# Inverse temperature and Lennard-Jones model

I am a mathematician and I am trying to understand the intension behind the inverse temperature and the activity (or intensity).

Consider e.g. the Hamiltonian [with (12,6)-Lennard-Jones pair potential] with some intensity parameter $$z > 0$$ and interaction parameters $$\sigma > 0,$$ $$\varepsilon > 0,$$ that is,

$$H_{\Lambda}(\omega) := \sum_{x\in\omega_{\Lambda}} z + 4 \sigma^{12} \varepsilon \sum_{\substack{\{x,y\} \subset \omega\\ \{x,y\} \cap \Lambda \neq \emptyset }} \frac{1}{\|x-y\|^{12}} - 4 \sigma^{6} \varepsilon \sum_{\substack{\{x,y\} \subset \omega\\ \{x,y\} \cap \Lambda \neq \emptyset }} \frac{1}{\|x-y\|^{6}},$$

where $$\Lambda \subset \mathbb{R}^k$$ is some compact set. Then, to study uniqueness (or phase transitions) of a corresponding (grand canonical) Gibbs measure one consider additionally the inverse temperature $$\beta$$ and tries to understand the low temperature behaviour.

My question: Where exactly must the inverse temperature be placed and why?

Some authors study particle models with $$\beta$$ acting on the whole Hamiltonian, that is, $$\beta H_{\Lambda}.$$

Some authors consider the intensity part $$z |\omega_{\Lambda}| = \sum_{x\in\omega_{\Lambda}} z$$ separately. Then, the inverse temperature $$\beta$$ is considered to act on the terms of pair potentials only, that is,

$$\beta \cdot \left(4 \sigma^{12} \varepsilon \sum_{\substack{\{x,y\} \subset \omega\\ \{x,y\} \cap \Lambda \neq \emptyset }} \frac{1}{\|x-y\|^{12}} - 4 \sigma^{6} \varepsilon \sum_{\substack{\{x,y\} \subset \omega\\ \{x,y\} \cap \Lambda \neq \emptyset }} \frac{1}{\|x-y\|^{6}}\right).$$

Is there a specific reason to avoid considering an inverse temperature before the whole Hamiltonian (for example in the Lennard-Jones model)?

I've never heard of "intensity" but it looks something like the chemical potential $$\mu$$. The grand canonical partition function is usually defined as $$Z= \sum_{{\rm states},N}\exp\{\beta(\mu N -H)\}= \exp\{-\beta(E-TS-\mu N)\}.$$ For a gas the Grand Potential (or the Landau free energy) is $$E-TS-\mu N=-PV$$.