Discrimination of physical states in quantum systems

Discrimination of physical states in quantum systems

Physica A 293 (2001) 115–129 www.elsevier.com/locate/physa Discrimination of physical states in quantum systems Mayumi Shingu-Yanoa; b; ∗ , Fumiaki ...

186KB Sizes 2 Downloads 50 Views

Physica A 293 (2001) 115–129

www.elsevier.com/locate/physa

Discrimination of physical states in quantum systems Mayumi Shingu-Yanoa; b; ∗ , Fumiaki Shibataa; b

b Department

a Graduate School of Humanities and Sciences, Japan of Physics, Faculty of Science, Ochanomizu University, Otuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, Japan

Received 4 August 2000

Abstract Quantum mechanical relaxation and decoherence processes are studied from a view point of discrimination problem of physical states. This is based on an information statistical mechanical method, where concept of a probability density and an entropy is to be generalized. We use a quasi-probability density of Q-function (Husimi function) and the corresponding entropy (Wehrl– Lieb entropy) and apply the method to a Brownian motion of an oscillator and a non-linear spin relaxation process. Our main concern lies in obtaining a discrimination probability Pd as a function of time and temperature. Quantum mechanical 7uctuation causes profound e8ects than c 2001 Elsevier Science B.V. All rights reserved. the thermal 7uctuation.  PACS: 02.50.Ey; 05.10.Gg; 05.30:−d; 05.40.Jc Keywords: Discrimination of states; Entropy; Information criterion; Non-linear spin relaxation; Quantum Brownian motion; Quantum 7uctuation

1. Introduction In a series of papers we have treated problems of signal detection in noise and discrimination of physical processes [1,2]. In this type of problems, we are required to make decision on the physical state using the received data. For instance, in the signal detection we must decide whether the data contain the target signal or not [3]. In the previous paper [2] we further treated dynamically time-evolving non-linear systems like laser. We could discriminate clearly the di8erent states of laser action. Namely, the dynamical phase transition from the initial transient state was found and fully ∗

Corresponding author. Tel.: +81-3-5978-5320; fax: +81-3-5978-5326. E-mail address: [email protected] (M. Shingu-Yano).

c 2001 Elsevier Science B.V. All rights reserved. 0378-4371/01/$ - see front matter  PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 6 1 7 - 8

116

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

analyzed. This cannot be done by direct observation of the data and becomes possible with the aid of an information theoretical method of statistics called AIC [4]. Most of the problems discussed above can be formulated in the framework of ordinary probability theory. In this paper, we extend the method to include the quantum mechanical systems. At Krst glance, it seems hopeless to generalize the concept of the AIC to take into account quantum mechanical properties. In the quantum statistical mechanics, a density matrix (operator)  plays a role of the probability density. However, it is quite diLcult to reformulate the AIC formalism in terms of , and thus, we use a quasi-probability density deduced from , the Q-function [5] (the Husimi function) [6], which can be used to deKne a quantum mechanical entropy. This is sometimes called the Wehrl–Lieb entropy [7,8] and the corresponding quantum mechanical AIC can be obtained. Then a quantum mechanical Brownian motion of an oscillator and a Brownian motion of spins are treated by this formalism.

2. Preliminaries We Krst brie7y summarize derivation of AIC. Let f(x| ) be the probability density of a stochastic variable X ; is a parameter characterizing the probability distribution. When a set of data {x(1) ; x(2) ; : : : ; x(N ) } is obtained, we can calculate a logarithmic likelihood function l( ) =

N 

ln f(x(l) | ) :

(2.1)

l=1

According to the law of large number, an asymptotic form of l( ) is found to be N 1 1  N →∞ l( ) = ln f(x(l) | ) → I2 ( ) ; N N

(2.2)

l=1

where

 I2 ( ) =



−∞

W (x) ln f(x| ) d x

≡ ln f(X | ) : In the above expression, W (x) being the true probability density of X . The function f(x| ) reduces to W (x) only when the parameter coincides with the true value 0 : f(x| 0 ) = W (x) : DeKning a quantity  ∞ I1 = W (x) ln W (x) d x = ln W (X ) ; −∞

(2.3)

(2.4)

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

117

we can introduce Kullback–Leibler’s (K–L) information I by I ≡ I1 − I2 ( ) ; 



W (x) dx f(x| ) −∞   W (X ) ≡ −S ; = ln f(X | )

=

W (x) ln

(2.5) (2.6) (2.7)

which characterizes a “distance” between the two distributions. The quantity S is Boltzmann’s relative entropy. ˆ and A value of maximizing l( ) is called the “maximum likelihood estimate”, , is determined by @

@ ˆ

ˆ =0: l( )

(2.8)

Expanding l( ) and I2 ( ) around ˆ and 0 , respectively, and taking an asymptotic limit (N → ∞), we Knd a relation of the form ˆ = I2 ( )

1 ˆ − 1} ; {l( ) N

(2.9)

where · means an average with respect to a distribution of a set of data {x(1) ; x(2) ; : : : ; x(N ) }. Introducing a quantity deKned by ˆ − 1] AIC ≡ −2[l( )

(2.10)

we have ˆ =− I2 ( )

1 AIC : 2N

(2.11)

Thus we have to only calculate the quantity AIC in order to minimize the K–L entropy. If there are many unknown parameters, (2.10) is generalized in terms of a vector  having k components: ˆ − k] : AIC = −2[l( )

(2.12)

According to (2.11), we note that the K–L information becomes small when the value of AIC is small giving a distribution in the neighborhood of the true distribution. On the basis of AIC, we can make decision between a null hypothesis H0 and an alternative hypothesis H1 . Then it is convenient to introduce a detection probability or a discrimination probability Pd by Pd = P(D1 |H1 ) ;

(2.13)

where P(D1 |H1 ) is the probability of making decision D1 when the hypothesis H1 is true.

118

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

In quantum statistical mechanics, it is usual to use the von Neumann entropy of the form S = −Tr  ln 

(2.14)

which corresponds to I1 of (2.4), where  is the density matrix. However, it is a diLcult task to introduce a quantum mechanical AIC starting from (2.14). Instead we use a quasi-probability density deKned by P(z; z ∗ ; t) ≡ z|(t)|z ;

(2.15)

where |z is a coherent state. The function P(z; z ∗ ; t) called Q-function is seen to be equivalent to the Hushimi function introduced many years ago [6] and has such a favorable property as the non-negativity. In contrast, if we use another quasi-probabilities like the P-function and the Wigner function [9,10], we immediately encounter a diLculty in deKning a corresponding entropy, because they are sometimes negative and=or non-existent. Then it is reasonable to introduce a quantum mechanical entropy by  2 d z S=− (2.16) P(z; z ∗ ; t) ln P(z; z ∗ ; t)  which has perfect correspondence to I1 of (2.4). When we treat spin systems, we have to only introduce a quasi-probability density P(z; z∗ ; t) ≡ z|(t)|z ;

(2.17)

where |z is a spin coherent state [11,12], or P (J ) ( ; ; t) ≡ J ; ; |(t)|J ; ;  ;

(2.18)

where |J ; ;  is a Bloch state [13,14], J being a magnitude of a spin. The corresponding entropies are, respectively, given by  2 d z S=− P(z; z∗ ; t) ln P(z; z∗ ; t) (2.19) 2 and 2J + 1 S=− 4

 0



 d sin

0

2

d P (J ) ( ; ; t) ln P (J ) ( ; ; t) :

(2.20)

Thus, we are naturally allowed to replace the probability density W (x) in the probability theory by its counterpart P’s in the quantum mechanical theory with suitable modiKcations. 3. Damping oscillator model The simplest nonetheless an important quantum mechanical system is a harmonic oscillator interacting with a reservoir [15,16]. Although a part of the study on this

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

119

system concerning the discrimination of dynamical time-evolving physical states was already reported [2], we make further detailed studies in this section. The Hamiltonian of the system is given by H = HS + HB + H1 ≡ H0 + H1 ;

(3.1)

HS = ˝!0 a† a ;

(3.2)

where

HB =



˝!j b†j bj

j

and H1 = ˝

 j

(gj bj a† + gj∗ b†j a) :

(3.3)

(3.4)

In this expressions, a and a† are the annihilation and creation operators of the oscillator, and bj and b†j are the corresponding operators of the reserver. The master equation of the reduced density matrix (t) is given by (t) ˙ = −i!0 [a† a; (t)] + #[2a(t)a† − (t)a† a − a† a(t)] † + 2#n[a(t)a T + a† (t)a − a† a(t) − (t)aa† ] ;

(3.5)

where we put T ij b†i bj  = n& and

 j

|gj |2

 0



(3.6)

d' exp{i(!0 − !j )'}nT ≡ #nT

(3.7)

with nT =

1 exp(˝!0 =kB T ) − 1

(3.8)

neglecting the shift of the frequency. As was mentioned in the introduction, it is convenient to use the Q-function deKned by P1 (z; z ∗ ; t) ≡ z|(a; a† ; t)|z ;

(3.9)

where |z is the coherent state satisfying a|z = z|z :

(3.10)

Applying transformation (3.9) to (3.5), we obtain the Fokker–Planck-type equation of the form @ @ @2 @ P1 (z; z ∗ ; t) = (# + i!0 ) (zP1 ) + (# − i!0 ) ∗ (z ∗ P1 ) + 2#(nT + 1) ∗ P1 ; @t @z @z @z @z (3.11)

120

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

which is solved to give P1 (z; z ∗ ; t|z0 ; z0∗ ; 0) =

  |z − z0 e−(i!0 +#)t |2 1 : exp − ((nT + 1)(1 − e−2#t ) (nT + 1)(1 − e−2#t ) (3.12)

The Langevin equation equivalent to (3.11) is seen to be d z(t) = −(i!0 + #)z(t) + R(t) ; dt where R(t) is assumed to be a white Gaussian noise, i.e.,

(3.13)

R(t) = Rx (t) + iRy (t) ;

(3.14)

Rx (t)Rx (t  ) = Ry (t)Ry (t  ) = 2#+2 &(t − t  ) ;

(3.15)

with

+ being deKned by +2 = nT + 1 :

(3.16)

Let us proceed to discriminate the process of the oscillator by the use of the AIC method. As an alternative hypothesis H1 , we take the process of (3.11) – (3.13). We choose a di8usion process as a hypothesis H0 , which is determined by the following equations: @2 @ P0 (z; z ∗ ; t) = 2#(nT + 1) ∗ P0 @z @z @t

(3.17)

and d z(t) = R(t) : dt The solution of (3.17) is given by P0 (z; z



; t|z0 ; z0∗ ; 0)

  1 |z − z0 |2 : = exp − 2#(nT + 1)t 2#(nT + 1)t

(3.18)

(3.19)

Using (3.12) and (3.19) we Knd the corresponding AIC1 and AIC0 : AIC0 − AIC1  

N 2tˆl ,|ˆz (l) − zˆ0 |2 1 |ˆz (l) − zˆ0 e−(i!ˆ 0 +1=,)tˆl |2 ln + = − 2 2tˆl ,(1 − e−2tˆl =, ) 1 − e−2tˆl =, l=1

(3.20) with the scaled variables and parameters as zˆ = z=+; !ˆ 0 = !0 =+; , = +=#, and tˆ = t+. We can now discriminate the two quantum mechanical processes by calculating (3.20). Especially, the discrimination probability Pd of (2.13) is evaluated: We show in Figs. 1 and 2 Pd as a function of , for the observation time Tˆ (=T+) = 10 and 100 by changing the parameter !ˆ 0 . As , is large, time variation of the damping process becomes slow and therefore, we need longer observation time T to see the

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

121

Fig. 1. Pd as a function of , for !ˆ 0 = 0:1 and for !ˆ 0 = 10 (observation time Tˆ = 10).

Fig. 2. Pd as a function of , for !ˆ 0 = 0:1 and for !ˆ 0 = 10 (observation time Tˆ = 100).

di8erences between the two processes. In contrast, when , is small, short observation time gives suLcient information to the discrimination. This is clearly seen in Fig. 3 where necessary observation time to satisfy Pd ¿ 0:8 is plotted by changing ,.

4. Non-linear spin relaxation process Now we proceed to study an important problem of the quantum mechanical processes, non-linear spin relaxation. The distribution function of the non-linear spin relaxation is characterized by magnitude of a spin, J . We want to discriminate the characteristics of the process with the use of the AIC method. In this process, the

122

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

Fig. 3. The parameter , versus observation time Tˆ to Knd Pd ¿ 0:8 keeping !ˆ 0 = 10.

master equation of the reduced density matrix  is given by [11,15] @ (t) = −i!0 [Sz ; (t)] + {∗+− (t)[S+ ; (t)S− ] + ∗−+ (t)[S− ; (t)S+ ] @t + ∗0 (t)[Sz ; (t)Sz ] + h:c} ;

(4.1)

where (t)s are the correlation functions of the reserve operators. Eq. (4.1) is mapped onto a c-number space by means of the quasi-probability density P (J ) ( ; ; t) = J ; ; |(t)|J ; ;  ;

(4.2)

|J ; ;  being the Bloch state, to Knd a Fokker–Planck-type equation of the form [17]   0!0 1 @ @ 0!0 @ (J ) (1 − z 2 ) T1−1 + z − (2 J + 1) P (J ) (z; t) ; P (z; t) = @t 2 @z @z 1 1 (4.3) where we put z = cos . In the following, we examine the non-linear spin relaxation process (J ¿1) versus the linear process (J = 12 ). We have thus the null hypotheses as H0 : The process determined by (4.3) with J = 12 . Explicitly the probability density is given by P0 (z; t) ≡ P (1=2) (z; t) 1 = Sz t z ; 2

(4.4)

where Sz t = Sz 0 e−t=T1 + Sz eq (1 − e−t=T1 )

(4.5)

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

123

and 1 Sz eq = − tanh x; 2

Sz 0 =

1 cos 0 2

(4.6)

with x = 4˝!0 =2; 4 being the inverse temperature. For simplicity we take the lowest non-linearity as an alternative hypothesis, namely, H1 : The process determined by (4.3) with J = 1. The corresponding probability density is given by P1 (z; t) ≡ P (1) (z; t)    2 3 1 z 3 2 2 2 ; (2 − Sz t ) + Sz t + z Sz t − = 4 3 2 4 2

(4.7)

where Sz t = Sz eq + (cT1 + cˆ1 )e−t='1 + (cT2 + cˆ2 )e−t='2 ;

(4.8)

Sz2 t = Sz2 eq + (bT1 + bˆ1 )e−t='1 + (bT2 + bˆ2 )e−t='2 ;

(4.9)

2(1 + tanh2 x) 3 + tanh2 x

(4.10)

Sz eq = −

4 tanh x ; 3 + tanh2 x

Sz2 eq = −

with 2 sinh x(1 ± cosh x) ; 3(2 cosh x ∓ 1) 2  1 2 2 Sz 0 (1 ± cosh x) ± Sz 0 − sinh x ; cˆ1 = 2 3 2 cT 1 =

bT 1 = ∓ 2

sinh x cT 1 ; 1 ± cosh x 2

and 1 1 = '1 T1

 2±

2

1 cosh x

bˆ 1 = ∓ 2

sinh x cˆ1 1 ± cosh x 2

(4.11) (4.12)

:

In the following, we use a scaled time variable tˆ ≡ t=T1 . From (4.4) and (4.7), AICs are derived as follows: N  1 (l) AIC0 = −2 ln Sz tˆl z 2

(4.13)

(4.14)

l

and AIC1 = −2

N  l

   3 1 z (l) 3 (l)2 2 2 2 : Sz tˆl − ln (2 − Sz tˆl ) + Sz tˆl + z 2 4 2 4 3 (4.15)

124

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

ˆ Fig. 4. Probability density of a sample data at various t(=t=T 1 ) for 4˝!0 =2 = 10 and 0 = 0:1: (a) tˆ = 0:3, (b) tˆ = 0:53, (c) tˆ = 2:3, (d) tˆ = 3.

The Langevin equation corresponding to (4.3) for J = 1 is derived by the use of the stochastic Liouville equation [18] and the TCL method in non-equilibrium statistical mechanics [19 –21]

d (tˆ) 1 5 = (1 − tanh x cos (tˆ)) + tanh x sin (tˆ) + 1 − tanh x cos (tˆ) 5(tˆ) ; 2 4 d tˆ (4.16) where 5(t) is a white Gaussian noise 5(tˆ)5(tˆ ) = &(tˆ − tˆ ) :

(4.17)

When an initial state is represented by the Bloch state, |J ; 0 ; 0 , the corresponding density matrix and the quasi-probability density are, respectively, given by 0 = |J ; 0 ; 0 J ; 0 ; 0 | and P (1) (t = 0) =

1 4



1 2 sin 2 sin2 2 0 + (1 + cos 2 cos 2 0 )2 2

(4.18)  :

(4.19)

In obtaining (4.19) we integrated over the azimuthal angle 0 . In Fig. 4 we show time evolution of simulated data of the probability density (J =1) for the relatively low temperature (4˝!0 =2 = 10) and the Kxed initial value of 0 = 0:1. These are generated by numerically integrating (4.16). In Fig. 5, we show the

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

125

Fig. 5. Probability density for the initial condition (4.19) with 0 = 0:1 and 4˝!0 =2 = 10. (a) a sample data at tˆ = 0:3, (b) a sample data at tˆ = 0:53, (c) the analytic solution (4.7) at tˆ = 0:3, (d) the analytic solution (4.7) at tˆ = 0:53.

probability density for the same parameters as in Fig. 4 but for the initial condition (4.19), where the quantum mechanical 7uctuations are taken into account. In the same Kgure we include the analytic solution (4.7) for the sake of comparison. Now we make detailed studies on the discrimination problem of the spin relaxation. Namely, in Fig. 6 we show the discrimination probability Pd deKned by (2.13) as a function of 4˝!0 =2 for the observation time interval Tˆ (=T=T1 ) = 1. This is done on the basis of (4.14) and (4.15). For this temperature region we Knd quite satisfactory result of Pd 1. Next, in Fig. 7 we show time variation of Pd for the relatively high-temperature condition of 4˝!0 =2=0:1. In the same Kgure, we include the corresponding sample data of cos (t) which shows the large 7uctuation of the spin in the intermediate region of time evolution. This is conKrmed by the behavior of the probability density at tˆ(=t=T1 ) = 2:3: In Fig. 8 we see the widely spread probability density due to the large 7uctuation of the spin. However, it is rather surprising to see that the discrimination probability Pd is greater than 0:7 in spite of the large 7uctuation. In Fig. 9 we show Pd as a function of time for 4˝!0 =2 = 1 and 10, keeping other parameters same as in Fig. 2(d) for 4˝!0 =2 = 0:1. In the intermediate region of time evolution, Pd becomes small when the temperature becomes low. This is due to the

126

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

Fig. 6. The discrimination probability Pd as a function of x(=4˝!0 =2) for 0 = 0:1 and Tˆ = 1.

Fig. 7. Pd and a sample data as a function of tˆ = t=T1 for 0 = 0:1 and 4˝!0 =2 = 0:1.

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

127

Fig. 8. Probability density of a sample data corresponding to Fig. 7 at tˆ = 2:3.

quantum mechanical 7uctuation which makes obscure di8erences in the non-linear process versus the linear process.

5. Summary and concluding remarks In this paper, we have treated discrimination problems in dynamically time-evolving physical states. Our main concern lies in quantum mechanical systems. In the previous papers [1,2] we could treat the signal detection in noise, discrimination of stochastic processes and physical states even for non-linear phenomena like laser oscillation. These were done on the basis of the information theoretical statistical method called AIC. The theoretical framework is formulated with the aid of the Kullback–Leibler’s information, which is written in terms of a probability density or probability function. In extending the formalism into quantum mechanical systems, we have a diLculty in obtaining the probability density, although a density operator plays a role of the probability density. Among several possibilities, we used the Q-function (or the normally mapped c-number equivalent of the density operator) for the probability density. This function is essentially the same as the Husimi function. For spin systems, we could also introduce the quasi-probability density which corresponds to the Q-function on the basis of the spin coherent state representation and the Bloch state. When these functions are employed, we can deKne the corresponding entropy (sometimes called the Wehrl–Lieb entropy). Thus, we can extend the K–L entropy and AIC formalism to the quantum mechanical systems. On the basis of the above-mentioned quasi-probability density, we could transform the quantum mechanical operator equations of the density matrices into the fully equivalent c-number equations of the Fokker–Planck type. Hence, we could use the same

128

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

Fig. 9. Pd as a function of tˆ for 0 = 0:1: (a) 4˝!0 =2 = 1, (b) 4˝!0 =2 = 10.

kind of computational techniques as for ordinary stochastic processes even for quantum mechanical systems. After a preliminary example of the damped oscillator, where , and T dependence of Pd is examined, we proceeded to the spin relaxation, which is the main theme in this paper. We Krst examined the generated data by showing the probability density which is consistent with the analytic solution. Next, we made detailed studies on the discrimination probability Pd to Knd temperature e8ect and its time dependence. Namely, as far as the temperature is high (say, 4˝!0 =2 = 0:1), Pd ∼ = 1 even for a relatively short interval of Tˆ = 1 (see, Fig 6). However, in an intermediate stage of time evolution Pd becomes very small when the temperature is low (4˝!0 =2=1; 10). This is entirely due to the quantum mechanical 7uctuation which gives rise to larger e8ects on the relaxation processes than the thermal 7uctuation.

M. Shingu-Yano, F. Shibata / Physica A 293 (2001) 115–129

129

In conclusion, we could formulate a quantum mechanical discrimination theory on the basis of AIC and applied the method to the quantum relaxation processes. References [1] M. Shingu-Yano, A. Kobayashi, Y. Nakamura, F. Shibata, J. Grad, School of Humanities and Sciences, Ochanomizu Univ., Tokyo, Vol. 2, 2000, p. 205. [2] M. Shingu-Yano, A. Kobayashi, F. Shibata, Physica A 293 (2001) 100. [3] R.N. McDonough, A.D. Whalen, Detection of Signals in Noise, 2nd Edition, Academic Press, Sandiego, New York, Tokyo, 1995. [4] H. Akaike, in: B.N. Petrov, F. Csaki (Eds.), Proceedings of the Second International Symposium on Information Theory, Academiai Kiado, Budapest, 1973, p. 267; H. Akaike, in: E. Parzen, K. Tanabe, G. Kitagawa (Eds.), Selected Papers of Hirotsugu Akaike, Springer, New York, 1998. [5] R.J. Glauber, Phys. Rev. 131 (1963) 2766. [6] K. Husimi, J. Phys.- Math. Soc. Japan 22 (1940) 264. [7] A. Wehrl, Rep. Math. Phys. 16 (1979) 353. [8] E.H. Lieb, Commun. Math. Phys. 62 (1978) 35. [9] E. Wigner, Phys. Rev. 40 (1932) 749. [10] R. Kubo, J. Phys. Soc. Japan 19 (1964) 2127. [11] Y. Takahashi, F. Shibata, J. Phys. Soc. 38 (1975) 656. [12] Y. Takahashi, F. Shibata, J. Stat. Phys. 14 (1976) 49. [13] J.M. Radcli8e, J. Phys. A 4 (1971) 313. [14] F.T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, Phys. Rev. A 6 (1972) 2211. [15] F. Haake, Springer Tract in Modern Physics, Vol. 66, Springer, Berlin, Heidelberg, New York, 1973, p. 98. [16] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Springer, Berlin, Heidelberg, New York, Tokyo, 1985. [17] F. Shibata, J. Phys. Soc. 49 (1980) 15. [18] R. Kubo, J. Math. Phys. 4 (1963) 174. [19] N. Hashitsume, F. Shibata, M. Shingu, J. Stat. Phys. 17 (1977) 144. [20] S. Chaturvedi, F. Shibata, Z. Phys. B 35 (1979) 297. [21] F. Shibata, T. Arimitsu, J. Phys. Soc. Japan 49 (1980) 891.