Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems

Abstract

Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called “exactness of the mean-field theory”. It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the α-Potts model with annealed vacancies and the α-Potts model with invisible states.

Appendix: Bordered Hessian Matrix

In this Appendix, we briefly explain the method of the bordered Hessian matrix. We want to find a maximum of a function $f(\overrightarrow{x})$ under the condition $g(\overrightarrow{x})=0$. The vector $\overrightarrow{x}$ is assumed to be a vector with n-components. We introduce the Lagrange function with a Lagrange multiplier λ,

$$ L(\lambda,\overrightarrow{x})\equiv f(\overrightarrow{x})-\lambda g( \overrightarrow{x}). $$

(67)

Candidates of maximum are obtained by

$$ \frac{\partial L}{\partial\lambda}=0,\qquad\frac{\partial L}{\partial \overrightarrow{x}}=0. $$

(68)

Solutions of Eq. (68) are denoted by λ ^∗ and $\overrightarrow{x}^{*}$.

The solution $(\lambda^{*},\overrightarrow{x}^{*})$ does not necessarily give a maximum point. In order to judge whether the solution gives the maximum, the method of the bordered Hessian is used. The bordered Hessian is defined as the following matrix,

$$ \mathcal{H}\equiv \begin{pmatrix} 0&\quad-\frac{\partial g}{\partial{x}_1^*}&\quad\cdots&\quad-\frac{\partial g}{\partial{x}_n^*} \\ -\frac{\partial g}{\partial{x}_1^*}&\quad\frac{\partial^2L}{\partial {x}_1^*\partial{x}_1^*}&\quad\cdots &\quad\frac{\partial^2L}{\partial{x}_1^*\partial{x}_n^*} \\ \vdots&\quad\vdots&\quad\ddots&\quad\vdots\\ -\frac{\partial g}{\partial{x}_n^*}&\quad\frac{\partial^2L}{\partial {x}_n^*\partial{x}_1^*}&\quad\cdots &\quad\frac{\partial^2L}{\partial{x}_n^*\partial{x}_n^*} \end{pmatrix} . $$

(69)

The k-th order minor determinant is defined as $\det\mathcal{H}^{(k)}$, where

$$ \mathcal{H}^{(k)}\equiv \begin{pmatrix} 0&\quad-\frac{\partial g}{\partial{x}_1^*}&\quad\cdots&\quad-\frac{\partial g}{\partial{x}_{k}^*} \\ -\frac{\partial g}{\partial{x}_1^*}&\quad\frac{\partial^2L}{\partial {x}_1^*\partial{x}_1^*}&\quad\cdots &\quad\frac{\partial^2L}{\partial{x}_1^*\partial{x}_{k}^*} \\ \vdots&\quad\vdots\quad&\ddots&\quad\vdots\\ -\frac{\partial g}{\partial{x}_{k}^*}&\quad\frac{\partial^2L}{\partial {x}_{k}^*\partial {x}_1^*}&\quad\cdots &\quad\frac{\partial^2L}{\partial{x}_{k}^*\partial{x}_{k}^*} \end{pmatrix} . $$

(70)

Whether the point $(\lambda^{*},\overrightarrow{x}^{*})$ is maximum can be judged from the sign of the minor determinants of the bordered Hessian matrix. The point $(\lambda^{*},\overrightarrow{x}^{*})$ is maximum if $(-1)^{k}\det \mathcal{H}^{(k)}>\nobreak0$ for all k=2,3,…,n. See [18] for more detail.