade

In the present contribution we justify and discuss the scaling laws characterizing the first phase of the energy transfer from large to small spatial scales in a chain of nonlinear oscillators (the so-called Fermi- Pasta-Ulam α-model). By means of qualitative estimates, we show that large scale initial excitations (long wavelength Fourier modes) produce injection of energy into smaller scales on times t > τ_c = ε^{−3/4} and up to a cutoff spatial scale ell_c = ε^{−1/4} , where ε is the energy per degree of freedom of the system

Soliton theory and the Fermi-Pasta-Ulam problem in the thermodynamic limit

We reconsider the Fermi-Pasta-Ulam problem from the point of view of soliton theory along the lines of the original work of Zabusky and Kruskal, but with attention to the thermodynamic limit. For a special class of long-wavelength initial data, we show that, in such a limit, the modal energy spectrum is determined by the solitons, and is given by an explicit analytic expression in terms of the specific energy, which does not correspond to equipartition of energy. This is shown to occur for specific energies below a certain nonvanishing threshold, within a time interval that also depends on the specific energy. A short discussion concerning the case of generic long-wavelength initial data is also given

On metastability in FPU

We present an analytical study of the Fermi-Pasta-Ulam (FPU) alpha-model with periodic boundary conditions. We analyze the dynamics corresponding to initial data with one low frequency Fourier mode excited. We show that, correspondingly, a pair of KdV equations constitute the resonant normal form of the system. We also use such a normal form in order to prove the existence of a metastability phenomenon. More precisely, we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes. Subsequently, the distribution remains unchanged up to the time scales of validity of our approximation. The phenomenon is controlled by the specific energy

Resonance, Metastability and Blow-up in FPU

We consider the FPU model with nonlinearity starting with terms of order n 65 3. We compute the resonant normal form in the region where only one low-frequency mode is excited and deduce rigorous results on the correspondence between the dynamics of the normal form and that of the complete system. As n varies, we give a criterion in order to deduce whether the FPU phenomenon (formation of a metastable packet of modes) is present or not. The criterion is that, if the normal form equation has smooth solutions then the FPU phenomenon is present, while it is absent if the solutions of the normal form equations have blow up in a finite time. In particular the phenomenon should be present for n 64 5 and absent for n 65 7

Energy cascade in Fermi-Pasta-Ulam models

We show that, for long-wavelength initial conditions, the FPU dynamics is described, up to a certain time, by two KdV-like equations, which represent the resonant Hamiltonian normal form of the system. The energy cascade taking place in the system is then qualitatively characterized by arguments of dimensional analysis based on such equations

On the numerical integration of the FPU-like systems

This paper concerns the numerical integration of systems of harmonic oscillators coupled by nonlinear terms, like the common FPU models. We show that the most used integration algorithm, namely leap-frog, behaves very gently with such models, preserving in a beautiful way some peculiar features which are known to be very important in the dynamics, in particular the “selection rules” which regulate the interaction among normal modes. This explains why leap-frog, in spite of being a low order algorithm, behaves so well, as numerical experimentalists always observed. At the same time, we show how the algorithm can be improved by introducing, at a low cost, a “counterterm” which eliminates the dominant numerical error