Canonical Quantum Observables Approximated by Molecular Dynamics for Matrix Valued Potentials

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1 Canonical Quantum Observables Approximated by Molecular Dynamics for Matrix Valued Potentials Anders Szepessy, KTH Stockholm Can molecular dynamics determine canonical quantum observables for any temperature? Which stress and heat flux in the conservation laws?

2 Conservation of mass, momentum and energy1 tρ(y, t) + k (ρuk ) = 0 t ρu ) + k (ρu uk σk ) = 0 te + k (Euk σk u + qk ) = 0 Jokkfall in Kalixa lven Stress tensor σ =? Heat flux q =? 1 Euler (1752), Laplace (1816)

3 Stress tensor and heat flux from molecular dynamics: Ẋ t = P t P t = k V k (X t ) pair potential interaction Figure 1: solid-liquid phase transformation, von Schwerin & Szepessy (2010)

4 and Irving & Kirkwood (1950), Hardy (1981) definition ρ(y, t) := η(y X t )f(x 0, p 0 )dx 0 dp 0 ρu(y, t) := p(y, t) := E(y, t) := R 3 η(y)dy = 1 R 2N R 2N R 2N η(y X t )P t f(x 0, P 0 ) dx }{{} 0 dp 0 initial density η(y X t )( P t k V k )fdx 0 dp 0 X2 x X1 X X3 Figure 2: support of η with red particle positions

5 yields stress tensor σ(y, t) = 1 2 where R 2N b k (X t ) := R 2N,k (X t X k t ) V k b k f dx 0 dp 0 η(y X t ) ( P t u(y, t) ) ( P t u(y, t) ) f dx 0 dp 0, 1 0 η ( y (1 s)x ) t sxt k ds

6 Potential k V k =? Schrödinger equation with Hamiltonian: Ĥ = 1 M + V (x) x R N nuclei coordinates, V : R N C d2 potential d = 1: Schrödinger observables for mass, momentum and energy satisfy the conservation laws 2. 2 Irving and Zwanzig (1951) for scalar smooth potentials

7 Constant temperature and several electron states Ĥ = 1 M + V (x) Schrödinger: ĤΦ n = E n Φ n Goal determine n Φ n, ÂΦ n e E n/t Observable A and Temperature T M = 12800, δ = M 0.25 ψ t, V (X t )ψ t E λ + (X t ) λ (X t ) t Electron eigenvalueproblem V (x)ψ (x) = λ (x)ψ (x)

8 If λ 2 λ 1 T lim τ τ 0 A(X t, P t ) dt τ approximates n Φ n,âφ n e E n/t n Φ n,φ n e E n/t using Langevin: Ẋ t = P t P t = λ 1 (X t ) κp t + 2κT Ẇt. All T possible?

9 All T Theorem 3 possible: There holds where n Φ n, Â ê H/T Φ n n Φ n, ê H/T Φ n = lim τ Zt k = (X t, P t ) with λ k, q k q k = d i=1 q, i d τ q k k=1 0 Ã kk (Z k t ) dt τ, τ q k = lim τ 0 e λ k (X1 t ) λ 1 (X1 t ) T dt τ, Ψ (x)h(x, p)ψ(x) = H(x, p) Ψ (x)a(x, p)ψ(x) = Ã(x, p) diagonal,... + O( 1 M 1/2 T ) for e Ĥ/T. diagonal, 3 C. Lasser, M. Sandberg, A. Szepessy, A. Kammonen in preparation

10 Proof uses Weyl quantization: Âφ(x) = RN ( M 1/2 2π )N e im 1/2 (x y) p A( x + y, p)dp φ(y)dy, } R N {{ 2 } L 2 -kernel V (x) = V (x), p 2 2 = 1 2M, p 2 so H(x, p) = 2 I + V (x), Weyl s law: Φ n, ÂΦ n = trace  n = trace(l 2 -kernel) = ( M 1/2 2π )N tracea(x, p)dxdp R 2N

11 In fact also Φ n, Â ˆBΦ n = ( M 1/2 n 2π )N R 2N trace ( A(z)B(z) ) dz Choosing B = e H/T : and trace(ae H/T ) = trace(ãe H/T ) d = Ã kk e H kk /T, k=1 H kk (z) = p λ k(x), ê H/T = e Ĥ/T + O(M 1/2 T 1 ), q k from normalization

12 The quantum density, momentum and energy observables satisfy the conservation laws (Irving & Zwanzig, 1951) ρ(y, t) := trace (ˆρ t f(ĥ)) = n Φ n, ˆρ t f(ĥ)φ n ˆρ 0 = ( N η(y x ) ) =1 ˆρ t = e it MĤ ˆρ 0 e it MĤ density operator time evolution ĤΦ n = E n Φ n Schrödinger eigensolutions ˆp 0 := ( η(y x )p ) momentum operator Ê 0 := ( η(y x )( p V k ) ) 2 2 k (scalar) energy operator

13 Why is quantum same as classical? If m 2: t  t = i M[Ĥ, Ât] Heisenberg i M[Ĥ, A(x)pm ] = {H, A(x)pm } d = dt A(x t)p m t = time evolution xt =x,p t =p  (x)p m = 0 A (x)p 2 V A(x) m = 1 A (x)p 3 2V A (x)p m = 2

14 Matrix valued potential? i M[Ĥ, Â] = O(M 1/2 ) {H, A} = O(1) Seek Ât = ˆΨ(x) ˆÃ t ˆΨ(x) then t ˆÃ t = im 1/2 [ ˆΨ (x)ĥ }{{ ˆΨ(x), } ˆÃ t ] diagonal? ˆΨ (x)ĥ ˆΨ(x) = (Ψ HΨ + 1 4M Ψ Ψ) Choose Ψ so that: diagonal + O k (M k ), any k.

15 Then ˆρ 0 := ˆΨ ( N η(y x )I ) ˆΨ =1 ˆp 0 := ˆΨ ( η(y x )p I ) ˆΨ Ê 0 := ˆΨ ( η(y x ) H ) ˆΨ with diagonal energy per particle partition N H = H =1 density operator Theorem 4 : The Schödinger observables for the density, momentum and energy solve the conservation laws O k (M k ) accurately, any k. 4 also in preparation: M. Sandberg and A. Szepessy