def calc_variables():
"""Calculates all variables of interest.
They are collected and returned as a list, for use in the main program.
"""
# In this example we simulate using the cut (but not shifted) potential
# but we only report results which have had the long-range corrections applied
# The value of the cut-and-shifted potential is not used, in this example
import numpy as np
import math
from averages_module import VariableType
from lrc_module import potential_lrc, pressure_lrc
# Preliminary calculations (n,r,total are taken from the calling program)
vol = box**3 # Volume
rho = n / vol # Density
kin = 1.5 * n * p * temperature # Average kinetic energy for NP-atom system
kin_q = kin - total_spr # Quantum estimator for kinetic energy
rad_g = rad_gyr(r)
# Variables of interest, of class VariableType, containing three attributes:
# .val: the instantaneous value
# .nam: used for headings
# .method: indicating averaging method
# If not set below, .method adopts its default value of avg
# The .nam and some other attributes need only be defined once, at the start of the program,
# but for clarity and readability we assign all the values together below
# Acceptance ratio of atomic moves
r_r = VariableType(nam='Atomic move ratio', val=r_ratio, instant=False)
# Acceptance ratio of centre-of-mass moves
c_r = VariableType(nam='COM move ratio', val=c_ratio, instant=False)
# Internal energy per atom for full potential with LRC
# LRC plus cut (but not shifted) PE already divided by factor P
# plus KE estimator: total classical KE for NP-atom system MINUS total spring potential
# all divided by N
e_f = VariableType(nam='E/N full',
val=potential_lrc(rho, r_cut) + (kin_q + total.pot) / n)
# Kinetic energy per atom, just for interest's sake
k_q = VariableType(nam='KE/N', val=kin_q / n)
# Pressure for full potential with LRC
# LRC plus ideal gas contribution plus total virial divided by V
kin_q = kin_q / 1.5 # Convert KE estimator to kinetic energy part of pressure
p_f = VariableType(nam='P full',
val=pressure_lrc(rho, r_cut) +
(kin_q + total.vir) / vol)
# Quantum spring energy per atom, just for interest's sake
e_q = VariableType(nam='Espring/N', val=total_spr / n)
# Quantum polymer radius of gyration, just for interest's sake
r_g = VariableType(nam='Radius of gyration', val=rad_g)
# Collect together into a list for averaging
return [r_r, c_r, e_f, p_f, e_q, k_q, r_g]
def calc_variables():
"""Calculates all variables of interest.
They are collected and returned as a list, for use in the main program.
"""
# In this example we simulate using the cut (but not shifted) potential
# The values of < p_c >, < e_c > and < density > should be consistent (for this potential)
# For comparison, long-range corrections are also applied to give
# estimates of < e_f > and < p_f > for the full (uncut) potential
# The value of the cut-and-shifted potential is not used, in this example
import numpy as np
import math
from averages_module import msd, VariableType
from lrc_module import potential_lrc, pressure_lrc, pressure_delta
from mc_lj_module import force_sq
# Preliminary calculations (n,r,total are taken from the calling program)
vol = box**3 # Volume
rho = n / vol # Density
fsq = force_sq(box, r_cut, r) # Total squared force
# Variables of interest, of class VariableType, containing three attributes:
# .val: the instantaneous value
# .nam: used for headings
# .method: indicating averaging method
# If not set below, .method adopts its default value of avg
# The .nam and some other attributes need only be defined once, at the start of the program,
# but for clarity and readability we assign all the values together below
# Move, creation, and destruction acceptance ratios
m_r = VariableType(nam='Move ratio', val=m_ratio, instant=False)
c_r = VariableType(nam='Create ratio', val=c_ratio, instant=False)
d_r = VariableType(nam='Destroy ratio', val=d_ratio, instant=False)
# Number
number = VariableType(nam='N', val=float(n))
# Density
density = VariableType(nam='Density', val=rho)
# Internal energy per atom for simulated, cut, potential
# Ideal gas contribution plus cut (but not shifted) PE divided by N
e_c = VariableType(nam='E/N cut', val=1.5 * temperature + total.pot / n)
# Internal energy per atom for full potential with LRC
# LRC plus ideal gas contribution plus cut (but not shifted) PE divided by N
e_f = VariableType(nam='E/N full',
val=potential_lrc(rho, r_cut) + 1.5 * temperature +
total.pot / n)
# Pressure for simulated, cut, potential
# Delta correction plus ideal gas contribution plus total virial divided by V
p_c = VariableType(nam='P cut',
val=pressure_delta(rho, r_cut) + rho * temperature +
total.vir / vol)
# Pressure for full potential with LRC
# LRC plus ideal gas contribution plus total virial divided by V
p_f = VariableType(nam='P full',
val=pressure_lrc(rho, r_cut) + rho * temperature +
total.vir / vol)
# Configurational temperature
# Total squared force divided by total Laplacian
t_c = VariableType(nam='T config', val=fsq / total.lap)
# Number MSD
n_msd = VariableType(nam='N MSD', val=float(n), method=msd, instant=False)
# Collect together into a list for averaging
return [m_r, c_r, d_r, number, density, e_c, p_c, e_f, p_f, t_c, n_msd]
def calc_variables ( ):
"""Calculates all variables of interest.
They are collected and returned as a list, for use in the main program."""
import numpy as np
import math
from averages_module import msd, cke, VariableType
from lrc_module import potential_lrc, pressure_lrc
from md_lj_module import hessian
# Preliminary calculations (n,r,v,f,total are taken from the calling program)
vol = box**3 # Volume
rho = n / vol # Density
kin = 0.5*np.sum(v**2) # Kinetic energy
tmp = 2.0 * kin / (3*n-3) # Remove three degrees of freedom for momentum conservation
fsq = np.sum ( f**2 ) # Total squared force
hes = hessian(box,r_cut,r,f) # Total Hessian
eng = kin + total.pot # Total energy for simulated system
# Variables of interest, of class VariableType, containing three attributes:
# .val: the instantaneous value
# .nam: used for headings
# .method: indicating averaging method
# If not set below, .method adopts its default value of avg
# The .nam and some other attributes need only be defined once, at the start of the program,
# but for clarity and readability we assign all the values together below
# Internal energy (cut-and-shifted) per atom
# Total KE plus total cut-and-shifted PE divided by N
e_s = VariableType ( nam = 'E/N cut&shifted', val = eng/n )
# Internal energy (full, including LRC) per atom
# LRC plus total KE plus total cut (but not shifted) PE divided by N
e_f = VariableType ( nam = 'E/N full', val = potential_lrc(rho,r_cut) + (kin+total.cut)/n )
# Pressure (cut-and-shifted)
# Ideal gas contribution plus total virial divided by V
p_s = VariableType ( nam = 'P cut&shifted', val = rho*tmp + total.vir/vol )
# Pressure (full, including LRC)
# LRC plus ideal gas contribution plus total virial divided by V
p_f = VariableType ( nam = 'P full', val = pressure_lrc(rho,r_cut) + rho*tmp + total.vir/vol )
# Kinetic temperature
t_k = VariableType ( nam = 'T kinetic', val = tmp )
# Configurational temperature
# Total squared force divided by total Laplacian with small Hessian correction
t_c = VariableType ( nam = 'T config', val = fsq/(total.lap-(2.0*hes/fsq)) )
# MSD of kinetic energy, intensive
# Use special method to convert to Cv/N
c_s = VariableType ( nam = 'Cv/N cut&shifted', val = kin/math.sqrt(n), method = cke, instant = False )
# Mean-squared deviation of conserved energy per atom
conserved_msd = VariableType ( nam = 'Conserved MSD', val = eng/n,
method = msd, e_format = True, instant = False )
# Collect together into a list for averaging
return [ e_s, p_s, e_f, p_f, t_k, t_c, c_s, conserved_msd ]
def calc_variables():
"""Calculates all variables of interest.
They are collected and returned as a list, for use in the main program.
"""
# In this example we simulate using the cut (but not shifted) potential
# Accordingly, < p_c > should match the input pressure and the values
# of < p_c >, < e_c > and density should be consistent (for this potential)
# For comparison, long-range corrections are also applied to give
# estimates of < e_f > and < p_f > for the full (uncut) potential
# The value of the cut-and-shifted potential is not used, in this example
import numpy as np
import math
from averages_module import msd, VariableType
from lrc_module import potential_lrc, pressure_lrc, pressure_delta
from mc_lj_module import force_sq
# Preliminary calculations (n,r,total are taken from the calling program)
vol = box**3 # Volume
rho = n / vol # Density
fsq = force_sq(box, r_cut, r) # Total squared force
# Variables of interest, of class VariableType, containing three attributes:
# .val: the instantaneous value
# .nam: used for headings
# .method: indicating averaging method
# If not set below, .method adopts its default value of avg
# The .nam and some other attributes need only be defined once, at the start of the program,
# but for clarity and readability we assign all the values together below
# Move and volume acceptance ratios
m_r = VariableType(nam='Move ratio', val=m_ratio, instant=False)
v_r = VariableType(nam='Volume ratio', val=v_ratio, instant=False)
# Density
density = VariableType(nam='Density', val=rho)
# Internal energy per atom for simulated, cut, potential
# Ideal gas contribution plus cut (but not shifted) PE divided by N
e_c = VariableType(nam='E/N cut', val=1.5 * temperature + total.pot / n)
# Internal energy per atom for full potential with LRC
# LRC plus ideal gas contribution plus cut (but not shifted) PE divided by N
e_f = VariableType(nam='E/N full',
val=potential_lrc(rho, r_cut) + 1.5 * temperature +
total.pot / n)
# Pressure for simulated, cut, potential
# Delta correction plus ideal gas contribution plus total virial divided by V
p_c = VariableType(nam='P cut',
val=pressure_delta(rho, r_cut) + rho * temperature +
total.vir / vol)
# Pressure for full potential with LRC
# LRC plus ideal gas contribution plus total virial divided by V
p_f = VariableType(nam='P full',
val=pressure_lrc(rho, r_cut) + rho * temperature +
total.vir / vol)
# Configurational temperature
# Total squared force divided by total Laplacian
t_c = VariableType(nam='T config', val=fsq / total.lap)
# Heat capacity (cut but not shifted)
# MSD of excess "enthalpy" divided by temperature and sqrt(N) to make result intensive
# NB this is not really the excess Cp/NkB, it simply omits the kinetic energy fluctuations
# i.e. we add the ideal gas part of Cv/NkB, 1.5, to get total Cp/NkB
enp = total.pot + pressure * vol
c_c = VariableType(nam='Cp/N cut',
val=enp / (temperature * math.sqrt(n)),
method=msd,
add=1.5,
instant=False)
# Heat capacity (full)
# MSD of excess "enthalpy" divided by temperature and sqrt(N) to make result intensive
# NB this is not really the excess Cp/NkB, it simply omits the kinetic energy fluctuations
# i.e. we add the ideal gas part of Cv/NkB, 1.5, to get total Cp/NkB
enp = n * potential_lrc(rho, r_cut) + total.pot + pressure * vol
c_f = VariableType(nam='Cp/N full',
val=enp / (temperature * math.sqrt(n)),
method=msd,
add=1.5,
instant=False)
# Volume MSD
vol_msd = VariableType(nam='Volume MSD',
val=vol,
method=msd,
instant=False)
# Collect together into a list for averaging
return [m_r, v_r, density, e_c, p_c, e_f, p_f, t_c, c_c, c_f, vol_msd]
def calc_variables():
"""Calculates all variables of interest.
They are collected and returned as a list, for use in the main program.
"""
import numpy as np
import math
from averages_module import msd, VariableType
from lrc_module import potential_lrc, pressure_lrc
# Preliminary calculations (n,r,total are taken from the calling program)
vol = box**3 # Volume
rho = n / vol # Density
fsq = np.sum(f**2) # Total squared force
# Variables of interest, of class VariableType, containing three attributes:
# .val: the instantaneous value
# .nam: used for headings
# .method: indicating averaging method
# If not set below, .method adopts its default value of avg
# The .nam and some other attributes need only be defined once, at the start of the program,
# but for clarity and readability we assign all the values together below
# Move acceptance ratio
m_r = VariableType(nam='Move ratio', val=m_ratio, instant=False)
# Internal energy per atom for simulated, cut-and-shifted, potential
# Ideal gas contribution plus total cut-and-shifted PE divided by N
e_s = VariableType(nam='E/N cut&shifted',
val=1.5 * temperature + total.pot / n)
# Internal energy per atom for full potential with LRC
# LRC plus ideal gas contribution plus total cut (but not shifted) PE divided by N
e_f = VariableType(nam='E/N full',
val=potential_lrc(rho, r_cut) + 1.5 * temperature +
total.cut / n)
# Pressure for simulated, cut-and-shifted, potential
# Ideal gas contribution plus total virial divided by V
p_s = VariableType(nam='P cut&shifted',
val=rho * temperature + total.vir / vol)
# Pressure for full potential with LRC
# LRC plus ideal gas contribution plus total virial divided by V
p_f = VariableType(nam='P full',
val=pressure_lrc(rho, r_cut) + rho * temperature +
total.vir / vol)
# Configurational temperature
# Total squared force divided by total Laplacian
t_c = VariableType(nam='T config', val=fsq / total.lap)
# Heat capacity (excess, cut-and-shifted)
# Total PE divided by temperature and sqrt(N) to make result intensive
# We add ideal gas contribution, 1.5, afterwards
c_s = VariableType(nam='Cv/N cut&shifted',
val=total.pot / (temperature * math.sqrt(n)),
method=msd,
add=1.5,
instant=False)
# Heat capacity (excess, full)
# Total PE divided by temperature and sqrt(N) to make result intensive; LRC does not contribute
# We add ideal gas contribution, 1.5, afterwards
c_f = VariableType(nam='Cv/N full',
val=total.cut / (temperature * math.sqrt(n)),
method=msd,
add=1.5,
instant=False)
# Collect together into a list for averaging
return [m_r, e_s, p_s, e_f, p_f, t_c, c_s, c_f]
def calc_variables():
"""Calculates all variables of interest.
They are collected and returned as a list, for use in the main program.
"""
from averages_module import msd, VariableType
from lrc_module import potential_lrc, pressure_lrc
import numpy as np
import math
# Preliminary calculations (n,v,f,total are taken from the calling program)
vol = box**3 # Volume
rho = n / vol # Density
kin = 0.5 * np.sum(v**2) # Kinetic energy
fsq = np.sum(f**2) # Total squared force
# Variables of interest, of class VariableType, containing three attributes:
# .val: the instantaneous value
# .nam: used for headings
# .method: indicating averaging method
# If not set below, .method adopts its default value of avg
# The .nam and some other attributes need only be defined once, at the start of the program,
# but for clarity and readability we assign all the values together below
# Internal energy (cut-and-shifted) per atom
# Total KE plus total cut-and-shifted PE divided by N
e_s = VariableType(nam='E/N cut&shifted', val=(kin + total.pot) / n)
# Internal energy (full, including LRC) per atom
# LRC plus total KE plus total cut (but not shifted) PE divided by N
e_f = VariableType(nam='E/N full',
val=potential_lrc(rho, r_cut) + (kin + total.cut) / n)
# Pressure (cut-and-shifted)
# Ideal gas contribution plus total virial divided by V
p_s = VariableType(nam='P cut&shifted',
val=rho * temperature + total.vir / vol)
# Pressure (full, including LRC)
# LRC plus ideal gas contribution plus total virial divided by V
p_f = VariableType(nam='P full',
val=pressure_lrc(rho, r_cut) + rho * temperature +
total.vir / vol)
# Kinetic temperature
# Momentum is not conserved, hence 3N degrees of freedom
t_k = VariableType(nam='T kinetic', val=2.0 * kin / (3 * n))
# Configurational temperature
# Total squared force divided by total Laplacian
t_c = VariableType(nam='T config', val=fsq / total.lap)
# Heat capacity (cut-and-shifted)
# Total energy divided by temperature and sqrt(N) to make result intensive
c_s = VariableType(nam='Cv/N cut&shifted',
val=(kin + total.pot) / (temperature * math.sqrt(n)),
method=msd,
instant=False)
# Heat capacity (full)
# Total energy divided by temperature and sqrt(N) to make result intensive; LRC does not contribute
c_f = VariableType(nam='Cv/N full',
val=(kin + total.cut) / (temperature * math.sqrt(n)),
method=msd,
instant=False)
# Collect together into a list for averaging
return [e_s, p_s, e_f, p_f, t_k, t_c, c_s, c_f]
cp = 2.5 - a_res[2,0]+(1.0+a_res[0,1]-a_res[1,1])*(1.0+a_res[0,1]-a_res[1,1])/(1.0+2.0*a_res[0,1]+a_res[0,2]) - 1.0
mu = temperature * ( np.log(density) + a_res[0,0] + a_res[0,1] )
z = density * np.exp ( a_res[0,0] + a_res[0,1] )
print('')
print ( "{:40}{:15.6f}".format('Pressure P', p ) )
print ( "{:40}{:15.6f}".format('Energy E/N', e ) )
print ( "{:40}{:15.6f}".format('Heat capacity Cv/NkB', cv ) )
print ( "{:40}{:15.6f}".format('Heat capacity Cp/NkB', cp ) )
print ( "{:40}{:15.6f}".format('Chemical potential mu', mu ) )
print ( "{:40}{:15.6f}".format('Activity z', z ) )
# Estimates for cut (but not shifted) potential by reverse-application of long-range & delta corrections
print('')
print('Lennard-Jones potential cut (but not shifted) at 2.5 sigma')
p = p - pressure_lrc ( density, r_cut ) + pressure_delta ( density, r_cut )
e = e - potential_lrc ( density, r_cut )
mu = mu - 2.0 * potential_lrc ( density, r_cut )
z = z * np.exp ( -2.0* potential_lrc ( density, r_cut ) / temperature )
print('')
print ( "{:40}{:15.6f}".format('Pressure P', p ) )
print ( "{:40}{:15.6f}".format('Energy E/N', e ) )
print ( "{:40}{:15.6f}".format('Chemical potential mu', mu ) )
print ( "{:40}{:15.6f}".format('Activity z', z ) )
# Results for cut-and-shifted potential from Thol et al (2015) fitting formula
print('')
print('Lennard-Jones potential cut-and-shifted at 2.5 sigma')
print('')
a_res = a_res_cutshift ( temperature, density )
def calc_variables ( ):
"""Calculates all variables of interest.
They are collected and returned as a list, for use in the main program.
"""
from averages_module import msd, VariableType
from lrc_module import potential_lrc, pressure_lrc
import numpy as np
import math
# Preliminary calculations (n,r,v,f,total are taken from the calling program)
vol = box**3 # Volume
rho = n / vol # Density
kin = 0.5*np.sum(v**2) # Kinetic energy
fsq = np.sum ( f**2 ) # Total squared force
ext = np.sum(0.5*p_eta**2/q) + np.sum(0.5*p_eta_baro**2/q_baro) + 0.5*p_eps**2/w_eps + temperature * (
g*eta[0] + np.sum(eta[1:])+ np.sum(eta_baro) ) # Extra terms for conserved variable
eng = kin + total.pot # Total energy (cut-and-shifted)
# Variables of interest, of class VariableType, containing three attributes:
# .val: the instantaneous value
# .nam: used for headings
# .method: indicating averaging method
# If not set below, .method adopts its default value of avg
# The .nam and some other attributes need only be defined once, at the start of the program,
# but for clarity and readability we assign all the values together below
# Internal energy (cut-and-shifted) per atom
# Total KE plus total cut-and-shifted PE divided by N
e_s = VariableType ( nam = 'E/N cut&shifted', val = eng/n )
# Internal energy (full, including LRC) per atom
# LRC plus total KE plus total cut (but not shifted) PE divided by N
e_f = VariableType ( nam = 'E/N full', val = potential_lrc(rho,r_cut) + (kin+total.cut)/n )
# Pressure (cut-and-shifted)
# Ideal gas contribution plus total virial divided by V
p_s = VariableType ( nam = 'P cut&shifted', val = rho*temperature + total.vir/vol )
# Pressure (full, including LRC)
# LRC plus ideal gas contribution plus total virial divided by V
p_f = VariableType ( nam = 'P full', val = pressure_lrc(rho,r_cut) + rho*temperature + total.vir/vol )
# Kinetic temperature
t_k = VariableType ( nam = 'T kinetic', val = 2.0*kin/g )
# Configurational temperature
# Total squared force divided by total Laplacian
t_c = VariableType ( nam = 'T config', val = fsq/total.lap )
# Density
density = VariableType ( nam = 'Density', val = rho )
# Conserved energy-like quantity per atom
# Energy plus PV term plus extra terms defined above
enp = eng + pressure*vol
conserved = VariableType ( nam = 'Conserved/N', val = (enp+ext)/n )
# Volume MSD
vol_msd = VariableType ( nam = 'Volume MSD', val = vol, method = msd, instant = False )
# Mean-squared deviation of conserved energy-like quantity
# Energy plus PV term plus extra terms defined above
enp = eng + pressure*vol
conserved_msd = VariableType ( nam = 'Conserved MSD', val = (enp+ext)/n,
method = msd, e_format = True, instant = False )
# Collect together into a list for averaging
return [ e_s, p_s, e_f, p_f, t_k, t_c, density, conserved, vol_msd, conserved_msd ]