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I am currently reading the book Statistical Mechanics of Lattice Systems by Friedli-Velenik. Given a finite set $\Lambda\subset\mathbb Z^d$ there are two definitions of the Hamiltonian (with empty boundary condition) appearing in the book and I think that they induce different probability distributions:

Let $$\mathscr{E}=\{\{i,j\}\subset\Lambda:i\sim j\}$$ be the set of edges in $\Lambda$.

Definition 1 (chapter 1.4, equation 1.44):

The Hamiltonian is given by $$H(\omega)=-\sum_{\{i,j\}\in\mathscr{E}}\omega_i\omega_j-h\sum_{i\in\Lambda}\omega_i$$ and the Gibbs distribution is defined by $$P(\omega)=\frac{1}{Z}e^{-\beta H(\omega)}$$

Definition 2 (see chapter 3):

The Hamiltonian is given by $$\tilde H(\omega)=-\beta\sum_{\{i,j\}\in\mathscr{E}}\omega_i\omega_j-h\sum_{i\in\Lambda}\omega_i$$ and the Gibbs distribution is defined by $$\tilde P(\omega)=\frac{1}{\tilde Z}e^{- \tilde H(\omega)}$$

Since we are talking about the Ising model, I would expect that $P=\tilde P$, but for this to be the case we would need to set $\tilde H:=\beta H$ instead, wouldn't we? Does someone know what is going on?

Filippo
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  • They are the same if you rescale the magnetic field $h$. – Meng Cheng Jun 28 '23 at 14:46
  • @MengCheng So you think that in the second definition one should replace $h$ by $\tilde h:=\beta h$? – Filippo Jun 28 '23 at 14:48
  • I think so, then they would become identical. – Meng Cheng Jun 28 '23 at 15:05
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    The first definition is what he refers to as the physicist’s convention and the second the mathematician’s convention. He talks about it in the first chapter and especially chapter 4. They are related by a rescaling of the parameters by $\beta$. – LPZ Jun 28 '23 at 16:26
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    @LPZ : Yes, this is the correct answer. The mathematicians' convention is to put $\beta$ only in front of the interaction energy. This does not change anything from a structural and conceptual point of view, while it makes life simpler when you're interested in convexity properties w.r.t. $(\beta,h)$, for instance. In Chapter 4, we treat the lattice gas in different ensembles and there the physicists' convention is more natural and convenient, especially in order to interpret the results (for instance, to recover the relevant equation of state). – Yvan Velenik Jun 28 '23 at 19:16
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    We explain this explicitly at the end of Section 1.6.1 and in Remarks 4.1 and 6.73, as well as just before Section 7.1. – Yvan Velenik Jun 28 '23 at 19:20

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