LE JOURNAL DE PHYSIQUE

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1 Nous The Tome 43 No 5 MAI 1982 LE JOURNAL DE PHYSIQUE J. Physique 43 (1982) MAi 1982, Classification Physics Abstracts Approach to equilibrium in a chain of nonlinear oscillators F. Fucito (*) (+), F. Marchesoni (**), E. Marinari (*) (+), G. Parisi (***), L. Peliti (*) (+ +), S. Ruffo (**) and A. Vulpiani (*) (*) Istituto di Fisica «G. Marconi», Università di Roma, Italy. (**) Istituto di Fisica, Università di Pisa, Italy, and INFN, Pisa. (***) Istituto di Fisica, Facoltà di Ingegneria, Università di Roma, Italy and INFN, Frascati. (+) INFN, Roma. (++) GNSMCNR, Unità di Roma. (Recu le 6 aout 1981, révisé le 30 décembre, accepté le 18 janvier 1982) 2014 Résumé. considérons l approche à l équilibre thermodynamique d une chaîne d oscillateurs non linéaires couplés où l énergie est initialement fournie aux modes de plus grande longueur d onde. La considération des effets des singularités dans le plan complexe de la solution des équations du mouvement permet de prédire qu à des temps relativement courts le spectre d énergie a une dépendance exponentielle en fonction du nombre d onde k lorsque celuici est grand. Elle permet également de distinguer entre deux régions temporelles : une à temps courts où la pente de l exponentielle dépend linéairement de ln t, et une à des temps intermédiaires où elle est proportionnelle à (In t)1/2. Ces prédictions sont en accord avec les simulations numériques que nous avons faites sur ce système. Abstract We consider the approach to thermodynamical equilibrium in a chain of coupled nonlinear oscillators when energy is initially fed only to the longest wavelength modes. Consideration of the effects of singularities in the complex plane of the solutions of the equations of motion allows us to predict that at not too long times the energy spectrum has an exponential tail in the wavenumber k and to distinguish between two time regions : a short time region where the slope of the exponential shows a linear dependence on In t, and an intermediate region where it is proportional to (In t)1/2. These predictions are successfully compared with the numerical simulations we have performed on the system. 1. Introduction. approach to equilibrium of a classical hamiltonian system is still an open and interesting problem. The main rigorous results are the KAM [1] and Sinai [2] theorems. They describe the behaviour of two opposite extreme cases of a nonintegrable hamiltonian system. The Sinai theorem shows the validity of the ergodic hypothesis for a gas of hard spheres. The main consequence of the KAM theorem is the permanence of invariant tori in presence of a small perturbation of an integrable system. In this case the long time behaviour of the system will not be described by a Boltzmann distribution. The range of validity of the KAM theorem for a system with a large number of degrees of freedom is not yet understood. Numerical simulations have been extensively examined for a linear chain of coupled nonlinear oscillators, starting from the famous work of Fermi, Pasta and Ulam [3]. More recent works [46] have shown that the regions of phase space where ordered motion takes place are much more extended than on the basis of the KAM theorem. Some expected authors [6] stress the existence of a threshold in energy which separates ordered from disordered behaviour. All these works study long times in order to distinguish between ordered and stochastic asymptotic behaviour. The present work analyses short times in order to understand the way chaotic behaviour sets in. Our system is a discrete version of a classical onedimensional field obeying a nonlinear Klein Article published online by EDP Sciences and available at

2 We 708 Gordon equation. This system has already been considered in reference [6]. We consider initial conditions such that energy is concentrated in the longest wavelength modes and we study how it propagates to shorter wavelengths. In the times we consider the bulk of the energy is still contained in the initially excited modes, and only a small quantity is fed to the shorter wavelengths. In order to allow for an approximate analytic treatment we focus our attention on the process by which this small amount is distributed. We are then able to compare our predictions with the results of the numerical simulations. as follows : the model is The paper is organized introduced in section 2, where the analytic treatment at short and intermediate times is given. The results of the numerical simulations as well as their comparison with our theoretical predictions are given in section 3. In section 4 we draw some conclusions and point out some possible directions for further work. 2. Analytical treatment. study the onedimensional nonlinear KleinGordon equation : where the real, onecomponent field p(x, t) is defined on the interval L/2 x L/2 with periodic boundary conditions. The corresponding hamiltonian density is given by : where the field 7r is canonically conjugate to po The system may be either considered as a onedimensional elastic string in an external anharmonic potential, or as a onedimensional classical field theory. We study the time evolution of the field ~ as a function of the initial condition p(x, 0), It is known that if H is finite at t 0, the solution of equation (2.1) exists and is unique [7]. It is useful to define the space Fourier transform of the field p as follows : We expect the field p to reach asymptotically a thermal equilibrium distribution, given by a Boltzmann factor exp( f3h) for some value of the inverse temperature f3 determined by the initial conditions. In this case, at values of the wavenumber k so large that the mass and nonlinear terms of H are negligible, one would have : This behaviour of W corresponds to functions p(x, t) which are not differentiable with respect to x. Now it is known that since T(x, 0) is analytical as a function of x, the solution T(x, t) will remain analytical at any finite time t [7]. Equation (2.6) can only be valid for infinite time. This means that, as time goes on, singularities of p(x, t) appear in the complex x plane which creep toward the real axis and accumulate onto it at infinite times. We show below that these singularities are simple poles. If we assume this to be the case, then we may relate these singularities to the large k behaviour of W by means of the theorem of residues. Let us consider the integral defining 0 (eq. (2.3)) in the complex x plane. At positive (negative) k we close the contour in the lower (upper) complex half plane. We then obtain the following expression for 0(k, t) : where the sum runs over all poles located in the relevant half plane, Rj being their residue and xj + iy; their location. We therefore obtain the following asymptotic behaviour of W(k, t ) at large k : where ys(t) is the imaginary part of the location of the pole which lies nearest to the real axis. Our strategy is then to evaluate the most likely value of ys(t) by extending the approach of Frisch and Morf [8] to a deterministic partial differential equation. We should in principle evaluate the analytic continuation to complex x of the solution p(x, t) of equation (2.1). We prefer instead to consider the solution 9(x + iy, t ) of the complex partial differential equation : We shall consider as a typical initial condition : We are interested in the behaviour, in the region I k I > ki, of the spectrum W(k, t), defined by : where ~ is an analytical function of z x + iy. Such an equation is the complexification of equation (2.1) with the hypothesis that T(x, t ) can be analytically continued. In other words we follow the commutative diagram given below :

3 Force 709 We first consider the case of an ordinary differential equation obtained from equation (2.9) by neglecting the laplacian >> term a2paz2. Let us consider the solution 4>«fJo, t) of this equation at real times t, as a function of its (complex) initial condition (po : If we write 4> 4>R + i4>i we have the following differential equations for 4>R 4>1 : imaginary axisx we may introduce a potential according to : The time needed to reach infinity starting from the point (po i(p" 0 may be then easily computed from the conservation of «energy» in the onedimensional motion : The force field associated to this system is drawn in figure 1 (only the first quadrant is shown). Although the origin is a stable point, we find a pair of unstable points on the imaginary axis, from which a trajectory escapes to infinity. The point at infinity appears as a saddle point, and may be reached in a finite times from points on the imaginary axis. This is the origin of the singularities of 4>(cpo, t). It is then clear that these singularities are simple poles, located at that value of To from which infinity may be reached in time t. We may estimate the location of these poles by computing the time needed to reach infinity starting from a given point on the imaginary 45 axis. Since the motion is in this case onedimensional (all along the At large ~o the time T behaves as : On the contrary, no singularity will appear at any time if for the time T diverges (although slowly) as Im (po I approaches this value from above. From equation (2.14) we therefore obtain the location of the pole of 4>( CPo, t) at large CPo s, therefore at short t s : If the effect of the laplacian term in equation (2.9) may be neglected at short enough times, the above analysis may be also applied to the onedimensional chain as follows : the solution T(z, t) of equation (2. 9) will be given by 4>(cpo(z), t), where CPo(z) is the analytical continuation of the initial condition T(x, 0) to the complex x plane. In particular, a typical initial condition yields in the complex x plane We have therefore at large y I : z Fig. 1. field of equation (2.11). Only the upper right quadrant is shown. The position of the instable point on the imaginary axis (1m cp m/gl/2) is marked. According to our hypothesis, the singularity of CPo(z, t) will be located where Im CPo, as estimated

4 A We 710 from equation (2. 19), will be of the order of that given by equation (2.16). Therefore We now consider the effect of the previously neglected laplacian term. We claim that the analysis we just made retains its validity at very short times. The laplacian term will not in fact be able to prevent a 9(z, t) from escaping to infinity if its initial imaginary part is large enough. Localized o bursts» will therefore appear in (p(x, t), whose spatial coherence is only due to correlations present in the initial conditions. The laplacian term will on the contrary be essential in introducing singularities in the part of the complex x plane which corresponds to small initial values of Im cpo If the term were not present, Im T would remain bounded at all times. If however! Im cp [ were to become larger than m/gi/2, the mechanism we just described would drive it to infinity in a very short time, and a singularity will be produced. We must now estimate the time needed for Im T to become larger than m/g 1/2. The analysis becomes simpler if we consider a harmonic chain ( g 0) in the limit of infinite length (L oo). We expect this analysis to be also valid for our case if g is not too large. It is known that the onepoint probability distribution function of a classical harmonic field in onedimensional is gaussian [9]. The variance Q2 of the gaussian will be then given at small y s by : cannot therefore draw conclusions about the behaviour at very long times before thermal equilibrium is reached. We summarize the results for W(k, t) at short and intermediate times : 3. Numerical results. present in this section the results of numerical simulations of equation (2.1 ) for different initial conditions, in order to verify the validity of the analytical results we obtained in the previous section. We consider a chain of N points with periodic boundary conditions which represents a discretization of equation (2.1 ). We set L (the length of the chain) equal to 1 ; and we consider the field cp(x) to be represented by gj cpu Ax 1/2), where Ax 1/N. The periodic boundary conditions then imply 9jlN 9j The Fourier transform of the discrete field Pj is defined as follows : The probability p, for I 1m cp to be larger than m/g 1/2 will be therefore given by : The time needed for this to happen will therefore be of order 1/pc. This yields the following estimate. of YS : The main effect of the nonlinear terms in this regime will be to change the value of (12. If a2 were time independent, equation (2.21) will be essentially correct. Let us distinguish between the role of short and long wavelength modes. At the times we are interested in, most of the energy is contained in the long wavelength modes, which may be assumed to be in a kind of thermal equilibrium among themselves. Their contribution to (J2 may be then considered as essential modes will ly constant in time. The short wavelength however also contribute to (J2. As long as W(k, t) is small in the large k region, their contribution is negligible. As time goes on, however, W(k, t ) will start increasing, what will increase the value of Q2 and fasten therefore the transfer of energy to short wavelength modes. This triggers a catastrophic process which our analytical tools are unable to handle. We Fig. 2. typical spectrum, In Wn(t) vs. n for A 5, g 5, N64, t2, T 1.

5 Slope 711 The quantity corresponding to the W(k, t ) of the previous section will be : We have performed simulations for N 64 and N 128. The initial conditions were chosen to be : significant differences have been seen in runs with different At. The results we report have been obtained via computations with 15 or 16 significant digits. Since part of the arguments in the previous section are probabilistic, it is convenient to introduce the average of Wn(t) centred around the time t : all other Fourier coefficient having been set to zero. The integration of equation (2.1 ) was performed by means of the central difference algorithm [10] : where the discretized force Fj({ T }) is given by : In the limit At 0 equation (3. 3) reduces to In order to eliminate oscillations, T must be chosen to be of the order of magnitude of the period of oscillation of a mode of wavenumber kn 2 nn/ L in a harmonic chain. Since m is quite small in our case, this yields comparatively large values of T, of order 0.1 : 0.2 in the short time region and 2 : 10 in the intermediate time region. The deviations of W,,(t) from its average may be understood on the basis of the statistical properties of the field (p(x, t) at large k, which is analysed in detail in a forthcoming paper [11]. We now present the results for short times. We have verified that the spectrum Wn(t ) can be very well described by an exponential in the region of not too small n. A typical example is shown in figure 2. It is easy to obtain the value of the slope S (t ) of this exponential from the data : The initial condition cpj(o) 0 corresponds to (pj( et) w;(0). We have done simulations keeping m and varying A and g in the range 0.5 : 10. At has been taken in the range x We have checked that the energy is conserved within 0.1 %; no This slope is related to ys(t) of the previous section by : We show in figure 3 the dependence of S (t ) on In (tagi/2) with different values of t, g and A. We see Fig. 3. S (arbitrary units) vs. In (Ag1/2 t), N 64, T 0.1 : 0.2.

6 Slope Slope 712 Fig. 4. S (arbitrary units) vs. (A2 g In t) 1/2, N 64, T 5. that the data gather on an universal curve, which is essentially a straight line, giving therefore a confirmation to the analysis of short times performed in section 2. This plot may be therefore considered as a check of equation (2.20). All error bars represent essentially uncertainties in the estimation of S(t). We have verified that the value of Wn(t ) coincides with its time average at short times. This is due to the dominance of a single complex singularity. As the time increases and we enter the intermediate time region, we expect that more than one complex singularity at roughly the same distance from the real x axis will Fig. 5. z S (arbitrary units) vs. (In t) 112, N 128, A 3, g 5, T 10 except for t 8 where T 2t

7 We We 713 appear. Oscillations will therefore be present in Wn(t) as a consequence of the interference among such singularities. In the intermediate time region we expect that S (t) oc (A2 g In t) 112. We have plotted S vs. this variable in figure 4, varying either t at fixed A and g, or g at fixed t and A. We see again that the data gather on a single straight line. Therefore the functional dependence of S (t ) is as expected on the basis of equation (2.23) for not too long times. The behaviour at longer times of a solution with a given value of A and g is shown in figure 5. One sees both the (In t) li2 behaviour and a break at longer times. This is connected with the observation done at the end of the previous section. 4. Conclusions. have seen that it is possible to give definite predictions about the behaviour of a nonintegrable hamiltonian system with an infinite number of degrees of freedom in the nonasymptotic time region. This has been achieved by the use of the analytical continuation of the equations of motion in the complex domain, a method introduced in a similar context by Frisch and Morf [8]. The role of the infinite volume limit has also been clarified. The free evolution of the system is indeed such that 9 may only become arbitrarily large in the infinite volume limit. In this case no linear perturbation may be considered small if one follows the system at long enough times. It is possible that intermittance concepts are relevant in this context [8, 11]. Further work aims at clarifying this connection [11]. One of our main results is that the system approaches equilibrium with a logarithmic dependence on t, so that the nonequilibrium spectrum may persist for extremely long times, and may be mistaken for a stationary state if the observation time is not sufficiently long. It is amusing to remark that this quasiequilibrium distribution is similar to Wien s law for black body radiation, with a slowly varying «Planck s constant». Acknowledgments. thank U. Frisch for having made available to us the paper in ref. [8] before publication. We also thank L. Galgani, A. Giorgilli and M. Vitaletti for helpful conversations and constructive criticism. References [1] KOLMOGOROV, A. N., Dokl. Akad. Nauk SSSR 98 (1954) 527. ARNOL D, V. I., Russ. Math. Surv. 18 (1963) 9. MOSER, J., Nachr. Akad. Wiss. Göttingen, Math. Phys. K1., 2, 1 (1962) 15. [2] SINAI, Y. G., Russ. Math. Surv. 25 (1970) 137. [3] FERMI, E., PASTA, J. and ULAM, S., Los Alamos Scientific Laboratory Report No. La1940 (1955). [4] HÉNON, H. and HEILES, C., Astron. J. 69 (1964) 73. [5] FORD, J., Adv. Chem. Phys. 24 (1973) 155. GUSTAVSON, F. G., Astron. J. 71 (1966) 670. BENETTIN, G., GALGANI, L. and STRELCYN, J. M., Phys. Rev. A14 (1976) BENETTIN, G., LO VECCHIO, G. and TENENBAUM, A., Phys. Rev. A22 (1980) [6] BUTERA, P., GALGANI, L., GIORGILLI, A., TAGLIANI, A. and SABATA, H., Nuovo Cimento 59B (1980) 81. [7] BREZIS, H., in : M. Atteia, D. Bancel and P. Gumovski (Eds.) : Nonlinear Problems of Analysis in geometry and Mechanics (Boston : Potman Adv. Publ. Program) 1981, p. 3. [8] FRISCH, U. and MORF, R., Phys. Rev. A 23 (1981) [9] VAN HEMMEN, J. L., Phys. Reports 65 (1980) 43. [10] See for example BENETTIN, G. et al. (1980) in ref. [5]. [11] FUCITO, F., MARCHESONI, F., SPERPEGLIONE, M. and VULPIANI, A., Pise preprint (1982).