VoLUME 74, NUMBER 3 PH YS ICAL REVIEW LETTERS 16 JANUARY 1995 Gaussian Model for Chaotic Instability of Hamiltonian Flows Lapo Casetti* Scuola Wormale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Roberto Livi~ Dipartimento di Fisica, Universita di Bologna, Via Irnerio 46, 40126 Bologna, Italy Marco Pettini~ Osservatorio Astroftsico di Arcetri, Largo E. Fermi 5, 50I25 Firenze, Italy (Received 2 November 1993) A general method to describe Hamiltonian chaos in the thermodynamic limit is presented which is based on a model equation independent of the dynamics. This equation is derived from a geometric approach to Hamiltonian chaos recently proposed, and provides an analytic estimate of the largest Lyapunov exponent A. The particular case of the Fermi-Pasta-Ulam P-model Hamiltonian is considered, showing an excellent agreement between the values of A predicted by the model and those obtained with computer simulations of the tangent dynamics. PACS numbers: 05. 45.+b, 03. 20.+I, 05. 20. — y N M(q, p) = g ' + V(q), (1) 2m where V(q) is a nonlinear interaction potential among N particles with mass I, positions q = q', . .. , q~, and momenta p = p', . . . , p~. For most of the interesting choices of V(q) — apart from some remarkable exceptions, like the Toda lattice model — Hamiltonian (1) is nonintegrable and the corre- sponding equations of motion d q' BV (2) dt2 Bq; ' exhibit chaotic, i.e. , unpredictable, behavior despite their deterministic nature. One of the most interesting features of these Hamiltonian models is the presence of a transition from a weakly to a strongly chaotic regime when the energy per particle is increased, and numerical results suggest that this transition is stable in the thermodynamic limit N ~ ~ [1]. The degree of chaoticity of Eqs. (2) can be quantified by the largest Lyapunov exponent A, „, which, roughly speaking, is a measure of the mean rate of exponential divergence between nearby orbits. The rigorous definition of Lyapunov exponents can be found elsewhere [2]. Here we rather deal with an estimate A of A, „which, for Hamiltonian systems of the form (1), is defined as follows: First the equations of motion (2) are linearized along a generic trajectory yielding the evolution equations d 8 V + gJ (3) dt2 Bq; Bqi for the variations g = g', ... , g~ of the coordinates (sum- mation over repeated indices is understood throughout the i =1, . .. , N, Many problems and applications ranging from the general theory of dynamical systems to plasma and condensed matter physics are represented in terms of many degrees of freedom Hamiltonian systems of the form paper and m = 1 is also assumed); then A is given by 114(t)II 114(0)ll ' where 11/11 is the Euclidean norm of g. An algorithm to compute A was developed by Benettin, Galgani, and Strelcyn [3]. Numerical analysis based on this algorithm shows that the transition between weak and strong chaos can be detected by a scaling crossover of A as a function of the energy per degree of freedom e = E/N [1]. Simple algebraic manipulations of Eqs. (3) lead to the equation 1 d'llgll' a'V, , dllgll &' 2 dt2 8 '8 ~ dt ) (5) where p, = g/11/11. The term containing dp/dt can be neglected [4], and by standard substitutions [5] we arrive at an evolution equation for the norm 11/11 having the form of a generalized Hill equation d2$ + Q(t)/=0, (6a) 8 V &q'&q' llgll llgll ' where I til I is proportional to 11/ 11. The present work aims at obtaining an analytic expres- sion for A in the limit N ~. At first sight this goal hits against a major obstacle: Q(t) has to be computed along a dynamical trajectory [Eqs. (6) are coupled to Eqs. (2) and (3)]. Hence Eq. (6a) can be useful to compute A only if we are able to model Q(t) by a function which is indepen- dent of the dynamics. The main point of this Letter is to show how such a model can be actually formulated using the differential geometric structure underlying dynamics. 0031-9007/95/74(3)/375(4)$06. 00 Oc 1995 The American Physical Society 375