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| Keywords: Hamiltonian flow classical.gif en Flow of a statistical ensemble in the potential x 6 + 4 x 3 - 5 x 2 - 4 x Over long times becomes swirled up and appears to become a smooth and stable distribution However this stability is an artifact of the pixelization the actual structure is too fine to perceive <br/>This animation is inspired by a discussion of Gibbs in his 1902 wikisource Elementary Principles in Statistical Mechanics Chapter XII p 143 Tendency in an ensemble of isolated systems toward a state of statistical equilibrium 2013-10-27 00 06 21 own Nanite Source Created with Matplotlib Created with Imagemagick Python source code Requires matplotlib ImageMagick Possibly does not run in Windows <source lang python > from pylab import import subprocess import sys import os figformat ' png' seterr divide 'ignore' rcParams'font size' 9 define color map that is transparent for low values and dark blue for high values weighted to show low probabilities well cdic 'red' 0 0 0 1 0 0 'green' 0 0 0 1 0 0 'blue' 0 0 7 0 7 1 0 7 0 7 'alpha' 0 0 0 0 1 0 4 0 4 0 2 0 6 0 6 0 4 0 8 0 8 0 6 0 9 0 9 1 1 1 cm_prob matplotlib colors LinearSegmentedColormap 'prob' cdic N 640 System dynamics potential is a polynomial potential_coefs array 1 0 0 4 -5 -4 0 'd' def potential x t return polyval potential_coefs x force function is its derivative force_coefs potential_coefs arange len potential_coefs -1 -1 -1 -1 def force x t derivative of potential x return polyval force_coefs x invmass 1 0 dt 0 03 def motion t x p returns dx/dt dp/dt return p invmass -force x t cur_x -0 1 cur_p 0 def rkky_step t x_i p_i dt kx1 kp1 motion t x_i p_i dt2 0 5 dt kx2 kp2 motion t+dt2 x_i+dt2 kx1 p_i+dt2 kp1 kx3 kp3 motion t+dt2 x_i+dt2 kx2 p_i+dt2 kp2 kx4 kp4 motion t+dt x_i+dt kx3 p_i+dt kp3 newx x_i + dt/6 0 kx1 + 2 0 kx2 + 2 0 kx3 + kx4 newp p_i + dt/6 0 kp1 + 2 0 kp2 + 2 0 kp3 + kp4 return newx newp Setup ensemble points most are randomly chosen x 0 + 0 5 rand 20000 p -1 0 + 2 0 rand 20000 the pilot points are set manually x0 0; p0 0 x1 0 4; p1 0 0 pilots 0 1 pilot_colors 0 0 0 7 0 1 0 7 0 0 E potential x 0 + 0 5 invmass p 2 set up plot limits and histogram bins xedges linspace -2 1 1 7 151 pedges linspace -7 5 7 5 151 Eedges linspace -9 9 151 pix 150 extent xedges0 xedges-1 pedges-1 pedges0 H histogram2d x p bins xedges pedges 0 transpose cmax amax H 0 8 extenten xedges0 xedges-1 Eedges-1 Eedges0 Hen histogram2d x E bins xedges Eedges 0 transpose cmaxen amax Hen 0 3 fig figure 1 ysize 2 6 xsize 1 3 fig set_size_inches xsize ysize Prepare lower plot axen axes 0 2/xsize 0 2/ysize 1 0/xsize 1 0/ysize frameon True axen xaxis set_ticks axen xaxis labelpad 2 axen yaxis set_ticks axen yaxis labelpad 2 xlim -2 1 1 7 ylim -9 9 xlabel 'position x ' ylabel 'energy' potx linspace -2 1 1 7 151 Prepare upper plot ax axes 0 2/xsize 1 5/ysize 1 0/xsize 1 0/ysize frameon True ax xaxis set_ticks ax xaxis labelpad 2 ax yaxis set_ticks ax yaxis labelpad 2 xlim -2 1 1 7 ylim -7 5 7 5 xlabel 'position x ' ylabel 'momentum p ' Start running simulation frames list delays list framemod 5 frame frames/background +figformat savefig frame dpi pix frames append frame delays append 16 print generating frames 0 sys stdout flush savesteps range 0 401 framemod + 600 1000 2000 6000 delays + 10 len savesteps delays1 200 delays-5 100 200 200 200 400 totalsteps max savesteps +1 for step in range totalsteps if step 20 0 print \b\b\b\b\b 0 3 format int round step 100 0/totalsteps sys stdout flush if step in savesteps Every several frames do a plot remlist list sca ax H histogram2d x p bins xedges pedges 0 transpose remlist append imshow H extent extent cmap cm_prob interpolation 'none' aspect 'auto' remlist-1 set_clim 0 cmax for i in pilots remlist + plot xi pi ' ' color pilot_colorsi markersize 3 E potential x step dt + 0 5 invmass p 2 sca axen pot potential potx step dt remlist + plot potx pot color 'r' zorder 0 Hen histogram2d x E bins xedges Eedges 0 transpose remlist append imshow Hen extent extenten cmap cm_prob interpolation 'none' aspect 'auto' zorder 1 remlist-1 set_clim 0 cmaxen for i in pilots remlist + plot xi Ei ' ' color pilot_colorsi markersize 3 frame frames/frame +str step +figformat savefig frame dpi pix frames append frame Clear out updated stuff for r in remlist r remove x p rkky_step step dt x p dt print \b\b\b\b\b done assert len delays len frames Assemble animation using ImageMagick calllist 'convert -dispose Background' split for delay frame in zip delays frames calllist + '-delay' str delay calllist + frame calllist + '-loop 0 -layers Optimize _animation gif' split f open 'anim_command txt' 'w' f write ' ' join calllist +'\n' f close print composing into animated gif sys stdout flush subprocess call calllist print done os rename '_animation gif' 'animation gif' </source> cc-zero Uploaded with UploadWizard 1D potential wells Statistical ensemble Phase space diagrams Images with Python source code | ||||
