"Transfer matrix in statistical mechanics"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
==introduction== | ==introduction== | ||
| − | |||
* transfer matrix is builtup from matrices of Boltzmann weights | * transfer matrix is builtup from matrices of Boltzmann weights | ||
* trace of monodromy matrix is the transfer matrix | * trace of monodromy matrix is the transfer matrix | ||
| 7번째 줄: | 6번째 줄: | ||
* partition function = trace of power of transfer matrices | * partition function = trace of power of transfer matrices | ||
* so the partition function is calculated in terms of the eigenvalues of the transfer matrix | * so the partition function is calculated in terms of the eigenvalues of the transfer matrix | ||
| + | * then the problem of solving the model is reduced to the computation of this trace | ||
| − | * | + | |
| + | ==정의== | ||
| + | * 스핀 $s_i\, i=1,\cdots, N$과 주기조건 $s_{N+1}=s_1$을 가정 | ||
| + | * 스핀 $s_i$과 $s_{i+1}$의 상호작용 $E(s_i,s_{i+1})$ | ||
| + | * 해밀토니안이 $H=\sum_{i=1}^{N} E(s_i,s_{i+1})$ 꼴로 쓰여지는 경우 | ||
| + | * 전달행렬을 $T_{s_i,s_{i+1}}=\exp(-\beta E(s_i,s_{i+1})$ 꼴로 쓸 수 있으며, 분배함수는 다음과 같이 주어진다 | ||
| + | $$ | ||
| + | Z_N=\sum_{s_1,\cdots,s_N}T_{s_1,s_2}\cdots,T_{s_N,s_1}=\operatorname{Tr} T^n | ||
| + | $$ | ||
| + | * 자유에너지는 다음과 같다 | ||
| + | $$ | ||
| + | f=-\frac{1}{\beta}\lim_{N\to \infty}\frac{\ln \Lambda_0^N}{N}=-\frac{1}{\beta}\ln \Lambda_0 | ||
| + | $$ | ||
==transfer matrix of the 1D Ising model== | ==transfer matrix of the 1D Ising model== | ||
* [[1d Ising model]] | * [[1d Ising model]] | ||
| + | |||
==transfer matrix of the 2D Ising model== | ==transfer matrix of the 2D Ising model== | ||
2013년 2월 3일 (일) 08:53 판
introduction
- transfer matrix is builtup from matrices of Boltzmann weights
- trace of monodromy matrix is the transfer matrix
- finding eigenvalues and eigenvectors of transfer matrix is crucial
- Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix
- partition function = trace of power of transfer matrices
- so the partition function is calculated in terms of the eigenvalues of the transfer matrix
- then the problem of solving the model is reduced to the computation of this trace
정의
- 스핀 $s_i\, i=1,\cdots, N$과 주기조건 $s_{N+1}=s_1$을 가정
- 스핀 $s_i$과 $s_{i+1}$의 상호작용 $E(s_i,s_{i+1})$
- 해밀토니안이 $H=\sum_{i=1}^{N} E(s_i,s_{i+1})$ 꼴로 쓰여지는 경우
- 전달행렬을 $T_{s_i,s_{i+1}}=\exp(-\beta E(s_i,s_{i+1})$ 꼴로 쓸 수 있으며, 분배함수는 다음과 같이 주어진다
$$ Z_N=\sum_{s_1,\cdots,s_N}T_{s_1,s_2}\cdots,T_{s_N,s_1}=\operatorname{Tr} T^n $$
- 자유에너지는 다음과 같다
$$ f=-\frac{1}{\beta}\lim_{N\to \infty}\frac{\ln \Lambda_0^N}{N}=-\frac{1}{\beta}\ln \Lambda_0 $$
transfer matrix of the 1D Ising model
transfer matrix of the 2D Ising model
transfer matrix of the six-vertex model
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expositions
- Anon.1980. The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics. Vol. 123. Berlin/Heidelberg: Springer-Verlag.[1]http://www.springerlink.com/content/f12j034740601kjx/.
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