Suppose we have a function $f_0:{\mathbb R}^3\rightarrow {\mathbb R}_+$ that satisfies the following property $$ \left\{\begin{eqnarray} &&{\bf v_1}^2+ {\bf v_2}^2 = {\bf v_1’}^2+ {\bf v_2’}^2 \\ &&{\bf v_1}+ {\bf v_2} = {\bf v_1’}+ {\bf v_2’} \end{eqnarray}\right. \Rightarrow f_0({\bf v}_1)f_0({\bf v}_2)=f_0({\bf v}’_1)f_0({\bf v}’_2) $$
Question: Under what conditions, such as continuity, smoothness, or even analyticity as assumed by physicists, can $\log f_0$ be written as a linear combination of $v^2$, the three components of $v$, and an arbitrary constant?
$f_0$ originates from Boltzmann’s distribution of particles in the velocity space, which specifies the equilibrium state in the absence of external forces in classical statistical mechanics. The two equalities represent the conservation laws of kinetic energy and momentum, respectively, in a collision between two perfectly elastic spheres. I encountered $f_0$ in the book Mathematical Statistical Mechanics by Colin J. Thompson.
Here is the original text from the book
Taking logarithms of both sides of Equation 6.1 we have $$ \log\,f_0({\bf v_1})+ \log\,f_0({\bf v_2}) = \log\,f_0({\bf v_1’}) + \log\,f_0({\bf v_2’}) $$ which has the form of a conservation law. Since for spinless molecules (e.g., hard spheres) the only conserved quantities are energy and momentum (and constants), it follows that must be a linear combination of $v^2$ and the three components of $v$, plus an arbitrary constant, i.e., $$ \log\,f_0({\bf v})= \log\,A-B({\bf v}-{\bf v_0})^2 $$
I cannot see a rigorous proof provided to explain it, nor are any reference bibliographies given, despite the book being titled "Mathematical Statistical Mechanics"…