Revisions to 2D random walk within a bounded area

added 73 characters in body

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edited Aug 18, 2014 at 7:33

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I sketched this up very quicklysuggest using Mod - so alla natural thing for looped boundary conditions on a torus.

Finite torus surface area is subject to checkyour bounded region.

2D random walk generally is simple:

walk = Accumulate[RandomReal[{-.1, .1}, {100, 2}]];
Graphics[Line[walk], Frame -> True]

enter image description here

Confinement to square region {{0,1},{0,1}} would be simple in principle with Mod[walk,1] (periodic boundary conditions) but visualizing will be hard:

Graphics[Line[Mod[walk, 1]], Frame -> True]

enter image description here

So I think logical, for periodic boundary conditions, to place it on a torus ( with arbitrary radiuses ):

map[φ_, θ_] = CoordinateTransformData["Toroidal" -> "Cartesian", 
              "Mapping", {r, θ, φ}] /. {\[FormalA] -> 1, r -> 2 Log[2]}

enter image description here

walk = Accumulate[RandomReal[{-.1, .1}, {10^4, 2}]];
Graphics3D[{Opacity[.5], Line[map @@@ walk]}, SphericalRegion -> True]

enter image description here

I sketched this up very quickly - so all is subject to check.

2D random walk generally is simple:

walk = Accumulate[RandomReal[{-.1, .1}, {100, 2}]];
Graphics[Line[walk], Frame -> True]

enter image description here

Confinement to square region {{0,1},{0,1}} would be simple in principle with Mod[walk,1] (periodic boundary conditions) but visualizing will be hard:

Graphics[Line[Mod[walk, 1]], Frame -> True]

enter image description here

So I think logical, for periodic boundary conditions, to place it on a torus ( with arbitrary radiuses ):

map[φ_, θ_] = CoordinateTransformData["Toroidal" -> "Cartesian", 
              "Mapping", {r, θ, φ}] /. {\[FormalA] -> 1, r -> 2 Log[2]}

enter image description here

walk = Accumulate[RandomReal[{-.1, .1}, {10^4, 2}]];
Graphics3D[{Opacity[.5], Line[map @@@ walk]}, SphericalRegion -> True]

enter image description here

I suggest using Mod - a natural thing for looped boundary conditions on a torus.

Finite torus surface area is your bounded region.

2D random walk generally is simple:

walk = Accumulate[RandomReal[{-.1, .1}, {100, 2}]];
Graphics[Line[walk], Frame -> True]

enter image description here

Confinement to square region {{0,1},{0,1}} would be simple in principle with Mod[walk,1] (periodic boundary conditions) but visualizing will be hard:

Graphics[Line[Mod[walk, 1]], Frame -> True]

enter image description here

So I think logical, for periodic boundary conditions, to place it on a torus ( with arbitrary radiuses ):

map[φ_, θ_] = CoordinateTransformData["Toroidal" -> "Cartesian", 
              "Mapping", {r, θ, φ}] /. {\[FormalA] -> 1, r -> 2 Log[2]}

enter image description here

walk = Accumulate[RandomReal[{-.1, .1}, {10^4, 2}]];
Graphics3D[{Opacity[.5], Line[map @@@ walk]}, SphericalRegion -> True]

enter image description here

added 3 characters in body

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edited Aug 17, 2014 at 23:52

Vitaliy Kaurov

73.4k
9
206
365

I sketched this up very quickly - so all is subject to check.

2D random walk generally is simple:

walk = Accumulate[RandomReal[{-.1, .1}, {100, 2}]];
Graphics[Line[walk], Frame -> True]

enter image description here

Confinement to square region {{0,1},{0,1}} would be simple in principle with Mod[walk,2]1] (periodic boundary conditions) but visualizing will be hard:

Graphics[Line[Mod[walk, 1]], Frame -> True]

enter image description here

So I think logical, for periodic boundary conditions, to place it on a torus ( with arbitrary radiuses ):

map[φ_, θ_] = CoordinateTransformData["Toroidal" -> "Cartesian", 
              "Mapping", {r, θ, φ}] /. {\[FormalA] -> 1, r -> 2 Log[2]}

enter image description here

walk = Accumulate[RandomReal[{-.1, .1}, {10^4, 2}]];
Graphics3D[{Opacity[.5], Line[map @@@ walk]}, SphericalRegion -> True]

enter image description here

I sketched this up very quickly - so all is subject to check.

2D random walk generally is simple:

walk = Accumulate[RandomReal[{-.1, .1}, {100, 2}]];
Graphics[Line[walk], Frame -> True]

enter image description here

Confinement to square region {{0,1},{0,1}} would simple in principle with Mod[walk,2] (periodic boundary conditions) but visualizing will be hard:

Graphics[Line[Mod[walk, 1]], Frame -> True]

enter image description here

So I think logical, for periodic boundary conditions, to place it on a torus ( with arbitrary radiuses ):

map[φ_, θ_] = CoordinateTransformData["Toroidal" -> "Cartesian", 
              "Mapping", {r, θ, φ}] /. {\[FormalA] -> 1, r -> 2 Log[2]}

enter image description here

walk = Accumulate[RandomReal[{-.1, .1}, {10^4, 2}]];
Graphics3D[{Opacity[.5], Line[map @@@ walk]}, SphericalRegion -> True]

enter image description here

I sketched this up very quickly - so all is subject to check.

2D random walk generally is simple:

walk = Accumulate[RandomReal[{-.1, .1}, {100, 2}]];
Graphics[Line[walk], Frame -> True]

enter image description here

Confinement to square region {{0,1},{0,1}} would be simple in principle with Mod[walk,1] (periodic boundary conditions) but visualizing will be hard:

Graphics[Line[Mod[walk, 1]], Frame -> True]

enter image description here

So I think logical, for periodic boundary conditions, to place it on a torus ( with arbitrary radiuses ):

map[φ_, θ_] = CoordinateTransformData["Toroidal" -> "Cartesian", 
              "Mapping", {r, θ, φ}] /. {\[FormalA] -> 1, r -> 2 Log[2]}

enter image description here

walk = Accumulate[RandomReal[{-.1, .1}, {10^4, 2}]];
Graphics3D[{Opacity[.5], Line[map @@@ walk]}, SphericalRegion -> True]

enter image description here

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created Aug 17, 2014 at 23:38

Vitaliy Kaurov

73.4k
9
206
365

I sketched this up very quickly - so all is subject to check.

2D random walk generally is simple:

walk = Accumulate[RandomReal[{-.1, .1}, {100, 2}]];
Graphics[Line[walk], Frame -> True]

enter image description here

Confinement to square region {{0,1},{0,1}} would simple in principle with Mod[walk,2] (periodic boundary conditions) but visualizing will be hard:

Graphics[Line[Mod[walk, 1]], Frame -> True]

enter image description here

So I think logical, for periodic boundary conditions, to place it on a torus ( with arbitrary radiuses ):

map[φ_, θ_] = CoordinateTransformData["Toroidal" -> "Cartesian", 
              "Mapping", {r, θ, φ}] /. {\[FormalA] -> 1, r -> 2 Log[2]}

enter image description here

walk = Accumulate[RandomReal[{-.1, .1}, {10^4, 2}]];
Graphics3D[{Opacity[.5], Line[map @@@ walk]}, SphericalRegion -> True]

enter image description here

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