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| Keywords: Ensemble classical 1DOF canonical.png en Ensemble canonically distributed over energy for a classical system consisting of one particle in a potential well 2013-10-30 22 45 06 own Nanite Source Created with Matplotlib Python source code Requires matplotlib <source lang python > from pylab import figformat ' png' saveopts 'dpi' 300 'transparent' True 'frameon' True 'bbox_inches' 'tight' seterr divide 'ignore' temp_canonical 4 1 energy_microcanonical -2 0 range_microcanonical 1 0 micro_e0 energy_microcanonical - 0 5 range_microcanonical micro_e1 energy_microcanonical + 0 5 range_microcanonical def potential x return x 6 + 4 x 3 - 5 x 2 - 4 x x linspace -2 5 2 5 2001 ; dx x1 - x0 mass 1 0 p linspace -15 15 2001 ; dp p1 - p0 psextent x0-0 5 dx x-1+0 5 dx p0-0 5 dp p-1+0 5 dp compute pixel edges used for pcolormesh xcorners zeros len x +1 xcorners len x x-0 5 dx xcorners-1 x-1 + 0 5 dx X P meshgrid x p E potential X + P 2/ 2 mass Hamiltonian make an energy range for plots vs energy Evals arange -8 10 0 1 phaseV array list sum E < Elim for Elim in Evals Evals2 Evals + 0 5 Evals1-Evals0 -1 phaseDOS diff phaseV also figure out the density of states function in position-energy xvals list phasesump array list sum E < Elim axis 0 for Elim in Evals phasedosp diff phasesump axis 0 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 def energyplot phaseDOS_E phaseDOS phasedosp ensemble doslighten 1 0 ensemblelighten 1 0 Plot the potential with density of states on sidebar Evals phaseDOS list of energies and DOS to plot on right panel fig figure energy-position plot ax axes 0 08 0 06 0 73 0 43 plot x potential x linewidth 2 color 'r' zorder 1 extent xcorners0 xcorners-1 Evals0 Evals-1 img imshow phasedosp cmap cm_prob extent extent interpolation 'none' aspect 'auto' origin 'lower' zorder 0 clim 0 amax phasedosp doslighten ax xaxis labelpad 2 ax yaxis labelpad -3 xlabel position x ylabel energy ax xaxis set_ticklabels ax yaxis set_ticklabels ylim -9 9 xlim -2 1 1 7 ax xaxis set_ticks -2 -1 0 1 density of states sidebar ax axes 0 83 0 06 0 14 0 43 axisbg 0 95 0 95 0 95 xlabel states ax xaxis set_ticks ax yaxis set_ticklabels ax yaxis set_ticks_position 'right' ylim -9 9 fill_betweenx phaseDOS_E phaseDOS linewidth 0 color 0 5 0 5 0 85 xlim -0 05 max phaseDOS max phaseDOS 1 1 phase space plot ax axes 0 08 0 50 0 73 0 455 img imshow ensemble cmap cm_prob extent psextent interpolation 'none' aspect 'auto' origin 'lower' zorder 0 clim 0 amax ensemble ensemblelighten ax xaxis labelpad 4 ax xaxis set_label_position 'top' ax xaxis set_ticklabels ax yaxis set_ticks ax xaxis set_ticks_position 'both' ax yaxis labelpad 0 xlabel position x ylabel momentum p ylim -7 5 7 5 xlim -2 1 1 7 ax xaxis set_ticks -2 -1 0 1 fig set_size_inches 3 4 5 fig patch set_alpha 0 allensemble E > -999 0 viewensemble E < 9 0 energyplot Evals2 phaseDOS phasedosp allensemble doslighten 0 8 ensemblelighten 16 0 savefig class_potential +figformat saveopts canonical phase space image canonical exp -E/temp_canonical print canonical T temp_canonical avg energy canonical_avgE sum E canonical /sum canonical print canonical_avgE energyplot Evals2 phaseDOS exp -Evals2/temp_canonical phasedosp exp -Evals2/temp_canonical newaxis canonical doslighten 0 3 sca gcf axes0 annotate \\langle E\\rangle -0 5 canonical_avgE textcoords None verticalalignment 'top' color 0 0 4 0 axhline canonical_avgE linestyle 'dotted' linewidth 1 color 0 0 4 0 annotate 1 2 7 -temp_canonical 1 2 7 arrowprops 'arrowstyle' '<->' text 1 15 7 -0 5 temp_canonical ' kT ' horizontalalignment 'right' verticalalignment 'center' sca gcf axes1 axhline canonical_avgE linestyle 'dotted' linewidth 1 color 0 0 4 0 savefig class_canonical_potential +figformat saveopts micro E < micro_e1 E > micro_e0 print microcanonical E0 energy_microcanonical Delta 0 5 range_microcanonical avg energy print sum E micro /sum micro tmp Evals2 < micro_e1 Evals2 > micro_e0 energyplot Evals2 phaseDOS tmp phasedosp tmp newaxis micro doslighten 0 5 ensemblelighten 3 0 sca gcf axes0 axhspan micro_e0 micro_e1 color 0 7 1 0 7 zorder -2 sca gcf axes1 axhspan micro_e0 micro_e1 color 0 7 1 0 7 zorder -2 savefig class_microcanonical_potential +figformat saveopts Position expectation values fig figure plot x sum micro axis 0 /float sum micro /dx label 'microcanonical' plot x sum canonical axis 0 /sum canonical /dx label 'canonical' xlim -2 1 1 7 fig get_axes 0 xaxis set_ticks -2 -1 0 1 xlabel position x ylabel PDF of position P x legend fig set_size_inches 4 4 fig patch set_alpha 0 savefig class_position_pdf +figformat saveopts Momentum expectation values fig figure plot p sum micro axis 1 /float sum micro /dp label 'microcanonical' plot p sum canonical axis 1 /sum canonical /dp label 'canonical' xlim -7 5 7 5 xlabel momentum p ylabel PDF of momentum P p legend fig set_size_inches 4 4 fig patch set_alpha 0 savefig class_momentum_pdf +figformat saveopts </source> cc-zero Uploaded with UploadWizard 1D potential wells Statistical ensemble Phase space diagrams Images with Python source code | ||||
