73 L. L. Gonοcalves et al. ments, each one containing n sites occupied by isotopes characterized by n dierent spin relaxation times. The Hamiltonian f

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1 Brazilian Journal of Physics, vol. 3, no. 4, December, 73 Nagel Scaling and Relaxation in the Kinetic Ising Model on an n-isotopic Chain L. L. Gonοcalves, Λ Departamento de Fisica, Universidade Federal do Ceará, Campus do Pici, C.P. 63, , Fortaleza, Ceará, Brazil M. López de Haro, y Centro de Investigación en Energ a, UNAM, Temixco, Morelos 658, México and J. Tagüe~na-Mart nez z Centro de Investigación en Energ a, UNAM, Temixco, Morelos 658, México Received on September, The kinetic Ising model on an n-isotopic chain is considered in the framework of Glauber dynamics. The chain is composed of N segments with n sites, each one occupied by a dierent isotope. Due to the isotopic mass dierence, the n spins in each segment have dierent relaxation times in the absence of interactions, and consequently the dynamics of the system is governed by multiple relaxation mechanisms. The solution is obtained in closed form for arbitrary n, by reducing the problem to a set of n coupled equations, and it is shown rigorously that the critical exponent z is equal to. Explicit results are obtained numerically for any temperature and it is also shown that the dynamic susceptibility satisfies the new scaling (Nagel scaling) proposed for glass-forming liquids. This is in agreement with our recent results (L. L. Gonοcalves, M. López de Haro, J. Tagüe~na-Mart nez and R. B. Stinchcombe, Phys. Rev. Lett. 84, 57 ()), which relate this new scaling function to multiple relaxation processes. Universal behavior in various physical model systems has been one of the clues to uncovering fundamental regularities of nature. About ten years ago, a scaling hypothesis (Nagel scaling) was proposed to describe experimental work on dielectric relaxation in glass-forming liquids[]. This hypothesis was meant to replace the more usual normalized Debye scaling, where the frequency is scaled with the inverse of the (single) relaxation time and the real and imaginary parts of the dynamic susceptibility are then divided by their values at zero and one, respectively. However, and in spite of its phenomenological success and apparent ubiquity, its physical origin is as yet not quite well understood. Very recently, we put forward what we believe to be the first bona fide microscopic model in which Nagel scaling is shown to arise[]. In this previous work, we computed the magnetization, the dynamic critical exponent z and the frequency and wavevector dependent susceptibility of the kinetic Ising model on an alternating isotopic chain with Glauber dynamics[3]. Our results indicated that as soon as the two relaxation times of this model became substantially dierent, the agreement with the Nagel scaling improved significantly. The question remained of whether the inclusion of more relaxation mechanisms would lead to the same sort of results. It is this issue that we mainly want to examine in this paper. Hence we present an extension of the alternating isotopic chain in which rather than only two relaxation times, n relaxation times are involved. The model consists of a linear chain with N seg- Λ lindberg@fisica.ufc.br y Also Consultant at Programa de Simulación Molecular del Instituto Mexicano del Petróleo; malopez@servidor.unam.mx z jtag@servidor.unam.mx

2 73 L. L. Gonοcalves et al. ments, each one containing n sites occupied by isotopes characterized by n dierent spin relaxation times. The Hamiltonian for the l th segment is the usual Ising Hamiltonian given by H l = J j= l;j l;j+ ; () where l;j is a stochastic (time-dependent) spin variable assuming the values ± and J the coupling constant. Note that if n =, this model reduces to the alternating isotopic chain treated in Ref. []. The configuration of the segment Φ isspecifiedby the set of values f l; ; l; ; l;n g l;nψ at time t. This configuration evolves in time due to interactions with a heat bath. We assume for the segments the usual Glauber dynamics so that the transition probabilities are given by w li ( l;i )= i fl ( l;i l;i + l;i l;i+ ) ; () where fl = tanh (J=k B T ), k B being the Boltzmann constant and T the absolute temperature, and i is the inverse of the relaxation time fi i of spin i in the absence of spin interactions. Φ The time dependent probability P l;nψ ;t for a given spin configuration satisfies the master equation c dp Φ l;nψ ;t = + i= w li ( l;i ) P Φ l;nψ ;t Φ Ψ w li ( l;i ) P T li l;n ;t ; (3) i= where T li Φ l;n Ψ f l; ; l; ; l;i ; l;i ; l;i+ ; l;n g. The dynamical properties we are interested in, namely the magnetization, the dynamic critical exponent and the susceptibility, require the knowledge of some moments of the probability P Φ l;nψ ;t. Hence, we introduce the following expectation values defined as q li (t) =h li (t)i = X f l;n g l;i P Φ l;nψ ;t ; (4) where the sum runs over all possible configurations. Using the master equation (3) and this definition of q li (t), one can easily derive the result dq l; dq l;j dq l;n = q l; fl (q l ;n + q l; ) ; = j q l;j fl (q l;j + q l;j+ ) = n q l;n fl (q l;n + q l+; ) Introducing now thefourier transform eq Q;l (l =; ; n) defined by where Q = ßm Nd ; (» j» n ) (5) eq Q;l = p X N exp(iqld)q l;j (6) N (m =; ; N), d = na, a being a lattice parameter and the vector Ψ Q given by one can derive the following result Ψ Q = dψ Q eq Q; eq Q;n C A ; (7) = M Q Ψ Q (8)

3 Brazilian Journal of Physics, vol. 3, no. 4, December, 733 where the matrix M Q has a rather suggestive structure given by M Q = fl fl eiqd fl fl 3fl 3fl 3 n fl n fl n e iqd nfl n nfl The solution to Eq. (8), which yields the magnetization, is straightforward, namely C A (9) Ψ Q (t) =em Qt Ψ Q () () The relaxation process of the wave-vector dependent magnetization is determined by the eigenvalues of M Q denoted as lq (l = ; ; n) and which for n 5 have to be computed numerically. The same applies to the dynamic critical exponent z, which is obtained by imposing the scaling relation fi Q ο ο z f(οq) for the critical mode, Q ; with the (Q-dependent ) relaxation time fi Q = Q ; in the region T! and Q! where Q!, and the correlation length ο is ο ο exp(j=k B T ). This yields ln where C is an irrelevant constant. By plotting ln Q = C + Jz k B T ; () vs. J k BT and considering the limit T!, we have checked analytically for n =[]andnumerically for n =3; 4 and 5 and various values of the s that z =, so that this model belongs to the same universality class as the uniform Ising chain. Let us now introduce the spatial Fourier transform bc Q (t ;t + t) of the time-dependent correlation defined by bc Q (t ;t + t) = nn NX NX l = i= j= e iqdl e iqdl h l;i (t ) l ;j (t + t)i () Then, the t!limit of the temporal Fourier transform of bc Q (t ;t + t) ; denoted by e CQ (!); is given by ec Q (!) = lim t! ßn i= j= Z he Q;i (t ) e Q;j (t + t)i exp( i!t); (3) with e Q;m = p N N P exp(iqld) l;m. After some lengthy and tedious but not very diicult algebra, using eq. (4) and the above definitions one may derive the result where g lq = ec Q (!) = i= j= m= a ml a lj n g lq i! lq (4) he Q;i e Q;j i eq (5) Here, the static correlation functions he Q;i e Q;l i eq are given by he Q;i e Q;l i eq = 8 >< > u jl ij u n u n cos(nq)+u n (i = l) eiqn u n e iqn u + u jl ij n e iqn u n e iqn u n +u jl ij (i 6= l) (6)

4 734 L. L. Gonοcalves et al. while u =tanh( J k BT ), a ml denotes the (m; l) element of the matrix A formed with the eigenvectors of M Q and a lj the (l; j) element of the matrix A. Finally, by using the fluctuation dissipation theorem[4], the response function S Q (!) turns out to be given by S Q (!) k B T n = k B T It should be noted that the frequency dependent susceptibility isχ (!) k B TS (!)( u)=( + u). Thus, χ (!) = u u + i= j= l g l i! l (8) he Q;i e Q;l i eq i! e CQ (!) A lq g lq i! lq (7) d If we set all the s equal in Eq. (7), i.e., we take the uniform chain, then of course the resulting susceptibility has the simple Debye form. In any case, its general structure is a linear combination of n Debye-like terms. In order to investigate whether the Nagel scaling holds for this susceptibility, it is convenient to recall that in the so-called Nagel plot the abscissa is ( + W )log (!=! p ) =W and the ordinate is log (χ (!)! p =! χ) =W. Here, χ is the imaginary part of the susceptibility, W is the full wih at half maximum of χ,! p is the frequency corresponding to the peak in χ,and χ = χ() χ is the static susceptibility. For the sake of illustration in Figs. and we present Nagel plots for the cases n = (with = and =),andn = 3 (with = and = and 3 = 3), respectively, and dierent values of the reduced (dimensionless) temperature T Λ k B T=J. By comparing the two figures, we can see that even for this case, where the relaxation times are very close, there is a marked improvement ofthe scaling in the low temperature limit with the increase of the relaxation mechanisms. On the other hand, the Debye scaling, shown as inset in the figures, presents an opposite behaviour. This result gives further support to the hypothesis that Nagel scaling is related to multiple relaxation mechanisms, as discussed in our previous work []. Figure. Nagel plot for n = (with =and =) and for T Λ =5; 7; ; 5; and 5. There is reasonable agreement with the scaling form for this choice except for low T Λ. Insert χ (!)=χ (! p) vs.!=! p in order to test Debye-like behavior. Oneofus(L.L.Gonοcalves) wants to thank the Centro de Investigación en Energia (UNAM) for the hospitality during his visit, and the Brazilian agencies CNPq and FINEP for partial financial support. The work has also received partial support by DGAPA-UNAM under projects IN4598 and IN7798. Figure. The same as Fig. but with n = 3( =, = and 3 = 3) and T Λ =5; 7; ; 5; and 5. The improvement in the agreement with Nagel scaling is rather noticeable, while the opposite trend is observed with respect to Debye scaling.

5 Brazilian Journal of Physics, vol. 3, no. 4, December, 735 References [] P. K. Dixon, L. Wu and S. R. Nagel, B. D. Williams and J. P. Carini, Phys. Rev. Lett. 65, 8 (99); L. Wu, P. K. Dixon, S. Nagel, B. D. Williams, and J. P. Carini, J. Non-Cryst. Solids 3-33, 3 (99); D. L. Lesley-Pelecky and N. O. Birge, Phys. Rev. Lett. 7, 3 (99); M. D. Ediger, C. A. Angell and S. R. Nagel, J. Phys. Chem., 3-3 (996). [] L. L. Gonοcalves, M. López de Haro, J. Tagüe~na- Martínez, and R. B. Stinchcombe, Phys. Rev. Lett. 84, 57 (). [3] R. J. Glauber, J. Math. Phys. 4, 94 (963). [4] M. Suzuki and R. Kubo, J. Phys. Soc. Jpn. 4, 5 (968).