1 watanabe@cc.u-tokyo.ac.jp 1 1.1 Nosé Hoover 1. ( 1) (a) (b) 1:
T ( f(p x, p y, p z ) exp p x + p y + p ) z (1) mk B T p x p y p = = z = 1 m m m k BT () k B T = 1.3 0.04 0.03 0.0 0.01 0-5 -4-3 - -1 0 1 3 4 5 Momentum : exp( p /(k B T )) T = 1.3 1.3 Γ = (q 1, q,, p 1, p, ) fdγ = 1 (3) f(γ) dγ dq 1 dq dp 1 dp f A A A Af dγ (4)
3 S S k B ln f = k B f ln fdγ (5) H(Γ) 1 H = E δs = 0 f = 1 Ω = δ(h E)dΓ (6) Ω H = E 0 H = U (7) β I = βu S (8) δi = 0 f f = Z 1 exp( βh) Z = exp( βh)dγ (9) f 0 exp( βh) (5) (7) f (9) S U ds = βk B du (10) ds = du T (11) β β = 1 k B T (1) T (1) H H = p x + p y + p z m + V (q x, q y, q z ) H p x = p x p x m (13) 1
4 H p x = Z 1 H p x exp( βh)dγ p x p x = Z 1 p x (exp( βh)) dγ ( β) p x = 1 px exp( βh)dγ β p x = 1 β p y p z () p x = m p y = m p z = 1 m β = k BT (14) (Local Equilibrium) 1.4 p /m = k B T q H = k B T (15) q (Kinetic Temperature) (Configuration Temeprature)
5.1 (Molecular Dynamics method, MD) T H E = H T C C = E T T E E = T 0 (16) CdT (17) C(T ) Z H = ln Z β = k BT ln Z T (16) (18) ( ). (Canonical Distribution) (Time Reversiblity) (Autonomous) (18)
6 (Ergodicity) (Efficiency) ( ) L. V. Woodcock Velocity Scaling method [1] Hoover Evans (Gaussian Thermostat )[] Extended System [3] Hoover Nosé Hoover[4] Nosé Hoover Berendsen [5] Nosé Poincaré[6] Nosé Poincaré.3 Nosé Hoover Nosé Hoover H(p, q) Nosé Hoover q = H p ṗ = H ζ = 1 τ p pζ ( p H p 1 β ββ = 1/(k B T )β τ (p, q, ζ) (p, q, ζ)dγ dpdqdζ f(p, q, ζ)dγ ) (19)
7 f f f t = divj ( ) = div Γf = (ṗf) p ( qf) q ( ζ)f ζ J Γf f f ( ṗ p + q q + ζ ) f + f ζ p ṗ + f f q + q ζ ζ = 0 ṗ q f ( ) H f ζf = p pζ p + H f q q + 1 ( τ p H p 1 ) f β ζ f = Z 1 exp [ β ( )] H + ζ τ 3 (p, q)f 0 f ζ 4 f 0 = fdζ = Z 1 0 exp( βh) f 0 (p, q) β Nosé Hoover (p, q) (p, q, ζ)f exp( β(h + ζ /(τ ))) 3 3.1 5 0 3 4 (p, q, ζ)(p, q) 5
8 H = p / + V (q) 1 { ṗ = V q q = p (0) L { ṗ = ilp 6 il = V q q = ilq p + p q (1) { p(t) = e itl p(0) q(t) = e itl q(0) (1) () (3) e itl = 1 + itl + (itl) + (4) p(0) q(0) p(t) q(t)u(t) = e ilt t tu( t) Ũ( t) { p(n t) = Ũ( t) n p(0) (5) q(n t) = Ũ( t)n q(0) Ũ( t) t U( t) L L K L U il K = p q il U = V q 6 i 1 i p (6)
9 il K q (il K ) q = il K ( p q q ) = il K p = p p q = 0 (il K ) = 0 (il U ) = 0 { e itl K = 1 + itl K e itl U = 1 + itl U (7) exp(ilt) = exp(i(l K + L U )t) exp(il K t) exp(il U t) exp(i t(l K + L U )) = exp(i tl K ) exp(i tl U ) + O( t) exp(i t(l K + L U )) = exp(i t L K) exp(i tl U ) exp(i t L K) + O( t ) 7 (Symplectic Integration) 8 H n O( t n ) 9 Ũ( t) = exp(il K t) exp(il U t) exp(il U t) exp(il K t) e i tl K = 1 + i tl K = 1 + tp q e i tl U = 1 + i tl U = 1 t V q p p p V q t q q + p t 7 1 1/ ( ) 8 9 1 (8)
10 q q + p t p p V q t q q + p t exp(i t L K) exp(i tl U ) exp(i t L K) exp(i t L U) exp(i tl K ) exp(i t L U) H Nosé Hoover ṗ = H q pζ q = H p ζ = 1 τ ( p 1 β L [ L = i p q q p pζ p + 1 ( τ p 1 ) ] β ζ pζ p ( exp pζ ) = p exp ( ζ) p p ) (9) (30) (31) q q + H t p (3) p p H t q (33) p p exp ( ζ t) (34) ζ ζ + 1 ( τ p 1 ) t (35) β
11 q q + H t (36) p p p exp ( ζ t/) (37) ζ ζ + 1 ( τ p 1 ) t (38) β p p H t (39) q ζ ζ + 1 ( τ p 1 ) t (40) β p p exp ( ζ t/) (41) q q + H p exp(i t L U) RESPA (REversible System Propagator Algorithm) [7] RESPA 3. Nosé Hoover H = 1 m t (4) p i + V (q 1, q, ) (43) i V Nosé Hoover ṗ i = V p i ζ q i q i = p i m ζ = 1 ( p ) (44) i τ m Nk BT i τ τ ( ) τ 3 Lennard-Jones N = 64000 Nosé Hoover
1 1.4 1.35 Without Thermostat With thermostat.5 Without Thermostat With thermostat 1.3 1.5 Kinetic Energy 1. 1.15 1.1 Power Spectra 1.5 1 1.05 1 0.5 0.95 0.9 40 50 60 70 80 90 100 110 10 130 Time 0 0 40 60 80 100 10 140 Frequency 3: Lennard-Jones 64000 Nose-Hoover NVE Nose-HooverNVT T = 1 ( ) T = 1.33 T = 1 τ τ [8] ( p ) T = 1 k B = 1 ṗ = q pζ q = p ζ = 1 τ (p 1) ζ ṗ = q pζ ζ p (45) τ ζ = pṗ (46) τ ζ = p( q pζ) = p ζ pq τ ζ = ζ(τ ζ + 1) pq (47) τ ζ + τ ζ ζ + ζ = pq (48) τ ζ pq() pq0 τ ζ + τ ζ ζ + ζ = 0 (49)
13 ζ q p 4: Nosé Hoover p ζ ζ(t) = τ 1 ζ(t/τ) (50) (49) ζ + ζ ζ + ζ = 0 (51) τ τ τ x = ζ, y = ζ ẋ = y (5) ẏ = x(y + 1) (53) E = x + y log(y + 1) x + y log(y + 1) = E ζ τ 3 Nosé Hoover Nosé Hoover Chain Kinetic-Moments (49) Nosé Poincaré Gaussian-Thermostat 3 Lennard-Jones
14 1.1 Tau=1 Tau=10 Tau=100 1.05 1 0.95 0.9 0.85 0 50 100 150 00 50 300 350 400 450 500 5: τ 3 Lennard-Jones T = 0.90.7 N = 68147 τ = 1, 10, 100 m T q i = p i m p i = V ζ = 1 τ ζp i q i ( 1 3N 3N i p i k B T ) (54) (44) 3N (44)τ (54) (54)5τ 1, 10, 100 τ τ 10 3.3 Nosé Hoover ( 6 ) Nosé Hoover Chain [9] Kinetic-Moments [10] Nosé Hoover Nos e Hoover [11, 1] 1 T A = lim A(t)dt (55) T T 0 10 exp( γt) sin(ωt) t/τ
15 0.8 q 0.4 0 ζ 0.08 0-0.08-0.4-0.8-0.8-0.4 0 0.4 0.8 p -0.8-0.4 p 0 0.4-0.4 0 q 0.4 0.8 6: Nosé Hoover (p, q) (p, q, ζ) E exp( E/k B T ) H = p / + q / Nosé Hoover (45) ṗ = q pζ q = p ζ = 1 τ (p 1) 3. p = r cos θ q = r sin θ ṙ = rζ cos θ ζ = 1 τ (r cos θ 1) τ r ζ θ cos θ 0 < t < π ṙ = πrζ ζ = π τ (r )
16 r + τ ζ ln r = C (56) C 6 τ ζ / 0 H = r /H H 1 ln H C (57) C C + ln (57) H H 0exp( βh) (3.3) Nosé Hoover Kinetic Moments Nosé Hoover Chain( ) 3.4 Nosé Hoover ( ) 1.. Nosé Hoover 3. 4. ( 7) / Langevin Langevin
17 熱を受け取りやすい場所 熱を受け取りにくい場所 熱流 7: 4 [1] L. V. Woodcock, Chem. Phys. Lett., 10, 57 (1971). [] W. G. Hoover, A. J. C. Ladd, and B. Moran, Phys. Rev. Lett., 48, 1818 (198); D. J. Evans, J. Chem. Phys, 78, 397 (1983). [3] S. Nosé, Mol. Phys. 5 55 (1984). [4] W. G. Hoover, Phys. Rev. A, 31, 1695 (1985). [5] H. J. C. Berendsen et al., J. Chem. Phys., 81, 3684 (1984). [6] S. D. Bond, B. J. Leimkuhler, and B. B. Lairdy, J. Comput. Phys., 151, 114 (1999). [7] M. Tuckerman, B. J. Berne, and G. J. Martyna, J. Chem. Phys., 97, 1990 (199) [8] H. Watanabe and H. Kobayashi, Molecular Simulation, 33, 77 (007). [9] G. J. Martyna, M. L. Klein, and M. Tuckerman, J. Chem. Phys., 97, 635 (199). [10] W. G. Hoover and B. L. Holian, Phys. Lett. A, 11, 53 (1996).
18 [11],, 6 10, 785 (007) [1] H. Watanabe and H. Kobayashi, Phys. Rev. E, 75, 04010(R), (007).