I understand Boltzmann distribution and Ising model are related through their energy functions. Boltzmann takes variables $\{0,1\}$ and Ising $\{-1,1\}$. How do you rewrite a Boltzmann distribution into Ising by mapping $0$ to $-1$? I don't know how to prove they are the same when mapping.
A good link would be much appreciated. Thanks
I have some problems with your terminology. The Boltzmann distribution for any discrete statistical mechanics model gives each state $s$ with energy $E(s)$ the probability $\exp(-\beta E(s))/ \sum_{s'} \exp(-\beta E(s'))$. It is not restricted to "variables $\{0,1\}$". An Ising model (with pairwise interactions) is a particular type of model where the variables $\sigma_j$ take values $\{-1,1\}$ and the energy is of the form $$E_{Ising}(\sigma) = \sum_{j} h(j) \sigma_j + \sum_{j\ne k} J(j,k) \sigma_j \sigma_k$$ What you're thinking of as "Boltzmann" is a boolean model where the variables $x_j$ take the values $\{0,1\}$ and the energy is of the form $$ E(x) = \sum_{j} q(j) x_j + \sum_{j \ne k} Q(j,k) x_j x_k $$
If you take $\sigma_j = 2 x_j - 1$, these are mapped to $\{-1,1\}$, and the energy functions are related:
$$\eqalign{E_{Ising}(\sigma) &= E(x) + constant\cr Q(j,k) &= 4 J(j,k)\cr q(j) &= 2 h(j) - 2\sum_{k} J(j,k) - 2\sum_{k} J(k,j)\cr}$$