= Virial theorem for a homogeneous pair potential
{title2=$2\langle K\rangle=q\langle W\rangle$}
For a differentiable total pair <potential energy> homogeneous of degree $q$ in all particle positions, the <Euler theorem for homogeneous functions> gives $\sum_i\boldsymbol r_i\cdot\nabla_iW=qW$. Since $\boldsymbol F_i=-\nabla_iW$, the force virial is $-qW$. The bounded-motion time-averaged <virial theorem> consequently gives the displayed equation. For $q\ne0$, total energy is $E=(q+2)\langle K\rangle/q$. A positive kinetic-temperature coefficient makes its <heat capacity> negative exactly when $-2<q<0$. For a pair potential proportional to $r^{p+1}$ this means $-3<p<-1$.
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