Lennard-Jones fluid simulation website

4. EOS-TMMC: Equations of state (pressure as a function density)

Equations of state (pressure as a function density) generated by grand-canonical transition-matrix Monte Carlo over the temperature range 0.70 to 1.50 at temperature increments of 0.05. For a given density, the mean pressure and its standard deviation are reported.

METHOD: Grand-canonical transition-matrix Monte Carlo [1]
V/σ3: 512
TRUNCATION: 3σ + standard long range corrections
Prob. of Disp. Move: 0.75
Prob. of Ins/Del. Move: 0.25
Biasing Function Update Frequency: 1.0E6 trials
Simulation Length: 8.0E9 trials
plot of density versus pressure
Equation of State Data:
Choose a temperature to jump to table of pressure versus density data
kBT/ε = 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50


RAW Simulation Data:
For each temperature, five raw macrostate (particle number) distributions are provided (gzipped tar files). Also included is the average total potential energy (dimensionless) of a given macrostate. The first column in each file represents N, the number of particles, and the second column represents ln Π (N), the natural logarithm of the probability of observing N particles given the conditions of the simulation (fixed chemical potential, volume, and temperature). The information required to manipulate the macrostate distributions are listed below.


Raw ln Π(N) and Total Potential Energy Data Obtained by TMMC
kBT/ε Nmin Nmax ln zsim
0.70 0 475 -5.943376
0.75 0 475 -5.400601
0.80 0 470 -4.937419
0.85 0 470 -4.540504
0.90 0 465 -4.200008
0.95 0 465 -3.867387
1.00 0 425 -3.823078
1.05 0 415 -3.436560
1.10 0 400 -3.245568
1.15 0 400 -3.068218
1.20 0 390 -2.902929
1.35 0 380 -0.9114399
1.40 0 375 -0.7669414
1.45 0 370 -1.940905
1.50 0 370 -1.568214

z is the so-called activity which is defined as
z = Λ-3 exp ( βμ )

where Λ is the de Broglie wavelength, β is 1/kBT (kB is Boltzmann's constant), and μ is the chemical potential. 

Remarks:

Uncertainties were obtained from 5 independent simulations and represent 67% confidence limits (one standard deviation). Here, an equation of state (pressure as a function of density, at a given temperature) is constructed from the particle number probability distribution (determined by grand-canonical transition-matrix Monte Carlo) by exploiting the fact that the Helmholtz free energy is related to the grand potential by a Legendre transform with respect to particle number. The details of this construction can be found it Ref. 2. It should be pointed out that the apparent van der Waals loops in the isotherms are system-size dependent [2]. Therefore, only the points outside the liquid-vapor coexistence region are expected to be reproduced reliably using system sizes other than that used here.

[1] J. R. Errington, J. Chem. Phys. 118, 9915 (2003).
[2] V. K. Shen and J. R. Errington, J. Phys. Chem B 108, 19595 (2004).

CPU Timings:

Below, we report the CPU times required to obtain each isotherm. CPU times were obtained using the ETIME function, an intrinsic function of the Intel Fortran Compiler v8.0 *. Codes were compiled using the "-fast" optimization level and executed on AMD Opteron processors* running Red Hat Linux*. We report mean and standard deviations of the CPU times required to produce the data above. Uncertainties were obtained from 5 independent simulations and represent 67% confidence limits (one standard deviation)

* This identification does not imply recommendation or endorsement by NIST, nor does it imply that it is the best available for the purposes described.


Mean CPU Time Required to Complete 8 Billion Trials
kBT/ε CPU Time (hrs) +/- (hrs) Architecture
0.70 64.5 5.3 AMD Opteron 248 2.2 GHz
0.75 70.0 5.6 AMD Opteron 246 2.0 GHz
0.80 64.2 1.7 AMD Opteron 248 2.2 GHz
0.85 60.8 1.1 AMD Opteron 248 2.2 GHz
0.90 58.9 0.6 AMD Opteron 248 2.2 GHz
0.95 55.0 0.1 AMD Opteron 248 2.2 GHz
1.00 53.0 0.4 AMD Opteron 248 2.2 GHz
1.05 51.3 0.5 AMD Opteron 248 2.2 GHz
1.10 49.7 0.3 AMD Opteron 248 2.2 GHz
1.15 56.6 0.5 AMD Opteron 246 2.0 GHz
1.20 53.8 0.3 AMD Opteron 246 2.0 GHz
1.35 49.8 1.2 AMD Opteron 246 2.0 GHz
1.40 47.4 0.2 AMD Opteron 246 2.0 GHz
1.45 46.7 0.1 AMD Opteron 246 2.0 GHz
1.50 46.7 0.3 AMD Opteron 246 2.0 GHz


kBT/ε= 0.70

ρσ33/ε             +/-

kBT/ε= 0.75

ρσ33/ε             +/-

kBT/ε= 0.80

ρσ33/ε             +/-

kBT/ε= 0.85

ρσ33/ε             +/-

kBT/ε= 0.90

ρσ33/ε             +/-

kBT/ε= 0.95

ρσ33/ε             +/-

kBT/ε= 1.00

ρσ33/ε             +/-

kBT/ε= 1.05

ρσ33/ε             +/-

kBT/ε= 1.10

ρσ33/ε             +/-

kBT/ε= 1.15

ρσ33/ε             +/-

kBT/ε= 1.20

ρσ33/ε             +/-

kBT/ε= 1.35

ρσ33/ε             +/-

kBT/ε= 1.40

ρσ33/ε             +/-

kBT/ε= 1.45

ρσ33/ε             +/-

kBT/ε= 1.50

ρσ33/ε             +/-