deduce with (\[def\_phi\]) $$\label{relation_entropy_production4}
\frac{1}{T}\int_{\mathbb{R}}m(\mathbb R\times[0,T],a)\,da=\mathcal{P}_v\,$$ where the combined entropy production $\mathcal{P}_v$ is taken for the entropy-entropy flux pair defined in (\[part\_entropy\_pair\]). Since $T>0$ is arbitrary and $\mathcal{P}_v$ independent of $T$ it follows from (\[relation\_entropy\_production4\]) $$\label{relation_entropy_production5}
\frac{d}{dt}\int_{\mathbb{R}}m(\mathbb R\times[0,t],a)\,da=\mathcal{P}_v\quad\mbox{for all}\quad t>0\,,$$ which finally relates (\[combined\_entropy\]) to (\[entropy\_production\]).

Then a weak solution $u=v\left(\frac{x}{t}\right)$ of (\[scl\]) with initial data (\[riemann\]) is said to satisfy the *entropy rate admissibility criterion* if it satisfies the following optimality criterion of the entropy production $$\mathcal{P}_v\leq\mathcal{P}_{\tilde{v}}$$ or equivalently $$\dot{\mathcal{H}}_v\leq\dot{\mathcal{H}}_{\tilde{v}}$$ holds, for any other weak solution $\tilde{u}=\tilde{v}\left(\frac{x}{t}\right)$ of (\[scl\]) with initial condition (\[riemann\]).

Using (\[relation\_entropy\_production5\]) one can express the entropy rate admissibility criterion for the particular entropy-entropy flux pair in (\[part\_entropy\_pair\]) in terms of the entropy production (\[entropy\_production\]): A solution $u=v\left(\frac{x}{t}\right)$ with initial data (\[riemann\]) and defect measure $m(x,t,a)$ satisfies *entropy rate admissibility criterion* if $$\label{entropy_rate}
\frac{d}{dt}\int_{\mathbb{R}}m(\mathbb R\times[0,t],a)\,da\leq \frac{d}{dt}\int_{\mathbb{R}}\tilde{m}(\mathbb R\times[0,t],a)\,da\quad\mbox{for all}\quad t>0\,$$ for any other weak solution $\tilde{u}=\tilde{v}\left(\frac{x}{t}\right)$ of (\[scl\]) with initial condition (\[riemann\]) and defect measure $\tilde{m}(x,t,a)$. One can also integrate (\[entropy\_rate\]) and obtains the equivalent condition $$\label{entropy_rate2}
\int_{\mathbb{R}}m(\mathbb R\times[0,t],a)\,da\leq \int_{\mathbb{R}}\tilde{m}(\mathbb R\times[0,t],a)\,da\quad\mbox{for all}\quad t>0\,.$$

Therefore (\[entropy\_rate\]) and (\[entropy\_rate2\]) show, that the entropy rate admissibility criterion can be interpreted as a growth condition of the entropy production (\[entropy\_production\]), which 