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You need to assume that there are only three linearly independent summation invariants,$^\ast$ $Q_1(\mathbf{v})=1$, $Q_2(\mathbf{v})=\mathbf{v}$, and $Q_3(\mathbf{v})=|\mathbf{v}|^2$. Then by definition, if the function $\log f(\mathbf{v})$ is a summation invariant, it must be a linear combination of $Q_1$, $Q_2$, and $Q_3$.

This assumption that $Q_1$, $Q_2$, and $Q_3$ are the only linearly independent summation invariants is phrased in the cited text as "the only conserved quantities are energy and momentum and constants".

$^\ast$ A summation invariant is a function $F(\mathbf{v})$ which is conserved upon collision, $F(\mathbf{v}_1)+F(\mathbf{v}_2)=F(\mathbf{v}'_1)+F(\mathbf{v}'_2)$.

The cited authors assume that there are only three linearly independent summation invariants,$^\ast$ $Q_1(\mathbf{v})=1$, $Q_2(\mathbf{v})=\mathbf{v}$, and $Q_3(\mathbf{v})=|\mathbf{v}|^2$. Then if the function $\log f(\mathbf{v})$ is a summation invariant, it must be a linear combination of $Q_1$, $Q_2$, and $Q_3$ [otherwise, $\log f$ would be a fourth independent summation invariant, counter to the assumption].

This assumption that $Q_1$, $Q_2$, and $Q_3$ are the only linearly independent summation invariants is phrased in the cited text as "the only conserved quantities are energy and momentum and constants". If the physics of the problem allows for a fourth summation invariant, for example $Q_4=|\mathbf{v}|$, then $\log f$ could also contain terms linear in $|\mathbf{v}|$.

$^\ast$ A summation invariant is a function $F(\mathbf{v})$ which is conserved upon collision, $F(\mathbf{v}_1)+F(\mathbf{v}_2)=F(\mathbf{v}'_1)+F(\mathbf{v}'_2)$.

You need to assume that there are only three linearly independent summation invariants,$^\ast$ $Q_1(\mathbf{v})=1$, $Q_2(\mathbf{v})=\mathbf{v}$, and $Q_3(\mathbf{v})=|\mathbf{v}|^2$. Then by definition, if the function $\log f(\mathbf{v})$ is a summation invariant, it must be a linear combination of $Q_1$, $Q_2$, and $Q_3$.

This assumption that $Q_1$, $Q_2$, and $Q_3$ are the only three summation invariants is phrased in the cited text as "the only conserved quantities are energy and momentum and constants".

$^\ast$ A summation invariant is a function $F(\mathbf{v})$ which is conserved upon collision, $F(\mathbf{v}_1)+F(\mathbf{v}_2)=F(\mathbf{v}'_1)+F(\mathbf{v}'_2)$.

You need to assume that there are only three linearly independent summation invariants,$^\ast$ $Q_1(\mathbf{v})=1$, $Q_2(\mathbf{v})=\mathbf{v}$, and $Q_3(\mathbf{v})=|\mathbf{v}|^2$. Then by definition, if the function $\log f(\mathbf{v})$ is a summation invariant, it must be a linear combination of $Q_1$, $Q_2$, and $Q_3$.

This assumption that $Q_1$, $Q_2$, and $Q_3$ are the only linearly independent summation invariants is phrased in the cited text as "the only conserved quantities are energy and momentum and constants".

$^\ast$ A summation invariant is a function $F(\mathbf{v})$ which is conserved upon collision, $F(\mathbf{v}_1)+F(\mathbf{v}_2)=F(\mathbf{v}'_1)+F(\mathbf{v}'_2)$.

The proof is not analytic, you need to assume that there are only three linearly independent summation invariants,$^\ast$ $Q_1(\mathbf{v})=1$, $Q_2(\mathbf{v})=\mathbf{v}$, and $Q_3(\mathbf{v})=|\mathbf{v}|^2$. Then by definition, if the function $\log f(\mathbf{v})$ is a collision invariant, it must be a linear combination of $Q_1$, $Q_2$, and $Q_3$.

This assumption that you know all summation invariants is essential.

$^\ast$ A summation invariant is a function $F(\mathbf{v})$ which is conserved upon collision, $F(\mathbf{v}_1)+F(\mathbf{v}_2)=F(\mathbf{v}'_1)+F(\mathbf{v}'_2)$.

You need to assume that there are only three linearly independent summation invariants,$^\ast$ $Q_1(\mathbf{v})=1$, $Q_2(\mathbf{v})=\mathbf{v}$, and $Q_3(\mathbf{v})=|\mathbf{v}|^2$. Then by definition, if the function $\log f(\mathbf{v})$ is a summation invariant, it must be a linear combination of $Q_1$, $Q_2$, and $Q_3$.

This assumption that $Q_1$, $Q_2$, and $Q_3$ are the only three summation invariants is phrased in the cited text as "the only conserved quantities are energy and momentum and constants".

$^\ast$ A summation invariant is a function $F(\mathbf{v})$ which is conserved upon collision, $F(\mathbf{v}_1)+F(\mathbf{v}_2)=F(\mathbf{v}'_1)+F(\mathbf{v}'_2)$.