English: Diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (magenta line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. Top: A single molecule moves around randomly. Middle: With more molecules, there is a clear trend where the solute fills the container more and more evenly. Bottom: With an enormous number of solute molecules, the randomness is gone: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas, following Fick's laws.
Image is made in Mathematica, source code below.
Εγώ, ο κάτοχος των πνευματικών δικαιωμάτων αυτού του έργου, δημοσιεύω αυτό το έργο ως κοινό κτήμα. Αυτό ισχύει σε παγκόσμια κλίμακα. Σε ορισμένες χώρες αυτό μπορεί να μην είναι νομικά εφικτό. Αν ναι: Παραχωρώ σε οποιονδήποτε το δικαίωμα να χρησιμοποιήσει αυτό το έργο "για οποιονδήποτε σκοπό", χωρίς κανέναν όρο, εκτός και αν τέτοιοι όροι τίθενται από την νομοθεσία
<< Mathematica source code >>
(* Source code written in Mathematica 6.0, by Steve Byrnes, 2010.
I release this code into the public domain. Sorry it's messy...email me any questions. *)
(*Particle simulation*)
SeedRandom[1];
NumParticles = 70;
xMax = 0.7;
yMax = 0.2;
xStartMax = 0.5;
StepDist = 0.04;
InitParticleCoordinates = Table[{RandomReal[{0, xStartMax}], RandomReal[{0, yMax}]}, {i, 1, NumParticles}];
StayInBoxX[x_] := If[x < 0, -x, If[x > xMax, 2 xMax - x, x]];
StayInBoxY[y_] := If[y < 0, -y, If[y > yMax, 2 yMax - y, y]];
StayInBoxXY[xy_] := {StayInBoxX[xy[[1]]], StayInBoxY[xy[[2]]]};
StayInBarX[x_] := If[x < 0, -x, If[x > xStartMax, 2 xStartMax - x, x]];
StayInBarY[y_] := If[y < 0, -y, If[y > yMax, 2 yMax - y, y]];
StayInBarXY[xy_] := {StayInBarX[xy[[1]]], StayInBarY[xy[[2]]]};
MoveAStep[xy_] := StayInBoxXY[xy + {RandomReal[{-StepDist, StepDist}], RandomReal[{-StepDist, StepDist}]}];
MoveAStepBar[xy_] := StayInBarXY[xy + {RandomReal[{-StepDist, StepDist}], RandomReal[{-StepDist, StepDist}]}];
NextParticleCoordinates[ParticleCoords_] := MoveAStep /@ ParticleCoords;
NextParticleCoordinatesBar[ParticleCoords_] := MoveAStepBar /@ ParticleCoords;
NumFramesBarrier = 10;
NumFramesNoBarrier = 50;
NumFrames = NumFramesBarrier + NumFramesNoBarrier;
ParticleCoordinatesTable = Table[0, {i, 1, NumFrames}];
ParticleCoordinatesTable[[1]] = InitParticleCoordinates;
For[i = 2, i <= NumFrames, i++,
If[i <= NumFramesBarrier,
ParticleCoordinatesTable[[i]] = NextParticleCoordinatesBar[ParticleCoordinatesTable[[i - 1]]],
ParticleCoordinatesTable[[i]] = NextParticleCoordinates[ParticleCoordinatesTable[[i - 1]]]];];
(*Plot full particle simulation*)
makeplotbar[ParticleCoord_] :=
ListPlot[{ParticleCoord, {{xStartMax, 0}, {xStartMax, yMax}}}, Frame -> True, Axes -> False,
PlotRange -> {{0, xMax}, {0, yMax}}, Joined -> {False, True}, PlotStyle -> {PointSize[.03], Thick},
AspectRatio -> yMax/xMax, FrameTicks -> None];
makeplot[ParticleCoord_] :=
ListPlot[ParticleCoord, Frame -> True, Axes -> False, PlotRange -> {{0, xMax}, {0, yMax}}, Joined -> False,
PlotStyle -> PointSize[.03], AspectRatio -> yMax/xMax, FrameTicks -> None]
ParticlesPlots =
Join[Table[makeplotbar[ParticleCoordinatesTable[[i]]], {i, 1, NumFramesBarrier}],
Table[makeplot[ParticleCoordinatesTable[[i]]], {i, NumFramesBarrier + 1, NumFrames}]];
(*Plot just the first particle in the list...Actually the fifth particle looks better. *)
FirstParticleTable = {#[[5]]} & /@ ParticleCoordinatesTable;
FirstParticlePlots =
Join[Table[makeplotbar[FirstParticleTable[[i]]], {i, 1, NumFramesBarrier}],
Table[makeplot[FirstParticleTable[[i]]], {i, NumFramesBarrier + 1, NumFrames}]];
(* Continuum solution *)
(* I can use the simple diffusion-on-an-infinite-line formula, as long as I correctly periodically replicate the
initial condition. Actually just computed nearest five replicas in each direction, that was a fine approximation. *)
(* k = diffusion coefficient, visually matched to simulation. *)
k = .0007;
u[x_, t_] := If[t == 0, If[x <= xStartMax, 1, 0], 1/2 Sum[
Erf[(x - (-xStartMax + 2 n xMax))/Sqrt[4 k t]] - Erf[(x - (xStartMax + 2 n xMax))/Sqrt[4 k t]], {n, -5, 5}]];
ContinuumPlots = Join[
Table[Show[
DensityPlot[1 - u[x, 0], {x, 0, xMax}, {y, 0, yMax},
ColorFunctionScaling -> False, AspectRatio -> yMax/xMax,
FrameTicks -> None],
ListPlot[{{xStartMax, 0}, {xStartMax, yMax}}, Joined -> True,
PlotStyle -> {Thick, Purple}]],
{i, 1, NumFramesBarrier}],
Table[
DensityPlot[1 - u[x, tt], {x, 0, xMax}, {y, 0, yMax},
ColorFunctionScaling -> False, AspectRatio -> yMax/xMax,
FrameTicks -> None],
{tt, 1, NumFramesNoBarrier}]];
(*Combine and export *)
TogetherPlots =
Table[GraphicsGrid[{{FirstParticlePlots[[i]]}, {ParticlesPlots[[i]]}, {ContinuumPlots[[i]]}},
Spacings -> Scaled[0.2]], {i, 1, NumFrames}];
Export["test.gif", Join[TogetherPlots, Table[Graphics[], {i, 1, 5}]],
"DisplayDurations" -> {10}, "AnimationRepititions" -> Infinity ]
{{Information |Description={{en|1=Diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill