Table of Contents
- Monte Carlo: particle in box (NVT)
- Monte Carlo: particle in box (CFCMC)
- Monte Carlo: Gibbs Vapor-Liquid Equilibrium
- Monte Carlo: Gibbs Vapor-Liquid Equilibrium (CFCMC)
Monte Carlo: particles in box (NVT)
A lennard-Jones fluid is simple model of a fluid interacting via the reduced Lennard-Jones potential $$U(r) = 4\left[\frac{1}{r^{12}} - \frac{1}{r^6}\right]$$
Here, we use a cutoff of 4.0 \(\sigma\) and tail-corrections and compute via Widom insertion the chemical potential.
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The average of the total energy:
Total energy
-------------------------------------------------------------------------------
Block[ 0] -2.838221e+03
Block[ 1] -2.838781e+03
Block[ 2] -2.838190e+03
Block[ 3] -2.838883e+03
Block[ 4] -2.838060e+03
---------------------------------------------------------------------------
Average -2.838427e+03 +/- 4.670014e-01 [ε]
The average pressure is computed as:
Average pressure tensor:
-------------------------------------------------------------------------------
7.0379e-01 3.4238e-04 -1.6270e-03 +/- 4.7707e-03 2.4653e-03 1.6942e-03 [ε.σ⁻³]
3.4238e-04 7.0388e-01 -5.1091e-05 +/- 2.4653e-03 4.6656e-03 1.3557e-03 [ε.σ⁻³]
-1.6270e-03 -5.1091e-05 6.9996e-01 +/- 1.6942e-03 1.3557e-03 4.7725e-03 [ε.σ⁻³]
Block[ 0] 8.400000e-01
Block[ 1] 8.400000e-01
Block[ 2] 8.400000e-01
Block[ 3] 8.400000e-01
Block[ 4] 8.400000e-01
---------------------------------------------------------------------------
Ideal gas pressure 8.400000e-01 +/- 0.000000e+00 [ε.σ⁻³]
Block[ 0] -1.351949e-01
Block[ 1] -1.408517e-01
Block[ 2] -1.367764e-01
Block[ 3] -1.402155e-01
Block[ 4] -1.342423e-01
---------------------------------------------------------------------------
Excess pressure -1.374562e-01 +/- 3.674814e-03 [ε.σ⁻³]
Block[ 0] 7.048051e-01
Block[ 1] 6.991483e-01
Block[ 2] 7.032236e-01
Block[ 3] 6.997845e-01
Block[ 4] 7.057577e-01
---------------------------------------------------------------------------
Pressure average 7.025438e-01 +/- 3.674814e-03 [ε.σ⁻³]
Using Widom insertion, we find the chemical potential is given by:
Widom insertion Rosenbluth weight statistics:
---------------------------------------------------------------------------
Block[ 0] 6.489275e+00
Block[ 1] 6.457452e+00
Block[ 2] 6.460533e+00
Block[ 3] 6.487067e+00
Block[ 4] 6.494927e+00
---------------------------------------------------------------------------
Average Rosenbluth weight: 6.477852e+00 +/- 2.170840e-02 [-]
Widom insertion chemical potential statistics:
---------------------------------------------------------------------------
Block[ 0] -1.8701508415534316
Block[ 1] -1.8652347611211533
Block[ 2] -1.8657118770613563
Block[ 3] -1.8698104999091023
Block[ 4] -1.8710213520189722
---------------------------------------------------------------------------
Beta * Excess chemical potential: -1.868389e+00 +/- 3.352543e-03 [-]
Beta * Ideal chemical potential: -3.566749e-01 +/- 1.832108e-17 [-]
Beta * Total chemical potential: -2.225064e+00 +/- 3.352543e-03 [-]
Beta * Imposed chemical potential: -inf [-]
---------------------------------------------------------------------------
Excess chemical potential: -2.242067e+00 +/- 4.023052e-03 [ε]
Ideal chemical potential: -4.280099e-01 +/- 2.198529e-17 [ε]
Total chemical potential: -2.670077e+00 +/- 4.023052e-03 [ε]
Imposed chemical potential: -inf [ε]
---------------------------------------------------------------------------
Imposed fugacity: 0.000000e+00 [ε.σ⁻³]
Measured fugacity: 1.296726e-01 +/- 4.349127e-04 [ε.σ⁻³]
Monte Carlo: particles in a box (CFCMC)
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Lambda histogram and bias:
---------------------------------------------------------------------------
0-0.000000 (lambda) P: 1.30774e+00 +/- 7.14095e-02 bias: 2.21994e+00 [-]
1-0.025000 (lambda) P: 1.30331e+00 +/- 6.80667e-02 bias: 2.21503e+00 [-]
2-0.050000 (lambda) P: 1.28486e+00 +/- 4.81727e-02 bias: 2.21321e+00 [-]
3-0.075000 (lambda) P: 1.29142e+00 +/- 6.08460e-02 bias: 2.24468e+00 [-]
4-0.100000 (lambda) P: 1.25968e+00 +/- 5.46987e-02 bias: 2.33087e+00 [-]
5-0.125000 (lambda) P: 1.32110e+00 +/- 7.34593e-02 bias: 2.67301e+00 [-]
6-0.150000 (lambda) P: 1.24415e+00 +/- 4.60494e-02 bias: 3.20426e+00 [-]
7-0.175000 (lambda) P: 1.14816e+00 +/- 6.80581e-02 bias: 3.71065e+00 [-]
8-0.200000 (lambda) P: 1.04607e+00 +/- 5.85775e-02 bias: 3.78275e+00 [-]
9-0.225000 (lambda) P: 9.88961e-01 +/- 5.01299e-02 bias: 3.56545e+00 [-]
10-0.250000 (lambda) P: 9.60097e-01 +/- 5.06641e-03 bias: 3.18626e+00 [-]
11-0.275000 (lambda) P: 9.93020e-01 +/- 1.36496e-02 bias: 2.73752e+00 [-]
12-0.300000 (lambda) P: 9.58129e-01 +/- 1.76886e-02 bias: 2.19360e+00 [-]
13-0.325000 (lambda) P: 9.51282e-01 +/- 1.46261e-02 bias: 1.66696e+00 [-]
14-0.350000 (lambda) P: 9.38367e-01 +/- 2.47256e-02 bias: 1.18065e+00 [-]
15-0.375000 (lambda) P: 9.27297e-01 +/- 3.68598e-02 bias: 7.80146e-01 [-]
16-0.400000 (lambda) P: 9.21885e-01 +/- 2.43490e-02 bias: 4.63859e-01 [-]
17-0.425000 (lambda) P: 9.25534e-01 +/- 1.99206e-02 bias: 2.43098e-01 [-]
18-0.450000 (lambda) P: 9.13562e-01 +/- 1.29640e-02 bias: 9.40928e-02 [-]
19-0.475000 (lambda) P: 9.21106e-01 +/- 3.09024e-02 bias: 3.13973e-02 [-]
20-0.500000 (lambda) P: 9.27051e-01 +/- 2.34937e-02 bias: 1.18339e-02 [-]
21-0.525000 (lambda) P: 9.25903e-01 +/- 2.87938e-02 bias: 2.85258e-02 [-]
22-0.550000 (lambda) P: 9.16145e-01 +/- 8.62114e-03 bias: 1.48056e-02 [-]
23-0.575000 (lambda) P: 9.35169e-01 +/- 2.43952e-02 bias: 9.52742e-03 [-]
24-0.600000 (lambda) P: 9.28978e-01 +/- 2.50483e-02 bias: 1.15441e-02 [-]
25-0.625000 (lambda) P: 9.19015e-01 +/- 1.88459e-02 bias: 0.00000e+00 [-]
26-0.650000 (lambda) P: 9.14177e-01 +/- 1.72043e-02 bias: 2.82292e-03 [-]
27-0.675000 (lambda) P: 9.02533e-01 +/- 1.27092e-02 bias: 6.07650e-04 [-]
28-0.700000 (lambda) P: 9.21598e-01 +/- 1.68121e-02 bias: 8.97147e-03 [-]
29-0.725000 (lambda) P: 9.35292e-01 +/- 2.23957e-02 bias: 1.08080e-02 [-]
30-0.750000 (lambda) P: 9.13931e-01 +/- 2.73706e-02 bias: 3.08510e-03 [-]
31-0.775000 (lambda) P: 9.32832e-01 +/- 3.29775e-02 bias: 1.53255e-02 [-]
32-0.800000 (lambda) P: 9.26190e-01 +/- 1.82922e-02 bias: 1.63149e-02 [-]
33-0.825000 (lambda) P: 9.23935e-01 +/- 2.22604e-02 bias: 1.12784e-02 [-]
34-0.850000 (lambda) P: 9.17867e-01 +/- 1.99313e-02 bias: 9.48316e-03 [-]
35-0.875000 (lambda) P: 9.21516e-01 +/- 2.50365e-02 bias: 1.29228e-02 [-]
36-0.900000 (lambda) P: 9.27461e-01 +/- 2.29343e-02 bias: 7.83733e-03 [-]
37-0.925000 (lambda) P: 9.24714e-01 +/- 1.77014e-02 bias: 1.15458e-02 [-]
38-0.950000 (lambda) P: 9.25534e-01 +/- 3.27156e-02 bias: 6.93352e-03 [-]
39-0.975000 (lambda) P: 9.24181e-01 +/- 3.17705e-02 bias: 1.63701e-02 [-]
40-1.000000 (lambda) P: 9.30249e-01 +/- 1.97409e-02 bias: 3.75212e-03 [-]
Lambda statistics:
---------------------------------------------------------------------------
0-0.000000 (lambda) Free energy: 1.016884e-02 +/- 5.468281e-02 [-]
1-0.025000 (lambda) Free energy: 1.356059e-02 +/- 5.115551e-02 [-]
2-0.050000 (lambda) Free energy: 2.781803e-02 +/- 3.695043e-02 [-]
3-0.075000 (lambda) Free energy: 2.272540e-02 +/- 4.665128e-02 [-]
4-0.100000 (lambda) Free energy: 4.760534e-02 +/- 4.287716e-02 [-]
5-0.125000 (lambda) Free energy: 0.000000e+00 +/- 5.537236e-02 [-]
6-0.150000 (lambda) Free energy: 6.001769e-02 +/- 3.677614e-02 [-]
7-0.175000 (lambda) Free energy: 1.403021e-01 +/- 5.927641e-02 [-]
8-0.200000 (lambda) Free energy: 2.334221e-01 +/- 5.567939e-02 [-]
9-0.225000 (lambda) Free energy: 2.895666e-01 +/- 5.020972e-02 [-]
10-0.250000 (lambda) Free energy: 3.191872e-01 +/- 5.282070e-03 [-]
11-0.275000 (lambda) Free energy: 2.854707e-01 +/- 1.375505e-02 [-]
12-0.300000 (lambda) Free energy: 3.212391e-01 +/- 1.839641e-02 [-]
13-0.325000 (lambda) Free energy: 3.284110e-01 +/- 1.536563e-02 [-]
14-0.350000 (lambda) Free energy: 3.420804e-01 +/- 2.648885e-02 [-]
15-0.375000 (lambda) Free energy: 3.539476e-01 +/- 3.963937e-02 [-]
16-0.400000 (lambda) Free energy: 3.598010e-01 +/- 2.675731e-02 [-]
17-0.425000 (lambda) Free energy: 3.558506e-01 +/- 2.152976e-02 [-]
18-0.450000 (lambda) Free energy: 3.688703e-01 +/- 1.418260e-02 [-]
19-0.475000 (lambda) Free energy: 3.606464e-01 +/- 3.405690e-02 [-]
20-0.500000 (lambda) Free energy: 3.542129e-01 +/- 2.543852e-02 [-]
21-0.525000 (lambda) Free energy: 3.554520e-01 +/- 3.124870e-02 [-]
22-0.550000 (lambda) Free energy: 3.660469e-01 +/- 9.385207e-03 [-]
23-0.575000 (lambda) Free energy: 3.454943e-01 +/- 2.611788e-02 [-]
24-0.600000 (lambda) Free energy: 3.521365e-01 +/- 2.709814e-02 [-]
25-0.625000 (lambda) Free energy: 3.629191e-01 +/- 2.065218e-02 [-]
26-0.650000 (lambda) Free energy: 3.681973e-01 +/- 1.884513e-02 [-]
27-0.675000 (lambda) Free energy: 3.810163e-01 +/- 1.412701e-02 [-]
28-0.700000 (lambda) Free energy: 3.601124e-01 +/- 1.822148e-02 [-]
29-0.725000 (lambda) Free energy: 3.453627e-01 +/- 2.403513e-02 [-]
30-0.750000 (lambda) Free energy: 3.684664e-01 +/- 2.978111e-02 [-]
31-0.775000 (lambda) Free energy: 3.479964e-01 +/- 3.583179e-02 [-]
32-0.800000 (lambda) Free energy: 3.551421e-01 +/- 1.978602e-02 [-]
33-0.825000 (lambda) Free energy: 3.575798e-01 +/- 2.398551e-02 [-]
34-0.850000 (lambda) Free energy: 3.641690e-01 +/- 2.160428e-02 [-]
35-0.875000 (lambda) Free energy: 3.602014e-01 +/- 2.741583e-02 [-]
36-0.900000 (lambda) Free energy: 3.537708e-01 +/- 2.475013e-02 [-]
37-0.925000 (lambda) Free energy: 3.567370e-01 +/- 1.905506e-02 [-]
38-0.950000 (lambda) Free energy: 3.558506e-01 +/- 3.577566e-02 [-]
39-0.975000 (lambda) Free energy: 3.573136e-01 +/- 3.427250e-02 [-]
40-1.000000 (lambda) Free energy: 3.507692e-01 +/- 2.122766e-02 [-]
---------------------------------------------------------------------------
Excess chemical potential: (ln(P(lambda=1))-ln(P(lambda=0)))
Block[ 0] -1.9425521171520197
Block[ 1] -1.9020588353830536
Block[ 2] -1.7945046752415006
Block[ 3] -1.864956751792043
Block[ 4] -1.877161937515691
---------------------------------------------------------------------------
Beta * Excess chemical potential: -1.875589e+00 +/- 6.764247e-02 [-]
Beta * Ideal gas chemical potential: -3.566749e-01 +/- 4.313206e-14 [-]
Beta * Total chemical potential: -2.232264e+00 +/- 6.764247e-02 [-]
Beta * Imposed chemical potential: -inf [-]
---------------------------------------------------------------------------
Excess chemical potential: -2.250707e+00 +/- 8.117096e-02 [ε]
Ideal gas chemical potential: -4.280099e-01 +/- 5.175848e-14 [ε]
Total chemical potential: -2.678716e+00 +/- 8.117096e-02 [ε]
Imposed chemical potential: -inf [ε]
---------------------------------------------------------------------------
Imposed fugacity: 0.000000e+00 [ε.σ⁻³]
Measured fugacity: 1.287423e-01 +/- 8.800834e-03 [ε.σ⁻³]
Monte Carlo: Gibbs Vapor-Liquid Equilibrium
Shifted potential at a cutoff of \(r_c=2.5\sigma\).
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| Property | Units | Vapor | Liquid |
| Density \(\left\langle\rho\right\rangle\) | \(N\cdot\sigma^{-3}\) | \(0.001928 \pm 0.000059\) | \(0.838342 \pm 0.000317\) |
| Molecules \(\left\langle N\right\rangle\) | - | \(06.96\pm 0.21\) | \(249.04 \pm 0.21\) |
| Energy \(\left\langle U\right\rangle\) | \(\epsilon\) | \(-0.17 \pm 0.012\) | \(-1321.05 \pm 0.71\) |
| Pressure \(\left\langle p\right\rangle\) | \(\epsilon\sigma^{-3}\) | \(0.001132 \pm 0.000035\) | \(0.001369 \pm 0.000728\) |
| Excess chemical potential \(\left\langle \mu^{ex}\right\rangle\) | \(\epsilon\) | \(-0.02528\pm0.00079\) | \(-3.67556\pm 0.0261\) |
| Ideal chemical potential \(\left\langle \mu^{id}\right\rangle\) | \(\epsilon\) | \(-3.7509\pm 0.0185\) | \(-0.105798\pm 0.000227\) |
| Chemical potential \(\left\langle \mu\right\rangle\) | \(\epsilon\) | \(-3.7762\pm 0.0177\) | \(-3.7814\pm 0.0261\) |
Liquid
Gibbs swap (CBMC) all: 509958010
Gibbs swap (CBMC) total: 1019276656
Gibbs swap (CBMC) constructed: 659678014
Gibbs swap (CBMC) accepted: 1223173
Gibbs swap (CBMC) fraction: 0.001200
Gibbs swap (CBMC) max-change: 0.000000
Vapor
Gibbs swap (CBMC) all: 509911407
Gibbs swap (CBMC) total: 1019276656
Gibbs swap (CBMC) constructed: 659678014
Gibbs swap (CBMC) accepted: 1223173
Gibbs swap (CBMC) fraction: 0.001200
Gibbs swap (CBMC) max-change: 0.000000
Monte Carlo: Gibbs Vapor-Liquid Equilibrium (CFCMC)
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| Property | Units | Vapor | Liquid |
| Density \(\left\langle\rho\right\rangle\) | \(N\cdot\sigma^{-3}\) | \(0.001946 \pm 0.000058\) | \( 0.836678 \pm 0.000218\) |
| Molecules \(\left\langle N\right\rangle\) | - | \(7.02\pm 0.21\) | \(248.98 \pm 0.21\) |
| Energy \(\left\langle U\right\rangle\) | \(\epsilon\) | \(-0.19 \pm 0.01\) | \(-1322.98 \pm 1.48\) |
| Pressure \(\left\langle p\right\rangle\) | \(\epsilon\sigma^{-3}\) | \(0.001139 \pm 0.000034\) | \(0.000494\pm 0.000366\) |
| Chemical potential Widom \(\left\langle \mu\right\rangle\) | \(\epsilon\) | \(-3.7721 \pm 0.0171\) | \(-3.76212 \pm 0.0358\) |
| Chemical potential CFCMC \(\left\langle \mu\right\rangle\) | \(\epsilon\) | \(-3.7676 \pm 0.0226\) | \(-3.77704 \pm 0.0212\) |
Liquid
Gibbs swap (CFCMC) all: 433767283
Gibbs swap (CFCMC) total: 108435452 108433519 216898312
Gibbs swap (CFCMC) constructed: 104794521 108411050 160644912
Gibbs swap (CFCMC) accepted: 69696354 17629208 109049917
Gibbs swap (CFCMC) fraction: 0.642745 0.162581 0.502770
Gibbs swap (CFCMC) max-change: 1.000000 0.100000 0.466139
Vapor
Gibbs swap (CFCMC) all: 501025563
Gibbs swap (CFCMC) total: 125272078 125241165 250512320
Gibbs swap (CFCMC) constructed: 125267752 113611272 128364421
Gibbs swap (CFCMC) accepted: 69696354 17629207 126271863
Gibbs swap (CFCMC) fraction: 0.556360 0.140762 0.504055
Gibbs swap (CFCMC) max-change: 0.299122 0.100000 1.000000
The fractional molecule was 46.4058% in the liquid phase and 53.5942% in the vapor phase.
1.9.8