Nosé Hoover 1.2 ( 1) (a) (b) 1:

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1 Nosé Hoover 1. ( 1) (a) (b) 1:

2 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 = Momentum : exp( p /(k B T )) T = 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 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 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 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 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 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 ))) (p, q, ζ)(p, q) 5

8 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 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)

10 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 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 = Nosé Hoover

12 Without Thermostat With thermostat.5 Without Thermostat With thermostat Kinetic Energy Power Spectra Time Frequency 3: Lennard-Jones 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 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 Tau=1 Tau=10 Tau= : τ 3 Lennard-Jones T = N = 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 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 q ζ p p q : 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 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 ( 7) / Langevin Langevin

17 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 (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 18 [11],, 6 10, 785 (007) [1] H. Watanabe and H. Kobayashi, Phys. Rev. E, 75, 04010(R), (007).

構造と連続体の力学基礎

構造と連続体の力学基礎 II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われるすぐ横に道路用桁橋有りしかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton