What is Density and particle current?

For a species of particles (molecules, ions, atoms in a solid) the number density $n(M,t)$ is the number of particles per unit volume around $M$ (in $\mathrm{m}^{-3}$; the molar concentration is $c = n/N_A$). The particle current density $\vect{j}_N$ is the vector such that the number of particles crossing an oriented surface element $\dd\vect S$ in $\dd t$ is $\vect{j}_N\cdot\dd\vect S\,\dd t$ (in $\mathrm{m}^{-2}\,\mathrm{s}^{-1}$): the particle flux through a surface $S$ is $\Phi_N = \iint_S \vect{j}_N\cdot\dd\vect S$, the number of particles per second through $S$. For particles carried by a fluid moving at $\vect v$, $\vect{j}_N = n\vect v$ (convection); diffusion is the transport that remains in a fluid at rest.