 Solution

Let ${\mathbf{A}}_i={\sum }_{j\ne i}{K}_{ij}({\mathbf{v}}_j-{\mathbf{v}}_i)$. Conservation of probability in the full phase space gives the dissipative Liouville equationNormalize $\int \varrho {\prod }_i{d}^3{x}_i{d}^3{v}_i=1$. Decay at infinity makes all boundary fluxes vanish, so the equation preserves this normalization of a probability distribution.

Unlike the Hamiltonian Liouville equation, this transport has nonzero phase-space divergence. In three dimensions,The probability density increases along a contracting characteristic flow map; setting its material derivative to zero would lose conservation of probability. The $j=i$ force is identically zero, including its derivative, which is why the last sums exclude it.