File:Ensemble classical 1DOF canonical.png
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![File:Ensemble classical 1DOF canonical.png](http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Ensemble_classical_1DOF_canonical.png/400px-Ensemble_classical_1DOF_canonical.png)
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Summary
DescriptionEnsemble classical 1DOF canonical.png |
English: Ensemble canonically distributed over energy, for a classical system consisting of one particle in a potential well. |
Date | |
Source | Own work |
Author | Nanite |
Source
![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/01/Created_with_Matplotlib-logo.svg/22px-Created_with_Matplotlib-logo.svg.png)
This plot was created with Matplotlib.
Python source code. Requires matplotlib.
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 = x[1] - x[0]
mass = 1.0
p = linspace(-15,15,2001) ; dp = p[1] - p[0]
psextent = (x[0]-0.5*dx, x[-1]+0.5*dx, p[0]-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*(Evals[1]-Evals[0]))[:-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 = (xcorners[0], xcorners[-1], Evals[0], 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().axes[0])
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().axes[1])
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().axes[0])
axhspan(micro_e0, micro_e1, color=(0.7,1,0.7),zorder=-2)
sca(gcf().axes[1])
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)
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.
http://creativecommons.org/publicdomain/zero/1.0/deed.enCC0Creative Commons Zero, Public Domain Dedicationfalsefalse |
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30 October 2013
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current | 21:51, 30 October 2013 | ![]() | 900 × 1,350 (174 KB) | Nanite | User created page with UploadWizard |
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