O Gendelman

L I Manevitch

A F Vakakis

R M'closkey

Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian Systems

The systems considered in this work are composed of weakly coupled, linear and essen-tially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) ''channeling'' of externally imparted energy from the linear to the nonlin-ear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

Introduction

In this and a companion paper we study nonlinear energy pumping in coupled mechanical oscillators. By this terminology, we denote the controlled spatial transfer of vibrational energy from the point of its initial generation to a different ͑predeter-mined͒ point where it eventually localizes. In essence, the energy pumping phenomenon corresponds to the controlled one-way channeling of the vibrational energy to a passive nonlinear ''sink'' where it localizes and diminishes in time due to damping dissipation. There exist numerous studies in the literature on ''static'' mode localization, spatial motion confinement, and on energy transfer due to internal resonances in coupled mechanical oscillators. The nonlinear energy pumping phenomenon discussed herein is a distinct nonlinear mechanism of energy transfer since it is realized through resonance capture ͓͑1͔͒.

Linear and nonlinear passive ''static'' mode localization and spatial motion confinement in periodic and nonperiodic coupled oscillators have been studied extensively in the literature ͓͑2-9͔͒. In these studies, linear and nonlinear standing wave motions were analyzed in ordered and disordered periodic coupled oscillators, and the existence of spatially localized free and forced standing waves was rigorously proven by means of theoretical, numerical, and experimental techniques. The standing wave localization considered in these previous works can be classified as ''static'' since it does not involve any controlled spatial transfer ͑transition͒ of energy through the system; indeed linear or nonlinear mode localization can be realized through appropriate selection of the initial conditions of the system, and does not involve any spatial ''flow'' of energy through the system.

Nonlinear transfer of energy between nonlinear modes in inter-nal resonance has also been studied extensively ͑cf. ͓10͔͒.In addition, as recently shown by Nayfeh and co-workers, under certain conditions energy transfer from high to low-frequency modes of a weakly nonlinear structure can also occur ͓͑11͔͒. However, these nonlinear energy exchanges are solely due to modal interactions and do not necessarily involve controlled, one-way spatial transfer of energy through the system.

To the authors' best knowledge the only previous study of the nonlinear energy pumping phenomenon is the one by Gendelman ͓12͔. In that work a system of two weakly coupled oscillators, a linear and an essentially ͑nonlinearizable͒ nonlinear one, was considered. Pumping of energy was demonstrated numerically by showing that, under certain conditions, energy initially imparted in the linear oscillator transfers to the essentially nonlinear one, even though this later oscillator is not directly excited. However, no rigorous analysis and explanation of this phenomenon is given in that work.

In Part I of this work we present numerical evidence of energy pumping in two and three-degrees-of-freedom coupled oscillators with essential nonlinearities and weak viscous damping. We then focus ͑for simplicity͒ in the two-degrees-of-freedom case, and analyze systematically the bifurcation structure of the free nonlinear periodic orbits of the underlying Hamiltonian system with no damping. We show that the occurrence ͑or lack of͒ energy pumping can be explained by considering the 1-1 and higher order resonant orbits of the Hamiltonian system. A direct analysis of the energy pumping phenomenon is carried in Part II by transforming the damped equations of motion using the action-angle variables of the underlying Hamiltonian system. We show that energy pumping is a resonance capture phenomenon on a 1-1 resonant manifold, and construct analytical approximations of energy pumping.

Nonlinear Energy Pumping: Numerical Evidence

Consider the following two-degrees-of-freedom system composed of two weakly coupled and weakly damped oscillators: y ¨1ϩy ˙1ϩCy 1 3 ϩ͑ y 1 Ϫy 2 ͒ϭ0

(1) y ¨2ϩy ˙2ϩ 2 2 y 2 ϩ͑ y 2 Ϫy 1 ͒ϭ0

Weak coupling is assured by requiring that Ӷ1, and all other variables are assumed to be O(1) quantities; dots denote differentiation with respect to the independent variable t ͑time͒. For ϭ0 the system decomposes into two uncoupled nonlinear and linear oscillators, labeled ''Oscillators 1 and 2,'' respectively. We note that oscillator 1 is essentially nonlinear ͑nonlinearizable͒.

In Fig. 1 we depict the transient responses of the two oscillators for ϭ0.5, 2 2 ϭ0.9, Cϭ5.0, ϭ0.1, and initial conditions y 1 (0) ϭy 2 (0)ϭ0, y ˙1(0)ϭ0, y ˙2(0)ϭͱ2h, where h ͑the energy of the system at tϭ0ϩ͒ varies; these initial conditions correspond to impulsive excitation of oscillator 2 at tϭ0. For hϭ0.5 ͑cf. Fig. 1͑a͒͒ both oscillators perform damped free oscillations and no energy pumping occurs, since most energy is stored in the directly excited oscillator 2. By increasing the initial energy level to h ϭ0.8 and 1.1.25 ͑cf. Figs. 1͑b,c͒͒, it is observed that energy transfer from the directly excited oscillator 2 to the unexcited oscillator 1 takes place; indeed, after an initial transient state most of the vibrational energy is irreversibly transferred ͑''pumped''͒ to oscillator 1. By further increasing the initial energy level the energy pumping phenomenon becomes less pronounced. This numerical simulation indicates that, for fixed system parameter values, energy pumping in the weakly coupled system takes place, above a specific value of the initial energy level (strength of the excitation).

Similar results are obtained for the three-degrees-of-freedom system governed by y ¨1ϩy ˙1ϩCy 1 3 ϩ͑ y 1 Ϫy 2 ͒ϭ0 y ¨2ϩy ˙2ϩ 2 2 y 2 ϩ͑ y 2 Ϫy 1 ͒ϩd͑ y 2 Ϫy 3 ͒ϭ0

(2) y ¨3ϩy ˙3ϩ 2 2 y 3 ϩd͑ y 3 Ϫy 2 ͒ϭ0 2 Motivated by these observations we now proceed to examine the periodic orbits of the underlying Hamiltonian system by eliminating damping from Eqs. ͑1͒. Since system ͑1͒ is weakly damped, one expects that, at least at the initial stages of the motion, the dynamics will be greatly influenced by the dynamics of the corresponding ͑undamped͒ Hamiltonian system. In turn, the undamped dynamics are dominated by periodic orbits. As a result, we expect that the topological structure of the periodic orbits ͑and their bifurcations͒ of the Hamiltonian system, will play a dominant role in the energy pumping phenomenon. representing two strongly coupled linear oscillators that are weakly attached to an essentially nonlinear oscillator. In Fig. 2 we depict the transient response of this system for ϭ0.5, 2 ϭ0.9, Cϭ5.0, dϭ1.0, ϭ0.1, and zero initial conditions except y ˙ 3 (0) 0. Whereas for low excitation no energy pumping to the nonlinear oscillator occurs ͑cf. Fig. 2͑a͒͒,as y ˙ 3 (0) increases energy pumping takes place ͑cf. Figs. 2͑b,c͒͒. Hence the nonlinear energy pumping phenomenon can also be realized in multi-degree-offreedom systems.

We now focus exclusively in the two-degrees-of-freedom system ͑1͒. Considering the transient responses depicted in Fig. 1 we note that, when energy pumping occurs the motion can be divided into two phases: In the initial phase energy is rigorously pumped from oscillator 2 to oscillator 1 in a one-way (irreversible) transfer, until oscillator 1 reaches a certain amplitude of oscillation; in the second phase of the motion, both oscillators perform decaying oscillations due to damping dissipation with oscillator 1 retaining most of the vibrational energy. Moreover, during the initial energy pumping phase ͑defined approximately for 0ϽtϽ40 in Fig. 1͑b͒, and 0ϽtϽ60 for Fig. 1͑c͒, the motion of oscillator 1 is composed of a ''fast'' oscillation with frequency nearly identical to the natural frequency of oscillator 2, and a ''slow'' envelope oscillation. This strongly suggests that a 1-1 internal resonance between oscillators 1 and 2 plays an important role in the energy pumping phase, although this still does not explain the one way energy pumping from oscillator 2 to oscillator 1.

Introducing the action-angle variables (I 2 , 2 )(R ϩ ϫS 1 ) for oscillator 2 defined by the relations y 2 ϭͱ2I 2 / 2 sin 2 , v 2 ϵy ˙2 ϭͱ2I 2 2 cos 2 , the Hamiltonian of the undamped system is ex- pressed as

H ϭF͑ y 1 ,v 1 ͒ϩG͑ I 2 ͒ϩ 2 H 1 ͑ y 1 ,v 1 , 2 ,I 2 ͒ (3)
where

F͑ y 1 ,v 1 ͒ϭ͑ v 1 2 /2͒ϩ͑Cy 1 4 /4͒, G͑I 2 ͒ϭ 2 I 2 , H 1 ͑ y 1 ,v 1 , 2 ,I 2 ͒ϭ͑ y 1 Ϫͱ2I 2 / 2 sin 2 ͒ 2 .
The equations of motion can then be placed in the following form:

y ˙1ϭ ץF ץv 1 ϩ 2 ץH 1 ץv 1 , v ˙1ϭϪ ץF ץy 1 Ϫ 2 ץH 1 ץy 1 , ˙2ϭ 2 ϩ 2 ץH 1 ץI 2 , I ˙2ϭϪ 2 ץH 1 ץ 2 (4)
where v 1 ϭy ˙1 . By fixing the Hamiltonian ͑total energy͒ to a constant level h, we can express the action I 2 in terms of the other variables of the system as follows:

H ϭF͑ y 1 ,v 1 ͒ϩG͑ I 2 ͒ϩ 2 H 1 ͑ y 1 ,v 1 , 2 ,I 2 ͒ ϭh⇒I 2 ϭL ͑ y 1 ,v 1 , 2 ,h ͒ (5)
where L is a complicated expression. As a reviewer pointed out, the inversion ͑5͒ is only possible if the system is nonsingular, i.e., if the condition ץH /ץI 2 0 is satisfied; clearly, this is the case in our problem. Taking into account ͑5͒, eliminating the time variable from ͑4͒, and combining the resulting first-order expressions into a single second-order one we obtain the reduced oscillator

y 1 Љϩ͑C/ 2 2 ͒y 1 3 ϭ 1 2 2 4 ͭ Ϫ2 2 2 y 1 ϩ4Cy 1 3 sin 2 2 ϩ 2 ͑ 4hϪ5Cy 1 4 ͒sin 2 ͱ2hϪ Cy 1 4 2 Ϫ 2 2 y 1 Ј 2 ϩ2 2 2 y 1 Ј ͫ Ϫsin 2 2 ϩ 2 y 1 cos 2 ͱ2hϪ Cy 1 4 2 Ϫ 2 2 y 1 Ј 2 ͬͮ ϩO͑ 2 ͒ ϵg͑ y 1 ,y 1 Ј , 2 ͒ϩO͑ 2 ͒. ( 6
)
We note that the derived expression is approximate since it neglects O( 2 ) terms; this approximation was imposed by the impossibility of finding an exact expression for L in ͑5͒. As a result, the following analysis is valid only for undamped systems ͑1͒ with sufficiently weak coupling. In ͑6͒, y 1 ϭy 1 ( 2 ), and primes denote differentiation with respect to 2 . In addition, the ''forcing term'' on the right-hand side is 2-periodic in 2 . This com- pletes the reduction process.

Employing the previous analysis, the problem of computing the periodic orbits of the undamped system ͑1͒ is equivalent to the problem of computing the periodic solutions of the reduced system ͑6͒. This equivalence holds since a periodic motion in y 1 and v 1 under a periodic change in 2 leads to a periodic motion for I 2 as well. Since the reduced system is essentially nonlinear we resort to an analytical/numerical technique to compute the periodic orbits and their bifurcations. In particular, we introduce a nonsmooth transformation of variables to transform the problem to a

Periodic Orbits of the Underlying Hamiltonian System

The underlying Hamiltonian two-degrees-of-freedom system is obtained by setting ϭ0in ͑1͒. At a fixed level of energy ͑Hamil-tonian͒ we employ the reduction method outlined in ͓13͔ to reduce the undamped system ͑1͒ to a single-degree-of-freedom nonautonomous oscillator with periodic forcing. This is a standard reduction process by which an (nϩ1)-degree-of-freedom Hamiltonian system with symmetry ͑symmetry of time translations͒ is reduced to an n-degree-of-freedom nonautonomous system with no symmetry.

set of nonlinear boundary value problems over finite domains. This technique was first developed by Pilipchuck ͓14,15͔ and then applied to smooth and nonsmooth problems in dynamics in a series of works ͓͑16,17͔͒. We refer the reader to these works for more technical details of the method.

We now compute the periodic solutions of ͑6͒ with period T ϭ4a ͑yet undetermined͒. We express the solution in the following form:

y 1 ͑ 2 ͒ϭX͑ ͑͒͒ϩe͑͒Y ͑ ͑͒͒, ϭ 2 /a (7)
where the new independent variables ͑͒ and e() are bounded nonsmooth functions of their argument :

͑͒ϭ 2 arcsin ͫ sin ͩ 2 ͪͬ , e͑ ͒ϭЈ͑ ͒. (8)
The derivative in ͑8͒ should be understood in the context of the theory of distributions. Both nonsmooth variables are periodic in with ͑normalized͒ period equal to 4. We note that by ͑7͒ the solution is expressed in terms of two components. X depends only on and is termed the R-component of the solution. Y also depends solely on and is multiplied by e; it is termed the I-component of the solution. Interestingly, expression ͑7͒ has a phenomenological resemblance to complex variable representation with e playing the role of the imaginary constant j ͑note that e 2 ϭϪj 2 ϭ1͒.

Employing the transformation ͑7͒ we express the derivatives and powers of y 1 in ͑6͒ in terms of X and Y, and set separately the R and I-components of the resulting expression equal to zero. We then obtain the following two subproblems governing the R and I-components of the solution:

Subproblem 1. Y Љϩ͑Ca 2 / 2 2 ͒Y 3 ϭa 2 g͑ y 1 ϭY ,y 1 ЈϭY Ј/a, 2 ϭm ͒ϩO͑ 2 ͒, aϭm, Xϭ0 Y ͑ Ϯ1 ͒ϭ0, mϭ1,2,3,.... ( 9
)
Subproblem 2.

XЉϩ͑Ca 2 / 2 2 ͒X 3 ϭa 2 g ͩ y 1 ϭX,y 1 ЈϭXЈ/a, 2 ϭ ͑ 2nϪ1 ͒ 2 ͪ ϩO͑ 2 ͒, aϭ ͑ 2nϪ1 ͒ 2 , Y ϭ0 XЈ͑Ϯ1 ͒ϭ0, nϭ1,2,3,.... ( 10
)
odic orbit with period equal to 2p, i.e., p-times the period of the nonhomogeneous term g. Finally, we note that the periodic solution y 1 (t) is obtained from the solutions of the nonlinear boundary value problems either as y 1 ( 2 )ϭe( 2 /a)Y (( 2 /a)) ͑Subproblem 1͒,o ra sy 1 ( 2 )ϭX(( 2 /a)) ͑Subproblem 2͒, where 2 ϭ 2 tϩ 20 ϩO(). Now, the nonlinear boundary value prob- lems above provide the solution only in the normalized halfperiod ͓Ϫ1,1͔. To extend the result over a full normalized period ͑equal to 4͒ we need to add the component of the solution in the interval ͓1,3͔; to perform this we take into account the symmetry properties of the nonsmooth variables and e, and add either the antisymmetric image of the solution about the point (Y ,)ϭ(0,1) ͑for Subproblem 1, cf. Fig. 3͑a͒͒, or the mirror image of the solution about the line ϭ1 ͑for Subproblem 2, cf. Fig. 3͑b͒͒.

The nonlinear boundary value problems ͑9͒ and ͑10͒ were solved using a single-point numerical shooting method. In Figs. 4 and 5 we depict the leading low-order subharmonic orbits and their bifurcations for the undamped two-degrees-of-freedom system with 2 2 ϭ0.9, Cϭ5.0, ϭ0.1 and varying values of the total energy h. In these figures we also present one-period representations of a number of subharmonic orbits. In the bifurcation plot of Fig. 4 we depict the values of Y Ј(Ϫ1) at the subharmonic orbits as functions of h ͑recall that the solution domain of the above nonlinear boundary value problems is Ϫ1рр1͒; in physical terms, each point denotes the initial slope ay 1 Ј of the subharmonic orbit, corresponding to zero initial displacement, y 1 ϭ0. In Fig. 5 we plot X(Ϫ1) as function of h; in physical terms, each point denotes the initial displacement y 1 of the subharmonic orbit, corresponding zero initial slope, y 1 Јϭ0. These plots depict only the responses of the unexcited oscillator 1; the corresponding responses of oscillator 2 are computed using relation ͑5͒ derived in We note that the above subproblems were obtained by setting either the R or I-component of the solution equal to zero. Then, the solutions of each subproblem provide a distinct class of subharmonic motions of the problem. In general, the problem obtained by applying the previous method leads to a coupled system of equations in X and Y, however, this case will not be considered here.

Since no analytical solution exists for these nonlinear boundary value problems we need to resort to a numerical method to solve them. Before we perform this numerical computation, however, we make the following remarks concerning the method of nonsmooth transformations. The boundary conditions in ͑9͒ and ͑10͒ impose smoothness on the transformed derivatives of y 1 ; these boundary conditions define the domain of the solutions of the nonlinear boundary value problems ͓Ϫ1,1͔. In addition, the quarter-period of the solution, a, for each subproblem is allowed a countable infinity of values. Taking into account that the period of the periodic solution is Tϭ4a, and that the nonhomogeneous term in ͑6͒ is 2-periodic in 2 , we conclude that Subproblem 1 computes the 2m:1, mϭ1,2,3, ... subharmonic orbits of ͑6͒, whereas Subproblem 2 computes the (2nϪ1):1, nϭ1,2,3, ... subharmonic orbits. A subharmonic orbit of order p:1 is a peri-system, the large regular region surrounding orbit A is expected to become a large region of attraction, with that orbit becoming an attractor. Additional stable and unstable 1:1 subharmonic orbits of the system are indicated in the Poincare ´maps, confirming the approximate asymptotic results of Fig. 5. At the small energy level hϭ0.05 there are two stable subharmonic orbits; both orbits correspond to localized motions, with orbit A localizing in oscillator 1 and orbit B in oscillator 2. At higher values of h the low-energy bifurcation of 1:1 subharmonic orbits ͑predicted in the plot of Fig. 5͒ has occurred and there exist four orbits, three stable and one unstable. Note that as h increases orbit A gradually delocalizes from oscillator 1 and localizes in oscillator 2. The regions of chaotic motion ͑the ''stochastic sea''͒ in the maps is a welldocumented feature in the dynamics of such strongly nonlinear systems.

The bifurcation diagrams and Poincare ´maps of the 1:1 subharmonic orbits lead to a preliminary qualitative explanation of the energy pumping phenomenon, which, as shown in the previous section, occurs only above a certain level of the initial energy h. When energy pumping occurs, an initial transfer of energy occurs from the directly excited ͑linear͒ oscillator 2 to the unexcited ͑nonlinear͒ oscillator 1; moreover, the ''fast'' oscillation during this initial phase of the motion has a frequency nearly identical to the linearized natural frequency 2 . Hence, it is logical to con- clude that the 1:1 subharmonic orbit A existing over the entire range of h and having a large domain of attraction, is mainly the course of the reduction process, and the relations y 2 ϭͱ2I 2 / 2 sin 2 , 2 ϭ 2 tϩ 20 ϩO(). Finally, we emphasize that, since the reduced system ͑6͒ neglects O( 2 ) terms, the results presented in Figs. 4 and 5 are approximate and valid only for sufficiently small values of . Of special interest are the 1:1 subharmonic orbits labeled A-D in Fig. 5. These orbits dominate the dynamics as shown below.

The domain of attraction and the stability of the 1:1 subharmonic orbits were determined by numerical Poincare ´ maps. These were constructed by considering the original undamped Eqs. ͑1͒. First, the four-dimensional phase space of the solutions of ͑1͒ was reduced to a three-dimensional isoenergetic manifold Ꭽ by fixing the total energy to a constant level, H (y 1 ,y ˙ 1 ,y 2 ,y ˙ 2 )ϭh; Ꭽ was then ''cut'' by the Poincare ´ section ⌺ϭ͕(y 1 ,y ˙ 1 ,y 2 )ʦᎭ/y 2 ϭ0,y ˙ 2 Ͼ0͖. The Poincare ´ map P was defined as, P :⌺→⌺, i.e., as a mapping of points on ⌺ to their images under the flow of the dynamical system on ⌺, under orientation preserving restrictions. Stable periodic orbits of ͑1͒ appear as centers in the Poincare ´ map, whereas, unstable periodic orbits appear as saddle points.

In Fig. 6 we depict the Poincare ´ maps of the undamped system ͑1͒ with Cϭ5.0, ϭ0.1 and varying values of the energy h. The 1:1 subharmonic orbits labeled A-D correspond to the ones of the approximate bifurcation plot of Fig. 5. A common feature of all these plots is a large region of regular motion ͑smooth quasiperiodic orbits͒ in the upper regions of the plots, surrounding the stable 1:1 subharmonic orbit A. When damping is added to the transients of oscillator 1 can not attain sufficiently large amplitudes, they cannot act as ''bridging'' orbits to excite the 1:1 subharmonic orbit A and no energy pumping can take place. Thus, for sufficiently low values of h no energy pumping is possible. For higher values of h the initial transients for oscillator 1 attain sufficiently large amplitudes to excite the 1:1 orbit and energy pumping can occur. This conjecture explains the lack of energy pumping for hϭ0.5 in the simulations of Fig. 1. Clearly, the previous arguments form merely a conjecture, but the more rigorous analysis in Part II of this work ͓͑1͔͒ validates these arguments.

Fig. 5 Leading "2nÀ1…:1 subharmonic orbits as functions of h:

X"À1… for nÄ1, ÀX"À1… for nÄ2, X"À1… for nÄ3; ÃÃÃÃ unstable 1:1 subharmonic orbits responsible for the energy pumping phenomenon. However, this family of orbits can not be directly excited at tϭ0 since it cannot satisfy pointwise the initial condition ((y 1 (0),y ˙ (0))ϭ(0,0) ͑this is the initial state of oscillator 1 when the energy pumping phenomenon is initiated͒; as a result, a transient ''bridging'' orbit must be initially excited, satisfying zero initial conditions and ultimately ''connecting'' with the 1:1 subharmonic orbit A. Under these conditions energy pumping occurs. Noting that the amplitude of the 1:1 orbit has a lower bound of approximately 0.47 ͑cf. Fig. 5͒ and considering the initial transients of the numerical simulations of Fig. 1, we conjecture that if the initial transient

Analytical Approximations

As a final note, we now present an analytical technique to approximate the transient responses of the Hamiltonian system; in contrast to most standard techniques which are based on the assumption of weak nonlinearity, the method used here deals with the strong ͑nonlinearizable͒ nonlinearity of oscillator 1. An extension of this technique for the damped system in Part II of this work ͓͑1͔͒ will enable us to analytically approximate the transient responses during the initial phase of energy pumping of Fig. 1.

To this end, we express system ͑1͒ in the following form:

y ¨1ϩy 1 ϩCy 1 3 Ϫy 2 ϭ0 (11) y ¨2ϩ 2 y 2 Ϫy 1 ϭ0 where 2 ϭ 2 2 ϩ. A transformation to complex variables is now introduced, 1 ϭy ˙1ϩ jy 1 , 2 ϭy ˙2ϩ jy 2 , (12) and ͑12͒ are rewritten as

˙1Ϫ j 2 ͑ 1 ϩ 1 *͒Ϫ j 2 ͑ 1 Ϫ 1 *͒ϩ jC 8 3 ͑ 1 Ϫ 1 *͒ 3 ϩ j 2 ͑ 2 Ϫ 2 *͒ϭ0 (13) ˙2Ϫ j 2 ϩ j 2 ͑ 1 Ϫ 1 *͒ϭ0
where the star denotes complex conjugate. An approximate solution of ͑13͒ is sought, based on the assumption of fast oscillations at frequency :

1 ϭ 1 e jt , 2 ϭ 2 e jt . ( 14
)
͑13͒, and averaging over the fast periodic terms e jt we obtain the following set of averaged equations governing the ͑complex͒ amplitudes i , iϭ1,2:

˙1ϩ j 2 ͩ Ϫ ͪ 1 Ϫ 3 jC 8 3 ͉ 1 ͉ 2 1 ϩ j 2 2 ϭ0 (15) ˙2ϩ j 2 1 ϭ0.
Interestingly, in contrast to ͑11͒, the transformed system ͑15͒ is completely integrable, with the following two first integrals of motion:

͉ 1 ͉ 2 ϩ͉ 2 ͉ 2 ϭN 2 , j 2 ͉ 1 ͉ 2 Ϫ 3 jC 16 3 ͉ 1 ͉ 4 ϩ j 2 ͑ 1 2 *ϩ 1 * 2 ͒ϭH. (16)
Employing these results, the complex amplitudes are expressed as 1 ϭN sin e j␦ 1 , 2 ϭN cos e j␦ 2 .

(17) Substituting ͑17͒ into ͑15͒, we obtain the final set of equations on the 2-Torus governing the angle-variables and ␦ϭ␦ 1 Ϫ␦ 2 :

␦ ˙ϩ 2 Ϫ 3CN 8 3 sin 2 ϩ cot 2 cos ␦ϭ0 (18) ˙ϩ 2 sin ␦ϭ0.
We note that the orbits of ͑18͒ can be analytically computed in terms of pseudo-elliptic quadratures by employing the change of variables ͑17͒ in ͑16͒, and then integrating the second of Eqs. ͑18͒ by quadratures.

By numerically integrating ͑18͒ we can study transient ͑nonpe-riodic͒ orbits in the neighborhoods of the 1:1 subharmonic orbits of the underlying Hamiltonian system. In Fig. 7 we present the phase plots of ͑18͒ for varying values of the energy-like first integral N, confirming the bifurcations of the 1:1 subharmonic orbits depicted in the analytical approximations of Fig. 5 and the numerical results of Fig. 6. Relations ͑14͒ indicate the presence of 1:1 internal resonance in the fast dynamics of oscillators 1 and 2. Substituting ͑14͒ into is above a critical level. Hence, energy can be ''pumped'' to a predetermined part of the system ͑the nonlinear oscillator͒, which, in essence acts as a passive nonlinear sink. Clearly, no such phenomenon is possible in linear systems.

A 1:1 stable subharmonic orbit of the underlying Hamiltonian system ͑obtained by setting damping equal to zero͒ was conjectured to be responsible for the energy pumping phenomenon. We conjecture that the reason for lack of energy pumping at low energies is due to the fact that the 1:1 subharmonic orbit can not be excited unless the energy of the system is above a critical level. The energy pumping phenomenon will be further studied in a companion paper ͓͑1͔͒ where it will be shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.

We remark that the energy pumping phenomenon, which in this work was purely passive, could be enhanced using active control. Utilizing this phenomenon one can introduce passive or active nonlinear sinks in predominantly linear extended periodic structures where externally imparted energy is directed and locally eliminated. This can lead to enhanced vibration and shock isolation designs of extended mechanical systems.

Discussion

We presented numerical evidence of energy pumping in nonlinear mechanical oscillators. The systems considered herein were composed of weakly coupled, linear, and essentially nonlinear ͑nonlinearizable͒ parts. In such systems it is possible to induce one-way ͑irreversible͒ ''channeling'' of vibrational energy from the linear to the nonlinear part, provided that the imparted energy

Fig. 1

 1 Fig. 1 Numerical transient responses y 1 "t… and y 2 "t… of system "1… for "a… hÄ0.5, "b… hÄ0.8, "c… hÄ1.125; oscillator 1, oscillator 2

Fig. 2

 2 Fig. 2 Numerical transient response of the three-degrees-offreedom system for "a… y ˙3"0…Ä2.0, "b… y ˙3"0…Ä3.0, and "c… y ˙3"0…Ä4.0; oscillator 1, """"oscillator 2, oscillator 3

Fig. 3

 3 Fig. 3 Construction of the solution y 1 over an entire normal- ized period "equal to 4… from the half-normalized period solutions "a… eY"…, and "b… X"…

Fig

  Fig. 4 Leading 2m:1 subharmonic orbits as functions of h:YЈ"À1… for mÄ1, YЈ"À1… for m Ä2,YЈ"À1… for mÄ3

Fig. 6

 6 Fig.6Poincare ´maps of the dynamics of the undamped system "1… at varying energy levels for 2 2 Ä0.9, CÄ5.0, Ä0.1: "a… hÄ0.05, "b… hÄ0.2, "c… hÄ0.8, "d… h Ä2.0

Fig. 7

 7 Fig. 7 Phase plots of the system of Eqs. "18… for 2 2 Ä0.9, C Ä5.0, Ä0.1, and varying values of the first integral of motion N: "a… NÄ0.4, "b… NÄ0.8, "c… NÄ1.9

Acknowledgments

This work was supported in part by NSF Young Investigator Award CMS-94-57750 ͑Dr. Allison Flatau is the Grant Monitor͒, and by an International Supplement to this Award ͑Ms. Cassandra Dudca is the Grant Monitor͒.


P Bonnet

N Nomède Martyr

H Bizot

J-P P Buisson

A Buleon

Olivier Chauvet

A Buelon

Photophysical comparative study of amylose and polyvinyle pyrrolidone / single walled carbon nanotubes complex

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INTRODUCTION:

The first report of near Infra-Red (NIR) fluorescence of Single Walled Carbon Nanotubes (SWNTs) by O'Connell et al. in 2002 1 has initiated a worldwide large research effort. Basic studies attempt to characterize and to understand the fundamental phenomena that drive the optical properties (electronic structure, excitonic decay and splitting…). Some other works are also using these peculiar optical properties for device applications (optoelectronic, sensors, etc.) 2,3,4 . As already shown by O'Connell et al, NIR fluorescence is only observed when the SWNTs are isolated: the interactions of the SWNTs with their environment but also an intrinsically low quantum yield (Q<10 -3 ) are critical for emission 1,5,6 . Indeed a controlled increase of the quantum yield may be very important for optoelectronic applications of the SWNTs. The elaboration of luminescent polymer/SWNT complexes or composites can be a way to modify the quantum yield, and multifunctional applications can be envisioned for these materials in the future. In order to prepare these composites, covalent or non-covalent functionalization of SWNTs by polymers may be used. Functionnalization of SWNTs is used for SWNTs dispersion, isolation and selectivity 7,8,9 , for production of SWNTs-based composites and for tailoring their properties 10,11 , for elaboration of sensors or devices for energy and optoelectronic applications 12,13,14 or for biological applications 13,15,16 . Non covalent functionnalization is especially interesting since it preserves the structural integrity of SWNTs and thus their physical properties, and also because it is reversible. The purpose of this paper is to show that the optical properties of polymer/SWNT complex are influenced by the complex supramolecular structure and by the interaction between the polymer chains and the SWNTs. Here we compare amylose derivative/SWNT and PVP/SWNT complexes where amylose is a polysaccharide biopolymer and PVP is (synthetic) polyvinyl pyrrolidone. These two different polymers have been chosen because they were supposed to wrap the SWNTs. Actually, these polymer/SWNT complexes exhibit quite different optical properties and we show that these is because they have different microstructures. Indeed, it shows that the supramolecular organization of the polymer/SWNT complex is a key point for potential applications based on the SWNTs optical properties.

EXPERIMENTAL SECTION:

As-produced HiPCO SWNTs (from Carbon Nanotech. Inc.) have been dispersed in D 2 O (0.5 mg.cm -3 ) in presence of Sodium Dodecyl Benzensulfonate (SDBS) (0.7wt%) following the procedure of ref. 17. After sonication and ultracentrifugation (120000g for 2h), isolated SWNTs suspended in SDBS micelles are obtained in the suspension. Hydroxypropylated Amylose (AmH) (Amylose from Aldrich, purified by butanol precipitation and hydroxypropylated at INRA Nantes 18 , M w =360000) or Polyvinyl Pyrrolidone (PVP) (from Aldrich, M w =360000) have been added to the suspensions. AmH and PVP chemical formula are given in Figure 1. Once the polymer is introduced in the suspension, the mixture is sonicated for 10 h in a water bath. The polymer content, [AmH] or [PVP] respectively, varies between 0 to 4.5 wt% as shown in Table 1. When needed, PVP/SWNT and AmH/SWNT suspensions have been dialyzed with a spectrapor MWCO 12-14000 membrane (from Spectrumlabs) against water and an ion retardation resin (AG11A8 from BioRad) for 48h in order to remove the SDBS surfactant.

Photoluminescence (PL) and Raman scattering spectra of the suspensions have been obtained with a RFS 100 Brucker FT Raman spectrometer using an excitation line at 1064 nm (1.16 eV). The detector of the spectrometer allows to detect emitted or scattered photons with energy between 0.85 and 1.16 eV.

RESULTS AND DISCUSSIONS:

1 -Structure:

Homogeneous and black suspensions are obtained for the PVP/SWNT mixtures whatever the amount of PVP. Homogeneous suspensions are also obtained with AmH as long as [AmH]< 1 wt % but a partial sedimentation is observed when [AmH]> 1 wt % 19 . Indeed, the stability of the suspensions suggests that the SWNT are isolated and protected from the aqueous environment in the solution. It has already been shown 1,7,20,21 that PVP can wrap individual SWNTs with several possible microstructures including double and triple helix 7 . Amylose is also considered as a wrapping agent of SWNTs 22,23 .

One way to probe the interaction of the SWNTs with their environment is to investigate their optical properties. Figure 2 shows the near Iinfra-Rred fluorescence spectrum of the SWNTs dispersed in SDBS before the introduction of the polymers in the solution, in the energy range 0.85-1.16 eV and with an excitation at 1.16 eV. NIR fluorescence of nanotubes is due to the radiative recombination of the S 11 singlet excitonic states 24,25 . The fluorescence peaks can be assigned to (9,4) (12,1) (11,3) (10,3) and (8,7) SWNTs following Bachilo et al 26 . The stars on the spectrum indicate the Raman G bands and RBM bands of the nanotubes. These SWNTs have diameters between 0.89 and 1.02 nm. The strong fluorescence observed in Figure 2 shows that the SWNTs are individualized by the strong ultrasonic and ultracentrifugation treatment in presence of SDBS before the introduction of the polymers.

Figure 3 shows the emission spectra of AmH/SWNT (left panel) and PVP/SWNT (right panel) suspensions (for 0.05, 2 and 4.5 wt %) before (curve 1) and after dialysis (curve 2). Before dialysis and for a polymer amount of 0.05 wt%, the spectra reveal both fluorescence bands (broad bands) and Raman bands (sharp bands, labelled with stars). The spectrum of AmH/SWNT is very close to the spectrum of free suspended SWNTs in SDBS micelles (cf Figure 2). When the amount of amylose increases, the PL bands loose their intensity and for [AmH]=4.5 wt%, fluorescence from (9,4) and (8,7) tubes becomes difficult to observe. Upon dialysis, the PL bands disappear for all [AmH] content and only the Raman bands remain visible. The situation is different for PVP/SWNT. For [PVP]=0.05 wt%, both fluorescence and Raman bands are visible but the fluorescence bands are shifted by comparison with Figure 2. In this case, dialysis leads to a frequency shift of the PL bands and quite a small reduction of their intensity. When the amount of PVP increases, there is no significant evolution of the emission spectra before or after dialysis. The intensity of the fluorescence bands does not seem to depend on [PVP] and dialysis results in quite a strong redshift of the PL bands.

The behaviour observed for PVP/SWNT is a signature of the wrapping ability of PVP. The addition of PVP to the SWNTs suspension in SDBS micelles induces a redshift of the PL peaks and a small loss of intensity. This shift is independent of the amount of PVP. It is due to the modification of the exciton energy with the wrapping of PVP directly around the SWNTs which causes a change of effective dielectric constant 27,28 . The microstructure of PVP wrapped SWNTs does not change with further additions of PVP unlike in the case of AmH addition. When SDBS is removed, a further 16 ± 4meV redshift and a loss of intensity are observed. It means that SDBS was still in contact with the nanotubes. When adding PVP into the SDBS stabilized SWNTs dispersion, it is likely that PVP has not been exchanged completely with SDBS. A part of the SWNTs surface is not covered by the polymer but by the surfactant. The dialysis removes the remaining SDBS, leaving SWNTs wrapped by PVP in the solution.

The situation is different in presence of AmH. There is an absence of shift for [AmH]=0.05% which shows that AmH is not in close contact with the SWNTs surface and conversely that SDBS is still covering a large part of the surface. With further addition of AmH, fluorescence is progressively quenched and it shifts progressively towards the red. The quenching of fluorescence is likely related to a (partial) sedimentation of the AmH/SWNT dispersion when more than 1 wt% of AmH is added into the SWNT suspension 19 . Please note also that a sedimentation of the suspensions is also observed when dialysis sets in AmH/SWNT dispersion. It means that AmH is not able to wrap the nanotubes, presumably because it interacts with the surfactant and three centers AmH/SWNTs/SDBS complexes are formed 17,19 .

PVP/SWNT and AmH/SWNT complexes may thus have different microstructure as reflected by the sedimentation and the fluorescence behaviour.

The supramolecular organization of the PVP/SWNT and AmH/SWNT complex are thus very different. In the first hand, PVP wraps individual SWNTs as described in the literature 7,20 and PVP wrapped SWNTs complex are soluble. In the second hand individual SWNTs are coated by AmH but SDBS is still necessary to stabilize water soluble complex. We suggest that SDBS interacts with the AmH/SWNT complex 0.85 0.90 0.95 1.00 1.05 1.10 through its alkyl part which fills the hydrophobic gap of the AmH helix 29 . This interaction leads to the formation of a three centers SDBS/AmH/SWNT complex 17,19 . The putative structures of PVP/SWNT and the SDBS/AmH/SWNT complex are sketched in Figure 4. Monte Carlo simulations 30 have shown that the flexible polymers chains, like PVP 31 , can wrap around individual SWNTs, while the helical conformation of AmH is not flexible enough to allow wrapping. SDBS/AmH inclusion complex are thus more easily formed. In situ polymerization of amylose around SWNTs, as proposed by Yang et al., may be a more efficient route for a direct wrapping 32 .

-Optical properties of the dispersions:

As shown in Figure 3, the optical characteristics of the SWNTs are very sensitive to their environment. In order to be more quantitative, the emission spectra have been fitted with lorentzian lines associated to the fluorescence peaks. An example is shown in Figure 5. It allows to extract the positions, the linewidths and the intensities of the fluorescence peaks. In Figure 6, the position of the (9,4) and (8,7) fluorescence peaks and the intensity of the (9,4), (12,1), (11,3), (10,3) and (8;7) peaks are plotted versus the polymer content. The peak intensities are normalized to the Raman G peaks and for each chirality (n,m) the maximal intensity is normalized to 1. According to Moore et al. 33 , the SWNT fluorescence peak to Raman Gpeak intensity ratio gives a qualitative measure of the energy transfer of the excited state to the surroundings.

In Figure 5, the PL peak position shifts about 20 meV for [AmH] = 4. 5wt% and about 9 meV for [PVP] = 4.5 wt % for (9,4) SWNTs. These values are consistent with the shifts reported for PVP 7,20 , DNA 34 or gelatine 35 . This red shift is a direct consequence of the dielectric screening of the coulomb interaction in SWNT by its environment. The exciton energy is the sum of two main terms: the direct Coulomb interaction which drives the binding energy of the exciton (E B ∝ε -α where ε is the effective dielectric constant and α ~1.4 27 ) and the exchange interaction which drives the exchange energy (E ex ∝ε -β with β=1 28 ).

These two terms have opposite contributions on the value of the exciton energy, a positive one for the exchange interaction and a negative one for the Coulomb interaction 28 . When the effective dielectric constant is modified, the exciton energy changes as ∆E=(∆ε/ε)×(αE B -βE ex ).

(1) In SWNTs the exchange interaction is stronger than the Coulomb interaction, and thus an increase of the dielectric constant leads globally to a red shift of the PL peaks 28 .

Dukovic et al. 36 . have estimated the binding energy of the SWNT exciton from two-photon photoluminescence excitation spectroscopy (PLE). For a (9,4) SWNT with a diameter of 0.92 nm, E B is found to be of the order of 370 meV. Lefebvre et al. 28 have estimated the exchange energy of a nanotube with the same diameter and they have found about 660 meV. Using these values in formula (1) and the shifts extracted from Figure 5 leads to a relative change of the dielectric constant of about ∆ε/ε ~14% and 6% for 4.5% of AmH and PVP respectively. The red shifts for the (8,7) SWNTs are smaller because E B and E ex decrease when the SWNT diameter increases 28,36,37 .

It is worth noting the important differences in the dependence of the PL peaks positions with the amount of AmH and PVP. This is the signature of the different microstructure of the PVP/SWNTs or AmH/SWNTs complex. Addition of only 0.05 wt% of PVP results in a strong shift of about 8 meV while further addition does not shift any more the lines. The rapid shift of the line suggests that a small quantity of PVP is sufficient to wrap the individual SWNTs.

Conversely, adding less than 1% of AmH does not shift the fluorescence lines. Below 1wt% the interaction between AmH and SWNTs seems thus to be weak. It is attributed to a supra molecular structure where the SWNT is still in contact with the SDBS surfactant, the AmH helix being essentially in contact with the surfactant alkyl queue. As a consequence the direct environment, and thus the effective dielectric constant, of the SWNTs is not modified. The shift of the PL line starts to happen above [AmH]=1%. It corresponds to the quantity above which sedimentation occurs. We argue that this concentration plays an equivalent role to a critical micellar concentration: isolated AmH chains complexified by SWNTs/SBDS are no more stable in water and AmH chains tend to aggregate. The SWNTs/SBDS complex are glued in a AmH polymer network similar to a composite. The transition from free AmH/SDBS/SWNT complex to embedded SWNTs causes an important change of dielectric constant (~14%).

The evolution of the fluorescence line position is also reflected in the evolution of the normalized fluorescence intensities of PVP/SWNT and AmH/SWNT suspensions (see lower panel of Figure 6). The introduction of 0.05wt% of PVP reduce by ~ 40% the PL intensity relative to free micelles suspended SWNTs, after which no important modifications are noticed for higher concentrations in PVP. When SWNTs are wrapped by PVP, the change of dielectric constant changes also the exciton energy. It results in a modification in the resonance conditions which leads to a decrease in the PL intensities. It can be noticed that within our experimental accuracy, this process does not seem to be selective in chiralities since the fiveour fluorescence lines behave quite in the same way.

It is very different with AmH where the fluorescence intensity remains constant below 1% of AmH and it decreases strongly at 2%. Fluorescence is fully quenched at 4.5%. Quenching of fluorescence cannot be only attributed to the modification of the resonance conditions as discussed above. First, addition of PVP changes also the resonance conditions and the fluorescence survives. Furthermore, important line shifts have also been reported in the literature without quenching: Chou et al. report a ~28 meV red shift relative to SDS stabilized SWNTs for (9,4) SWNTs engaged in DNA/SWNT and the fluorescence is still effective 34 . In our case and as shown in Figure 3 after dialysis, the (9,4) SWNTs PL peak is red shifted by 33 meV with [PVP]=4.5wt% of PVP by comparison with SDBS/SWNT dispersion. Two other mechanisms can be invoked to explain the loss of fluorescence. A self quenching process may occur by exciton energy transfer (EET) between close enough nanotubes. The exciton desexcitation may also be mediated in a non radiative way via the AmH chains. The EET process consists of the transfer of the exciton energy from an excitated SWNT (donor) to a neighbouring SWNT (acceptor). It requires that the two SWNTs are close enough together and that there is a spectral overlap between the donor emission and the acceptor absorption bands. Usually the EET is associated with an increase of the emission intensity of the acceptor. Förster resonance energy transfer (FRET) is the mechanism which is the most commonly invoked for SWNTs 38,39,40 . The FRET efficiency is proportional to R DA -6

where R DA is the intertube distance. Indeed the formation of the AmH/SWNT "composite" structure increases the local concentration of SWNTs and thus it may enhance the probability of EET. In table 2, we have listed some possible donor-acceptor couples concerned by this process for semiconducting couples: the SWNT donor bandgap must be larger than the SWNT acceptor one. It should also be added that EET to a metallic tube is likely which results in a absence of fluorescence. As can be seen in Figure 3 and Figure 5, there is a general decrease of the fluorescence and it does not seem that there is a transfer from one kind of semiconducting tubes to another. It suggests that the FRET mechanism is not responsible for the quenching of the fluorescence and the quenching is mainly due to a non radiative desexcitation of the exciton through the polymer chains. Such a process is likely related to the broadening of the PL lines that is observed when [AmH] increases. For instance the (8,7) PL line broadens by about 8.5 meV when 3.3% of AmH is introduced (it corresponds to a relative broadening of 46%). The broadening is the signature of the coupling to the vibrational modes of AmH. In summary, we argue that below 2%, the optical properties of the AmH/SWNT dispersion are mainly driven by the dielectric screening, while above 3% quenching of the fluorescence is mainly caused by a strong coupling with the vibrational modes of the AmH network.

CONCLUSION:

In Summary, PVP/SWNT and AmH/SWNT complex have been prepared in aqueous solutions in presence of SDBS surfactant. The NIR fluorescence of the PVP/SWNT complex is quite independent of the amount of PVP, unlike the case of the AmH/SWNT complex. We attribute these different behaviours to the two distinct microstructures of the complexes. With PVP, a wrapping of the SWNTs occurs while with AmH, the situation is more intricate: at low content, isolated three-body complexes are formed while at high content a composite structure is obtained. The influence of polymers molecular weight and the functionalisation degree of amylose on the polymer/SWNT complex microstructures is under investigation.

The emission properties of the PVP/SWNT complex are driven by the dielectric screening resulting from the addition of PVP to the SWNT suspensions. This is also the same for AmH/SWNT complex below [AmH] = 2%. Above this quantity of AmH, the quenching of the fluorescence is observed. We argue this is due to the coupling of the excitonic states with the vibrational modes of the polymer network and the AmH/SWNT composite microstructure.

This paper shows that the optical properties are efficient probes of the microstructures of polymers/SWNT complexes. It can be helpful for optoelectronic or sensing applications. ********** REFERENCES:

FIGURE 1 :

 1 FIGURE 1: Chemical structures of (a): PVP; (b): Hydroxypropylated amylose (AmH). The circle shows the hydroxypropyl side group.

Table 1 :

 1 Compositions of the AmH/SWNT and PVP/SWNT complex.

1 FIGURE 3 :

 13 FIGURE 3:(Left panel) Emission spectra of AmH/SWNT complex with [AmH]= 0.05, 2.3 and 4.5 wt % from top to bottom respectively. (Right panel) Emission spectra of PVP/SWNT complex with [PVP]= 0.05, 2.0 and 4.5 wt % from top to bottom respectively. Spectra labelled "1" correspond to the as produced complex. Spectra labelled "2" are obtained after the dialysis of the dispersion. They are free of SDBS. The stars indicate the Raman G band and the RBM band, the arrows underline the PL peak shifts induced by the dialysis treatment.

FIGURE 6 :

 6 FIGURE 6: (Upper panel): Shift of the PL peak positions of the (9,4) and (8,7) SWNT versus the polymer content for AmH (left) and PVP (right). (Lower panel): Normalized PL intensity evolution versus the polymer content for AmH (left) and PVP (right). The dashed lines are guides for the eyes.

Table 2 :

 2 Possible Donors-Acceptors couples in an exciton energy transfer process.

	Donors	Acceptors
	(10,2)	(9,4); (12.1); (11.3); (10,3); (8,7)
	(9,4)	(12.1); (11.3); (10,3); (8,7)
	(12,1)	(11.3); (10,3); (8,7)
	(11,3)	(10,3); (8,7)
	(10,3)	(8,7)


Vincent Bragard

Thomas Pellegrini
email: thomas.pellegrini@irit.fr

Julien Pinquier
email: julien.pinquier@irit.fr

Pyc2Sound: a Python tool to convert images into sound

Keywords:

This article reports ongoing work on a user interface dedicated to generate sound from pictures and hand drawings.

If we imagine what sound would correspond to a given image, on what parameters of the image do we focus and what would the result sound like? In this paper, we try to answer this question by giving a model transforming images into sound based on chosen parameters extracted from the image. For this, an input image is first binarized, then its skeleton is extracted and 'tracks' are identified and used to generate chirps in an additive synthesis approach.

INTRODUCTION

A tool capable of transforming an image into sound could have several possible applications. For example, it could serve as an artistic tool, which artists use to draw sound; it could also serve as a mean to teach children to correctly form letters and numbers; as a last example, it could allow blind people to hear a sound metaphor of a painting. Here, we propose a first prototype of such a tool. In the domain of sound synthesis, several pieces of software proposing a way to transform images into sound already exist, each with their own approach on the subject. For example, Audiosculpt, created by the IRCAM, filters white noise with an input image [START_REF]Audiosculpt: a visual and "sculptural" approach to sound manipulation[END_REF]. Another software, Photosounder, considers the input image as a magnitude spectrogram and applies an inverse FFT on it to synthesize a sound [START_REF]Photosounder[END_REF][START_REF] Sturmel | Signal reconstruction from stft magnitude: a state of the art[END_REF]. The limits of these approaches are that we can only treat the image as a whole instead of being able to treat different objects in the image differently. Another problem is to determine Figure 1: Input image the time domain of the resulting signal, different methods exist, but choosing which one to use is difficult and depends on the type of application we wish to make [START_REF] Seung-Yeo | Application of image sonification methods to music[END_REF]. A solution some softwares like VOSIS have adopted is to treat a video instead of a still image [START_REF] Mcgee | Vosis: a multi-touch image sonification interface[END_REF]. The tool presented in this article takes a different approach for which, instead of taking the image directly to synthesize a sound, a mathematical model is used to extract a skeleton of shapes contained in the image with a Voronoï diagram [START_REF] Aurenhammer | Voronoi diagrams: a survey of a fundamental geometric data structure[END_REF]. A graphical user interface will allow the user to generate the sound of an image. The user can also draw on a canvas and the drawing will be the basis for synthesis. In this paper, the three main steps (image skeletonization, track identification and synthesis) will be presented first, then the graphical user interface of the tool will be shown. In order to describe all the processing steps, we will consider the simple image depicted in figure 1.

IMAGE SKELETONIZATION

In an image, what we, as humans, perceive first, are the general shapes of it. Moreover, we tend to read them from left to right, and we also tend to correlate height in an image to the pitch's height in a sound. Since we want to imitate this kind of perception, images are taken as time-frequency representations, where time is represented on the x axis and frequency on the y axis. In order to generate a sound from these shapes (in our case a handwritten '3'), the shapes' skeleton contained in the image has to be extracted. To do so, the size of the image is doubled so that even if the shapes in it were one-pixel wide lines, they become two-pixel wide and have separated edges on each side. Then, the new image is binarized with a threshold given by the user, in order to precisely identify the shapes. Once it is done, the shapes' edges, i.e. shapes' pixels directly neighboring the background, are isolated and fed to a Voronoï algorithm. It gives the result shown in figure 2.

The Voronoï diagram contains too much information since it contains shapes' skeleton, but also edges in the space between shapes. It needs to be cleaned up. Several steps are needed in order to do so. Firstly, every edges that have one or both of their extremities outside a shape are removed. Secondly, all the vertical edges are 'rewired', as they can not be synthesized due to their infinitesimal time range. To rewire them, the weights of both the extremities of a vertical edge are calculated, the weight of a vertex being the sum of the length of all the edges connected to it. The extremity having the smallest weight is removed and every edge that was connected to it is rerouted to the other extremity of the vertical edge. At this point, there are still few unwanted edges at some places. In order to remove them, successive prunings are done on the skeleton, with a length threshold and a number of successive prunings given by the user. Finally, all the vertices which are not connected to any edges are removed and a clean skeleton is obtained as shown in figure 3. Note that in figure 3, the values of the axes have changed: the skeleton's vertices are now spread from 0 to 5 seconds in the time axis and from 800 to 1600 Hz in the frequency axis. These values are given by the user and were arbitrarily chosen for this example.

SEGMENTATION IN TRACKS

Now that the skeleton is properly scaled in the frequency and time domains, each edge can be considered as a linear chirp that will be synthesized.

But first, the different 'tracks' of the skeleton must be separated: since each edge is synthesized separately, we have to be careful of phase alignments at each vertex. A track is defined as a series of connected edges which only go forward in time and with only one simultaneous edge at a time. Figure 4 shows how our example would be split.

To achieve this, all the skeleton's edges are sorted in the

e d l i s t o f a l l t h e e d g e s i n i t i a l i z e L i s t T r a c k s a s an empty l i s t w h i l e L i s n ' t empty : i n i t i a l i z e CurTrack a s an empty l i s t t a k e t h e f i r s t edge o f L and put i t i n CurTrack w h i l e L c o n t a i n s an edge whose s t a r t i n g p o i n t i s t h e e n d i n g p o i n t o f t h e l a s t edge i n CurTrack : t a k e t h a t edge from L and put i t a t t h e end o f CurTrack s a v e CurTrack i n L i s t T r a c k s

After that, ListTracks contains the lists of each isolated track.

SYNTHESIS

Each track is now correctly isolated, and they are synthesized separately so that the final sound can be generated by additive synthesis. We could have chosen a sample based synthesis approach, where a sample sound is taken then modified, but we have chosen the additive synthesis approach so that we have more control over the synthesized sound. By construction, the edge list of each track is still sorted in the time domain, and the coordinates of its extremities give the starting/ending times and frequencies of the chirp to be synthesized. For each edge, a linear sine chirp is synthesized, then its ending phase is conveyed to the next edge in order for them to be aligned. Once the track is synthesized, a fade-in and a fade-out are applied to it, so that it does not start/end abruptly. The length of the fades are given by the user through the interface. The user can also decide to add harmonics to the synthesized sound and their strength.

USER INTERFACE

The user interface is a simple graphical interface. It contains two utilization modes: Harmonic melody, shown in figure 5, and Sound drawing, shown in figure 6. Harmonic melody allows the user to load an image, and asks for different parameters to generate the corresponding sound, like the binarization threshold, the desired frequency range, the pruning parameters and the length of fades. The user can also add harmonics to the generated sound thanks to sliders. Each time the user generates a sound, the graph of the clean skeleton is shown. It helps the user to tune parameters to his needs. If he is satisfied with the audio result, it can be saved as a file. An illustration of our example is available at https://goo.gl/QyMuO2. The second mode, Sound drawing is almost identical. However, instead of a text field to generate a sound from an image on the disk, a canvas is displayed, on which the user can draw with his mouse. This canvas supports basic features like undoing/redoing actions or drawing straight lines.

CONCLUSIONS

In this paper, a methodology has been proposed, allowing to extract relevant information of objects' shape in the image in order to synthesize a sound representative of these information. The sound itself is generated by additive synthesis, and is composed of different linear sine chirps.

We plan to extend Pyc2Sound's synthesis capabilities. To do this, other pertinent parameters in images will be identified and extracted in order to enrich the complexity of the generated sounds. For example, the signal amplitude will be influenced by the thickness of the shape, and color will influence the harmonics' power. We could also make the sound richer by using the image's spectra as a basis for synthesis.

In order to reach the possible applications we proposed in the introduction, we plan to port the code on more mobile devices, like touchscreen tablets, and improve the graphical user interface, so that it becomes simple and more appealing to use.

Figure 2 :

 2 Figure 2: Voronoï diagram

Figure 3 :

 3 Figure 3: Final skeleton

Figure 4 :

 4 Figure 4: Track identification

Figure 6 :

 6 Figure 6: GUI: Sound Drawing


Cécile Monthus

Thomas Garel

Statistics of low energy excitations for the directed polymer in a 1 + d random medium (d = 1, 2, 3)

We consider a directed polymer of length L in a random medium of space dimension d = 1, 2, 3. The statistics of low energy excitations as a function of their size l is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities ρ bulk L (E = 0, l) and ρ boundary L (E = 0, l). We find that both densities follow the scaling behavior ρ bulk,boundary

, where θ d is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value θ1 = 1/3 in one dimension). In the limit x = l/L → 0, both scaling functions R bulk (x) and R boundary (x) behave as R bulk,boundary (x) ∼ x -1-θ d , leading to the droplet power law ρ bulk,boundary L (E = 0, l) ∼ l -1-θ d in the regime 1 ≪ l ≪ L. Beyond their common singularity near x → 0, the two scaling functions R bulk,boundary (x) are very different : whereas R bulk (x) decays monotonically for 0 < x < 1, the function R boundary (x) first decays for 0 < x < xmin, then grows for xmin < x < 1, and finally presents a power law singularity R boundary (x) ∼ (1 -x) -σ d near x → 1. The density of excitations of length l = L accordingly decays as ρ boundary L (E = 0, l = L) ∼ L -λ d where λ d = 1 + θ d -σ d . We obtain λ1 ≃ 0.67, λ2 ≃ 0.53 and λ3 ≃ 0.39, suggesting the possible relation λ d = 2θ d .

I. INTRODUCTION

The model of a directed polymer in a random medium has attracted a lot of attention in the last twenty years for two main reasons. On the one hand, it is directly related to non-equilibrium growth models [1]; on the other hand, it plays the role of a 'baby spin glass' model in the field of disordered systems [1,2,3,4,5]. This model presents a low temperature disorder dominated phase, where the order parameter is an 'overlap' [2,4,6,7]. This low temperature phase displays an extreme sensitivity with respect to temperature or disorder changes [5,8,9,10,11,12,13], and aging properties for the dynamics [14]. In finite dimensions, a scaling droplet theory was proposed [5,15], in direct correspondence with the droplet theory of spin glasses [16], whereas in the mean-field version of the model on the Cayley, a freezing transition very similar to the one occurring in the Random Energy Model was found [2]. The phase diagram as a function of space dimension d is the following [1] : for d > 2, there exists a phase transition between the low temperature disorder-dominated phase and a free phase at high temperature [17,18]. This phase transition has been studied numerically in d = 3 [19,20], exactly on a Cayley tree [2] and on hierarchical lattice [21]. On the contrary, in dimension d ≤ 2, there is no free phase, i.e. any initial disorder drives the polymer into the strong disorder phase. In this paper, we will be interested in the low energy excitations above the ground state in dimensions d = 1, 2, 3.

In disordered systems, there can be states that have an energy very close to the ground state energy but which are very different from the ground state in configuration space. For spin glasses, the debate between the droplet and replica theories concerns the probabilities and the properties of these states. In the droplet theory [16], the low temperature physics is described in terms of rare regions with nearly degenerate excitations which appear with a probability that decays with a power law of their size. In the replica theory [START_REF] Mézard | Spin glass theory and beyond[END_REF], the replica symmetry breaking is interpreted as the presence of many pure states in the thermodynamic limit, i.e. the nearly degenerate ground states appear with a finite probability for arbitrary large size. More generally, the statistical properties of the nearly degenerate excitations (their numbers, their sizes, their geometric properties, the barriers separating them, etc...) are interesting in any disordered system, since they govern all properties at very low temperature. In particular, a linear behavior in temperature of the specific heat C(T ) = bT + O(T 2 ) seems rather generic for a large class of disordered models, including (i) spin glasses where this behavior is measured experimentally [START_REF] Binder | [END_REF] and numerically [24] (ii) disordered elastic systems [25] (iii) one-dimensional spin models where this behavior can be exactly computed via the Dyson-Schmidt method [START_REF] Luck | Systèmes désordonnés unidimensionnels[END_REF]. For the last case, the coefficient b of the linear term of the specific heat can be put in direct correspondence with the density ρ(E = 0, l) of two-level low energy excitations of size l [START_REF] Doussal | [END_REF]28,29] via the simple formula b = (π 2 /6) dlρ(E = 0, l). Other integrals like dl l k ρ(E = 0, l) with k = 1, 2, ... determine the low temperature behavior of other observables. The explicit computation of the density ρ(E = 0, l) of excitations as a function of their size l has been possible only for one-dimensional models, such as the case of one particle in a random potentials [START_REF] Doussal | [END_REF]28] and the random field Ising chain via strong disorder renormalization [28]. For higher dimensional disordered systems, the statistics of excitations can only be studied numerically. In particular, there has been a lot of efforts to characterize the distribution and the topology of the low energy excitations in of spin glasses [30]. For the directed polymer model in finite dimensions, we are only aware of the work of Tang [31], where the probability of two degenerate non overlapping ground states with binary disorder in 1 + 1 was found to decay as L -2/3 . Our aim in this paper is to measure the statistics ρ L (E = 0, l) of low energy excitations of length l, for a directed polymer of length L in a Gaussian random potential, in dimensions d = 1, 2, 3. We compare our results with the droplet scaling theory in finite dimensions [5,15]. We also try to make the connection with the exact results [2] on the Cayley tree (d = ∞).

The paper is organized as follows. In section II, we briefly recall known results on the directed polymer in a random medium, concerning exact results in d = 1 and d = ∞ (Cayley tree), as well as the spin glass inspired droplet scaling theory. After presenting the parameters of our numerical study in section III, we briefly present our results on ground state energies in section IV, before focusing on low energy excitations. The statistics of boundary and bulk excitations are respectively studied in sections V and VI. We finally summarize and discuss our results in section VII.

II. SUMMARY OF PREVIOUS WORK ON DIRECTED POLYMERS AT LOW TEMPERATURE

A.

Ground state properties

The probability distribution of the ground state energy E 0 of a directed polymer of length L in dimension 1 + d is expected to follow a scaling form

P d (E 0 , L) ∼ 1 L θ d P d E = E 0 -Le 0 L θ d (1)
where e 0 represent the ground state energy density per monomer. The exponent θ d then governs both the fluctuation and the correction to extensivity of the mean value (Note that this is not always the case in disordered systems, see e.g. [32]). This result has been proven in d = 1 with the exact value of the exponent [33,34,35,36]

θ 1 = 1/3 (2)
For the mean-field version on the Cayley tree (d = ∞) one has formally θ ∞ = 0 [2] : the width is of order O(1), whereas the correction to the extensive term Le 0 in the averaged value E 0 (L) is of order O(ln L) [2]. In finite dimensions d = 2, 3, 4, 5, ..., the exponent θ d has been numerically measured [37,38,39,40]. The values of the KPZ exponent χ d measured in [40] for dimensions d = 2, 3 translate into the following values for the directed polymer exponent θ d θ 2 = 0.244 (3) θ 3 = 0.186 (4) through the correspondence θ d = χ d /(2-χ d ). Note that the existence of a finite upper critical dimension has remained a very controversial issue between the numerical studies [37,38,39,40] and various theoretical approaches [41,42,43].

Beyond the exponent θ, the scaling function P d itself is also of interest : it is exactly known in d = 1 [35,36] (as well as in other geometries [44]), and has been studied on the Cayley tree, with the conclusion that the distribution is not universal but depends on the disorder distribution [45]. Another important property of the ground state in finite dimensions d is the probability distribution of its end-point position R 0 that follows the scaling form

Q d (R 0 , L) ∼ 1 L ζ d d Q d r 0 = R 0 L ζ d (5)
where the exponent ζ d is directly related to the previous exponent θ d via the simple relation [46]

ζ d = 1 + θ d 2 (6)
This corresponds to a superdiffusive behavior ζ > 1/2 as soon as θ > 0, and in particular in one dimension ζ 1 = 2/3. The probability distribution Q d (R 0 , L) has been studied for the case d = 1 in [47].

B. Exact identities from the statistical tilt symmetry

In their continuum version, directed polymers in random media belong to a special class of disordered models for which exact remarkable identities for thermal fluctuations can be derived [48]. The Hamiltonian of these models have a deterministic part which consists in quadratic interactions and a random part whose statistics is translation invariant. For the directed polymer in dimension 1 + 1, this so-called statistical tilt symmetry leads in particular to the following identity for the fluctuation of the end point r L [4,5,15

] < r 2 L > -< r L > 2 = T L (7)
Other identities giving relations between higher cumulants can be similarly derived [5,15,48]. The identity (7) is rather surprising at first sight, since the fluctuations of the end-point are found to independent of the disorder and to be exactly the same as in the absence of disorder! The exact result (7) is particularly interesting at very low temperature, since it predicts a linear behavior in T of the second cumulant, and thus puts constraints on the statistics of nearly-degenerate excitations. In simpler 1D models of one particle in random potentials one can explicitly relate [START_REF] Doussal | [END_REF]28] the linear behavior of the position fluctuations to the rare configurations with two nearly degenerate minima ∆E ∼ T . For directed polymer in random media, any theory of low energy excitations has to reproduce the result (7), and we will now describe its interpretation within the droplet theory [5,15].

C. Droplet theory for low energy excitations in finite dimensions

The droplet theory for directed polymers [5,15], is very similar to the droplet theory of spin glasses [16]. It is a scaling theory that can be summarized as follows. The exponent θ involved in the fluctuations of the ground state energy E 0 over the samples (see eq. ( 1)) also governs the fluctuations of the energy within one sample as the end point varies. As a consequence, if one assumes a scaling distribution analogous to (1), the probability to find a nearly degenerate ground state ∆E ∼ T of order L behaves as T /L θ , so it is rare, but it corresponds to a very large fluctuation of the end-point of order ∆r ∼ L ζ with the spatial exponent ζ ( see eq. ( 5)). The contribution of these rare nearly degenerate paths to the disorder averaged fluctuations ( 7) is thus of order (T /L θ ) × (L ζ ) 2 = T L 2ζ-θ = T L using the scaling relation (6).

This naive 'zero-order' argument has to be refined if one is interested into the density ρ L (E = 0, l) of excitations that involve an arbitrary length l for a polymer of length L. For 1 ≪ l ≤ L, the droplet theory assumes that the same scalings apply : the probability of an excitation of length l is of order T /l θ and leads to a fluctuation of the end-point of order ∆r ∼ l ζ . However now, the crucial notion of 'independent excitations' [5] has to be introduced to obtain a consistent picture. Both for spin glasses [16] and for directed polymers [5], the idea is that in a given sample, droplets with neighbouring sizes tend to have a big overlap. More precisely for the polymer, two excitations of lengths l 1 and length l 2 with l 1 ∼ l 2 will typically merge and then follow the same path to join the ground state, as shown by the tree structure of optimal paths to all end-points [1]. As a consequence in [5], droplets are considered to be independent only if there is a factor of order 2 between their sizes, and this gives a factor d ln l = dl/l in all integrations over droplets

dlρ indep (E = 0, l) ∼ dl l θ+1 (8)
A more intuitive view of the dl/l factor is to remark that independent excitations stem from a branching process along the ground state path, and that the infinitesimal number of branches between l and l + dl is precisely dl/l. In summary, the droplet theory predicts a power law distribution of independent excitations, with exponent (1 + θ). Note that the absence of any characteristic scale in l means that there exists some infinite correlation length in the system in the whole low temperature phase. This is in contrast with simpler models like the random field Ising chain [28] or in the spin glass chain in external field [29], where the density of excitations was found to decay exponentially in l, the correlation length being the Imry-Ma length.

D. Exact results for low energy excitations on the Cayley tree

Many exact results for directed polymers on the Cayley tree have been derived by Derrida and Spohn [2]. From the point of view of excitations, the most important result concerns the distribution of the overlap in the thermodynamic limit, which is simply the sum of two delta peaks at q = 0 and q = 1 in the whole low temperature phase [2] :

π(q) = (1 -Y )δ(q) + Y δ(q -1) (9)
and the distribution of Y over the samples is the same as in the Random Energy Model [49]. In particular, the disorder average of eq. ( 9) is [2] :

π(q) = T T c δ(q) + 1 - T T c δ(q -1) (10)
i.e. the overlap is zero with probability T /T c , and one otherwise. This means that for a polymer of length L, the important excitations are those of length l ∼ L, and that these excitations keep a finite weight in the limit L → ∞.

To understand the origin of this surprising result, Fisher and Huse [5] have computed that the probability to find an excitation of length l which branches off at a distance s = Ll ≪ L from the root behaves as s -3/2 , i.e. using the length notation l = Ls

ρ L (E = 0, l) ∼ 1 (L -l) 3/2 (11)
Finally, Tang has studied numerically the overlap distribution P L (q) for a polymer of finite-size L [31] to characterize how the two delta peaks develop in (10) : the data for 0 < q < 1 follow the scaling behavior (see Figs 4(a) and 4(b) of [31])

π L (q) ∼ L -1/2 π(q) ( 12
)
where the scaling function π(q) present the same singularity with exponent 3/2 near q → 0 and q → 1 π(q) ∝ q→0

1 q 3/2 (13) π(q) ∝ q→1 1 (1 -q) 3/2 (14)
so that, in the limit L → ∞, π ∞ (q) only contains two δ peaks at q = 0 and q = 1 (see eq. ( 10)).

III. MODEL AND NUMERICAL DETAILS

In this paper, we present numerical results for the random-bond version of the directed polymer model on a 1 + d hypercubic lattice. The partition function satisfies the following recursion

Z L+1 ( r) = 2d j=1 e -βǫL( r+ ej , r) Z L ( r + e j ) (15)
The bond-energies ǫ L ( r + e j , r) are random independent variables, drawn with the Gaussian distribution

ρ(ǫ) = 1 √ 2π e -ǫ 2 2 (16)
For each dimension d, we now give the typical lengths L we have studied, with the corresponding number n s of disordered samples :

(i) For d = 1, L = 50, 100, 200, 300, 400, 600, with respective n s /10 We first briefly describe our results on the ground state energy statistics, before we turn to the measure of the density of low energy excitations on which we focus our attention in this paper.

IV. DISTRIBUTION OF THE GROUND STATE ENERGY

A.

Scaling distribution

For the sizes L we have considered, the distribution of the ground state energy E 0 follows the scaling form

P d (E 0 , L) ≃ 1 ∆E 0 (L) F d x = E 0 -E av 0 (L) ∆E 0 (L) (17

B.

Behavior of the width ∆E0(L) and average E av 0 (L)

The exponent θ d of eq.( 1) is expected to govern both the width ∆E 0 (L) and the correction to extensivity of the average

∆E 0 (L) ∼ L θ d (18) E av 0 (L) L ∼ e 0 (d) + L θ d -1 e 1 (d) + ... ( 19
)
Our measures of the exponent θ d from the width ∆E 0 (L) yield

θ 1 ∼ 0.33 (20) θ 2 ∼ 0.24 (21) θ 3 ∼ 0.18 (22)
are in agreement with the exact value θ 1 = 1/3 and with the previous numerical measures quoted in Eq. ( 4) for d = 2, 3.

The fits of the average yields

d = 1 : E av 0 (L) L ∼ -0.95 + 0.84 L θ1-1 -1.19 L -1 (23) d = 2 : E av 0 (L) L ∼ -1.53 + 1.48 L θ2-1 -0.94 L -1 (24) d = 3 : E av 0 (L) L ∼ -1.81 + 2.05 L θ3-1 -1.43 L -1 (25)
V. STATISTICS OF BOUNDARY EXCITATIONS A. for each sample, we generate the ground state (bold line) of energy E0, and consider all end-points Ai where the energy E(Ai) of the best path ending at Ai satisfies E(Ai) -E0 < T (light lines). These best paths tend to cluster into families. In our counting procedure of independent excitations, two excitations are independent if they have no bond in common. For instance on the Figure, end-points A1, A2, A3 and A4 count for a single excitation associated to the branch point B1. Similarly A7 and A8 count for a single excitation associated to the branch point B4.

Measure of independent excitations

In this Section, we are interested into the density ρ boundary L (E = 0, l) of boundary excitations defined as

ρ boundary L (E = 0, l) = lim T →0 1 T N boundary L (E < T, l) (26)
where N boundary L (E < T, l) is the disorder averaged number of independent boundary excitations of energy 0 < E < T → 0 and of length l existing for a directed polymer of length L. This number N indep L (E < T, l) is measured as follows. For each sample, we consider all end-points r different from the ground state r 0 : if the energy E min ( r) of the best path ending at r satisfies E min ( r) -E 0 < T , we construct the best path ending at r to measure its length l, i.e. the length over which it is different from the ground state. However, as explained already above when summarizing the droplet theory (8), one is interested into the number of independent excitations. So here we have used the following criterion : two excitations are independent if they have no bond in common. In d = 1, since we use polymers with (±) steps, this means that two excitations are independent if they join the ground state at different points (see Fig. 3). In d = 2 and d = 3, two excitations are allowed to join the ground state at the same point provided they branch along two different directions.

In [START_REF] Luck | Systèmes désordonnés unidimensionnels[END_REF], the parameter T is simply a cut-off and we have checked the independence of the density ρ boundary (E = 0, l) with respect to T for T small enough. For instance in d = 2 and d = 3, we have checked that the cut-offs T = 0.1 and T = 0.05 yield the same density ρ boundary (E = 0, l).

B.

Boundary excitations for d = 1 On Fig. 4(a), the density ρ boundary L (E = 0, l) is shown in a log-log plot for various sizes L. These curves can be rescaled according to 28) so that in the regime 1 ≪ l ≪ L, the statistics of independent excitations

ρ boundary (E = 0, l) = 1 L 1+θ1 R boundary x = l L (
R boundary (x) ∝ x→0 1 x 1+θ1 (
ρ boundary L (E = 0, l) ∼ 1 l 1+θ1 for 1 ≪ l ≪ L (29)
follows the droplet power law ( 8).

(ii) The function R boundary (x) is minimum at some finite value x min (d = 1) ∼ 0.74, and then grows for x min < x < 1 (iii) In the regime x = l/L → 1, the scaling function R boundary (x) diverges. To describe the regime of these very large excitations, let us first consider the extreme case l = L of an excitation that branches off at the origin. Our result for l = L follow the scaling behavior

ρ boundary L (E = 0, l = L) ∼ 1 L λ1
with λ 1 ∼ 0.67 (30) in agreement with the exponent measured by Tang [31] for the probability of two degenerate ground states in the case of binary disorder. More generally, we find that ρ boundary L (E = 0, l = Ly) with finite y also decays as in (30), see Fig. 5(b). Since the exponent λ 1 ∼ 0.66 is different from the exponent 1 + θ 1 = 4/3 appearing in the scaling form [START_REF] Doussal | [END_REF], the singularity of the scaling function R boundary (x) near x → 1 is given by In d = 2, we find the same properties for the statistics of boundary excitations with the appropriate exponent θ 2 ∼ 0.24. The density ρ boundary L (E = 0, l) shown in a log-log plot for various sizes L on Fig. 6(a) follow the scaling form

R boundary (x) ∝ x→1 1 (1 -x) σ1 with σ 1 = 1 + θ 1 -λ 1 ∼ 0.66 (31)
ρ boundary (E = 0, l) = 1 L 1+θ2 R boundary x = l L ( 32
)
as shown on Fig. 6(b). The master curve R boundary (x) shown on Fig. 7(a) has the same three important properties as in the d = 1 case (i) In the region x → 0, the scaling function follows the power law (see the log-log plot on Fig. 6(b))

R boundary (x) ∝ x→0 1 x 1+θ2 (33)
leading to a statistics of independent excitations that follows the droplet power law ( 8).

ρ boundary L (E = 0, l) ∼ 1 l 1+θ2 for 1 ≪ l ≪ L (34)
(ii) The function R boundary (x) is minimum at some finite value x min (d = 2) ∼ 0.64, and then grows for x min < x < 1 (iii) The density of excitations of length l ∼ L that branches off at the origin decays with the power law

ρ boundary L (E = 0, l = L) ∼ 1 L λ2 with λ 2 ∼ 0.53 (35)
as shown on Fig. 7(b). So the singularity of the scaling function R boundary (x) near x → 1 is given by In d = 3, we find again the same properties for the statistics of boundary excitations with the appropriate exponent θ 3 ∼ 0.18. The density ρ boundary L (E = 0, l) shown in a log-log plot for various sizes L on Fig. 8(a) follow the scaling form as shown on Fig. 8(b). The master curve R boundary (x) shown on Fig. 9(a) has the same three important properties as in the previous cases (i) In the region x → 0, the power law (see the log-log plot on Fig. 8(b))

R boundary (x) ∝ x→1 1 (1 -x) σ2 with σ 2 = 1 + θ 2 -λ 2 ∼ 0.71 (36)
ρ boundary (E = 0, l) = 1 L 1+θ3 R boundary x = l L (37)
R boundary (x) ∝ x→0 1 x 1+θ3 (38)
leads to a statistics of independent excitations

ρ boundary L (E = 0, l) ∼ 1 l 1+θ3 for 1 ≪ l ≪ L (39)
that follows the droplet power law (8).

(ii) The function R boundary (x) is minimum at some finite value x min (d = 3) ∼ 0.56, and then grows for x min < x < 1 (iii) The density of excitations of length l ∼ L that branches off at the origin decays with the power law

ρ boundary L (E = 0, l = L) ∼ 1 L λ3 with λ 3 ∼ 0.39 (40)
as shown on Fig. 9(b). So the singularity of the scaling function R boundary (x) near x → 1 is given by

R boundary (x) ∝ x→1 1 (1 -x) σ3 with σ 3 = 1 + θ 3 -λ 3 ∼ 0.79 (41)
VI.

STATISTICS OF BULK EXCITATIONS

A.

Measure of independent excitations

We now turn to the density ρ bulk L (E = 0, l) of bulk excitations defined as

ρ bulk L (E = 0, l) = lim T →0 l T L N bulk L (E < T, l) (42)
where N bulk L (E < T, l) is now the disorder averaged number of independent bulk excitations of energy 0 < E < T → 0 and of length l existing in the bulk for a directed polymer of length L. Here the additional normalisation factor (l/L) with respect to the analog definition of boundary excitations [START_REF] Luck | Systèmes désordonnés unidimensionnels[END_REF] ensures a coherent normalization between the two densities ρ boundary and ρ bulk : ρ boundary (E = 0, l) represents the probability that the end-monomer belongs to an excitation of length l, whereas ρ bulk represents the probability that a bulk-monomer belongs to an excitation of length l. The number N indep L (E < T, l) is measured as follows. For each sample, we consider all points r 0 (t) of the ground FIG. 10: Notion of independent bulk excitations (in d = 1) : for each sample, we generate the ground state (bold line) and consider all points Ai of the ground state, with partial energy E0(Ai). When the energy E1(Ai) of the second best path ending at Ai satisfies E1(Ai) -E0(Ai) < T , we construct these bulk excitations (light lines) that join again the ground state at some branch point Bj . These bulk excitations tend to cluster into families. In our counting procedure of independent excitations, two excitations are independent if they have no bond in common. For instance on the Figure, end-points A4, A5, A6 and A7 count for a single excitation associated to the branch point B5. The other independent excitations are (A1, B1), (A2, B2), (A3, B3) and (A8, B4).

state with t = L, L -1, .., 2 as possible end points of bulk excitations. We consider the best paths joining the ground state at r 0 (t) but arriving from different points than r 0 (t -1), to see if they have an relative energy with respect to the ground state smaller than the cut-off T . If this is the case, we have found a bulk excitation, and we measure its length l, i.e. the length over which it is different from the ground state. Again, as for boundary excitations, we are interested into independent excitations, and we use the criterion according to which two excitations are independent if they have no bond in common. In d = 1, since we use polymers with (±) steps, this means that two excitations are independent if they join the ground state at different points (see Fig. 10). In d = 2 and d = 3, two excitations are allowed to join the ground state at the same point provided they branch along two different directions. In dimension d = 1, 2, 3, we find that the density of bulk excitations follows the scaling form

ρ bulk (E = 0, l) = 1 L 1+θ d R bulk x = l L (43)
See Figs 11, 12, 13 for d = 1, 2, 3 respectively. As d increases, the quality of the rescaling gets weaker, because of the smaller sizes L that can be studied via transfer matrix.

In contrast with the scaling function R boundary (x) of boundary excitations, the scaling function R bulk (x) decays monotonically for 0 < x < 1. In the region x → 0, the scaling function follows the power law

R bulk (x) ∝ x→0 1 x 1+θ d (44)
so that in the regime 1 ≪ l ≪ L, the statistics of independent excitations

ρ bulk L (E = 0, l) ∼ 1 l 1+θ d for 1 ≪ l ≪ L (45)
follows the droplet power law (8).

VII. SUMMARY AND CONCLUSIONS

We have studied the low energy excitations of a directed polymer in a 1 + d random medium. For dimensions d = 1, 2, 3, we find that the densities of bulk and boundary excitations follow the scaling behavior ρ bulk,boundary L (E = 0, l) = L -1-θ d R bulk,boundary (x = l/L). In the limit x = l/L → 0, both scaling functions R bulk (x) and R boundary (x) behave as R bulk,boundary (x) ∼ x -1-θ d , leading to the droplet power law ρ bulk,boundary L (E = 0, l) ∼ l -1-θ d in the regime 1 ≪ l ≪ L. Beyond their common singularity near x → 0, the two scaling functions R bulk,boundary (x) are very different (this shows the importance of boundary conditions) : whereas R bulk (x) decays monotonically for 0 < x < 1, the function R boundary (x) first decays for 0 < x < x min , then grows for x min < x < 1, and finally presents a power law singularity R boundary (x) ∼ (1x) -σ d near x → 1. The density of excitations of length l ∼ L decays as ρ boundary L (E = 0, l = L) ∼ L -λ d where λ d = 1 + θ dσ d . Our numerical estimates λ 1 ≃ 0.67, λ 2 ≃ 0.53 and λ 3 ≃ 0.39 suggest the relation λ d = 2θ d , although we are not aware of any simple argument to justify it. However, if it holds, this would mean that the scaling function R boundary (x) has singularities with exponents (1 + θ d ) and (1θ d ) near x → 0 and x → 1 respectively, i.e. these singularities tend to become the same as θ d decreases, i.e. as the dimension d increases (see Fig. 14 ). This trend is reminiscent of the result on the Cayley tree discussed in Eq. (12,14) where the singularities have the same exponent on both sides, even if the value (3/2) seems specific to the tree structure and cannot be obtained as the limit θ d = 0 in our results. Let us now mention how one recovers the identity (7) of the statistical tilt symmetry. A boundary excitation of length l is expected to give rise to a fluctuation of the end-point of order ∆r ∼ l ζ (we temporarily drop the dimension dependence of the exponents), so that at order T , one gets

< ∆r 2 > ∼ T L 1 dl l 2ζ ρ boundary L (E = 0, l) (46)
The contribution of excitations of length l with x = l/L finite reads

< ∆r 2 > 0<x<1 = T L 2ζ-θ 1 0 dx x 2ζ R boundary (x) (47)
whereas the contribution of very large excitations of length l = Ly with finite y reads

< ∆r 2 > x∼1 ∼ T L 2ζ-λ (48)
Since λ > θ, the leading contribution is the first one. Using the scaling relation (6), this contribution is of order L 2ζ-θ = L, as it should to recover (7). Finally, since the directed polymer model plays the role of a ' baby spin glass', and since various numerical studies on spin glasses [30] find both the droplet scaling behavior for small excitations and system-size excitations governed by another 'global' exponent, one may wonder whether both types of excitations can be understood within a single scaling function R(x) of the volume fraction x = v/V , where the droplet exponent describes the power law in the

  ) as shown on Figs 1(a), 1(b), 2(a) for d = 1, 2, 3 respectively. We have checked that the function F 1 (u) of Fig. 1(a) agrees with the numerical tabulation of the exact result given on the web site [50]. The three functions F 1 , F 2 , F 3 are shown together for comparison on Fig. 2(b).

FIG. 1 :FIG. 2 :

 12 FIG. 1: (a) Rescaled distribution F1(u) of the d = 1 ground state energy (see equation (17)), for L = 50, 100, 200, 300, 400, 600. (b) Rescaled distribution F2(u) of the d = 2 ground state energy (see equation (17)), for L = 10, 20, 40, 80, 120.

4 FIG. 3 :

 43 FIG.3: Notion of independent boundary excitations (in d = 1) : for each sample, we generate the ground state (bold line) of energy E0, and consider all end-points Ai where the energy E(Ai) of the best path ending at Ai satisfies E(Ai) -E0 < T (light lines). These best paths tend to cluster into families. In our counting procedure of independent excitations, two excitations are independent if they have no bond in common. For instance on the Figure, end-points A1, A2, A3 and A4 count for a single excitation associated to the branch point B1. Similarly A7 and A8 count for a single excitation associated to the branch point B4.

FIG. 4 :

 4 FIG. 4: (a) Log-log plot of the density ρ boundary L (E = 0, l) of boundary excitations of length l in d = 1 for L = 50, 100, 200, 300, 400, 600. (b) Same data after the rescaling of equation (27) with θ1 = 1/3.

FIG. 5 :

 5 FIG. 5: d = 1: (a) Rescaled distribution R boundary (x = l/L) (see equation (27)), that is singular for x → 0 and x → 1. (b) Log-log plot of the density ρ boundary L

  27) as shown on Fig. 4(b). The master curve R boundary (x) shown on Fig. 5(a) has three important properties (i) In the region x → 0, the scaling function R boundary (x) follows the power law (see the log-log plot on Fig. 4(b))

FIG. 6 :

 6 FIG. 6: (a) Log-log plot of the density ρ boundary L (E = 0, l) of boundary excitations of length l in d = 2 for L = 20, 40, 60, 80, 100. (b) Same data after the rescaling of equation (32) with θ2 ∼ 0.24.

FIG. 7 :

 7 FIG. 7: d = 2:(a) Rescaled distribution R boundary (x = l/L) (see equation (32)), that is singular for x → 0 and x → 1. (b) Log-log plot of the density ρ boundary L

FIG. 8 :

 8 FIG. 8: (a) Log-log plot of the density ρ boundary L (E = 0, l) of boundary excitations of length l in d = 3 for L = 18, 24, 30, 36, 42 (b) Same data after the rescaling of equation (37) with θ3 ∼ 0.18.

FIG. 9 :

 9 FIG. 9: d = 3 :(a) Rescaled distribution R boundary (x = l/L) (see equation (37)), that is singular for x → 0 and x → 1. (b) Log-log plot of the density ρ boundary L

FIG. 11 :

 11 FIG. 11: (a) Log-log plot of the density ρ bulk L (E = 0, l) of boundary excitations of length l in d = 1 for L = 50, 100, 200, 300, 400. (b) Same data after the rescaling of equation (43) with exponent θ1 = 1/3.

FIG. 12 :FIG. 13 :

 1213 FIG. 12: (a) Log-log plot of the density ρ bulk L (E = 0, l) of boundary excitations of length l in d = 2 for L = 20, 40, 60, 80, 100. (b) Same data after the rescaling of equation (43) with exponent θ2 = 0.24.

FIG. 14 :

 14 FIG. 14: Comparison of the scaling function R boundary (x) for d = 1 (thin line), d = 2 (dashed line) and d = 3 (thick line).

  6 = 160, 40, 10, 4.7, 2.5, 4.5. (ii) For d = 2, L = 20, 40, 60, 80, 100, with respective n s /10 6 = 30, 4.4, 5.8, 2.2, 1. (iii) For d = 3, L = 12, 18, 24, 30, 36, 42, with respective n s /10 6 = 12.5, 2.8, 0.9, 1.6, 0.76, 0.4.

regime x → 0, whereas the statistics of system-size excitations depends on the global properties of the scaling function R(x).