線形回帰による定常状態の勾配の特定時間の経過とともに自然に変化する動作を持つプロセスと、一部のデータをシミュレートするための JSL の作成方法について考えさせられました。 この JSL は、1 次元のチューブの端の間で跳ね返る 30 点のシミュレーションを実行します。ポイントは互いに通り過ぎることができず、お互いやチューブの端で跳ね返ることができません。 チューブの端には少しの摩擦があり、時間の経過とともに速度が減衰し、他の場所では速度が平均化されます。 シミュレーション ループはかなり効率的です。 30 分で 1 億サイクルを実行します。 1 次元の制約により、高速な行列演算で 30 個すべてのオブジェクトの衝突をチェックできます。
グラフは、サイクル数として X 軸に時間を示します。 下部のカラフルなグラフは、Y 軸上の 30 ポイントの位置を追跡しています。 ポイントの色を見ると、ポイントが上下に移動し、当たったときに跳ね返ることがわかります。 上のグラフは、絶対速度の合計の減衰を示しています。 ポイントは跳ね返るとまったく同じ反対になるため、動作が同期するにつれて、グラフの右端には多くの同様のバンディングが表示されます。
Rainbow graph of point positions.ポイント位置のレインボー グラフ。
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 ) )}})
)
);
これは単なる遊びであり、実際のシステムを表すものではありません。 物理専攻ではありません。 楽しみ。
