The diffusive flow of particles can be studied by applying a constant force f to a system using the microscopic equations of motion
which have the conserved energy
Since the force is applied in the 
direction, there will be
a net flow of particles in this direction, i.e., a current 
.
Since this current is a thermodynamic quantity, there is an estimator
for it:
and 
. The constant force can be considered as
arising from a potential field
The potential gradient 
will give rise to
a concentration gradient 
which is opposite to the
potential gradient and related to it by
However, Fick's law tells how to relate the particle current to the
concentration gradient
where D is the diffusion constant. Solving for D gives
Let us apply the linear response formula again to the above nonequilibrium average. Again, we make the identification:
Thus,
In equilibrium, it can be shown that there are no cross correlations between different particles. Consider the initial value of the correlation function. From the virial theorem, we have
which vanishes for 
. In general,
Thus,
In equilibrium, 
being linear in the velocities (hence momenta).
Thus, the diffusion constant is given by, when the limit 
is taken,
However, since no spatial direction is preferred, we could also choose to apply the external force in the y or z directions and average the result over the these three. This would give a diffusion constant
The quantity
is known as the velocity autocorrelation function, a quantity we will encounter again in other contexts.