通过线性回归识别稳态斜率让我想到了随着时间的推移会有自然变化的行为的流程，以及如何制作一些 JSL 来模拟一些数据。 这个 JSL 模拟了 30 个点在一维管的两端之间弹跳；这些点不能相互通过，也不能相互反弹，也不能从管子的末端反弹。 在管子的末端有一点摩擦，使速度随着时间的推移而衰减，并在其他地方平均速度。 模拟循环相当有效；它在半小时内运行 1 亿个周期。 由于一维约束，它可以在快速矩阵运算中检查所有 30 个对象的碰撞。
该图在 X 轴上显示时间，作为循环数。 底部的彩色图形跟踪 Y 轴上的 30 个点位置。 观察点的颜色，您可以看到点上下移动并在击中时弹跳。 上图显示了绝对速度总量的衰减。 因为点在弹跳时变成了相同的对立面，所以图表的右端显示了许多类似的条带，因为行为同步。
Rainbow graph of point positions.点位置的彩虹图。
dt = New Table( "particles" );
n = 30; // the graph builder is hard-coded for 30, probably don't change this
dt << addmultiplecolumns( "pos", n, "Numeric" );
dt << addmultiplecolumns( "vel", n, "Numeric" );
dt << newcolumn("time");
dt << newcolumn("energy");
dt << addrow( 1 );
// create the particles in a 1-dimensional tube, 1 unit long.
// the n particles are sandwiched between two 50% reflective end walls.
allpos = 1 :: n; // column indexes for data table subscripting: the position columns
allposDown = 1 :: n - 1; // all but the last. when this is used below, it is shifted over by adding the 0-wall.
dt[N Rows( dt ), allpos] = Sort List( J( 1, n, Random Uniform( 0, .01 ) ) ); // initial positions. Note the sortlist.
allvel = allpos + n; // the velocity columns come after the n position columns
dt[N Rows( dt ), allvel] = J( 1, n, Random Normal( 0, .0001 ) );// initial velocities
// run the simulation. pre-load the vectors from the table.
newvel = dt[N Rows( dt ), allvel];
newpos = dt[N Rows( dt ), allpos] + newvel;
limit=1e8; // 1e8: ~1500 seconds
start = tickseconds();
For( i = 1, i < limit, i += 1,
change = 0; // only record to the table when particles bounce.
newpos += newvel; // update all particle positions
// collisions happened where no longer sorted. The particles can't pass
// through each other. make them bounce with some energy sharing and loss.
// loc() returns indexes of flipped sort positions; the subtraction's
// right hand side is shifted over with the zero wall added.
seq1 = Loc( newpos - (0 || newpos[0, allposDown]) < 0 );
For Each( {iPos}, seq1, // repair (by bouncing) each flipped sort
If( ipos > 1, // 1 is the zero wall, leave it alone
where = Mean( newpos[iPos - 1], newpos[iPos] ); // both points move to 'where'
speed = Mean( Abs( newvel[iPos - 1] ), Abs( newvel[iPos] ) ); // and get 'speed'
newpos[iPos - 1] = Max( 0, where );
newvel[iPos - 1] = -speed; // speed is positive, low point moves lower
, //
where = -newpos[iPos] / 1000; // point that went below 0...flip sign, move back in range, barely
speed = Abs( newvel[iPos] ) / 2; // the energy leaks off the end; the other half of this average is lost.
);
newpos[iPos] = Min( 1, where );
newvel[iPos] = speed; // high point gets positive speed
change = 1;
);
if( newpos[n]>1, // handle point 30 (last point). 29 and 30 were lested above. now test 30 against one wall.
newpos[n] = 1 - (newpos[n] - 1) / 1000;
newvel[n] = -newvel[n]/2;
change = 1;
);
If( change, // add a row to the table. there are only ~20,000 rows for 1e8 iterations
dt << addrows( 1 );
dt[N Rows( dt ), allvel] = newvel;
dt[N Rows( dt ), allpos] = newpos;
dt:time[N Rows( dt )] = i;
dt:energy[N Rows( dt )] = sum(abs(newvel));
);
);
stop = tickseconds();
speed = "Wall clock time: "||char(stop-start,9,1)||" seconds.";
dt << Graph Builder( Size( 1754, 804 ), Show Control Panel( 0 ), Show Legend( 0 ),
Variables( X( :time ), Y( :energy ), Y( :pos 01 ), Y( :pos 02, Position( 2 ) ), Y( :pos 03, Position( 2 ) ), Y( :pos 04, Position( 2 ) ), Y( :pos 05, Position( 2 ) ), Y( :pos 06, Position( 2 ) ), Y( :pos 07, Position( 2 ) ), Y( :pos 08, Position( 2 ) ), Y( :pos 09, Position( 2 ) ), Y( :pos 10, Position( 2 ) ), Y( :pos 11, Position( 2 ) ), Y( :pos 12, Position( 2 ) ), Y( :pos 13, Position( 2 ) ), Y( :pos 14, Position( 2 ) ), Y( :pos 15, Position( 2 ) ), Y( :pos 16, Position( 2 ) ), Y( :pos 17, Position( 2 ) ), Y( :pos 18, Position( 2 ) ), Y( :pos 19, Position( 2 ) ), Y( :pos 20, Position( 2 ) ), Y( :pos 21, Position( 2 ) ), Y( :pos 22, Position( 2 ) ), Y( :pos 23, Position( 2 ) ), Y( :pos 24, Position( 2 ) ), Y( :pos 25, Position( 2 ) ), Y( :pos 26, Position( 2 ) ), Y( :pos 27, Position( 2 ) ), Y( :pos 28, Position( 2 ) ), Y( :pos 29, Position( 2 ) ), Y( :pos 30, Position( 2 ) )),
Relative Sizes( "Y", [79 337] ),
Elements( Position( 1, 1 ), Points( X, Y, Legend( 5 ) ) ),
Elements( Position( 1, 2 ), Points( X, Y( 1 ), Y( 2 ), Y( 3 ), Y( 4 ), Y( 5 ), Y( 6 ), Y( 7 ), Y( 8 ), Y( 9 ), Y( 10 ), Y( 11 ), Y( 12 ), Y( 13 ), Y( 14 ), Y( 15 ), Y( 16 ), Y( 17 ), Y( 18 ), Y( 19 ), Y( 20 ), Y( 21 ), Y( 22 ), Y( 23 ), Y( 24 ), Y( 25 ), Y( 26 ), Y( 27 ), Y( 28 ), Y( 29 ), Y( 30 ), Legend( 3 ))),
SendToReport( Dispatch( {}, "time", ScaleBox, {Scale( "Log" ), Min( 99 ), Max( 101000000 ), Inc( 1 ), Minor Ticks( 1 ), Label Row( {Show Minor Labels( 0 )})} ),
Dispatch( {}, "energy", ScaleBox, {Scale( "Log" ), Format( "Best", 10 ), Min( 0.0000009 ), Max( 0.003 ), Inc( 1 ), Minor Ticks( 1 ), Label Row( {Show Major Grid( 1 ), Show Minor Labels( 0 )})}),
Dispatch( {}, "pos 01", ScaleBox, {Min( -0.008 ), Max( 1.03 ), Inc( 0.2 ), Minor Ticks( 3 )} ),
Dispatch( {}, "400", ScaleBox, {Legend Model( 5, Properties( 0, {Line Color( 0 )}, Item ID( "energy", 1 ) ) )} ),
Dispatch( {}, "graph title", TextEditBox, {Set Text( "energy remaining in system (top), 30 points spreading out (bottom)" )} ),
Dispatch( {}, "Graph Builder", FrameBox, {Background Color( 32 ), Marker Size( 0 ), Marker Drawing Mode( "Fast" )} ),
Dispatch( {}, "Y 1 title", TextEditBox, {Set Text( "points" )} ),
Dispatch( {}, "Graph Builder", FrameBox( 2 ), {Marker Size( 0 ), Marker Drawing Mode( "Fast" ), {Add Text Annotation(Text("the points are initially near y==0 and are confined to a 1-dimensional Y axis from 0 to 1. As they bump each other they exchange some velocity/energy and bounce so they never exchange positions in the 1-D column. When the first or last point bumps the end, half of its energy is lost, so the energy curve drops toward zero and the points slow down. The initial velocities are random normal. The fish-net appearance around 10,000 shows the zig-zag bouncing if the colors are examined rather than the long straight lines. Each iteration adds 1 to the time axis for 100 million iterations."),Fixed Size( 0 ),Text Box( {44, 47, 392, 614} ),Filled( 0 )), Add Text Annotation( Text( speed ), Fixed Size( 0 ), Text Box( {56, 459, 278, 496} ), Filled( 0 ) )}})
)
);
这只是为了好玩，并不代表任何现实世界的系统。 不是物理专业的。 享受。
