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Let's define $n=\frac{N}{V}$ the number density of communicative civilizations in the Galaxy, where $N$ is the total number of such civilizations existing at any given time in the Milky Way and $V$ is the total galactic volume. Assuming an homogeneous distribution across the Galaxy, the average separation $r$ between two neighbour communicative civilization is $r\sim n^{-1/3}$. In order for two civilization to communicate, their separation needs necessarily to be smaller than their communicative phase lifetime $L$. This is assuming communication technology that is limited by the speed of light. We can then ask what would be the minimum value for $L$ such that communication can occur (we call this value $L_c$. $L_c$).  That is we impose $r=L$. We then obtain $L\approx (N/V)^{-1/3}$. On the other hand our discussion of the Drake equation led to the conclusion that, roughly speaking, $N \sim L$. And so we can further simplify and write $L_c \sim \left(\frac{V}{L_c}\right)^{1/3}$ = V^{1/4}