Dispersionless pulse transport in mass-spring chains: All possible perfect Newton's cradles
DDispersionless pulse transport in mass-spring chains:All possible perfect Newton’s cradles
Ruggero Vaia
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, I-50019 Sesto Fiorentino, Italy andIstituto Nazionale di Fisica Nucleare, Sezione di Firenze, I-50019 Sesto Fiorentino, Italy (Dated: August 5, 2020)A pulse traveling on a uniform nondissipative chain of N masses connected by springs is soondestructured by dispersion. Here it is shown that a proper modulation of the masses and the elasticconstants makes it possible to obtain a periodic dynamics and a perfect transmission of any kindof pulse between the chain ends, since the initial configuration evolves to its mirror image in thehalf period. This makes the chain to behave as a Newton’s cradle. By a known algorithm basedon orthogonal polynomials one can numerically solve the general inverse problem leading from thespectrum to the dynamical matrix and then to the corresponding mass-spring sequence, so yieldingall possible “perfect cradles”. As quantum linear systems obey the same dynamics of their classicalcounterparts, these results also apply to the quantum case: for instance, a wavefunction localizedat one end would evolve to its mirror image at the opposite chain end.FIG. 1. A perfect 5-mass chain. The masses are proportionalto the sequence (35, 20, 18, 20, 35) and the springs’ elasticconstants to (7 , , , I. INTRODUCTION
By mass-spring chain one means a sequence of N masses { m i } connected by N − { K i } .The system’s Hamiltonian reads H = N (cid:88) i =1 P i m i + 12 N − (cid:88) i =1 K i ( Q i − Q i +1 ) , (1)where Q i is the displacement (measured from the equi-librium position) and P i the momentum of the i -th mass.This physical model, depicted in Fig. 1, is very general:for instance, it can be realized by the electric LC circuitof Fig. 2, where springs and masses are replaced by ca-pacitors and inductors, respectively, while displacementsare represented by the capacitors’ charges.Nanoscopic realizations of the model (1) are relevantfor their end-to-end transport properties [2, 3] and even FIG. 2. An electric circuit equivalent to the spring-masschain (1): the masses and the elastic constants are replacedby the inductances and inverse capacitances, respectively: m i = L i and K i = C − i . The charges Q i flowing in theconductances ( ∂ t Q i = I i ) play the role of the displacementsand the momenta are P i = L i I i . The charge on capacitor C i is Q i − Q i +1 . atomic chains, which can be suitably described by themass-spring chain model, have been created and charac-terized [4], e.g., studying the transmission of a momen-tum pulse given to an extremal atom.The purpose of this paper is to demonstrate the ex-istence, for any chain length N , of an infinite numberof sequences { m i } and { K i } which can yield the perfectend-to-end transmission of such a pulse. For this reasonsuch mass-spring chains behave in analogy to the popu-lar Newton’s cradle [5, 6], a mechanical device that dis-plays an almost perfect transfer of momentum betweenthe endpoints of an array of metallic spheres. Dubbingthem “mass-spring Newton’s cradles” is therefore natu-ral, also looking at Fig. 1, and justified by the wide use ofthis terminology in physical systems where one can ob-serve the perfect transfer of a localized conserved quan-tity over a one-dimensional structure, e.g., the “quantumNewton’s cradle” of Ref. [7].It is well known that a uniform mass-spring chain,made of identical masses m i = m and identical springs K i = K , cannot efficiently transfer a pulse, even in theabsence of dissipation, due to the effect of dispersion . In-deed, its normal modes, which coincide with the Fourier a r X i v : . [ phy s i c s . c l a ss - ph ] A ug components of the coordinates and momenta, do haveincommensurate frequencies [8], ω n = 2 (cid:114) Km sin π ( n − N , , n = 1 , ..., N , (2) n being the normal-mode label. It follows that their timeevolution, ruled by the phase factors e − iω n t , can neverlead to coherently recombining their amplitudes, i.e., withequal phases.It is worth to mention that, historically, the re-search for coherent transmission of signals along one-dimensional systems preferred to renounce to the linear-ity (i.e., using non-Hooke springs), while keeping trans-lation invariance, i.e., uniformity. Starting from the fa-mous Fermi-Pasta-Ulam numerical simulation [9], moreand more studies enlightened the properties of nonlin-ear models and showed that nonlinearity can be an an-tidote against dispersion; several examples of localizedand coherently propagating excitations, the “solitons”,were discovered, such as in the Toda lattice , or in thecontinuous systems described by the
Korteweg-De Vries equation (known since the 19th century) and the
Non-linear Schr¨odinger equation, just to mention the mostwell-known ones [10].At variance with this way to reach coherence, in thispaper Hooke’s law is kept and translation invariance isinstead abandoned, so dealing with nonuniform chains.Conceding a limited degree of nonuniformity, it was pos-sible [11] to enhance the transmission properties of thelinear chain (1) keeping it uniform in the bulk and sym-metrically tuning two masses and their spring at bothends, in order to maximize the transmission of a pulsegiven to the first mass. Such a tuning has two maineffects, namely, modulation of the amplitudes of the ex-cited normal modes and deformation of the frequencyspectrum: the best trade-off between them produces ahuge improvement in the end-to-end transmission effi-ciency, attaining 98.7 % in the asymptotic limit of infi-nite length [11]. The explanation is that only few nor-mal modes with nearly equally-spaced frequencies are in-volved in the dynamics of the initial kick. However, thisapproach is very particular: it yields optimal, but notperfect, transmission and only for excitations localizedon the first mass; a kick given, say, to the second masswould not coherently reach the opposite end.Allowing for full nonuniformity, here a different ques-tions are raised: Can one obtain “magic” mass-springsequences that yield an exactly periodic and perfectlytransmitting dynamics? How many of those sequencesexist?An obvious requirement for coherence is that thenormal-mode frequencies be commensurate, i.e., integermultiples of a finite frequency ω , say ω n = ω k n , (3)with { k n } any sequence of integers (with no commonfactors, to be absorbed into ω ). Indeed, the dynamical phases, evolving as e iω k n t , would become unity after atime period 2 π/ω (and integer multiples of it); then, thenormal modes would coherently recombine to exactly re-produce the initial configuration. It is again obvious that,in order to yield the spectrum (3), one must renouncethe assumption of uniform masses and springs along thechain, as they give (2). However, in order to be effectivefor transmission, the chain must allow for pulses travel-ing from one extremity to reproduce themselves at theopposite one without changes in shape: this entails thatthe chain has to be at least mirror symmetric , i.e., thetransformation of inverting the sequence of masses andsprings is a symmetry, m i = m N +1 − i , K i = K N − i ; (4)the 2 N − { m i , K i } are therefore reducedto N independent ones. With this assumption, it willbe shown that if the above defined integers { k n } are al-ternating in parity, then at the half period t ∗ = π/ω (and odd multiples of it) the chain configuration becomesthe mirror image of the initial one, so that any pulse atone end would be transferred to the opposite end withidentical shape. It will be also proven that the “magic”mass-spring sequences exist and are in one-to-one corre-spondence with the distinct successions of the N integers { k n } defined in Eq. (3), attaining the goal of finding andclassifying all perfect chains, i.e., those with full 100 %transmission efficiency whatever the shape of the initialpulse.As for the mathematical methods, the task of findingthe normal modes and the frequencies of the mass-springchain (1) can be reduced, by using mass-weighted canoni-cal variables, to the diagonalization of a tridiagonal sym-metric matrix, usually dubbed a Jacobi matrix [12]. Thechain’s mirror symmetry further entails it to be symmet-ric also with respect to the second diagonal, so one dealswith a persymmetric
Jacobi matrix. The goal pursuedhere is however the inverse problem , namely one startsfrom the desired eigenvalue succession (3) and wishes toobtain the corresponding Jacobi matrix and, in turn, therelated mass-spring sequence. Fortunately, it is knownthat the inverse problem of calculating the elements ofa persymmetric Jacobi matrix, such that its eigenval-ues be a given nondegenerate [13] sequence, is well-posedand the solution has been proven to exist [14] and tobe unique [15]. Moreover, the matrix elements can becalculated, at least numerically, by means of efficient al-gorithms [16]. The remaining task of relating these ele-ments to the values of the masses and of the spring con-stants, can also be unambiguously solved [17].Remarkably, an explicit analytic solution to the inverseproblem was recently found [18] for any N when k n = n −
1, i.e., the frequencies are proportional to the sequenceof the first N integers, ω n = ω ( n − , n = 1 , ..., N . (5)For instance, with N = 5 one gets the chain of Fig. 1. For N ≤ t ∗ to its mirror-symmetric counter-part, which amounts to state that a localized wavepacketat one end would be perfectly transmitted to the oppositeend of the chain.The paper is organized as follows. Basic notations anddefinitions are briefly recalled in Section II, as well as thedynamics of the chain in terms of its normal modes. InSection III the transmission amplitude is defined as indi-cator of the pulse-transfer efficiency. The inverse problemand the solution algorithm are the subject of Section IV.Eventually, in Section V the dynamics of different mass-spring chains are compared and discussed, including theuniform, the optimized quasi-uniform [11], and the per-fect chains with the spectrum (5) and more selected per-fect chains. II. THE FREE MASS-SPRING CHAIN
Consider the chain described by the Hamiltonian (1);the absence of external springs ( free-free boundary con-ditions, i.e., K = K N = 0) entails translation invari-ance, so that the system is expected to possess a zero-frequency normal mode (translation mode). In terms ofthe displacement and momentum vectors, Q ≡ { Q i } and P ≡ { P i } , the same Hamiltonian can be written in ma-trix form, H = 12 P T M − P + 12 Q T KQ , (6)where the “mass matrix” M is diagonal, its elementsbeing { M ij = m i δ ij } , and K = K − K · · ·− K K + K − K · · · − K K + K ... ... . . . N , (7)is the symmetric tridiagonal “elastic matrix”. Itsrows sum up to zero, so that K has the eigenvector(1 , , ...,
1) corresponding to the translation mode with eigenvalue zero, so det K = 0. The canonical transfor-mation to mass-weighted coordinates, q = M / Q and p = M − / P , turns the Hamiltonian into H = 12 p T p + 12 q T Aq . (8)The N × N matrix A = M − / KM − / is a Jacobi ma-trix, i.e., a tridiagonal symmetric matrix, A = a − b · · · − b a − b ...0 − b a . . . 0... . . . . . . − b N − · · · − b N − a N N , (9)whose nonzero elements are a i = K i − + K i m i , i = 1 , ..., N ,b i = K i √ m i m i +1 , i = 1 , ..., N − . (10)The assumption of mirror symmetry (4) entails thatthe matrices M and K are persymmetric (symmetricwith respect to the antidiagonal), and the same holdsfor A , a i = a N +1 − i , b i = b N − i . (11)Note that as det A = 0 only N − N independent parameters { m i , K i } , apart from anoverall factor: indeed, Eq. (10) shows that such a factordoes not affect A , reflecting the fact that scaling massesand spring constants by the same factor does not affectthe system’s frequencies and normal modes. This choiceis arbitrary, e.g., one can fix the first mass m or the to-tal mass [19], and fully completes the mapping between A and the pair ( M , K )As all b ’s are nonzero the eigenvalues of the matrix A are distinct [12]; moreover, they are nonnegative since A is positive semi-definite, (cid:88) ij A ij q i q j = (cid:88) i K i ( Q i − Q i +1 ) ≥ , ∀{ q i } . (12)Denoting by U = { U ni } the orthogonal matrix that diag-onalizes A , (cid:88) ij U ni A ij U mj = λ n δ nm , (13)and introducing the normal-mode coordinates and mo-menta, ˜ q n = N (cid:88) i =1 U ni q i , ˜ p n = N (cid:88) i =1 U ni p i , (14)the Hamiltonian (8) becomes a sum of independent os-cillators, H = 12 N (cid:88) n =1 (cid:0) ˜ p n + ω n ˜ q n (cid:1) . (15)where the eigenfrequencies are the positive square rootsof the eigenvalues, ω n = √ λ n . They are assumed inincreasing order, ω n +1 > ω n , starting from ω = 0.The chain’s time evolution is a superposition ofnormal-mode motions, q i ( t ) = N (cid:88) n =1 U ni N (cid:88) j =1 U nj (cid:104) q j (0) cos ω n t + p j (0) sin ω n tω n (cid:105) . (16)All frequencies are positive, except that of the translationmode, ω = 0: its corresponding component is to beunderstood as the overall translation U i (cid:80) j U j (cid:2) q j (0) + p j (0) t (cid:3) . III. PERFECT PULSE TRANSMISSION
The transmission of a pulse between the chain ends canbe described as follows. Assume the first mass is givenan instantaneous kick, i.e., a given momentum ¯ p , as inexperiments with ion chains [4], q (0) = (0 , , ..., , p (0) = (¯ p, , ..., . (17)One seeks for values of the chain parameters (4) suchthat the dynamics leads in a certain time τ as close aspossible to the mirror-symmetric momentum distribu-tion p ( τ ) = (0 , , , ..., ¯ p ). With these initial conditionsEq. (16) gives p i ( t ) = ∂ t q i ( t ) = ¯ p N (cid:88) n =1 U ni U n cos ω n t . (18)The transmission amplitude α N ( t ) ≡ p N ( t ) / ¯ p is de-fined as the ratio between the momentum of the lastmass at time t and the input momentum of the firstmass. Using the fact that the eigenvectors of A alternatebetween mirror-symmetric and -antisymmetric [12, 20], U n,N +1 − i = ( − ) n − U ni , one has α N ( t ) = N (cid:88) n =1 U n cos[ π ( n − − ω n t ] . (19)The numbers U n weigh the contributions from the nor-mal modes and can be regarded as a normalized probabil-ity density, since (cid:80) n U n = 1. The same parameter α N ( t )characterizes the transmission of an initial elongation ofthe first mass, q (0) = (¯ q, , ..., p (0) = : indeed, Eq. (16) yields q N ( t ) = ¯ q α N ( t ). Perfect transmission occurs when at some time instant t ∗ all phases are coherent, i.e., they are equal or differ byinteger multiples of 2 π : π ( n − − ω n t ∗ = π × even integer . (20)This amounts to require that the frequencies are integermultiples of a characteristic frequency ω = π/t ∗ , ω n = ω k n , (21)with the (different) coprime integers k n having the sameparity of n −
1, since k = 0; it is equivalent to requirethat the increments δ n ≡ k n +1 − k n , i = 1 , ..., N − , (22)be odd positive integers with no common factors (i.e., co-prime). In the next section it is shown that that any givenfrequency spectrum corresponds to a particular mass-spring sequence, so it follows that for any N ≥ t ∗ the chain returns to the initial state, as,e.g., p (2 t ∗ ) = ¯ p (cid:80) Nn =1 U n cos(2 π k n ) = ¯ p , showing aperfectly periodic dynamics [21]. The time evolutionconsists in the propagation of the initial pulse along thechain, with a shape involving displacements of all masses.Actually, perfect behavior does not require the partic-ular initial configuration (17): Eq. (16) with the spec-trum (21)-(22) tells that any initial shape of the chainevolves to its exact mirror image at t ∗ and is restoredat 2 t ∗ . Such a perfect mass-spring cradle is depicted inFig. 1: to make it resemble a Newton’s cradle, it is imag-ined to involve two auxiliary hanging masses that pe-riodically transmit/receive momentum by instantaneoushard-sphere collision with the chain extrema.The upcoming sections are devoted to the calculationof the “magic” mass-spring sequences that determine α N ( t ∗ ) = 1, i.e., 100 % transmission amplitude for anyvalue of N . IV. THE INVERSE PROBLEM
The task of finding the matrix elements of the tridiag-onal symmetric and mirror-symmetric matrix (9) start-ing from the requirement that it have a given spectrum { λ n , n = 1 , ..., N } is an “inverse problem”. It is well-posed, as in the case considered here (free-free boundaryconditions) the number of independent matrix elementsto be determined, N −
1, is equal to the number of inputvariables, the positive eigenvalues.In the previous Section it was shown that the neces-sary and sufficient condition to yield perfect transmissionis that the mode frequencies be given by Eq. (21), wherethe increasing sequence of integers { k n } obeys the con-straint (22). A. Simplest case: analytic solution
The simplest choice for the odd-number sequence (22)is δ n = 1, or k n = n −
1, corresponding to the spec-trum (5). In this case an analytic solution was recentlyobtained [18], based on the following result, whose proofis sketched in Appendix A. Let A be the N × N matrix (9)with the mirror-symmetric entries a i = N − i − N − i ) , i = 1 , ..., N ,b i = (cid:112) i (2 i −
1) ( N − i ) (2 N − − i ) , i = 1 , ..., N − λ n = 2( n − , n = 1 , ..., N . (24)Both a i and b i are of order N in the matrix bulk, anddecrease almost parabolically toward the matrix edges,where they are of order N . Their imbalance, namely theratio between largest and smallest entries, is of order N .Hence, the frequency sequence ω n = ω ( n −
1) canbe obtained by imposing a factor ω / A through the trans-formation (10) admit closed expressions [18] in terms ofbinomial coefficients, m i = m (cid:18) N − i − (cid:19) (cid:18) N − i − (cid:19) − K i = m ω ( N − (cid:18) N − i − (cid:19) (cid:18) N − i − (cid:19) − . (25)It turns out that for i +1 < n/ M i +1 < M i , K i +1 > K i , (26)implying that the smallest masses and the largest elasticconstants lie in the middle of the chain. For large n onefinds that the ratio between largest and smallest valuesis of order √ N . The binomials being rational numbers,one can choose m and ω in such a way that all { m i } and { K i } be expressed by coprime integers: a few of these“magic numbers” are shown in Table 1 of Ref. [18]. Thesequences of masses and elastic constants are graphicallyreported in Fig. 3 for N = 11 and N = 41. B. General case: numerical solution
For a general choice of the eigenvalues (21), the inverseproblem has to be faced numerically. To this purpose agood algorithm was proposed by de Boor and Golub [16](BG). It constructs the sequence of characteristic poly-nomials { χ i ( λ ) } ( i = 0 , ..., N ) of the matrix A and itssubmatrices, using their orthogonality with respect to theinternal product (cid:10) χ, ˜ χ (cid:11) ≡ N (cid:88) n =1 w n χ ( λ n ) ˜ χ ( λ n ) , (27) FIG. 3. The sequence of masses { m i } and elastic con-stants { K i } as given by Eqs. (25) for chains of size N = 11(squares) and N = 41 (bullets). Choosing m = (cid:112) ( N − /π , ω = π/ ( N − i − / ( N −
1) allowsto appreciate the overall scaling behavior [18]. As the springs K i connect m i and m i +1 , for clarity they are plotted at i + . where the weights are defined in terms of the requiredeigenvalues, w n = w N (cid:89) m =1 m (cid:54) = n (cid:12)(cid:12) λ n − λ m (cid:12)(cid:12) − , (28) w being an arbitrary positive constant. The propertyof orthogonality allows for the sequential construction ofthe polynomials, χ i +1 ( λ ) = ( λ − a i +1 ) χ i ( λ ) − b i χ i − ( λ ) , (29)starting from χ ( λ ) = 1, b ( λ ) ≡
0, with the coefficients a i +1 = (cid:10) λ χ i , χ i (cid:11)(cid:10) χ i , χ i (cid:11) , b i = (cid:10) χ i , χ i (cid:11)(cid:10) χ i − , χ i − (cid:11) (30)corresponding to the matrix elements one is looking for.The numerical procedure starts by calculating andstoring the weights (28) and proceeds in a simple mannerwith the iteration of Eqs. (29) and (30). It takes few linesof code, a clear example being found in Ref. [22].Once the matrix A is known, one has to reconstructthe sequence of masses and springs [17]. The massesare related to the components of the translation-modeeigenvector, v = { U i } , since the identity0 = v t A v = N − (cid:88) i =1 K i (cid:18) v i √ m i − v i +1 √ m i +1 (cid:19) (31)entails the ratios v i / √ m i to be equal, hence m i = c v i ,with c a constant that is determined by the choice of m . The components of the equation A v = define arecursion relation, v i +1 = v i a i − v i − b i − b i , (32)that can be used to obtain the mass sequence startingfrom the given first mass (see Sec. II); in this way onehas v = a v /b , then v = ( a a − v b ) / ( b b ), and soon. Eventually, the elastic constants follow from Eq. (10), K i = c v i v i +1 b i . V. RESULTS AND COMPARISONS
The aim of this Section is to propose examples ofperfectly transmitting chains, besides the exactly solvedchain described by Eqs. (25), and compare their dy-namics with that of the uniform chain and two kinds ofquasi-uniform chains described in Ref. [11]: the first onewith optimized m and the second one with optimized m , m and K . These optimized chains were shown tobe able to rise the asymptotic transmission amplitude,which vanishes for the uniform chain, to α ∞ = 0 . α ∞ = 0 . m i = 1 and K i = 1) has the normal-modefrequencies ω n = 2 sin k n , k n = π ( n − N , n = 1 , ..., N , (33)which are not commensurate, of course: the only low- n modes are approximately spaced by ω = π/N , while thespacing decreases for higher n . The parameters k n arequasi-wavevectors and lead to defining an analog of thegroup velocity [11], ∂ω/∂k = cos k , which is almost oneat low k : indeed, it turns out that the maximal trans-mission amplitude occurs after the pulse has been trav-eling along the chain for a time t ∗ > ∼ π/ω that is slightlylarger than N . The fact that low- k modes are almostcommensurate means that long-wavelength pulses travelmore coherently: this corresponds to the vibrating-stringlimit, N → ∞ with ω k → k . A travel time of order N isalso obtained when the only chain ends are modified inorder to improve the transmission performance [11].Table I reports data concerning three chains of 11masses, i.e., the uniform and the optimized quasi-uniformchains. The first row gives the “arrival time” t ∗ where α N ( t ) attains its maximum value α = α N ( t ∗ ): thesequantities were calculated numerically. Besides the cor-responding mirror-symmetric sequence of masses andsprings, the table columns report the square-amplitudesof the normal modes, which are determined by the ini-tial condition (17), as well as the “coherence factors” atarrival, c n = cos[ π ( n − − ω n t ∗ ] , n = 1 , ..., N ; (34)the latter represent which fraction of the initial ampli-tude of the n th mode contributes to the overall transmit-ted amplitude α = (cid:80) n U n c n . It appears that higher- n modes are less efficient (or less coherent), which explainsthe strategy used in the optimized chains: these per-form better because the initial configuration gives largerweight to low- n modes, which are reciprocally more co-herent [11]. This improvement can be appreciated bylooking at the dynamical evolution shown in the firstthree panels of Fig. 4, where the instantaneous momentaof all masses are reported at equal time intervals between t = 0 and t ∗ (as said in Sec. III, one can equivalently thinkof the elongations of each mass). See the SupplementalMaterial [ ? ] for animations.The same analysis applies to the longer chains, N = 41,reported in Table III and in Fig. 6; there, the differencebetween the three cases is more evident. During the evo-lution the initial pulse appears to propagate along thechain with almost unit velocity, while at the arrival timean increasing amount of energy is transferred to the lastmass, and not “dispersed” along the chain.As discussed in Sec. III, all perfectly transmittingchains can be characterized by the sequence of odd co-prime integers (22) which identifies the spectrum (21). Inorder to make a reasonable comparison with the abovequasi-uniform chains, one can conveniently choose theparameter ω = π/ ( N − t ∗ = N −
1, equal to the chain length, hence withunit (average) velocity. Tables II and IV report, besidesthe sequence (22), the corresponding values of massesand springs. In both tables, column (D) refers to thecase of Eq. (5), yielding the analytic recipe (25), and thecorresponding dynamics is shown in the fourth panels ofFigs. 4 and 6. Columns (E) and (F) of the same ta-bles report particular choices of the frequency sequence,among the infinite possible ones, which display less im-balanced (or “more uniform”) chains, as quantified in thelast row. The reported data were calculated numericallyby the method described in Sec. IV.From the dynamics of these chains, Figs. 5 and 7, itappears that the behavior is more complex, since a fewof the normal modes are tuned with a frequency spacing δ >
1. As a matter of fact, if all δ n were equal to δ , theywould not be coprime and one should set ω δ = ˜ ω , gettinga shorter arrival time ˜ t ∗ = t ∗ /δ . In the cases shown inFigs. 4 and 6, with δ = 3 (E) and δ = 5 (F), one indeedobserves a faster propagation of the initial pulse, as theincipient behavior is to yield large transmission at theearlier time ˜ t ∗ ; however, at this time not all modes arecoherent yet: this is particularly appreciable in Fig. 7,panel (F) at t = t ∗ /
5; of course, perfect coherence amongall modes will occur later at t ∗ . VI. CONCLUSIONS
Historically, the research aimed at obtaining efficientpulse transmission along one-dimensional mass-spring ar-rays mostly assumed translation-invariant (i.e., uniform)chains with nonelastic springs, as many nonlinear dy-namic equations have been known to admit localizedsoliton-like solutions able to travel along the chain whilepreserving their shape.At variance with this approach, in this paper unifor-mity is renounced instead of linearity, and it is shown howto characterize all mass-spring chains that yield perfectend-to-end pulse transmission, thus showing a dynamicsanalogous to that of a Newton’s cradle. In particular,all perfectly transmitting arrays of N masses pairwiseconnected by N − δ , ..., δ N − ). For the simplest sequence (1 , , ..., N . For instance, one can think of a desk toy that couldreplace the Newton cradle, like that shown in Fig. 1, themain difference being in the finite time required for apulse to travel along the chain: in the ideal case wheredissipation is neglected a pulse starting from one end canbounce back and forth indefinitely.In the electrical circuit of Fig. 2, provided thecapacitance-induction sequence is a “magic” one, a cur-rent pulse, generated by an external inductance coupledto L , would also bounce back and forth along the array.One can imagine other applications, spanning from themacroscopic to the microscopic world [2–4]. The versa-tility of the model (1) allows for many alternative imple-mentations: basically, one could state that what is pre-sented here constitutes a solution looking for a problem.A suggestion may be that some biologically active poly-mer chains could be close to some “magic” mass-springsequence, enhancing their ability to transfer energy. ACKNOWLEDGMENTS
The author thanks L. Banchi, T. J. G. Apollaro,A. Cuccoli, and P. Verrucchi for having driven his in-terest towards these topics.
Appendix A: Eqs. (23) entail Eq. (24)
Following Ref. [18], the proof by induction on N startsfrom N = 1, where the statement is trivially true, as A = [0] has the eigenvalue λ = 0. Assuming that thestatement holds true for dimension N , one has to showthat the validity follows for N +1, i.e., that A N +1 has the N eigenvalues of A N plus the eigenvalue λ N +1 = 2 N .The entries of A N +1 are a i = N + 4( i − N +1 − i ) , i = 1 , ..., N +1 ,b i = i (2 i −
1) ( N +1 − i ) (2 N +1 − i ) , i = 1 , ..., N . (A1)It is simple algebra to verify that the tridiagonal matrix2 N − A N +1 , with diagonal elements2 N − a i = N ( N −
1) + ( N +2 − i ) , (A2)factorizes as 2 N − A N +1 = HH T . The matrix H = h · · · r h ...... . . . 00 · · · r N h N +1 N +1 (A3)is lower bidiagonal and has positive elements given by h i = ( N +1 − i )(2 N +1 − i ) , i = 1 , ..., N +1 ,r i = i (2 i − , i = 1 , ..., N . (A4)The matrix A N +1 = 2 N − HH T , has the same spec-trum of the matrix2 N − H T H = A N ...00 · · · N N +1 ; (A5)hence, its eigenvalues are those of A N plus the ( N + 1)theigenvalue 2 N . [1] F. Herrmann and P. Schm¨alzle, Simple explanation of awell-known collision experiment, Am. J. Phys. , 761(1981).[2] D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan,A. Majumdar, H. J. Maris, R. Merlin, and S. R. Phillpot,Nanoscale thermal transport. J. Appl. Phys. , 793(2003).[3] P. M. Norris, N. Q. Le, and C. H. Baker, Tuning PhononTransport: From Interfaces to Nanostructures, J. HeatTransfer , 061604 (2013).[4] M. Ramm, T. Pruttivarasin, and H. H¨affner, Energytransport in trapped ion chains, New J. Phys. , 063062 (2014).[5] F. Herrmann and M. Seitz, How does the ball-chainwork?, Am. J. Phys. , 977 (1982).[6] S. Hutzler, G. Delaney, D. Weaire, and F. MacLeod,Rocking Newton’s cradle, Am. J. Phys. , 1508 (2004).[7] T. Kinoshita, T. Wenger, and D. S. Weiss, A quantumNewton’s cradle, Nature , 900 (2006).[8] C. Kittel, Introduction to Solid State Physics (Wiley, NewYork, 1953).[9] E. Fermi, J. Pasta, and S. Ulam, Studies of nonlinearproblems, Los Alamos Document LA-1940 (1955).[10] T. Dauxois, M. Peyrard,
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10 between 0 and t ∗ . The ordinate representsthe momenta p i ( t ) of each mass as they evolve starting from the configuration (17). The first three panels correspond to thecolumns of Table I, namely the uniform chain (A), the quasi-uniform chains with optimal extremal mass m (B) and with 2optimized extremal masses, m , m , and their spring K (C); the last panel is the perfect chain (D) reported in Table II.Animations are available as Supplemental Material [ ? ]. FIG. 5. Snapshots of the dynamics at equal time intervals of t ∗ /
10 between 0 and t ∗ = 10 for the two alternative N = 11perfectly transmitting chains (E) and (F) described in Table II. Animations are available as Supplemental Material [ ? ]. FIG. 6. N = 41 chain, snapshots of the dynamics at equal time intervals of t ∗ /
10 between 0 and t ∗ . The first three panelscorrespond to the columns of Table III, namely the uniform chain (A), the quasi-uniform chains with optimal extremal mass m (B) and with 2 optimized extremal masses, m , m , and their spring K (C); the last panel is the perfect chain (D) reportedin Table IV. Animations are available as Supplemental Material [ ? ]. FIG. 7. Snapshots of the dynamics at equal time intervals of t ∗ /
10 between 0 and t ∗ = 10 for the two alternative N = 41perfectly transmitting chains (E) and (F) described in Table IV. Animations are available as Supplemental Material [ ??