File:Entropy Time Arrow.gif
Summary
| Description |
English: Microscopically, Physics has no preference for time to flow forward or backward.
But (on average) both cases will evolve toward a situation with higher entropy, which provides us with a macroscopic arrow of time. |
| Date | |
| Source | https://twitter.com/j_bertolotti/status/1513453337676595200 |
| Author | Jacopo Bertolotti |
| Permission (Reusing this file) |
https://twitter.com/j_bertolotti/status/1030470604418428929 |
Mathematica 13.0 code
(**Velocity Verlet**)
Clear[evo, pos, posold, posoldold, accelerations, oldaccelerations,
velocities, halfvelocities, tableForces];
L = 10; (*(Half) Size of the box*)
\[Sigma] = 0.8; (*particle radius*) \[Sigma]plot = 0.55 \[Sigma];(*particle radius for graphing*)
dt = 0.005;
LJforce[p1_, p2_] := 4*2 (12 \[Sigma]^12/Max[Norm[p1 - p2], \[Sigma]]^13) Normalize[p1 - p2];
pos = Flatten[Table[N[{i, j}], {i, -L + 1, 0, L/6}, {j, 0, L - 1, L/6}], 1];
nparticles = Dimensions[pos][[1]]
m = Table[1, {nparticles}];
initialvelocities = Table[0, {j, 1, nparticles}]; initialvelocities[[15]] = {1, 10}; initialvelocities[[4]] = {5, 5};
tableForces = Chop@Table[ LJforce[pos[[i]], pos[[j]]], {i, 1, nparticles}, {j, 1, nparticles}];
MatrixForm[tableForces];
accelerations = Table[Sum[m[[i]]*tableForces[[i, j]], {j, 1, nparticles}], {i, 1, nparticles}];
posold = pos;
oldaccelerations = accelerations;
pos = posold + initialvelocities*dt + accelerations/2 dt^2;
velocities = initialvelocities + (accelerations + oldaccelerations)/2 dt;
nsteps = 1000;
evo1 = Reap[Do[
tableForces = Chop@Table[ LJforce[pos[[i]], pos[[j]]], {i, 1, nparticles}, {j, 1, nparticles}];
oldaccelerations = accelerations;
posold = pos;
halfvelocities = velocities + accelerations/2 dt;
For[j = 1, j <= nparticles, j++,
If[pos[[j, 2]] >= L,
halfvelocities[[j, 2]] = -Abs[halfvelocities[[j, 2]] ] ];
If[pos[[j, 2]] <= -L,
halfvelocities[[j, 2]] = Abs[halfvelocities[[j, 2]] ] ];
If[pos[[j, 1]] >= L,
halfvelocities[[j, 1]] = -Abs[halfvelocities[[j, 1]] ] ];
If[pos[[j, 1]] <= -L,
halfvelocities[[j, 1]] = Abs[halfvelocities[[j, 1]] ] ];
];
pos = posold + halfvelocities dt;
accelerations = Chop@Table[ Sum[m[[i]]*tableForces[[i, j]], {j, 1, nparticles}], {i, 1, nparticles}];
velocities = halfvelocities + (accelerations + oldaccelerations)/2 dt;
Sow[pos];
, nsteps];][[2, 1]];
dt = -0.005;
pos = Flatten[Table[N[{i, j}], {i, -L + 1, 0, L/6}, {j, 0, L - 1, L/6}], 1];
accelerations = Table[Sum[m[[i]]*tableForces[[i, j]], {j, 1, nparticles}], {i, 1, nparticles}];
posold = pos;
oldaccelerations = accelerations;
pos = posold + initialvelocities*dt + accelerations/2 dt^2;
velocities = initialvelocities + (accelerations + oldaccelerations)/2 dt;
evo2 = Reap[Do[
tableForces = Chop@Table[ LJforce[pos[[i]], pos[[j]]], {i, 1, nparticles}, {j, 1, nparticles}];
oldaccelerations = accelerations;
posold = pos;
halfvelocities = velocities + accelerations/2 dt;
For[j = 1, j <= nparticles, j++,
If[pos[[j, 2]] >= L,
halfvelocities[[j, 2]] = Abs[halfvelocities[[j, 2]] ] ];
If[pos[[j, 2]] <= -L,
halfvelocities[[j, 2]] = -Abs[halfvelocities[[j, 2]] ] ];
If[pos[[j, 1]] >= L,
halfvelocities[[j, 1]] = Abs[halfvelocities[[j, 1]] ] ];
If[pos[[j, 1]] <= -L,
halfvelocities[[j, 1]] = -Abs[halfvelocities[[j, 1]] ] ];
];
pos = posold + halfvelocities dt;
accelerations = Chop@Table[Sum[m[[i]]*tableForces[[i, j]], {j, 1, nparticles}], {i, 1, nparticles}];
velocities = halfvelocities + (accelerations + oldaccelerations)/2 dt;
Sow[pos];
, nsteps];][[2, 1]];
frames3 = Table[
Graphics[{
Disk[#, \[Sigma]plot] & /@ evo1[[j]],
Disk[#, \[Sigma]plot] & /@ (({2.5 L, 0} + #) & /@ evo2[[j]])
,
Line[{{-L - \[Sigma]plot, -L - \[Sigma]plot}, {-L - \[Sigma]plot, L + \[Sigma]plot}, {L + \[Sigma]plot, L + \[Sigma]plot}, {L + \[Sigma]plot, -L - \[Sigma]plot}, {-L - \[Sigma]plot, -L - \[Sigma]plot}}],
Line[({2.5 L, 0} + # &) /@ {{-L - \[Sigma]plot, -L - \[Sigma]plot}, {-L - \[Sigma]plot, L + \[Sigma]plot}, {L + \[Sigma]plot, L + \[Sigma]plot}, {L + \[Sigma]plot, -L - \[Sigma]plot}, {-L - \[Sigma]plot, -L - \[Sigma]plot}}]
,
Text[ Style["Time flows forward", Bold, Black, FontSize -> 15], {0, 1.2 L}],
Text[ Style["Time flows backward", Bold, Black, FontSize -> 15], {2.5 L, 1.2 L}],
Text[ Style["Identical initial conditions", Bold, Black, FontSize -> 16], {1.2 L, L + 4.5}]
}, PlotRange -> {{-L - 1, 3.5 L + 1}, {-L - 1, L + 5}},
ImageSize -> Large]
, {j, 1, nsteps}];
ListAnimate[frames3]
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
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This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication. |
| The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.
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