Michel Peyrard

Controlling the Energy Flow in Nonlinear Lattices: A Model for a Thermal Rectifier

We address the problem of heat conduction in 1D nonlinear chains; we show that, acting on the parameter which controls the strength of the on-site potential inside a segment of the chain, we induce a transition from conducting to insulating behavior in the whole system. Quite remarkably, the same transition can be observed by increasing the temperatures of the thermal baths at both ends of the chain by the same amount. The control of heat conduction by nonlinearity opens the possibility to propose new devices such as a thermal rectifier.

Nonlinear dynamics and statistical physics of DNA

DNA is not only an essential object of study for biologists—it also raises very interesting questions for physicists. This paper discuss its nonlinear dynamics, its statistical mechanics, and one of the experiments that one can now perform at the level of a single molecule and which leads to a non-equilibrium transition at the molecular scale.After a review of experimental facts about DNA, we introduce simple models of the molecule and show how they lead to nonlinear localization phenomena that could describe some of the experimental observations. In a second step we analyse the thermal denaturation of DNA, i.e. the separation of the two strands using standard statistical physics tools as well as an analysis based on the properties of a single nonlinear excitation of the model. The last part discusses the mechanical opening of the DNA double helix, performed in single molecule experiments. We show how transition state theory combined with the knowledge of the equilibrium statistical physics of the system can be used to analyse the results.

Kink dynamics in the highly discrete sine-Gordon system

Abstract We study kink dynamics in a very discrete sine-Gordon system where the kink width is of the order of the lattice spacing. Numerical simulations exhibit new properties of kinks in this case: they lose the memory of their initial velocity and propagate preferentially at well-defined velocities which correspond to quasi-steady states, while a kink moving at other velocities suffers relatively high rates of radiation of small amplitude oscillations. When a small external driving force is applied to the system, the same velocities appear as plateus in the strongly nonlinear mobility of the kink. The energy radiated by the kink is calculated for a simple model that preserves the discrete character of the system, and the preferential velocities for the kink are obtained to good accuracy. Similar results may be expected to be valid for other discrete systems manifesting topological solitons. The numerical simulations reveal also new stable “multiple-kink” excitations which can propagate almost freely in extremely discrete systems where “ordinary” simple kinks are pinned to the lattice by discreteness. The stability of the “multiple-kinks” is discussed.

Kink-antikink interactions in the double sine-Gordon equation

We study numerically the interactions of a kink (K) and an antikink (K) in the double sine-Gordon equation with potential V(φ) = −4[ - cos(φ/2) + η cos φ]/[1 + |4η|]. As a function of initial velocity in the KK collisions, we observe and analyze quantitatively a rich structure of “resonances” in most ranges of η.

Helicoidal model for DNA opening

Abstract We present a new dynamical model of DNA. This model has two degrees of freedom per base-pair: one radial variable related to the opening of the hydrogen bonds and an angular one related to the twisting of each base-pair responsible for the helicoidal structure of the molecule. The small amplitude dynamics of the model is studied analytically: we derive small amplitude envelope solutions made of a breather in the radial variables combined with a kink in the angular variables, showing the role of the topological constraints associated to the helicoidal geometry. We check the stability of the solutions by numerical integration of the motion equations.

The design of a thermal rectifier

The idea that one can build a solid-state device that lets heat flow more easily in one way than in the other, forming a heat valve, is counter-intuitive. However, the design of a thermal rectifier can be easily understood from the basic laws of heat conduction. Here we show how it can be done. This analysis exhibits several ideas that could in principle be implemented to design a thermal rectifier, by selecting materials with the proper properties. In order to show the feasibility of the concept, we complete this study by introducing a simple model system that meets the requirements of the design.

Kink-antikink interactions in a modified sine-Gordon model

We study numerically the interactions of a kink (K) and an antikink (K) in a parametrically modified sine-Gordon model with potential V(φ) = (1 − r)2(1 − cosφ)/(1 + r2 + 2rcosφ). As the parameter r is varied from the pure sine-Gordon case (r = 0) to values for which the model is not completely integrable (r ≠ 0), we find that a rich structure arises in the KK collisions. For some regions of r(−0.20⪅r<0) this structure is very similar to that observed in KK interactions in the φ4 model, and we show that the theory recently suggested for these collisions also applies quantitatively to the modified sine-Gordon model. In other regions of r we observe new scattering phenomena, which we present in detail numerically and discuss in a qualitative manner analytically.

The pathway to energy localization in nonlinear lattices

Abstract Intrinsic localized modes have been shown to exist as exact solutions in nonlinear lattices [R.S. MacKay, S. Aubry, Nonlinearity 7 (1994) 1623]. We investigate the mechanisms that can lead to their formation in a physical system. We show that they can emerge from a uniform energy distribution in several steps. First modulational instability can generate small breathers and then their interaction leads to the growth of the largest excitations. This mechanism, first investigated in simple one-dimensional lattices is shown to be valid in a multicomponent lattice as well as in a two-dimensional system. The signature of the discrete breathers in the dynamical structure factor of the lattice is determined and the consequences for their experimental detection are discussed. Finally some suggestions for indirect observations of the discrete breathers are presented.

Order of the phase transition in models of DNA thermal denaturation.

We examine the behavior of a model which describes the melting of double-stranded DNA chains. The model, with displacement-dependent stiffness constants and a Morse on-site potential, is analyzed numerically; depending on the stiffness parameter, it is shown to have either (i) a second-order transition with nu( perpendicular) = -beta = 1,nu(||) = gamma/2 = 2 (characteristic of short-range attractive part of the Morse potential) or (ii) a first-order transition with finite melting entropy, discontinuous fraction of bound pairs, divergent correlation lengths, and critical exponents nu( perpendicular) = -beta = 1/2,nu(||) = gamma/2 = 1.

Nonlinear modes in coupled rotator models

Abstract We compare the properties of generalized nonlinear Klein-Gordon equations with a sinusoidal coupling (sine-lattice equations) to those of the usual nonlinear Klein-Gordon equation with harmonic coupling for three equations leading to the same sine-Gordon equation in the continuum limit. Using Hirotas bilinear method we find a strong analogy between the properties of nonlinear oscillatory modes in the three equations. Numerical simulations confirm this analogy even for very discrete cases. However we point out a fundamental difference between the harmonic and sinusoidal coupling by exhibiting a new class of localized nonlinear excitations which are specific to the sinusoidal coupling, the rotating modes. They can be created thermally. An approximate analytical solution is obtained for these modes which are intrinsically discrete and appear to be extremely stable, even in collisions with breathers. Possible implications for physics are discussed.