File:Smoluchowski equation.gif
Summary
| Description |
English: The Smoluchowski equation generalizes the diffusion equation to the presence of an external potential.
β is the thermodynamic beta D is the diffusion coefficient U is the external potential (Notice that there is more than one equation known as "Smoluchowski equation") |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1279010141254410243 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
| GIF development |
Mathematica 12.1 code
d = 1; \[Beta] = 10;
U[x_] := 0;
sol1 = NDSolve[{D[p[x, t], t] == D[d E^(-\[Beta] U[x]) D[E^(\[Beta] U[x]) p[x, t], x], x], p[x, 0] == E^(-(x^2/(2 0.1^2))), p[50, t] == 0, p[-50, t] == 0}, p, {t, 0, 10}, {x, -50, 50}]
U[x_] := 0.1 x;
sol2 = NDSolve[{D[p[x, t], t] == D[d E^(-\[Beta] U[x]) D[E^(\[Beta] U[x]) p[x, t], x], x], p[x, 0] == E^(-(x^2/(2 0.1^2))), p[50, t] == 0, p[-50, t] == 0}, p, {t, 0, 10}, {x, -50, 50}]
U[x_] := 0.5 x^2;
sol3 = NDSolve[{D[p[x, t], t] == D[d E^(-\[Beta] U[x]) D[E^(\[Beta] U[x]) p[x, t], x], x], p[x, 0] == E^(-(x^2/(2 0.1^2))), p[50, t] == 0, p[-50, t] == 0}, p, {t, 0, 10}, {x, -50, 50}]
U[x_] := 0.3 x^4 + 0.2 Cos[5 \[Pi] x] + 0.2;
sol4 = NDSolve[{D[p[x, t], t] == D[d E^(-\[Beta] U[x]) D[E^(\[Beta] U[x]) p[x, t], x], x], p[x, 0] == E^(-(x^2/(2 0.1^2))), p[50, t] == 0, p[-50, t] == 0}, p, {t, 0, 5}, {x, -50, 50}]
p0 = Table[
Grid[{
{Text[Style["\!\(\*FractionBox[\(d\\\ p\), \(d\\\ t\)]\)= \[Del].(D \!\(\*SuperscriptBox[\(e\), \(\(-\[Beta]\)\\\ U\)]\) \[Del] \!\(\*SuperscriptBox[\(e\), \(\[Beta]\\\ U\)]\) p)", Bold]], SpanFromLeft, SpanFromLeft, ""},{
Plot[{0, Evaluate[p[x, t] /. sol1] /. {t -> \[Tau]}}, {x, -1.5, 1.5}, PlotRange -> {-0.1, 1.2}, Axes -> False, Filling -> Axis, PlotStyle -> {Orange, Purple}]
,
Plot[{0.1 x + 0.2, Evaluate[p[x, t] /. sol2] /. {t -> \[Tau]}}, {x, -1.5, 1.5}, PlotRange -> {-0.1, 1.2}, Axes -> False, Filling -> Axis, PlotStyle -> {Orange, Purple}]
,
LineLegend[{Purple, Orange}, {"Concentration p(x,t)", "Potential U(x)"}]
}, {
Plot[{0.5 x^2, Evaluate[p[x, t] /. sol3] /. {t -> \[Tau]}}, {x, -1.5, 1.5}, PlotRange -> {-0.1, 1.2}, Axes -> False, Filling -> Axis, PlotStyle -> {Orange, Purple}]
,
Plot[{0.3 x^4 + 0.2 Cos[5 \[Pi] x] + 0.2, Evaluate[p[x, t] /. sol4] /. {t -> \[Tau]}}, {x, -1.5, 1.5}, PlotRange -> {-0.1, 1.2}, Axes -> False, Filling -> Axis, PlotStyle -> {Orange, Purple}]
, SpanFromAbove
}}]
, {\[Tau], 0, 0.5, 0.002}];
ListAnimate[p0]
Licensing
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