def investigate_autocorrelation(
lattice_sizes,
max_t: int = 10**6,
temps=None,
):
"""
Plots autocorrelation for different temperatures and sizes as a
function of time.
Parameters:
-----------
lattice_sizes: iterable
max_t: int
Maximum time to run simulations for.
temps: iterable
take_every: int
Only run for 'take_every'-th lattice size.
Returns:
--------
List $\tau_e$ , labelled by the temperature, and size.
"""
if temps is None:
temps = linspace(1.8, 3.0, 50)
time = linspace(0, max_t, 50, dtype=int)
fig, axes = plt.subplots(nrows=len(lattice_sizes),
sharex='col',
sharey='col')
tau_e, print_interval = [], int(len(temps) / 5)
for ax, l in zip(axes, tqdm(lattice_sizes)):
t_e, counter = [], 0
for temp in tqdm(temps):
s = Simulation(duration=max_t, lattice_size=l, temperature=temp)
autocorr, sigma_autocorr = generate_autocorr_data(s, time)
t_e.append(find_tau_e(autocorr, time, max_t))
if counter % print_interval == 0:
plot_autocorrelation(autocorr, sigma_autocorr, ax, s, time)
counter += 1
tau_e.append(t_e)
tau_e = array(tau_e)
plt.xlabel(r'$\tau$')
fig.set_size_inches(10.5, 10.5)
save_plot('Autocorrelation', legend_loc='upper right')
plt.errorbar(temps, mean(tau_e, axis=0), yerr=std(tau_e, axis=0), fmt='-')
plt.axvline(x=t_c)
plt.xlabel('temperature / (J / $k_B$)')
plt.ylabel(r'$\tau_e$ / time steps')
save_plot('Coherence lifetime vs. temperature')
plt.errorbar(lattice_sizes,
mean(tau_e, axis=1),
yerr=std(tau_e, axis=1),
fmt='.')
plt.xlabel('Lattice size / arb. units')
plt.ylabel(r'$\tau_e$ / time steps')
save_plot('Coherence lifetime vs. lattice size')
return tau_e
def generate_data(
temps: iter,
size=32,
hs=0,
r1=3,
r2=4,
**kwargs,
):
global use_disk
use_disk = False
simulations = [
Simulation(magnetic_field=hs,
lattice_size=size,
temperature=t,
dry_run=True) for t in temps
]
data = parmap(lambda s: multi_run(s, **kwargs),
tqdm(simulations, desc='Temperature samples'))
sigmas = array(data).T[r1]
meta_sigmas = array(data).T[r2]
if r1 == 3:
data = array([
sigma**2 / (temp**2) * size**2
for temp, sigma in zip(temps, sigmas)
])
elif r1 == 1:
data = array([
sigma**2 / (temp**2) * size**2
for temp, sigma in zip(temps, sigmas)
])
else:
data = sigmas
sigma_data = data[:] * meta_sigmas[:]
return data, sigma_data
def investigate_magnetisation_critical_transitions(lattice_sizes,
temps=None,
**kwargs):
"""
Plot the magnetisation and mean energy as a function of temperature.
Determine critical temperature by approximating the transition to a
Fermi-Dirac distribution.
Returns:
--------
List of critical temperatures.
"""
if temps is None:
temps = linspace(1.2, 3, 32)
fig, ax = plt.subplots(nrows=2, sharex='col')
ax[0].set_ylabel('magnetisation / spin')
ax[1].set_ylabel('energy per spin/ J')
ax[0].axvline(x=t_c, color='k', ls='--')
ax[1].axvline(x=t_c, color='k', ls='--')
res = []
for l in tqdm(lattice_sizes):
simulations = [
Simulation(lattice_size=l,
temperature=t,
duration=smart_duration(l, t),
dry_run=True) for t in temps
]
data = parmap(lambda s: multi_run(s, **kwargs), tqdm(simulations))
terminal_magnetisations = array(data).T[0]
# The data.T[1] holds the relative errors, not absolute ones.
sigma_magnetisations = array(data).T[1]
sigma_magnetisations = terminal_magnetisations * sigma_magnetisations
terminal_energies = array(data).T[2]
ax[0].errorbar(temps,
terminal_magnetisations,
yerr=sigma_magnetisations,
fmt='-',
label='N = ' + str(l))
ax[1].plot(temps, terminal_energies, '-', label='N =' + str(l))
popt, _ = curve_fit(magnetisation_fit, temps, terminal_magnetisations)
res.append(popt[0])
plt.xlabel('temperature / J')
plt.xticks(
append(linspace(min(temps), t_c, 3), linspace(t_c, max(temps), 3)))
save_plot('Critical temperature from magnetisation')
return res
def investigate_chi(temps, sizes):
"""
Produce a plot of magnetic susceptibility vs temperature.
Returns list of critical transition temperatures.
"""
plot_magnetic_response(temps)
t_cs = []
for size in tqdm(sizes):
chi_data = array([chi(t, size) for t in tqdm(temps)]).T
chis, chierrs = chi_data[0], chi_data[1]
plt.errorbar(temps, chis, yerr=chierrs, label=str(size), fmt='+')
t_cs.append(temps[argmax(chis)])
plt.axvline(x=t_c)
plt.ylabel('susceptibility / spin')
plt.xlabel('temperature / (J/$k_B$)')
save_plot('Susceptibility vs Temperature')
return t_cs
def investigate_heat_capacity(lattice_sizes, temps=None, skip=3, **kwargs):
"""
Plot the heat capacity versus temperature for different lattice sizes.
Find the critical temperature for each size by doing a spline smoothing
and finding the peak.
Parameters:
-----------
lattice_sizes: iterable
temps: iterable
**kwargs: keyword_arguments
Arguments to be passed to multi_run()
Returns:
List of critical temperatures.
"""
if temps is None:
temps = append(linspace(1.6, 4.25, 12), linspace(1.8, 2.8, 24))
temps.sort()
fig, _ = plt.subplots()
crit_temps, counts = [], 0
for l in tqdm(lattice_sizes, desc='Lattice sizes'):
for t in temps:
pass
caps, cap_errs = generate_data(
temps,
l,
**kwargs,
)
up_temps, up_caps = upscale(caps, temps)
main_peak = argmax(up_caps)
crit_temps.append(up_temps[main_peak])
counts += 1
if counts % skip == 0:
plt.errorbar(temps,
caps,
fmt='+',
yerr=cap_errs,
label='N = ' + str(l))
plt.plot(
up_temps[main_peak],
up_caps[main_peak],
'bo',
)
plt.plot(up_temps, up_caps, label='N = ' + str(l) + ' smoothed')
plt.xticks(
append(linspace(min(temps), t_c, 3),
linspace(t_c, max(temps), 4)))
plt.ylabel(r'$C_V/ k_B$')
plt.axvline(x=t_c, ls='-.', color='k')
fig.set_size_inches([10.5, 10.5])
plt.xlabel('temperature / $J/ k_B$')
save_plot('Heat capacity', )
return crit_temps
def plot_magnetic_response(temps):
"""Plot magnetisation vs H, for different temperatures."""
counts = 0
p = int(len(temps) / 6)
for t in tqdm(temps):
chi(t, sz=64, plot_q=(counts % p == 0))
counts += 1
plt.ylabel('Magnetisation / spin')
plt.xlabel('Applied field / (J/$\mu$)')
save_plot('Magnetisation vs. external field')
def investigate_magnetic_fluctuations(hs, temps=None, skip=3, **kwargs):
"""
Plot the magnetic susceptibility versus temperature for different
lattice sizes. Find the critical temperature for each H by spline
interpolation and finding the maximum.
Parameters:
-----------
hs: iterable
temps: iterable
**kwargs: keyword_arguments
Arguments to be passed to multi_run()
Returns:
List of critical temperatures.
"""
if temps is None:
temps = append(linspace(1.6, 3.9, 12), linspace(1.8, 2.8, 24))
temps.sort()
fig, _ = plt.subplots()
crit_temps, counts = [], 0
for h in tqdm(hs, desc='External fields.'):
# get upscaled and smoothed capacity vs temperature data.
mags, _ = generate_data(temps, hs=h, r1=1, r2=1, **kwargs)
up_temps, up_mags = upscale(mags, temps)
main_peak = argmax(up_mags)
plt.plot(
up_temps[main_peak],
up_mags[main_peak],
'bo',
)
crit_temps.append(up_temps[main_peak])
counts += 1
if counts % skip == 0:
plt.plot(temps, mags, 'b+', label=r'$\vec{H}$ = ' + str(h))
plt.plot(up_temps, up_mags, label='N = ' + str(h) + ' smoothed')
plt.ylabel(r'$\chi/ (\mu/J)')
plt.axvline(x=t_c, ls='-.', color='k')
plt.xticks(
append(linspace(min(temps), t_c, 3),
linspace(t_c, max(temps), 6)))
fig.set_size_inches(8.5, 8.5)
fig.tight_layout()
plt.xlabel('temperature / (J/$k_B$')
save_plot('Magnetic susceptibility')
return crit_temps