The time dependent form of the Nernst–Planck equation is a conservation of mass equation used to describe the motion of a charged chemical species in a fluid medium. It describes the flux of ions under the influence of both an ionic concentration gradient and an electric field . It extends Fick's law of diffusion for the case where the diffusing particles are also moved with respect to the fluid by electrostatic forces:
- t is time,
- D is the diffusivity of the chemical species,
- c is the concentration of the species, and u is the velocity of the fluid,
- z is the valence of ionic species,
- e is the elementary charge,
- is the Boltzmann constant
- T is the temperature.
- u is relative velocity of the observer to the ionic system
In the context of Neuroscience, this equation is best known in its steady-state form, where there is a balance of diffusion and drift. Setting time derivatives to zero, and noting that the term represents a current flux
Integrating the divergence over an arbitrary surface, one obtains the steady state Nernst-Planck equation
Finally, in units of and the gas constant R, one obtains the more familiar form (e.g. 
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