# How should one check an instant-equilibration algorithm for conservation?

We have a system of equations that describes the equilibration between several phases $A, B, ...$. If we define the total concentration $T = A + B + ...$, then $\partial_t T=0$ must hold.

We are further assuming equilibrium at all times, so we have devised an algorithm for this that transforms $A\rightarrow A'$, $B\rightarrow B'$ etc. Finally, the total concentration must be conserved, so we must check if $T'=T$. But should one also check if $\partial_t T'=0$, i.e. the conservation equation with the equilibrated variables?

Does it make sense to test both $T'-T=0$ and $\partial_t T'=0$?

I would think one of those conditions would always be sufficient. If you have shown that $T'-T=0$ for all $t$ and that $\partial_tT=0$, then $\partial_tT'=0$ must be true by default. That is $\partial_tT=0$ shows that $T$ is a constant and if $T'-\text{constant}=0$ for all $t$, then $T'$ is also a constant.
• Clear answer. Indeed, if for all $t$, $\partial_t T=0$ and $T' = T$, then $\partial_t T'=\partial_t T=0$. The converse is not true, so it appears that $T'=T$ is a stronger condition than $\partial T'=0$. – Marco van Hulten Oct 8 '17 at 16:52