Nagel scaling and relaxation in the kinetic Ising model on an n-isotopic chain

Nagel scaling and relaxation in the kinetic Ising model on an n-isotopic chain

L. L. Gonçalves,^*

Departamento de Fisica, Universidade Federal do Ceará,

Campus do Pici, C.P. 6030,

60451-970, Fortaleza, Ceará, Brazil

M. López de Haro,^†

Centro de Investigación en Energía, UNAM,

Temixco, Morelos 62580, México

and J. Tagüeña-Martínez^‡

Centro de Investigación en Energía, UNAM,

Temixco, Morelos 62580, México

Received on 1 September, 2000

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 different isotope. Due to the isotopic mass difference, the

n

spins in each segment have different 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 2. 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. Gonçalves, M. López de Haro, J. Tagüeña-Martínez and R. B. Stinchcombe,

Phys. Rev. Lett.

84 , 1507 (2000)), 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 [1]. 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 [2]. 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 different, 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 segments, each one containing n sites occupied by isotopes characterized by n different spin relaxation times. The Hamiltonian for the l^th segment is the usual Ising Hamiltonian given by

where s_{l, j} is a stochastic (time-dependent) spin variable assuming the values ±1 and J the coupling constant. Note that if n = 2, this model reduces to the alternating isotopic chain treated in Ref. [2]. The configuration of the segment is specified by the set of values {s_{l, 1}, s_{l, 2},...s_{l, n}} º {s^{l, n}} 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

where g = tanh(2J/k_BT), k_B being the Boltzmann constant and T the absolute temperature, and a_i is the inverse of the relaxation time t_i of spin i in the absence of spin interactions.

The time dependent probability P({s^l,n},t) for a given spin configuration satisfies the master equation

where T_li{s^l,n} º {s_l,1, s_l,2,... s_l,i-1, -s_l,i, s_l,i+1, ...s_l,n}. 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({s^l,n}, t) . Hence, we introduce the following expectation values defined as:

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

Introducing now the Fourier transform

_Q,l (l = 1, 2,...n) defined by

where Q = (m = 0, 1, 2...N), d = na, a being a lattice parameter and the vector Y_Q given by

one can derive the following result

where the matrix M_Q has a rather suggestive structure given by

The solution to Eq. (8), which yields the magnetization, is straightforward, namely

The relaxation process of the wave-vector dependent magnetization is determined by the eigenvalues of M_Q denoted as l_lQ (l = 1, 2, ...n) and which for n ³ 5 have to be computed numerically. The same applies to the dynamic critical exponent z, which is obtained by imposing the scaling relation t_Q ~ x^z f(xQ) for the critical mode, l_1Q, with the (Q-dependent ) relaxation time t_Q = -, in the region T ® 0 and Q ® 0 where l_1Q ® 0, and the correlation length x is x ~ exp(2J/k_BT). This yields

where C is an irrelevant constant. By plotting ln (-) vs. and considering the limit T® 0, we have checked analytically for n = 2 [2] and numerically for n = 3, 4 and 5 and various values of the a¢s that z = 2, so that this model belongs to the same universality class as the uniform Ising chain.

Let us now introduce the spatial Fourier transform

_Q(t¢, t¢ + t) of the time-dependent correlation defined by

Then, the t¢® ¥ limit of the temporal Fourier transform of _Q( t¢, t¢ + t) , denoted by _Q(w), is given by

with

_Q,m = exp(iQld)s_l,m. After some lengthy and tedious but not very difficult algebra, using eq. (4) and the above definitions one may derive the result

where

Here, the static correlation functions á

_-Q,i

_Q,l ñ_eq are given by

while u = tanh(), a_ml denotes the (m, l)- element of the matrix A formed with the eigenvectors of M_Q and the (l, j)- element of the matrix A^-1. Finally, by using the fluctuation dissipation theorem [4], the response function S_Q( w) turns out to be given by

It should be noted that the frequency dependent susceptibility is c(w) º k_BTS₀(w)(1 - u)/(1 + u). Thus,

If we set all the a¢s equal in Eq. (17), 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 (1 + W)log₁₀(w/w_p) /W² and the ordinate is log₁₀( c^¢¢(w)w_p/wDc) /W. Here, c^¢¢ is the imaginary part of the susceptibility, W is the full width at half maximum of c^¢¢, w_p is the frequency corresponding to the peak in c^¢¢, and Dc = c(0) - c_¥ is the static susceptibility. For the sake of illustration in Figs. 1 and 2 we present Nagel plots for the cases n = 2 (with a₁ = 1 and a₂ = 2), and n = 3 (with a₁ = 1 and a₂ = 2 and a₃ = 3), respectively, and different values of the reduced (dimensionless) temperature T^*º k_BT/2J. 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 of the 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 [2].

One of us (L. L. Gonçalves) 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 IN104598 and IN117798.

References

[2] L. L. Gonçalves, M. López de Haro, J. Tagüeña-Martínez, and R. B. Stinchcombe, Phys. Rev. Lett. 84, 1507 (2000).

[3] R. J. Glauber, J. Math. Phys. 4, 294 (1963).

[4] M. Suzuki and R. Kubo, J. Phys. Soc. Jpn. 24, 51 (1968).

† Also Consultant at Programa de Simulación Molecular del Instituto Mexicano del Petróleo; e-mail: [e-mail]

‡ E-mail: [e-mail]

• [1] P. K. Dixon, L. Wu and S. R. Nagel, B. D. Williams and J. P. Carini, Phys. Rev. Lett. 65, 1108 (1990);
• L. Wu, P. K. Dixon, S. Nagel, B. D. Williams, and J. P. Carini, J. Non-Cryst. Solids 131-133, 32 (1991);
• D. L. Lesley-Pelecky and N. O. Birge, Phys. Rev. Lett. 72, 1232 (1992);
• M. D. Ediger, C. A. Angell and S. R. Nagel, J. Phys. Chem. 100, 13200 -13212 (1996).
• [2] L. L. Gonçalves, M. López de Haro, J. Tagüeña-Martínez, and R. B. Stinchcombe, Phys. Rev. Lett. 84, 1507 (2000).
• [3] R. J. Glauber, J. Math. Phys. 4, 294 (1963).
• [4] M. Suzuki and R. Kubo, J. Phys. Soc. Jpn. 24, 51 (1968).

Publication Dates

• Publication in this collection
  09 Jan 2002
• Date of issue
  Dec 2000

History

• Received
  01 Sept 2000