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This directory has 2 scripts that compute the diffusion coefficient of
a 2d Lennard-Jones fluid using 2 different methods. See the
discussion in Section 6.22 of the manual for an overview of the
methods and pointers to doc pages for the commands which implement
them.
These scripts are provided for illustration purposes. No guarantee is
made that the systems are fully equilibrated or that the runs are long
enough to generate good statistics and highly accurate results.
-------------
These are the 2 methods for computing the diffusion coefficient. The
first computes the slope of the mean-squared displacement (MSD) of the
atoms. The second time-integrates the velocity-auto-correlation
function (VACF) for the system. In both cases a single time origin is
used. This allows the diffusion coefficient to be estimated
on-the-fly using variables in the LAMMPS input scripts. For better
statistics you typically want to use multiple time origins,
post-processing the data from dump files.
in.msd.2d = mean-squared displacement (MSD)
in.vacf.2d = velocity auto-correlation function (VACF)
Both systems have 3200 atoms and run for 100000 timesteps, after
equilibration.
The scripts were both run on 8 processors. They both run in about 10
seconds and produce the accompanying log files.
The state point of the LJ fluid is rho* = 0.6, T* = 1.0, and Rcut =
2.5 sigma.
-------------
Here is how to extract the diffusion coefficient from the log file
output for each method.
(1) in.msd.2d
The 3rd column of output is the instantaneous mean-squared
displacment, which grows over time. The 4th column estimates the
slope of the MSD from its two end-points, and uses it to compute the
diffusion coefficient. The 5th column fits a straight line to the
entire (growing) set of MSD data and uses the slope of the line to
compute the diffusion coefficient. You can see that both measures
give roughly the same answer and rapidly become roughly constant for
the 100K step simulation.
Dcoeff = 0.36
(2) in.vacf.2d
The 3rd column of output is the instantaneous velocity
auto-correlation function (VACF). The 4th column is the
time-integration of the VACF (for every timestep), integrated up to
that point in time, converted into the diffusion coefficient. You can
see the VACF approach gives a more noise, fluctuating value for the
diffusion coefficient, compared to the MSD approach.
Dcoeff = 0.25 to 0.45

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# sample LAMMPS input script for diffusion of 2d LJ liquid
# mean-squared displacement via compute msd
# settings
variable x equal 40
variable y equal 40
variable rho equal 0.6
variable t equal 1.0
variable rc equal 2.5
# problem setup
units lj
dimension 2
atom_style atomic
neigh_modify delay 0 every 1
lattice sq2 ${rho}
region simbox block 0 $x 0 $y -0.1 0.1
create_box 1 simbox
create_atoms 1 box
pair_style lj/cut ${rc}
pair_coeff * * 1 1
mass * 1.0
velocity all create $t 97287
fix 1 all nve
fix 2 all langevin $t $t 0.1 498094
fix 3 all enforce2d
# equilibration run
thermo 1000
run 5000
unfix 2
# data gathering run
reset_timestep 0
# factor of 4 in 2 variables is for 2d
compute msd all msd com yes
variable twopoint equal c_msd[4]/4/(step*dt+1.0e-6)
fix 9 all vector 10 c_msd[4]
variable fitslope equal slope(f_9)/4/(10*dt)
thermo_style custom step temp c_msd[4] v_twopoint v_fitslope
# only need to run for 10K steps to make a good 100-frame movie
#dump 1 all custom 1 tmp.dump id type vx vy vz
#dump 2 all image 100 image.*.jpg type type zoom 1.6 adiam 1.2
thermo 1000
run 100000

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# sample LAMMPS input script for diffusion of 2d LJ liquid
# mean-squared displacement via compute msd
# settings
variable x equal 40
variable y equal 40
variable rho equal 0.6
variable t equal 1.0
variable rc equal 2.5
# problem setup
units lj
dimension 2
atom_style atomic
neigh_modify delay 0 every 1
lattice sq2 ${rho}
region simbox block 0 $x 0 $y -0.1 0.1
create_box 1 simbox
create_atoms 1 box
pair_style lj/cut ${rc}
pair_coeff * * 1 1
mass * 1.0
velocity all create $t 97287
fix 1 all nve
fix 2 all langevin $t $t 0.1 498094
fix 3 all enforce2d
# equilibration run
thermo 1000
run 5000
unfix 2
# data gathering run
reset_timestep 0
compute vacf all vacf
fix 4 all ave/time 1 1 1 c_vacf[4] #file tmp.vacf
# factor of 0.5 is for 2d
fix 5 all vector 1 c_vacf[4]
variable vacf equal 0.5*dt*trap(f_5)
thermo_style custom step temp c_vacf[4] v_vacf
# only need to run for 10K steps to make a good 100-frame movie
#dump 1 all custom 1 tmp.dump id type vx vy vz
#dump 2 all image 100 image.*.jpg type type zoom 1.6 adiam 1.2
thermo 1000
run 100000

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LAMMPS (12 Feb 2014)
# sample LAMMPS input script for diffusion of 2d LJ liquid
# mean-squared displacement via compute msd
# settings
variable x equal 40
variable y equal 40
variable rho equal 0.6
variable t equal 1.0
variable rc equal 2.5
# problem setup
units lj
dimension 2
atom_style atomic
neigh_modify delay 0 every 1
lattice sq2 ${rho}
lattice sq2 0.6
Lattice spacing in x,y,z = 1.82574 1.82574 1.82574
region simbox block 0 $x 0 $y -0.1 0.1
region simbox block 0 40 0 $y -0.1 0.1
region simbox block 0 40 0 40 -0.1 0.1
create_box 1 simbox
Created orthogonal box = (0 0 -0.182574) to (73.0297 73.0297 0.182574)
4 by 2 by 1 MPI processor grid
create_atoms 1 box
Created 3200 atoms
pair_style lj/cut ${rc}
pair_style lj/cut 2.5
pair_coeff * * 1 1
mass * 1.0
velocity all create $t 97287
velocity all create 1 97287
fix 1 all nve
fix 2 all langevin $t $t 0.1 498094
fix 2 all langevin 1 $t 0.1 498094
fix 2 all langevin 1 1 0.1 498094
fix 3 all enforce2d
# equilibration run
thermo 1000
run 5000
Memory usage per processor = 2.06238 Mbytes
Step Temp E_pair E_mol TotEng Press
0 1 -1.56492 0 -0.5652325 -1.5346995
1000 0.97537833 -1.5723957 0 -0.5973222 0.92877783
2000 0.99008371 -1.5748206 0 -0.58504633 1.0809416
3000 1.0111412 -1.5848987 0 -0.57407352 1.0174297
4000 1.0055417 -1.5857581 0 -0.58053054 0.95647691
5000 0.97069909 -1.5851114 0 -0.61471562 0.90108296
Loop time of 0.548328 on 8 procs for 5000 steps with 3200 atoms
Pair time (%) = 0.242437 (44.2139)
Neigh time (%) = 0.0589295 (10.7471)
Comm time (%) = 0.0797399 (14.5424)
Outpt time (%) = 0.00014773 (0.0269419)
Other time (%) = 0.167074 (30.4697)
Nlocal: 400 ave 406 max 394 min
Histogram: 1 1 0 1 0 2 1 0 1 1
Nghost: 202.5 ave 212 max 191 min
Histogram: 1 0 0 0 3 1 0 2 0 1
Neighs: 2800.88 ave 2903 max 2690 min
Histogram: 1 1 0 0 1 2 1 0 1 1
Total # of neighbors = 22407
Ave neighs/atom = 7.00219
Neighbor list builds = 599
Dangerous builds = 0
unfix 2
# data gathering run
reset_timestep 0
# factor of 4 in 2 variables is for 2d
compute msd all msd com yes
variable twopoint equal c_msd[4]/4/(step*dt+1.0e-6)
fix 9 all vector 10 c_msd[4]
variable fitslope equal slope(f_9)/4/(10*dt)
thermo_style custom step temp c_msd[4] v_twopoint v_fitslope
# only need to run for 10K steps to make a good 100-frame movie
#dump 1 all custom 1 tmp.dump id type vx vy vz
#dump 2 all image 100 image.*.jpg type type zoom 1.6 adiam 1.2
thermo 1000
run 100000
Memory usage per processor = 2.29126 Mbytes
Step Temp msd[4] twopoint fitslope
0 0.97069909 5.6797985e-31 1.4199496e-25 5e+20
1000 0.98259938 4.018182 0.20090906 0.18018405
2000 0.98866167 9.1771673 0.22942916 0.20819132
3000 0.98066292 14.377973 0.23963287 0.2226381
4000 0.97384109 19.666022 0.24582526 0.23154108
5000 0.97843004 25.570023 0.25570022 0.23873417
6000 0.98703537 31.182236 0.25985196 0.2446473
7000 0.97169186 36.915546 0.26368246 0.24892937
8000 0.9823893 42.723242 0.26702026 0.25284158
9000 0.9998457 49.159596 0.27310886 0.25648405
10000 0.98782869 55.95808 0.27979039 0.26026001
11000 0.99481941 63.387246 0.28812384 0.26436008
12000 0.9760981 70.325901 0.29302458 0.26836823
13000 0.97774297 77.590396 0.2984246 0.2723266
14000 0.96752038 85.055703 0.30377036 0.27627449
15000 0.98697717 92.723433 0.3090781 0.28007928
16000 0.98385955 100.89787 0.31530583 0.28396785
17000 0.9839555 108.44716 0.31896223 0.28772551
18000 0.98000182 115.932 0.32203333 0.29120847
19000 0.96633821 123.99378 0.32629942 0.2945701
20000 0.98641362 131.63233 0.32908082 0.29781576
21000 0.9763779 140.20546 0.33382253 0.30091009
22000 0.97984221 149.3632 0.33946182 0.30410181
23000 0.98384014 157.80278 0.34304952 0.30727461
24000 0.97601102 165.78405 0.34538343 0.31026343
25000 0.99623187 173.8015 0.347603 0.31311142
26000 0.98943691 182.06075 0.35011683 0.31582297
27000 0.98992058 188.30114 0.34870582 0.31827868
28000 0.97957751 196.93202 0.35166432 0.32053702
29000 0.99513582 206.48445 0.35600767 0.32277974
30000 0.99754125 213.2124 0.355354 0.32491307
31000 0.98894929 221.74208 0.35764851 0.32691848
32000 0.973493 229.28872 0.35826363 0.3288054
33000 0.99989261 236.7565 0.35872197 0.33061911
34000 0.97192218 243.57374 0.35819667 0.33225106
35000 0.99711642 250.65118 0.35807311 0.33369124
36000 0.97718259 258.94971 0.35965237 0.33507325
37000 0.98194807 265.83921 0.35924218 0.33634589
38000 0.97195138 272.94984 0.35914453 0.33755459
39000 0.98219017 280.07692 0.35907297 0.33864166
40000 0.98039694 288.4453 0.36055663 0.33967254
41000 0.97621359 295.85108 0.360794 0.3406849
42000 0.97460413 303.13769 0.3608782 0.34161073
43000 0.9799912 312.06356 0.3628646 0.34254857
44000 0.96313376 320.55123 0.36426276 0.34347747
45000 0.97643796 329.68761 0.36631956 0.34443057
46000 0.98641729 338.40463 0.36783111 0.34540729
47000 0.97811939 345.46683 0.3675179 0.34634877
48000 0.99744864 352.87823 0.36758148 0.34720815
49000 0.97480356 362.60002 0.37000002 0.34807624
50000 0.97841509 368.44194 0.36844193 0.34892214
51000 0.97865859 375.44238 0.36808076 0.34966222
52000 0.97415415 382.95453 0.36822551 0.35036495
53000 0.97984491 390.33776 0.36824317 0.35103186
54000 0.99289379 396.4986 0.36712833 0.35162842
55000 0.98606668 403.49336 0.36681214 0.35217399
56000 0.98585489 411.98283 0.36784182 0.35270646
57000 0.98507959 418.14272 0.36679186 0.35321673
58000 0.98030805 423.23791 0.36486026 0.353654
59000 0.98315137 429.40554 0.36390299 0.35400654
60000 0.98762585 437.17376 0.36431146 0.35433653
61000 0.96937507 442.38698 0.36261228 0.3546452
62000 0.97194792 450.91689 0.36364265 0.3549106
63000 0.99877655 460.61733 0.36556931 0.3552122
64000 0.98525211 469.26926 0.36661661 0.35554833
65000 0.9833149 477.52571 0.36732747 0.35589823
66000 0.97191797 485.72618 0.36797438 0.35624251
67000 0.98479592 492.83716 0.36778892 0.35657544
68000 0.98418943 498.90452 0.36684156 0.35688872
69000 0.96953928 507.57013 0.36780444 0.35718476
70000 0.98373337 515.05745 0.36789818 0.35749562
71000 0.98260952 522.33093 0.36783868 0.35778626
72000 0.98906053 527.99215 0.36666121 0.35804724
73000 0.99569597 534.99359 0.36643396 0.35827372
74000 0.97627362 540.94769 0.3655052 0.35847643
75000 0.97276792 546.45533 0.36430355 0.35864641
76000 0.97659072 554.59353 0.36486417 0.35879714
77000 0.9807196 562.96571 0.36556215 0.35896542
78000 0.97398601 571.22804 0.36617182 0.35914269
79000 0.98124212 577.92968 0.36577828 0.35930881
80000 0.98506783 586.73568 0.3667098 0.35948276
81000 0.97926561 596.68206 0.36832226 0.35968089
82000 0.97906184 604.24971 0.36844495 0.35988893
83000 0.96540502 610.7078 0.36789626 0.3600856
84000 0.98726761 619.32703 0.36864704 0.36027536
85000 0.98133061 627.20955 0.36894679 0.3604747
86000 0.99142106 634.68836 0.36900486 0.36067069
87000 0.97917566 641.73186 0.36881141 0.36086497
88000 0.99391197 649.76607 0.36918527 0.36105571
89000 0.98521911 655.50224 0.36825969 0.36123298
90000 0.97419059 664.0397 0.36891095 0.36139717
91000 0.98687774 671.19045 0.36878596 0.36156016
92000 0.97816545 677.37724 0.36813981 0.36171235
93000 0.98734859 684.56734 0.36804696 0.3618463
94000 0.99116168 691.47221 0.36780437 0.36197063
95000 0.99982024 698.01624 0.36737697 0.36208597
96000 0.99232404 703.53179 0.3664228 0.36218818
97000 0.97829693 707.91832 0.36490635 0.36225634
98000 0.99878715 713.77777 0.36417233 0.36230715
99000 0.97026354 717.24468 0.36224478 0.3623226
100000 0.98911409 722.1539 0.36107695 0.36230617
Loop time of 9.28795 on 8 procs for 100000 steps with 3200 atoms
Pair time (%) = 4.87621 (52.5003)
Neigh time (%) = 1.26444 (13.6138)
Comm time (%) = 1.60752 (17.3076)
Outpt time (%) = 0.00843725 (0.0908408)
Other time (%) = 1.53135 (16.4875)
Nlocal: 400 ave 410 max 389 min
Histogram: 1 1 1 0 1 1 0 0 1 2
Nghost: 205.375 ave 216 max 194 min
Histogram: 1 0 0 2 2 0 1 0 0 2
Neighs: 2818 ave 3010 max 2683 min
Histogram: 2 0 2 0 1 1 0 1 0 1
Total # of neighbors = 22544
Ave neighs/atom = 7.045
Neighbor list builds = 12748
Dangerous builds = 0

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LAMMPS (12 Feb 2014)
# sample LAMMPS input script for diffusion of 2d LJ liquid
# mean-squared displacement via compute msd
# settings
variable x equal 40
variable y equal 40
variable rho equal 0.6
variable t equal 1.0
variable rc equal 2.5
# problem setup
units lj
dimension 2
atom_style atomic
neigh_modify delay 0 every 1
lattice sq2 ${rho}
lattice sq2 0.6
Lattice spacing in x,y,z = 1.82574 1.82574 1.82574
region simbox block 0 $x 0 $y -0.1 0.1
region simbox block 0 40 0 $y -0.1 0.1
region simbox block 0 40 0 40 -0.1 0.1
create_box 1 simbox
Created orthogonal box = (0 0 -0.182574) to (73.0297 73.0297 0.182574)
4 by 2 by 1 MPI processor grid
create_atoms 1 box
Created 3200 atoms
pair_style lj/cut ${rc}
pair_style lj/cut 2.5
pair_coeff * * 1 1
mass * 1.0
velocity all create $t 97287
velocity all create 1 97287
fix 1 all nve
fix 2 all langevin $t $t 0.1 498094
fix 2 all langevin 1 $t 0.1 498094
fix 2 all langevin 1 1 0.1 498094
fix 3 all enforce2d
# equilibration run
thermo 1000
run 5000
Memory usage per processor = 2.06238 Mbytes
Step Temp E_pair E_mol TotEng Press
0 1 -1.56492 0 -0.5652325 -1.5346995
1000 0.97537833 -1.5723957 0 -0.5973222 0.92877783
2000 0.99008371 -1.5748206 0 -0.58504633 1.0809416
3000 1.0111412 -1.5848987 0 -0.57407352 1.0174297
4000 1.0055417 -1.5857581 0 -0.58053054 0.95647691
5000 0.97069909 -1.5851114 0 -0.61471562 0.90108296
Loop time of 0.556362 on 8 procs for 5000 steps with 3200 atoms
Pair time (%) = 0.242478 (43.5828)
Neigh time (%) = 0.0590148 (10.6073)
Comm time (%) = 0.0862918 (15.51)
Outpt time (%) = 0.000148952 (0.0267725)
Other time (%) = 0.168428 (30.2731)
Nlocal: 400 ave 406 max 394 min
Histogram: 1 1 0 1 0 2 1 0 1 1
Nghost: 202.5 ave 212 max 191 min
Histogram: 1 0 0 0 3 1 0 2 0 1
Neighs: 2800.88 ave 2903 max 2690 min
Histogram: 1 1 0 0 1 2 1 0 1 1
Total # of neighbors = 22407
Ave neighs/atom = 7.00219
Neighbor list builds = 599
Dangerous builds = 0
unfix 2
# data gathering run
reset_timestep 0
compute vacf all vacf
fix 4 all ave/time 1 1 1 c_vacf[4] #file tmp.vacf
# factor of 0.5 is for 2d
fix 5 all vector 1 c_vacf[4]
variable vacf equal 0.5*dt*trap(f_5)
thermo_style custom step temp c_vacf[4] v_vacf
# only need to run for 10K steps to make a good 100-frame movie
#dump 1 all custom 1 tmp.dump id type vx vy vz
#dump 2 all image 100 image.*.jpg type type zoom 1.6 adiam 1.2
thermo 1000
run 100000
Memory usage per processor = 2.29126 Mbytes
Step Temp vacf[4] vacf
0 0.97069909 1.9407915 0
1000 0.98259938 0.068345562 0.21424576
2000 0.98866167 0.028205164 0.24370013
3000 0.98066292 0.008852823 0.27439588
4000 0.97384109 0.011903564 0.3159275
5000 0.97843004 0.019295534 0.33596598
6000 0.98703537 0.01070801 0.34517088
7000 0.97169186 -0.0082938435 0.33784119
8000 0.9823893 0.038402174 0.32756669
9000 0.9998457 0.045317593 0.33379383
10000 0.98782869 0.027148675 0.34751149
11000 0.99481941 0.035783937 0.36009047
12000 0.9760981 -0.0010825507 0.36059769
13000 0.97774297 0.048323885 0.36103752
14000 0.96752038 0.0008189784 0.34988029
15000 0.98697717 -0.0339338 0.35841003
16000 0.98385955 0.03729417 0.36378228
17000 0.9839555 -0.0063931744 0.37486488
18000 0.98000182 0.065439765 0.39975619
19000 0.96633821 0.0034363237 0.39585239
20000 0.98641362 -0.020284 0.39696422
21000 0.9763779 0.013683539 0.36183593
22000 0.97984221 -0.0077108606 0.32642476
23000 0.98384014 -0.030550764 0.32775468
24000 0.97601102 -0.0047287909 0.3294707
25000 0.99623187 0.00653461 0.34586445
26000 0.98943691 -0.05047279 0.3283058
27000 0.98992058 -0.01720223 0.30840611
28000 0.97957751 0.020915942 0.3149002
29000 0.99513582 0.03712206 0.30525251
30000 0.99754125 -0.022509889 0.29511243
31000 0.98894929 0.015289267 0.25586423
32000 0.973493 0.015971435 0.25226411
33000 0.99989261 0.026989142 0.29050941
34000 0.97192218 0.043710515 0.29067058
35000 0.99711642 0.047231436 0.27989734
36000 0.97718259 0.0059969847 0.28843181
37000 0.98194807 0.017440303 0.30325718
38000 0.97195138 -0.040449666 0.29696592
39000 0.98219017 0.0063692991 0.28915832
40000 0.98039694 -0.0038122219 0.24799014
41000 0.97621359 0.012624961 0.2374766
42000 0.97460413 -0.0024187 0.24693474
43000 0.9799912 0.015928618 0.26238505
44000 0.96313376 -0.067284605 0.24397772
45000 0.97643796 -0.0039001998 0.22733962
46000 0.98641729 -0.026949085 0.24167989
47000 0.97811939 -0.050859011 0.2335985
48000 0.99744864 -0.008154229 0.2417371
49000 0.97480356 -0.011051498 0.25474033
50000 0.97841509 -0.00531423 0.25357072
51000 0.97865859 0.067722096 0.25086864
52000 0.97415415 0.015863025 0.2516842
53000 0.97984491 0.021332829 0.28226376
54000 0.99289379 -0.0192578 0.28300764
55000 0.98606668 0.029307891 0.27592186
56000 0.98585489 0.0062922121 0.2687204
57000 0.98507959 -0.0068688582 0.27868708
58000 0.98030805 0.013766115 0.31008116
59000 0.98315137 0.033068034 0.32211415
60000 0.98762585 0.050662295 0.30109082
61000 0.96937507 -0.02797113 0.29968066
62000 0.97194792 -0.0032022157 0.2866526
63000 0.99877655 -0.010154313 0.31701083
64000 0.98525211 -0.020415497 0.31774092
65000 0.9833149 0.0087192442 0.31256891
66000 0.97191797 0.0047184494 0.29880531
67000 0.98479592 -0.010779275 0.294197
68000 0.98418943 -0.035264623 0.29993828
69000 0.96953928 -0.028114432 0.27403611
70000 0.98373337 -0.057363336 0.25554163
71000 0.98260952 0.048742037 0.27102884
72000 0.98906053 0.010799224 0.26908376
73000 0.99569597 -0.0092675754 0.26927752
74000 0.97627362 0.013945821 0.2730712
75000 0.97276792 0.015036012 0.25847255
76000 0.97659072 -0.027078556 0.26783118
77000 0.9807196 -0.044553679 0.25993053
78000 0.97398601 -0.00027444729 0.26127735
79000 0.98124212 -0.012488833 0.27454966
80000 0.98506783 -0.015190822 0.2676633
81000 0.97926561 -0.012755191 0.27046398
82000 0.97906184 0.012564185 0.2835038
83000 0.96540502 -0.007372877 0.29622738
84000 0.98726761 0.021015365 0.32432233
85000 0.98133061 0.020043402 0.3540913
86000 0.99142106 0.025350024 0.36697674
87000 0.97917566 0.0061562414 0.37456288
88000 0.99391197 0.0068565008 0.39150922
89000 0.98521911 0.015500816 0.38121119
90000 0.97419059 0.00037609894 0.34181128
91000 0.98687774 -0.0073221495 0.35316892
92000 0.97816545 0.014057005 0.33181146
93000 0.98734859 0.016570523 0.34546487
94000 0.99116168 -0.0068196043 0.36316635
95000 0.99982024 0.076723346 0.38376316
96000 0.99232404 -0.048534983 0.38369657
97000 0.97829693 0.0041180664 0.38022523
98000 0.99878715 -0.00089133295 0.40289006
99000 0.97026354 -0.0039532716 0.43650647
100000 0.98911409 0.028647976 0.4499919
Loop time of 10.5771 on 8 procs for 100000 steps with 3200 atoms
Pair time (%) = 4.88134 (46.1499)
Neigh time (%) = 1.2657 (11.9664)
Comm time (%) = 1.78371 (16.8638)
Outpt time (%) = 0.0207323 (0.196011)
Other time (%) = 2.62565 (24.8239)
Nlocal: 400 ave 410 max 389 min
Histogram: 1 1 1 0 1 1 0 0 1 2
Nghost: 205.375 ave 216 max 194 min
Histogram: 1 0 0 2 2 0 1 0 0 2
Neighs: 2818 ave 3010 max 2683 min
Histogram: 2 0 2 0 1 1 0 1 0 1
Total # of neighbors = 22544
Ave neighs/atom = 7.045
Neighbor list builds = 12748
Dangerous builds = 0