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.
Yo, el titular de los derechos de autor de esta obra, lo libero al dominio público. Esto aplica en todo el mundo. En algunos países esto puede no ser legalmente factible; si ello ocurriese: Concedo a cualquier persona el derecho de usar este trabajo para cualquier propósito, sin ningún tipo de condición al menos que éstas sean requeridas por la ley.
<< 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 ]
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{{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