Editing Fick's laws of diffusion (section)

== Fick's second law ==
'''Fick's second law''' predicts how diffusion causes the concentration to change with respect to time. It is a [[partial differential equation]] which in one dimension reads
: <math>\frac{\partial \varphi}{\partial t} = D\,\frac{\partial^2 \varphi}{\partial x^2},</math>
where
* {{mvar|φ}} is the concentration in dimensions of <math>[\mathsf{N}\mathsf{L}^{-3}]</math>, example mol/m<sup>3</sup>; {{math|1=''φ'' = ''φ''(''x'',''t'')}} is a function that depends on location {{mvar|x}} and time {{mvar|t}},
* {{mvar|t}} is time, example s,
* {{mvar|D}} is the diffusion coefficient in dimensions of <math>[\mathsf{L}^2\mathsf{T}^{-1}]</math>, example m<sup>2</sup>/s,
* {{mvar|x}} is the position, example m.

In two or more dimensions we must use the [[Laplacian]] {{math|1=Δ = ∇<sup>2</sup>}}, which generalises the second derivative, obtaining the equation
: <math>\frac{\partial \varphi}{\partial t} = D\Delta \varphi . </math>

Fick's second law has the same mathematical form as the [[Heat equation]] and its [[fundamental solution]] is the same as the [[Heat kernel]]<!-- This is true for the case of an initial Gaussian distribution. Other problem geometries will lead to different solutions. (e.g. diffusion with a fixed boundary concentration, inter-penetration of two solids, etc) -->, except switching thermal conductivity <math>k</math> with diffusion coefficient <math>D</math>:
<math display="block">\varphi(x,t)=\frac{1}{\sqrt{4\pi Dt}}\exp\left(-\frac{x^2}{4Dt}\right).</math>

=== Derivation of Fick's second law ===
<!-- Not correct, see Talk
=== Fick's first law ===
In one dimension, the following derivation is based on a similar argument made in Berg 1977 (see [[#References|references]]).

Consider a collection of particles performing a random walk in one dimension with length scale {{math|Δ''x''}} and time scale {{math|Δ''t''}}. Let {{math|''N''(''x'',''t'')}} be the number of particles at position {{mvar|x}} at time {{mvar|t}}.

At a given time step, half of the particles would move left and half would move right. Since half of the particles at point {{mvar|x}} move right and half of the particles at point {{math|''x'' + Δ''x''}} move left, the net movement to the right is:
: <math>-\tfrac{1}{2}\bigl[N(x + \Delta x, t) - N(x, t)\bigr]</math>

The flux, {{mvar|J}}, is this net movement of particles across some area element of area {{mvar|a}}, normal to the random walk during a time interval {{math|Δ''t''}}. Hence we may write:
: <math>J = - \frac{1}{2} \left[\frac{ N(x + \Delta x, t)}{a \Delta t} - \frac{ N(x, t)}{a \Delta t}\right]</math>

Multiplying the top and bottom of the right hand side by {{math|(Δ''x'')<sup>2</sup>}} and rewriting, one obtains:
: <math> J = -\frac{\left(\Delta x\right)^2}{2 \Delta t}\left[\frac{N(x + \Delta x, t)}{a \left(\Delta x\right)^2} - \frac{N(x, t)}{a \left(\Delta x\right)^2}\right]</math>

Concentration is defined as particles per unit volume, and hence
: <math>\varphi (x, t) = \frac{N(x, t)}{a \Delta x}.</math>

In addition, {{math|{{sfrac|(Δ''x'')<sup>2</sup>|2Δ''t''}}}} is the definition of the one-dimensional diffusion constant, {{mvar|D}}. Thus our expression simplifies to:
: <math> J = -D \left[\frac{\varphi (x + \Delta x, t)}{\Delta x} - \frac{\varphi (x , t)}{\Delta x}\right]</math>

In the limit where {{math|Δ''x''}} is infinitesimal, the right-hand side becomes a space derivative:
: <math> J = - D \frac{\partial \varphi}{\partial x}  </math> This is only the case for the initial condition of a very of a initial gaussian distribution. Other problem geometries will lead to different solutions
-->
Fick's second law can be derived from Fick's first law and the [[mass conservation]] in absence of any chemical reactions:
: <math>\frac{\partial \varphi}{\partial t} + \frac{\partial}{\partial x}J = 0
\Rightarrow\frac{\partial \varphi}{\partial t} -\frac{\partial}{\partial x}\left(D\frac{\partial}{\partial x}\varphi\right)\,=0.</math>

Assuming the diffusion coefficient {{mvar|D}} to be a constant, one can exchange the orders of the differentiation and multiply by the constant:
:<math>\frac{\partial}{\partial x}\left(D\frac{\partial}{\partial x} \varphi\right) = D\frac{\partial}{\partial x} \frac{\partial}{\partial x} \varphi = D\frac{\partial^2\varphi}{\partial x^2},</math>
and, thus, receive the form of the Fick's equations as was stated above.

For the case of diffusion in two or more dimensions Fick's second law becomes
: <math>\frac{\partial \varphi}{\partial t} = D\,\nabla^2\varphi,</math>
which is analogous to the [[heat equation]].

If the diffusion coefficient is not a constant, but depends upon the coordinate or concentration, Fick's second law yields
: <math>\frac{\partial \varphi}{\partial t} =  \nabla \cdot (D\,\nabla\varphi).</math>

An important example is the case where {{math|'''φ'''}} is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant {{mvar|D}}, the solution for the concentration will be a linear change of concentrations along {{mvar|x}}. In two or more dimensions we obtain
: <math> \nabla^2\varphi = 0,</math>
which is [[Laplace's equation]], the solutions to which are referred to by mathematicians as [[harmonic functions]].