Microscopic View of Ohm's Law (courtesy of GSU)
The current density (electric current per unit area, J=I/A) can be expressed in terms of the free electron density as:

Current Density from Free Electrons
 J = nevd where: J = current density n = free electron density e = electric charge vd = drift velocity

The number of atoms per unit volume (and the number free electrons for atoms like copper that have one free electron per atom) is:

Atoms per Volume
 n = NAρ/A where: n = number of atoms NA = Avogadro's number ρ = density (kg / m3) A = atomic mass (kg / mole)

From the standard form of Ohm's law and resistance in terms of resistivity:

Electric Resistance
 R = ρL/A where: R = resistance (ohms) ρ = resistivity L = length A = area

Electric Current
 I = V / R where: I = current (amps) V = potential (volts) R = resistance (ohms)

Current Density from Volts & Ohms
J =
 VRA = V(ρLA)/A = ELρL = Eρ = σE
where:
J = current density (joules)
V = electric field (volts)
R = resistance (ohms)
A = area (meters2)
ρ = resistivity
L = length (meters)
E = electric field
σ = conductivity

The next step is to relate the drift velocity to the electron speed, which can be approximated by the Fermi speed:

Fermi Speed
 $$v_F = \sqrt{2E_F \over m}$$ where: vF = Fermi speed EF = electric field m = mass

The drift speed can be expressed in terms of the accelerating electric field E, the electron mass, and the characteristic time between collisions.

Drift Velocity of Electron
vd =
 eEm τ = eEm dvF
where:
vd = drift velocity
e =
E = electric field
m =
τ = electromagnetic decay
d = distance between atoms
vF = Fermi speed

The conductivity of the material can be expressed in terms of the Fermi speed and the mean free path of an electron in the metal.

Conductivity
σ =
 ne2dmvF
where:
σ = conductivity
n = free electron density
e =
d = distance between atoms
m =
vF = Fermi speed