The Logarithmic Energy of N Ordered Particles on the Weyl Chamber

In the setting of Second-Order Equations on Euclidean Open Sets, let $N\ge2$ be a natural number, let $W_{N}$ be the Weyl chamber, let $\beta\in\mathbb{R}$ be positive and let $\log:(0,\infty)\to\mathbb{R}$ be the natural logarithm. Let $\Pi_{N}=\{(i,j)\in[N]\times[N]:i<j\}$, a nonempty finite set: it contains $(1,2)$ and is a subset of the finite set $[N]\times[N]$. For $x\in W_{N}$ and $(i,j)\in\Pi_{N}$ the number $x_{i}-x_{j}$ is positive, by the definition of $W_{N}$ and claim 1 of Elementary Order Arithmetic in an Ordered Field.

The logarithmic energy of strength $\beta$¶ is the function $H_{\beta}:W_{N}\to\mathbb{R}$,

the sum over the finite index set $\Pi_{N}$.