Smoluchowski Diffusion Equation

The Smoluchowski diffusion equation is a special case of the Fokker–Plank equation. Both equations can be modeled with a conservative mass transport equation:

∂c(t,x)

∂t

+

masstransportmodel

︷

∇·(-d∇c(x)+βF(x))

=0

Set up the mass transport model variables

vars

:

In[1]:=

vars={c[t,x],t,{x}};

Set up a line domain with a width of 8 units:

In[2]:=

Ω=Line[{{-8},{8}}];

Specify the model parameters species diffusivity

d

and a migration term that depends on a linear potential

U(x)

that relates to

F(x)

with

F(x)=-

∇

x

U(x)

:

In[3]:=

U[x_]:=γ*xpars="ModelForm""Conservative","DiffusionCoefficient"{{d}},"MassConvectionVelocity"β*(-Grad[U[x],{x}]),d1,γ1,β1;

Set up the known analytical solution:

In[4]:=

solution[t_,x_]:=Evaluate[With[{x0=0,t0=0},1/Sqrt[4πd(t-t0)]*Exp[-(x-x0+dβγ(t-t0))^2/(4d*(t-t0))]]/.pars]

Set up the initial conditions at

t=1/10

from the analytical solution:

In[5]:=

ics=c[1/10,x]solution[1/10,x]

Out[5]=

c

1

10

,x

-

5

2

1

10

+x



5

2π

Set up the equation:

In[6]:=

eqn={MassTransportPDEComponent[vars,pars]0,ics}

Out[6]=



∇

{x}

·{{-1}}.

∇

{x}

c[t,x]+

∇

{x}

·{-c[t,x]}+

(1,0)

c

[t,x]0,c

1

10

,x

-

5

2

1

10

+x



5

2π



Solve the PDE:

In[7]:=

cfun=NDSolveValue[eqn,c,{t,1/10,1},{x}∈Ω]

Out[7]=

InterpolatingFunction

Domain: {{0.1,1.},{-8.,8.}}

Output: scalar



Visualize the solution:

In[8]:=

frames=Table[Plot[cfun[t,x],{x}∈Ω,PlotRange{0,1}],{t,1/10,1,1/10}];ListAnimate[frames,SaveDefinitionsTrue]

Out[8]=

Plot the error at

t=1

:

In[9]:=

Plot[{solution[1,x]-cfun[1,x]},{x}∈Ω,PlotRangeAll]

Out[9]=

Wolfram Language Example Repository

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Smoluchowski Diffusion Equation

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