修士論文物性研究電子版 Vol. 5, No. 2, (2016 年 5 月号 ) 27-2 F14A001B

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2 (Molecular Dynamics: MD) Verlet (Verlet Neighbor List: VNL ) (Cell Linked List: CLL ) (VNL-CLL ) VNL MD 10 VNL ( ) CLL ( ) CLL ( ) VNL O(N 2 ) CLL O(N 2 ) O(N) VNL-CLL VNL-CLL Verlet ( ). VNL ( ) i

3 i SKIN( ) VNL-CLL ii

4 1 (Molecular Dynamics: MD ) (Monte Carlo: MC ) MC MD MD [1 4] 1.1 MD MC 2 ( pair potential) 4 m N U N ({r i }) r i i {r i } 2 φ(r) / ε r / σ 1.1 Lennard-Jones(LJ) LJ 1

5 r 2 (pair potential) φ(r) U N ({r i })= 1 2 N i=1 i =j N φ( r i r j ) (1.1) 1/2 (i j) 2 U N i F i = U N (1.2) r i = 1 N φ(r ij ) 2 r i = 1 2 = = j =i N [ φ(rij ) j =i N j =i N j =i r i dφ(r ij ) dr ij dφ(r ij ) dr ij + φ(r ] ji) r i r ij r i r i r j r ij (r i r j )/r ij j i 2 Lennard-Jones(LJ) 1.1 LJ [ (σ ) 12 ( σ ) ] 6 φ(r) =4ɛ (1.) r r ɛ σ 1 2 r <σ r 1 F = φ(r) = dφ(r) dr r r = 24ɛ σ [ ( σ ) 1 ( σ ) ] 7 r 2 r r r LJ ( ) O(N 2 ) O(N 2 ) O(N) (1.4) 2

6 1.2 φ(r) LJ d 2 r dt 2 = F (t) m (1.5) MD Verlet [1 5] (1.5) Δt 2 t +Δt t Δt r(t ± Δt) r(t +Δt) =r(t)+δtv(t)+ (Δt)2 2 r(t Δt) =r(t) Δtv(t)+ (Δt)2 2 F (t) m + O((Δt) ) (1.6) F (t) m + O((Δt) ) (1.7) r(t +Δt)+r(t Δt) =2r(t)+(Δt) 2 F (t) m + O((Δt)4 ) (1.8) t +Δt t r(t +Δt) r(t Δt) =2Δtv(t)+O((Δt) ) (1.9) r(t +Δt) =2r(t) r(t Δt)+(Δt) 2 F (t) m + O((Δt)4 ) (1.10) v(t) = 1 2Δt {r(t +Δt) r(t Δt)} + O((Δt)2 ) (1.11) Verlet t +Δt t t Δt t =Δt r(δt) 1.7 r(0) r(2δt) (1.11) v(δt) (Δt Δt) (1.11) ( v t + Δt ) = 1 [r(t +Δt) r(t)] (1.12) 2 Δt (1.10) ( v t + Δt ) ( = v t Δt ) +Δt F (t) 2 2 m (1.1) (1.12) r(t +Δt) ( r(t +Δt) =r(t)+δtv t + dt ) 2 (1.14) Verlet (Verlet s leap frog) [1 5] (1.1), (1.14)

7 1.2 L = V 1/ V 1. ( ) CPU (Periodic Boundary Condition) 1.2 x, y, z L = V 1/ V ( 1.2 ) shadow 4

8 minimum image convention 1.4 MD r i (t) v i (t) [6 9] (a) 1 N N N E = φ( r i r j ) + 2 i=1 i =j i=1 1 2 mv2 i (1.15) (NVE ) (b) T (t) T (t) = 1 N mvi 2 (1.16) Nk B T (t) 1 T = T (t) t = lim T (t)dt (1.17) τ τ 0 t (c) Virial i W i F i m d2 r i dt 2 = F i + W i (1.18) i r i r i F i τ 0 i=1 τ [ r i d2 r i dt 2 dt = r i dr ] τ τ i dt 0 0 dr i dt dr i dt (1.19) dt 1 r i v i τ τ N ( ) m dri = 1 N r i (F i + W i ) dt 2 i=1 5 t i=1 t (1.20)

9 Clausiu Virial Virial Virial N N r i W i = Nk B T r i F i i=1 t i=1 t (1.21) Virial P i W i r ds n ds P nds Gauss (d) N r i W i i=1 P = Nk BT V = Nk BT V t + 1 V 1 V = S P n rds = PV (1.22) N r i F i i=1 N r i i U N ({r i }) i=1 t t (1.2) (1.24) 1 (r =0) (r, r + dr) dn(r) dn(r) =ρg(r)4πr 2 dr, ρ = N V (1.25) ρ g(r) r g(r) =1 r 0 g(r) =0 E = E id + E ex (1.26) E id = 2 Nk BT (1.27) E ex = U N ({r i }) = 1 Z N Z N = U N ({r i })exp( U N ({r i })/k B T )dr N (1.28) exp( U N ({r i })/k B T )dr N (1.29) g(r) E ex N =2πρ 6 0 φ(r)g(r)r 2 dr (1.0)

10 (1.24) (1.28) P ρk B T =1 2πρ k B T 0 φ (r)g(r)r dr (1.1) 1.5 ( ) LJ 2 O(N 2 ) O(N) O(N 2 ) 2 VNL-CLL (Verlet Neighbor List: VNL ) (Cell Linked List: CLL ) CLL (Exclusive CLL ) VNL CLL VNL O(N) 1 O(N 2 ) CLL O(N) VNL VNL-CLL VNL CLL 2 VNL VNL CLL 4 VNL CLL VNL-CLL 5 (VNL VNL CLL Exclusive CLL VNL-CLL ) CPU1 LJ NVE 100,000 Fujitsu PRIMERGY RX00 S7 CPU: Intel Xeon E Memory: 8GB Compiler: (R) 64 (R) C XE ( (R) 64 ) (C) Intel Corporation. σ m τ = mσ 2 /k B T ɛ 7

11 ɛ/k B Δt 0.005τ 8

12 2 2.1 r cut r r cut 4πρ rcut 0 drr 2 g(r) 4 πρr cut (2.1) ρ =1.0 LJ r cut = (1000 ) 2.1 SKIN = r list r cut (Verlet Neighbor List: VNL ) [10, 11] O(N 2 ) (VNL) SKIN = r list r cut r cut r list 9

13 O(N 2 ) O(N) VNL VNL Algorithm1 SKIN/2 ( ) r cut r list = r cut +SKIN point[i] i j i VNL j list[i] i=point[i] point[i+1] - 1 Algorithm2 j list[ ] 10 [12 14] Algorithm 2.2(a) v Algorithm 1 Make the Verlet Neighbor List 1: =====Save current configuration.===== 2: for i=0 to NumParticle do : rx0[i] rx[i] 4: ry0[i] ry[i] 5: rz0[i] rz[i] 6: end for 7: =====Make the Verlet Neighbor List.===== 8: (int)nlist 0 9: for i=0 to NumParticle - 1 do 10: point[i] nlist : for j=i+1 to NumParticle do 12: calculate r ij 1: if r ij <r list then 14: nlist nlist : list[nlist] j 16: if r ij <r cut then 17: calculate force 18: end if 19: end if 20: end for 21: end for 22: point[n-1] nlist

14 Algorithm 2 Use the Verlet Neighbor List 1: for i=0 to NumParticle - 1 do 2: j begin point[i] : j end point[i+1] - 1 4: nlist point[i] 5: if j begin <=j end then 6: for j nab =j begin to j end do 7: j list[j nab ] 8: calculate r ij 9: if r ij <r cut then 10: calculate force 11: end if 12: end for 1: end if 14: end for n 2.2(b) v ndt = SKIN/2 (2.2) SKIN/2 MD SKIN/2 VNL Algorithm4 (a) t = t 0 (b) t = t 0 + ndt 2.2 Verlet Neighbor List criterion (a)t = t 0 (b)n 11

15 Algorithm Check update of the Verlet Neighbor List 1: dispmx 0 2: for i=0 to NumParticle do : dispmx max ( rx[i] - rx0[i],dispmx) 4: dispmx max ( ry[i] - ry0[i],dispmx) 5: dispmx max ( rz[i] - rz0[i],dispmx) 6: dispmx.0 dispmx 7: end for 8: if dispmx > SKIN / 2 then 9: Make the Verlet Neighbor List 10: end if Algorithm 4 the Verlet Neighbor List 1: ==== MD loop start ===== 2: for step = 0 to nstep do : move r(t) r(t+dt) and v(t) v(t+dt/2) 4: Check update of the VNL 5: if Need update of the VNL then 6: Make the VNL and calculate force O(N 2 ) 7: else 8: Use the VNL and calculate force O(N) 9: end if 10: move v(t+dt/2) v(t+dt) 11: end for 2.2 SKIN( ) (2.2) max r(t 0 + ndt) r(t 0 ) > SKIN i 2 (2.) n dt t = t 0 SKIN/2 n n T ρ N r cut n+1 n (O(N)) n +1 (O(N 2 )) T VNL = 1 2 Nτ f [ N 1 n +1 + n ( rlist )] 4πρ drr 2 g(r) 1 n +1 0 τ f 1 2 VNL ( ) r list (N, T, ρ, r cut ) (2.) SKIN n (2.4) SKIN = 2n x (2.5) 12

16 x SKIN/2 x D l D x l D l D = Dδt l D r list = l th T 1 2 mv2 = 2 k BT ( ) v x l th l th = v δt v kb T x cδt (2.6) m c Maxwell c = ,000 10, ,000 g(r) =1 rlist 0 drr 2 g(r) r list 1 n +1 T VNL f VNL = 2τ f N(N 1) = 1 n +1 + n [ ] 1 4π n +1N 1 ρr list 1 = 1 n +1 + n n +1 1 N 1 [ 4π ρ(r cut +2n x ) 1 ] (2.7) (2.8) [15] 1 n +1 2 n r list VNL (2.2) n r list SKIN N = 1,000 N = 10,000 N = 100,000 7 p(v x ) v x 2. x T =0.772 ρ =0.8 N =1, 000, 10, 000, 100, kb T/m k B T/m 1

17 SKIN SKIN rlist 4πρ drr 2 g(r) 4 0 πρr list (2.9) O(N 2 ) O(N) ( O(N )) SKIN O(N 2 ) SKIN = N = 1,000 N = 10,000 N = 100,000 time [sec / N] SKIN = r list - r cut 5 6 (a) ( ) 1 ρ = 0.5 ρ = 0.8 ρ = T = 0.5 T = 1.0 T = time [sec / N] time [sec / N] time [sec / N] SKIN (b) SKIN (c) rcut = 2.5 rcut =.5 rcut = SKIN (d) 2.4 SKIN (2.8) (a) (b) (c) (d) (SKIN ) SKIN (O(N 2 )) SKIN 14

18 SKIN L O(N 2 ) SKIN SKIN (2.8) 2.4(a) (d) LJ NVE 100,000 1 SKIN (a) T =0.772 ρ =0.8 r cut =2.5 ( )N = 1, 000, 10, 000, 100, 000 SKIN SKIN = 0.5σ(N = 1, 000), 1.1σ(N =10, 000), 6.σ(N = 100, 000) n +1 VNL O(N 2 ) O(N) ( n ) n SKIN (b) N =1, 000 T =0.772 r cut =2.5 ρ =0.5, 0.8, 1.0 SKIN SKIN = 0.8σ(ρ =0.5), 0.6σ(ρ =0.8), 0.5σ(ρ =1.0).. O(N 2 ) SKIN (c) N =1, 000 ρ =0.8 r cut =2.5 T =0.500, 1.000, SKIN SKIN = 0.4σ(T =0.5), 0.5σ(T =1.0), 0.7σ(T =2.0) n SKIN (d) N =1, 000 T =1.000 ρ =0.8 r cut =2.5,.5, 4.5 SKIN SKIN = 0.6σ(r cut =2.5), 0.4σ(r cut =.5), 0.σ(r cut =4.5) ( O(N ) ) O(N 2 ) 15

19 SKIN 2.4(a) (d) SKIN (2.8) (2.6) c =.0 c =.0 Maxwell k B T/m ( 100,000 k B T/m 1/100, ,000 10, ) c SKIN (2.8) SKIN c =.0 N [1000, 100, 000] VNL O(N 2 ) (N >100, 000) 16

20 .1 L cell L cell r cut L cell.1 ( ) (Cell Linked List: CLL ) [10, 11] ( 100 ) O(N) r cut CLL Algorithm5 icell(ix, iy, iz) Algorithm6 imap (.2).1 (CLL) L cell r cut 17

21 Algorithm 5 Create the maps of the cell list (Cell Index) 1: int ix, iy, iz 2: int imap, mapsize, maps[mapsize] : int M ( ) 4: mapsize = 1 * M * M * M (1 M*M*M ) 5: ===== Initialize the array of the map ===== 6: for imap = 0 to mapsize do 7: map[imap] 0 8: end for 9: ===== create the maps of the cell list ===== 10: for iz=0 to M do 11: for iy=0 to M do 12: for ix=0 to M do 1: imap ( icellno(ix, iy, iz) ) * 1; 14: map[ imap + 1 ] icellno( ix+1, iy, iz ); 15: map[ imap + 2 ] icellno( ix+1, iy+1, iz ); 16: map[ imap + ] icellno( ix, iy+1, iz ); 17: map[ imap + 4 ] icellno( ix - 1, iy+1, iz ); 18: map[ imap + 5 ] icellno( ix+1, iy, iz - 1); 19: map[ imap + 6 ] icellno( ix+1, iy+1, iz - 1); 20: map[ imap + 7 ] icellno( ix, iy+1, iz - 1); 21: map[ imap + 8 ] icellno( ix - 1, iy+1, iz - 1); 22: map[ imap + 9 ] icellno( ix+1, iy, iz+1); 2: map[ imap + 10 ] icellno( ix+1, iy+1, iz+1); 24: map[ imap + 11 ] icellno( ix, iy+1, iz+1); 25: map[ imap + 12 ] icellno( ix - 1, iy+1, iz+1); 26: map[ imap + 1 ] icellno( ix, iy, iz+1); 27: end for 28: end for 29: end for Algorithm 6 function icellno(ix, iy, iz): Return the Index of the Cell 1: int ix, iy, iz 2: int icellno : int M ( ) 4: icellno mod(ix + M, M) +mod(iy+m,m)*m + mod(iz + M, M) * M * M (mod(a, b) a b ) 5: return(icellno) imap imap+1 imap N cell =( 1)/2 =1 N cell map[ ] MD imap (Algorithm7) [ L/2, L/2] [0, 1] i icell icell i head[icell]( 0 ) list[i] i head[icell] CLL 18

22 .2 imap imap+1 imap+1. icell i head[icell] i i list[i].4 head list position2 head[2] = 8 8 list[8] = list -1 O(N)..4 1 head[ ] 2 list[ ] head[2] = list[8] = Algorithm8 Algorithm5 Algorithm8 Algorithm9 19

Algorithm 7 Link the Particle Index to the Cell Index 1: int icell, i, j 2: int M ( ) : double Systemsize ( ) 4: double cellsize ( ) 5: cellsize Systemsize / (double)m 6: if cellsize < rcut then 7:

23 Algorithm 7 Link the Particle Index to the Cell Index 1: int icell, i, j 2: int M ( ) : double Systemsize ( ) 4: double cellsize ( ) 5: cellsize Systemsize / (double)m 6: if cellsize < rcut then 7: Error: Cell size is too small for cutoff radius. 8: end if 9: for i=0 to NumParticle do 10: dmyrxi rx[i] - floor( rx[i] + 0.5) 11: dmyryi ry[i] - floor( ry[i] + 0.5) 12: dmyrzi rz[i] - floor( rz[i] + 0.5) 1: icell (int)( ( dmyrxi ) / (double)m ) + (int)( ( dmyryi ) / (double)m ) * M +(int)((dmyrzi+0.5)/(double)m)*m*m 14: list[i] head[icell] 15: head[icell] i 16: end for.2 CLL O(N) ( 1, 000 ) VNL (a) (b) (c).5 : (a) 1 (f c =1,N cell = ) (b) 2 (f c =2,N cell =5 ) (c) (f c =,N cell =7 ) f c x, y, z N cell V ref V cutoff N cell 1 ( 1)/2 =1 2 (5 1)/2 =62 (7 1)/2 =

24 Algorithm 8 Use the cell linked List and calculate force 1: for icell=0 to Ncell do 2: ====== select i particle contain in each cells ====== : i head[icell] 4: while i >= 0do 5: ====== j particle contain in the current cell ====== 6: j list[i] 7: while j >= 0do 8: Caluculate r ij 9: if r ij <r cut then 10: Calculate force 11: end if 12: end while 1: j list[j] 14: ====== j particle contain in the neighbor cells ====== 15: jcell0 reference cell icell 16: for neighbor = 0 to reference cell do 17: jcell map[ jcell0 + neighbor ] 18: j head[ jcell ] 19: while j >= 0 do 20: Caluculate r ij 21: if r ij <r cut then 22: Calculate force 2: end if 24: j list[j] 25: end while 26: end for 27: i list[i] 28: end while 29: end for Algorithm 9 Cell Linked List 1: Create the map of the cell list. 2: ===== MD Loop Start ===== : for step = 0 to nstep do 4: move r(t) r(t+dt) and v(t) v(t+dt/2) 5: Link the particle index to the cell index. 6: Use the cell linked list and calculate force. 7: move v(t+dt/2) v(t+dt) 8: end for r cut ( V cutoff =4πr cut/ ) ( V ref = L cell ) L cell = r cut =2.5 V ref /V cutoff VNL VNL V ref =4πr list / r cut =2.5, r list = r cut VNL O(N 2 ) 21

25 Algorithm 10 Create the maps of the exclusive cell list (Cell Index) 1: int imap, mapsize, maps[mapsize], count 2: int M ( ) : int reference cell ( ), adjacent cell ( ) 4: refernce cell (2.0 adjacent cell+1) 1/2.0 5: =====Initialize the array of the map===== 6: for imap = 0 to mapsize do 7: map[imap] 0 8: end for 9: =====Create the array of the map===== 10: for iz=0 to M do 11: for iy=0 to M do 12: for ix=0 to M do 1: imap icell(ix, iy, iz) * reference cell 14: count 0 15: for jx= -adjacent cell to adjacent cell do 16: for jy= -adjacent cell to adjacent cell do 17: for jz= -adjacent cell to adjacent cell do 18: if jy > 0 then 19: count ++ 20: map[ imap + count] icell(ix + jx, iy + jy, iz + jz) 21: else if jy == 0 then 22: if jx >= 0 AND jx > 0 then 2: count ++ 24: map[ imap + count] icell(ix + jx, iy + jy, iz + jz) 25: else if jx >= 0 AND jz == 0 then 26: count ++ 27: map[ imap + count] icell(ix + jx, iy + jy, iz + jz) 28: else if jx == 0 AND jz < 0 then 29: count ++ 0: map[ imap + count] icell(ix + jx, iy + jy, iz + jz) 1: else 2:...nothing to do : end if 4: end if 5: end for 6: end for 7: end for 8: =====End of the adjacent cell loop===== 9: end for 40: end for 41: end for 42: =====End of the M loop===== L cell >r cut (f c = 1) (.1) r cut f c 1 >L cell > r cut f c (f c > 1) (.2) (.5) f c x, y, z f c =1 CLL (Exclusive cell linked list) [16,17]

26 .5(a) CLL 1 ( ) ( L cell ) r cut 2 2 r cut >L cell >r cut /2.5(b) L cell = L M M N =1, 000 ρ =0.8 T =1.0 r cut =2.5 L L cell /r cut [1.0772(M =4), 1.46(M =)] 1 (f c =1,.5(a)) L cell /r cut [0.478(M =9), (M =4)] 2 (f c =2,.5(b)) L cell /r cut [(M = 1), 0.478(M =9)] (f c =,.5(c)) M =4 L cell = L M = (.4) 4 M =5 L cell = L M = (.5) 5 CLL (f c =1) L cell r cut = (f c =2) 2 r cut >L cell r cut /2 2 (f c =2) 5 Algorithm10 (.). CLL, 5, 7 VNL 1. (Algorithm9) 2. (Algorithm7). (Algorithm8) 4. (Algorithm8) 5. (Algorithm8) 5 1,, 4, 5 2 2

27 1 1, 4, 5 1 T CLL = 1 2 (N cell +1) τ α (.6) ( ρl cell (N cell +1) 1 ) τ β πr cut + 4 N cell L cell τ γ [15] 1 2 τ α,τ β,τ γ 1 (.6) CLL 1 CLL (.6).6(a),(b) VNL LJ NVE 100,000.6(a) 2 L cell /r cut 0.478(9 ) (a) V ρ N L =(N/ρ) 1/ L cell (.6) 2 r cut =2.5 2 (b) r cut =4.5 (f c =) r cut =.5, (f c =2) r cut =1.5 1 (f c =1) (.6) r cut (.1) N =1, 000 ρ =0.8 L cell = L cell = L/M =10.772/ =.5906 r cut =4.5 r cut =

28 time [sec / N] time [sec / N] 10 1 ρ = 0.80 ρ = 0.60 ρ = L cell / r cut (a) r cut = 1.50 r cut = 2.50 r cut =.50 r cut = L cell / r cut (b).6 (.6) (a) (b) (SKIN) ( ) (a) 2 (b) r cut =1.5 (f c =) r cut =2.5 r cut =.5 2 (f c =2) r cut =4.5 1 (f c =1) 25

29 4 VNL-CLL 2 (Verlet Neighbor List: VNL ) (Cell Linked List: CLL ) VNL O(N 2 ) 10,000 n n r list (2.8) CLL ( ) CLL O(N) 10,000 (.6) VNL ( ) VNL O(N 2 ) CLL VNL-CLL 26

4.1 VNL-CLL VNL n n +1 (O(N 2 )) VNL CLL(O(N)) O(N) VNL VNL-CLL Algorithm11 Algorithm12, Algorithm1 4.1 VNL-CLL VNL CLL Verlet r list Algorithm 11 VNL-CLL 1: Create the map of the cell list.

30 4.1 VNL-CLL VNL n n +1 (O(N 2 )) VNL CLL(O(N)) O(N) VNL VNL-CLL Algorithm11 Algorithm12, Algorithm1 4.1 VNL-CLL VNL CLL Verlet r list Algorithm 11 VNL-CLL 1: Create the map of the cell list. 2: Make the VNL and calculate force. : ===== MD loop start ===== 4: for step = 0 to nstep do 5: move r(t) r(t+dt) and v(t) v(t+dt/2) 6: Check update of the VNL. 7: if Need update of the VNL then 8: =====Make the VNL using CLL===== 9: Link the particle index to the cell index. 10: Use the CLL and make the VNL. 11: Calculate force. 12: else 1: =====Use the VNL===== 14: Use the verlet neighbor list and calculate force. 15: end if 16: move v(t+dt/2) v(t+dt) 17: end for 27

31 Algorithm 12 VNL-CLL: Make the VNL using CLL 1: for icell = 0 to Ncell do 2: =====select i particle contain in each cell===== : i head[icell] 4: while i >= 0 do 5: =====j particle contain in the current cell===== 6: j cell list[i] 7: while j >= 0 do 8: calculate r ij 9: if r ij <r list then 10: neighbor list[i][point[i]] j 11: point[i] ++ 12: if r ij <r cut then 1: calculate force 14: end if 15: end if 16: j cell list[j] 17: end while 18: =====j particle contain in the neighbor cell===== 19: jcell0 reference cell * icell 20: while neighbor = 1 to reference cell do 21: jcell map[ jcell0 + neighbor ] 22: j head[jcell] 2: while j >= 0 do 24: calculate r ij 25: if r ij <r list then 26: neighbor list[i][point[i]] j 27: point[i] ++ 28: if r ij <r cut then 29: calculate force 0: end if 1: end if 2: j cell list[j] : end while 4: end while 5: i cell list[i] 6: end while 7: end for 8: for i=0 to NumParticle do 9: end point[i] point[i] 40: end for 1 CLL 2 Verlet VNL CLL VNL (Algorithm12) VNL (Algorithm1) VNL-CLL CLL CLL VNL-CLL r cut VNL r list L cell L cell >r list (f c = 1) (4.1) r list f c 1 >L cell > r list f c (r list = r cut + SKIN) (f c > 1) (4.2) 28

32 Algorithm 1 VNL-CLL: Use the VNL 1: for i=0 to NumParticle do 2: j begin 0 : j end end point[i] - 1 4: if j begin <=j end then 5: for j nab =j begin to j end do 6: j neighbor list[i][jnab] 7: if j > -1 then 8: calculate r ij 9: if r ij <r cut then 10: calculate force 11: end if 12: end if 1: end for 14: end if 15: end for 4.2 VNL CLL VNL-CLL 1 1 f VNL CLL = 1 T CLL n +1 N (τ CLL/τ VNL )+ n n +1 = 1 n +1 T CLL N CLL (Vref /V VNL ref )+ n n +1 ( 1 4 N 1 1 N 1 ) πρ(r cut +SKIN) 1 ( ) 4 πρ(r cut +2n x ) 1 VNL CLL 1 CLL T CLL τ CLL, τ VNL 1 CLL VNL r list VNL CLL 2r list (4.) τ CLL = V ref CLL τ VNL Vref VNL = (2r list) 4πrlist 1.91 (4.4) / VNL-CLL T ρ N r cut SKIN 4.2(b) 4.2(a) (4.) SKIN L cell SKIN VNL SKIN SKIN CLL O(N 2 ) CLL 29

33 r cut + SKIN L cell / r cut (a) r cut + SKIN L cell / r cut (b) (4.) 4.2 (SKIN ). N =1, 000 T =1.0, ρ =0.5, r cut =2.5 1 r list = r cut +SKIN r list =.1, L cell /r cut =0.42(f c = )((a) ) r list =.15, L cell /r cut =0.42(f c = )((b) 4.) SKIN SKIN 0

34 (a) ( 4.) T =1.0 ρ =0.8 r cut =2.5 ( )N = 500, 1, 000, 10, 000, 100, SKIN N L box n +1 1 VNL-CLL CLL O(N 2 ) O(N) VNL VNL O(N 2 ) CLL CLL (b) ( 4.4) N =1, 000 ρ =0.8 r cut =2.5 T =0.2, 0.5, 1.0, SKIN n SKIN VNL SKIN VNL-CLL SKIN SKIN VNL-CLL (c) ( 4.5) N =1, 000 T =1.0 r cut =2.5 ρ =0., 0.5, 0.8, SKIN L box = V/ρ O(N 2 ) SKIN SKIN (d) ( 4.6) N =1, 000 T =1.0 ρ =0.8 r cut =1.5, 2.0, 2.5,.0 4 SKIN SKIN r cut 1

35 rcut + SKIN L cell / rcut rcut + SKIN L cell / rcut N = 500 N =1, 000 rcut + SKIN L cell / rcut rcut + SKIN L cell / rcut N =10, 000 N = 100, (4.) SKIN T =1.0 ρ =0.8 r cut = rcut + SKIN rcut + SKIN L cell / rcut L cell / rcut 5 T =0.2 T = rcut + SKIN rcut + SKIN L cell / rcut L cell / rcut 5 T =1.0 T = (4.) SKIN N =1, 000 ρ =0.8 r cut =2.5 2

36 rcut + SKIN rcut + SKIN L cell / rcut L cell / rcut 5 ρ =0. ρ = rcut + SKIN rcut + SKIN L cell / rcut L cell / rcut 5 ρ =0.8 ρ = (4.) SKIN N =1, 000 T =1.0 r cut =2.5 rcut + SKIN L cell / rcut rcut + SKIN L cell / rcut r cut =1.5 r cut =2.0 rcut + SKIN L cell / rcut rcut + SKIN L cell / rcut r cut =2.5 r cut = (4.) SKIN N =1, 000 T =1.0 ρ =0.8

37 5 LJ Verlet (Verlet Neighbor List: VNL ) (SKIN) (Cell Linked List: CLL ) (Exclusive CLL ) VNL CLL VNL-CLL ( N, T, ρ, r cut ) (SKIN) SKIN= 0.5 VNL SKIN VNL CLL SKIN= 0.5 VNL-CLL SKIN VNL-CLL 5.1 (N <1000) O(N 2 ) CLL SKIN VNL VNL O(N 2 ) CLL O(N) VNL CLL VNL-CLL SKIN CLL 2 VNL O(N) VNL-CLL VNL CLL ( ) [18 20] VNL-CLL CLL (Pairwise Cell List) [21] ( ) [22 26] GPU [27, 28] VNL-CLL 4

38 Performance [atom steps / sec 10 6 ] VNL-CLL SKIN opt VNL-CLL SKIN = 0.5 CLL VNL SKIN opt VNL SKIN = Number of Particle 5.1 ( ) VNL-CLL ( )SKIN=0.5 VNL-CLL ( )CLL ( ) VNL ( )SKIN=0.5 VNL 5

39 6

40 [1] (2000). [2] (2001) [] (200) [4] (1988) [5] HOW TO (2004). [6] J. P. Hansen, I. R. Mcdonald, Theory of Simple Liquids, Fourth Edition, Elsevier (201). [7] D. J. Evans, Gary. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Cambridge University Press (2008). [8] M. E. Tuckerman, Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press (2010). [9] D. Frenkel, B. Smit, Understanding Molecular Simulation, Academic Press (2002). [10] M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press (2009). [11] D. C. Rapaport, The Art of Molecular Dynamics Simulation, Cambridge University Press (1995). [12] A. A. Chialvo and P. G. Debenedetti, On the use of the Verlet neighbor list in molecular dynamics, Computer Physics Communications 60 (1990) [1] A. A. Chialvo and P. G. Debenedetti, On the performance of an automated Verlet neighbor list alogritm for large systems on a vector processer, Computer Physics Communications 64 (1991) [14] A. A. Chialvo and P. G. Debenedetti, An automated Verlet neighbor list algorithm with a multiple time-step approach for the simulation of large systems, Computer Physics Communications 70 (1992) [15] G. Sutmann, V. Stegailov, Optimization of neighbor list techniques in liquid matter simulations, Journal of Molecular Liquids 125 (2006) [16] M. Isobe, Simple and efficient algorithm for large scale molecular dynamics simulation in hard disk sistem International Journal of Modern Physics C 10 (1999) [17] W. Mattson, B. M. Rice, Near-neighbor calculations using a modified cell-linked method, Computer Physics Communications 119 (1999) [18] OpenMP (2006) [19] C/C++ OpenMP (2009) 7

41 [20] (2010) [21] P. Gonnet, Pairwise Verlet Lists: Combining Cell Lists and Verlet Lists to Improve Memory Locality and Parallelism, Journal of Computational Chemistry (2012) [22] Z. Yao, J. S. Wang, G. R. Liu and M. Cheng, Improved neighbor list algorithm in molocular simulations using cell decomposition and data sorting method, Computer Physics Communications 161 (2004) [2] S. Meloni, M. Rosati, Efficient particle labeling in atomistic simulations, Journal of Computational Chemistry 126 (2007) [24] U. Welliing, G. Germano, Efficiency of linked cell algorithms, Computer Physics Communications 182 (2011) [25] H. Watanabe, M. Suzuki and N. Ito, Efficient Implementations of Molecular Dynamics Simulations for Lennard-Jones Systems, Progress of Theoretical Physics, 126 (2011) [26] H. Watanabe, M. Suzuki and N. Ito, Huge-scale Molecular Dynamics Simulation of Multibubble Nuclei, Computer Physics Communicaitons, 184 (201) [27] J. A. Anderson, C. D. Lorenz, A. Travesset, General purpose molecular dynamics simulations fully implemented on graphics processing units, Journal of Computational Physics, 227 (2008) [28] A. J. Proctor, C. A. Stevens and S. S. Cho, GPU-Optimized Hybrid Neighbor/Cell List Algorithm for Coarse-Grained MD simulations of Protein and RNA Folding and Assembly, Proceedings of the ACM Conference on Bioinformatics, Computational Biology and Biomedicine, (201)

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