Ising model

Boltzman distribution: \pi (\text{x}) = \frac{1}{Z}e^{-U(\text{x})/kT}

Partition function Z=Z(T) = \int_D e^{-U(\text{x})/kT}d\text{x} (also called normalizing constant)

Potential energy U(\text{x}) =-J\sum_{\sigma \thicksim \sigma^{'}} x_\sigma x_{\sigma^{'}} + \sum_\sigma h_\sigma x_\sigma

The symbol \sigma \thicksim \sigma^{'} means that they are neighboring pair, J is the interaction strength, h_\sigma is the external magnetic field.

In zero field we can also define the potential energy as

U(\text{x})=\sum_{\sigma \thicksim \sigma^{'}} 1_{x_\sigma \ne x_{\sigma^{'}}}

We define the expectation of U(\text{x}) with respect to \pi as internal energy:

<U>= E_\pi \{U(\text{x})\} = \int_D U(\text{x})\pi(\text{x})d\text{x}

The free energy of the system is F = -kT\text{log}Z

The specific heat of the system is C = \frac{\partial <U>}{\partial T}

The mean magnetization per spin is <m> = E_\pi \{\frac{1}{N^2} |\sum_{\sigma \in S}x_\sigma|\}

1.

Let \beta = 1/kT, we have

\partial \text{log}Z /\partial \beta = \frac{1}{Z}\partial Z/\partial \beta = \frac{1}{Z}\partial[\int_D e^{-\beta U(\text{x})}d\text{x}]/ \partial\beta

=\frac{1}{Z}\int_D \{\partial[e^{-\beta U(\text{x}) }] / \partial \beta\}d{\text{x}} =\frac{1}{Z}\int_D-U(\text{x}) e^{-\beta U(\text{x}) } d{\text{x}}

=\frac{1}{Z}\int_D \frac{\partial e^{-\beta U(\text{x}) }} {\partial \beta} d{\text{x}}

= -<U>

Thus \frac{\partial \text{log}Z}{\partial \beta} = -<U>

2. C = \frac{\partial <U>}{\partial T} = \frac{1}{kT^2}Var_\pi\{U(\text{x})\}

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