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Ringkasan
| Keterangan |
English: Pressure of classical ideal gas and quantum ideal gases (Fermi gas, Bose gas) as a function of temperature, for a fixed density of particles. This is for the case of non-relativistic (massive, slow) particles in three dimensions.
Русский: Давление классического и квантовых газов (Ферми и Бозе) в зависимости от температуры.
A few features can be seen:
The figure has been scaled in a way that the particle degeneracy factor, density, mass, etc. are all factored out and irrelevant. See also Quantum ideal gas entropy 3d.svg and Quantum ideal gas chemical potential 3d.svg. |
| Tarikh | |
| Sumber | Karya sendiri |
| Pengarang | Nanite |
| Versi lain |
|
| SVG genesis |  This plot was created with Matplotlib. |
| Kod sumber | Python code#!/usr/bin/env python3
import numpy as np
from matplotlib import pyplot as plt
import mpmath
import sys
def fixpoly(s,x):
# The mpmath polylog sometimes returns a tiny spurious imaginary part, and
# it throws exceptions for weird cases. Clean this up.
try:
return np.real(mpmath.fp.polylog(s,x))
except (OverflowError, ValueError):
return np.nan
polylog = np.vectorize(fixpoly, otypes=[float])
# assumed density of states G(E) = V * g_0 * (E - E_0) ^ (alpha - 1) / Gamma(alpha)
# as appropriate for 3D, nonrelativistic, massive particles.
# The prefactor g_0 includes things like spin degeneracy, mass, planck constant, factors of pi, etc.
# V is volume of the system (where applicable - for particles in harmonic well,
# just make sure V*g_0 is the right value).
# We will only plot things that are independent of V and g_0, and we assume E_0 = 0.
# The key parameter in this density of states is alpha.
# for massive particle in a box, alpha = dimensionality/2
# for particle in a harmonic well, alpha = dimensionality
# for hyperrelativistic particles in a box, alpha = dimensionality
if len(sys.argv) > 1:
# allow massive-particle-in-box dimensionality to be provided as command line arg
alpha = float(sys.argv[1]) / 2
else:
# default to 3D massive case:
alpha = 1.5
# Both Fermi gas and Bose gas are easily calculated in grand canonical ensemble.
# Fermi gas has grand potential given by:
# Omega = - V * g_0 * (kT)^(alpha + 1) * fdint(alpha, mu/kT)
# where fdint(s, t) is the complete fermi-dirac integral,
# fdint(s, t) = -polylog(s + 1, -exp(t))
# Bose gas has grand potential:
# Omega = - V * g_0 * (kT)^(alpha + 1) * beint(alpha, mu/kT)
# (only valid for mu <= 0) where beint(s, t) is
# beint(s, t) = +polylog(s + 1, +exp(t))
# The classical ideal gas is in-between these, with:
# Omega = - V * g_0 * (kT)^(alpha + 1) * exp(mu/kT)
# Quantities:
# Pressure P = -d(Omega)/d(V) = -Omega/V
# Particle number N = -d(Omega)/d(mu)
# Entropy S = -d(Omega)/d(T)
# Note that:
# d/dt polylog(s, exp(t)) = polylog(s-1, exp(t))
#
# So these derivatives are easy to compute, e.g., for Fermi:
# N = -V * g_0 * (kT)^(alpha) * fdint(alpha - 1 , mu/kT)
# S = k * N * ( (alpha + 1) * fdint(alpha, mu/kT) / fdint(alpha-1, mu/kT) - mu/kT )
# ( .. likewise substituting beint or exp for the bose/classical cases .. )
#
# Unfortunately, there is no analytic formula for mu in terms of N, only the other way.
# This is unfortunate since we want to plot temperature-dependence of a system
# with fixed N and fixed V.
# However we can see that both P and N are functions of kT and mu/kT. So, we can
# sweep the value of mu/kT, and then at each point, choose T such that N has the
# desired value.
# To nondimensionalize, we will use a characteristic temperature scale T' based on
# the fixed number N', using the classical case for mu=0:
# g_0 (kT')^(alpha) = N'/V
# Roughly speaking, this is the temperature at which the thermal de broglie
# wavelength is comparable to the distance between identical particles.
# Likewise, a characteristic pressure to match this temperature and density:
# P' = N' k T' / V
# = g_0 (kT')^(alpha + 1)
# With T' and P' in hand, we can rescale all our formulas to be independent of
# g_0, N', and volume, and also do our mu-scanning trick.
# e.g., in the fermi case we get:
# (T/T')^(alpha) = 1/fdint(alpha - 1, mu/kT)
# P/P' = (T/T')^(alpha + 1) * fdint(alpha, mu/kT)
# Variables used below:
# z = exp(mu/kT)
# T refers to T/T', where T' defined above.
# P refers to P/P', where P' defined above.
# S refers to S/(N.k)
# mu refers to mu/(kT')
# Pressure vs temperature graph
fig1 = plt.figure()
fig1.set_size_inches(3,2.3)
ax1 = plt.axes((0.09, 0.17, 0.90, 0.82))
# Entropy vs temperature graph
fig2 = plt.figure()
fig2.set_size_inches(3,2.3)
ax2 = plt.axes((0.18, 0.17, 0.81, 0.82))
# Chemical potential vs temperature graph
fig3 = plt.figure()
fig3.set_size_inches(3,2.3)
ax3 = plt.axes((0.15, 0.17, 0.84, 0.82))
# Fermi gas
color_fermi = '#1f77b4'
T_fermi = mpmath.fp.gamma(alpha+1) ** (1./alpha)
P_fermi = T_fermi / (alpha+1)
# sweep z; make the last point to be basically T=0
z = np.exp(np.linspace(-2, 20, 201))
z[-1] = 1e100
T = (-polylog(alpha, -z)) ** (-1./alpha)
P = (T)**(alpha + 1) * -polylog(alpha + 1, -z)
S = (alpha+1) * polylog(alpha + 1, -z)/polylog(alpha, -z) - np.log(z)
mu = np.log(z) * T
# extend traces to exactly T=0
T = np.concatenate((T, [0.]))
P = np.concatenate((P, [P_fermi]))
S = np.concatenate((S, [0.]))
mu = np.concatenate((mu, [T_fermi]))
ax1.plot(T,P, label="Fermi", color=color_fermi)
ax2.plot(T,S, label="Fermi", color=color_fermi)
ax3.plot(T,mu, label="Fermi", color=color_fermi)
# Indicate fermi temperature
P_at_T_fermi = np.interp(T_fermi, T[::-1], P[::-1])
S_at_T_fermi = np.interp(T_fermi, T[::-1], S[::-1])
mu_at_T_fermi = np.interp(T_fermi, T[::-1], mu[::-1])
ax1.plot([T_fermi, T_fermi], [P_at_T_fermi, -100 ], color=color_fermi, lw=1, ls=(0, (1,5)))
ax2.plot([T_fermi, T_fermi], [S_at_T_fermi, -100 ], color=color_fermi, lw=1, ls=(0, (1,5)))
ax3.plot([T_fermi, T_fermi], [mu_at_T_fermi, -100 ], color=color_fermi, lw=1, ls=(0, (1,5)))
# Ideal gas -- this is just a straight line T=P but for consistency, calculate
# it similarly to the bose and fermi cases.
color_classical = '#ff7f0e'
z = np.exp(np.linspace(-2, 20, 200))
T = (z) ** (-1./alpha)
P = (T)**(alpha + 1) * z
S = (alpha+1) - np.log(z)
mu = np.log(z) * T
# extend traces to exactly T=0
T = np.concatenate((T, [0.]))
P = np.concatenate((P, [0.]))
S = np.concatenate((S, [-np.inf]))
mu = np.concatenate((mu, [0.]))
ax1.plot(T,P, label="classical", color=color_classical)
ax2.plot(T,S, label="classical", color=color_classical)
ax3.plot(T,mu, label="classical", color=color_classical)
# Bose gas
color_bose = '#2ca02c'
# Approach mu=0 from below, making sure to include the last floating point
# number smaller than 1.
z = np.concatenate((np.linspace(0.01, 0.99, 99), [0.999, 0.9999, 1 - 1e-16]))
#z = 1 - np.exp(np.linspace(-0.1, -34, 200))
#z = np.exp(np.linspace(-2, -1e-15, 100))
T = (polylog(alpha, z)) ** (-1./alpha)
P = (T)**(alpha + 1) * polylog(alpha + 1, z)
S = (alpha+1) * polylog(alpha + 1, z)/polylog(alpha, z) - np.log(z)
mu = np.log(z) * T
if alpha > 1:
# In >2 dimensions, the bose gas starts condensing at nonzero temperature,
# right at the point when mu=0.
Tcrit = (polylog(alpha, 1)) ** (-1./alpha)
Pcrit = (Tcrit)**(alpha + 1) * polylog(alpha + 1, 1)
Scrit = (alpha+1) * polylog(alpha + 1, 1)/polylog(alpha, 1)
# What about T < Tcrit?
# Basically now mu is pinned to 0. It cannot go any higher because that would
# mean infinite particles in every state with energy below mu.
# Instead as temperature lowers, mu stays at 0 and the particle number in the
# continuum of states with energy above mu will drop accordingly.
# The continuum is called the 'noncondensed phase'; the particles that
# have disappeared have all necessarily gone into some lowest-energy state
# that is infinitesimally above mu, the 'condensed phase'.
# So, let's set mu=0, and drop our constraint on N, and calculate the pressure
# from the continuum phase just as before. (the condensed phase contributes
# no pressure, in macroscopic ideal gas).
T2 = np.linspace(Tcrit, 0, 101)
P2 = T2 ** (alpha + 1) * polylog(alpha + 1, 1)
# In the case of entropy "S/Nk", we have to be careful since now the total N' is
# distinct from our continuum N (excited).
S2 = T2 ** alpha * (alpha+1) * polylog(alpha + 1, 1)
mu2 = 0*T2
# concatenate to existing traces
T = np.concatenate((T, T2))
P = np.concatenate((P, P2))
S = np.concatenate((S, S2))
mu = np.concatenate((mu, mu2))
# Mark the critical temperature with a *
ax1.plot([Tcrit],[P2[0]], '*', color='k', ms=12, mew=0, alpha=0.6, zorder=2.5)
ax2.plot([Tcrit],[S2[0]], '*', color='k', ms=12, mew=0, alpha=0.6, zorder=2.5)
ax3.plot([Tcrit],[0], '*', color='k', ms=12, mew=0, alpha=0.6, zorder=2.5)
ax1.plot([Tcrit, Tcrit], [P2[0], -100 ], color=color_bose, lw=1, ls=(0, (4,2)))
ax2.plot([Tcrit, Tcrit], [S2[0], -100 ], color=color_bose, lw=1, ls=(0, (4,2)))
ax3.plot([Tcrit, Tcrit], [0, -100 ], color=color_bose, lw=1, ls=(0, (4,2)))
else:
# extend traces to 0
T = np.concatenate((T, [0.]))
P = np.concatenate((P, [0.]))
S = np.concatenate((S, [0.]))
mu = np.concatenate((mu, [0.]))
ax1.plot(T,P, label="Bose", color=color_bose)
ax2.plot(T,S, label="Bose", color=color_bose)
ax3.plot(T,mu, label="Bose", color=color_bose)
# format temperature axis nicely
for ax in [ax1, ax2, ax3]:
ax.set_xlim(0,1.8)
ax.set_xlabel('temperature', labelpad=0)
tl = []
tl.append((0, "0", 'k'))
if alpha > 1:
tl.append((Tcrit, r"$T_{\rm B}$", color_bose))
tl.append((T_fermi, r"$T_{\rm F}$", color_fermi))
ticks, labels, colors = zip(*tl)
ax.set_xticks(ticks)
ax.set_xticklabels(labels)
for label,color in zip(ax.xaxis.get_ticklabels(), colors):
label.set_color(color)
ax1.set_ylim(0,1.8)
ax1.set_yticks([0])
ax1.set_ylabel('pressure', labelpad=-5)
ax1.legend(loc='lower right')
fig1.savefig('quantum ideal gas pressure %gd.svg'%(alpha*2,))
yrange = 0.2 + 1 + (1 + np.log(1.8))*alpha # range to fit the right side
ax2.set_ylim(-0.4*yrange, +yrange)
ax2.set_ylabel('entropy per particle $S/Nk$')
ax2.legend(loc='lower right')
fig2.savefig('quantum ideal gas entropy %gd.svg'%(alpha*2,))
yrange = 1.5 + 0.7*alpha
ax3.set_ylim(-yrange, +0.6*yrange)
ax3.set_yticks([0])
ax3.set_yticklabels([r'$\varepsilon_0$'])
ax3.set_ylabel('chemical potential $\\mu$')
ax3.legend(loc='lower left')
fig3.savefig('quantum ideal gas chemical potential %gd.svg'%(alpha*2,))
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| Tarikh/Waktu | Gambar kenit | Ukuran | Pengguna | Komen | |
|---|---|---|---|---|---|
| semasa | 19:45, 19 Januari 2021 | | 270 × 207 (19 KB) | Nanite | add bose-critical and fermi temperature ticks |
| 05:16, 18 Januari 2021 |  | 270 × 207 (20 KB) | Nanite | fill closer to margins; mark origin as 0,0 | |
| 01:23, 23 Jun 2020 |  | 270 × 207 (18 KB) | Nanite | Uploaded own work with UploadWizard |
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