content_image/415352.jpg content_image/415366.jpg content_image/415363.jpg content_image/415364.jpg content_image/415359.jpg content_image/415354.jpg
<img src='content_image/415352.jpg'>

where 

In order to estimate τ$_{Q}$ in eq. (49), we measure the normalized time autocorrelation function defined by 

Q 2 − ( Q ) 2 where · · · denotes the average over a sequence of the data obtained by a MC simulation. Since we are interested in how correlation times in internal energy and heat ca- pacity measurements are affected by their methods, we calculate correlation functions of the following four ob- servables 

<img src='content_image/415366.jpg'>

where the subscripts U and N denote the usual measure- ment and the new measurement by using the formulae derived in § III, respectively. The results are shown in Fig. 3. All the four observables are measured with the SCO method. We see that correlation functions in new measurements (full symbols) are much smaller than those in usual measurements (open symbols). This probably comes from the fact that ˜ E and ∂ ˜ E ∂β can change without change in spin configurations since they depend on not only { S$_{i}$ } but also { g$_{ij}$ } . This result shows that large fluctuations in heat capacity measured with the new for- mula do not originate from an increase in the correlation time. 

We next estimate the correlation time τ$_{Q}$ from φ$_{Q}$ ( t ) as 

<img src='content_image/415363.jpg'>

<img src='content_image/415364.jpg'>

<img src='content_image/415359.jpg'>

In practice, we set the lower limit to be one and adjust the upper limit so that it is much larger than the correlation time. For example, the upper limit was set to 3 , 000 when we estimated the correlation time for E$_{U}$ . The estimated values are shown in Table I. We estimate the correlation time for C$_{N}$ to be zero because it is negligibly small. We calculated τ for C$_{N}$ with changing the upper limit from 2 to 10 , 000, and found that it does not exceed 0 . 2 for any upper limits. 

In Table I, we also show the values of relative variance and those of C$_{SE}$ . We find that the relative variances in new measurements are larger than those in usual mea- surements, especially in the heat capacity. Concerning the internal energy, we can explicitly show the increase in variance from eq. (25) as 

<img src='content_image/415354.jpg'>

where we have used the fact that ∂β is always positive. Since 〈 E$_{N}$ 〉 = 〈 E$_{U}$ 〉 , the inequality (53) shows that the relative variance of E$_{N}$ is larger than that of E$_{U}$ . We next turn to how C$_{SE}$ is affected by the measurement methods. We see from Table I that there is no significant difference between C$_{SE}$ for E$_{U}$ and that for E$_{N}$ . On the other hand, C$_{SE}$ for C$_{N}$ is four times larger than that for C$_{U}$ . This is the reason why the heat capac- ity in the case (c) fluctuates more than that in the other cases. It should be noted that the relative statistical er- ror is proportional to C$_{SE}$ (see eq. (48)). Equation (48) also tells us that the number of measurements with C$_{N}$ should be 16 times as large as that with C$_{U}$ to make both the statistical errors the same. 

In summary, to attain a certain accuracy, estimation of the heat capacity with C$_{N}$ requires larger number of measurements than that with C$_{U}$ since fluctuations in C$_{N}$ are larger than those in C$_{U}$ . On the other hand, the efficiency in the internal energy measurement with E$_{N}$ is almost the same as that with E$_{U}$ in the present case. When the heat capacity measurement with C$_{N}$ does not work well, it might be possible to estimate the heat capacity with lower statistical error by doing numerical differential of the internal energy which is evaluated with E$_{N}$ . 

To examine the validity of the formula derived for the replica exchange MC method, we again do simulations in the following three cases: 

(a) Usual MC method with usual replica exchange MC method. 

## C. Replica exchange method 

