A quantum heat equation 5th Spring School on Evolution Equations, TU Berlin

Transcription

1 A quantum heat equation 5th Spring School on Evolution Equations, TU Berlin Mario Bukal A. Jüngel and D. Matthes ACROSS - Centre for Advanced Cooperative Systems Faculty of Electrical Engineering and Computing University of Zagreb March 13, 2013

2 Introduction The equation The sixth order quantum diffusion equation tn = x (n x ( 1 2 ( 2 x log n) n 2 x(n 2 x log n)) ) on Ω (0, ), n(, 0) = n 0 on Ω. 1

3 Introduction The equation The sixth order quantum diffusion equation tn = x (n x ( 1 2 ( 2 x log n) n 2 x(n 2 x log n)) ) on Ω (0, ), n(, 0) = n 0 on Ω. Outline of the talk Origin (quantum background) Basic analysis existence of solutions, long time behaviour,... Other (interesting) properties formal gradient flow structure 1

4 Introduction Origin of the equation Quantum drift diffusion model [Degond et. al. 04.] tn = x(n x(a + V )) on R (0, ). [ A - quantum chemical potential in a nonlocal relation with particle density n; V - given external potential ] 2

5 Introduction Origin of the equation Quantum drift diffusion model [Degond et. al. 04.] tn = x(n x(a + V )) on R (0, ). [ A - quantum chemical potential in a nonlocal relation with particle density n; V - given external potential ] Approximation of A in terms of ε scaled Planck constant [ pseudo-differential calculus ] A = A 0[n] + ε 2 A 2[n] + ε 4 A 4[n] + O(ε 6 ), A 0[n] = log n, A 2[n] = 1 x 2 n (Bohm potential), 6 n A 4[n] = 1 ( ( 2 x log n) n 2 x(n x 2 log n)). 2

6 Introduction Origin of the equation Quantum drift diffusion model [Degond et. al. 04.] tn = x(n x(a + V )) on R (0, ). [ A - quantum chemical potential in a nonlocal relation with particle density n; V - given external potential ] Approximation of A in terms of ε scaled Planck constant [ pseudo-differential calculus ] A = A 0[n] + ε 2 A 2[n] + ε 4 A 4[n] + O(ε 6 ), A 0[n] = log n, A 2[n] = 1 x 2 n (Bohm potential), 6 n A 4[n] = 1 ( ( 2 x log n) n 2 x(n x 2 log n)). Approximated QDD model (for V = 0) tn = 2 xn heat eq. ( ( ε2 2 6 x n x n )) ( ( x + ε4 1 n 360 x n x 2 ( 2 x log n) )) n 2 x(n x 2 log n) DLSS eq. sixth order equation 2

7 Basic analysis of the equation Existence of solutions Semilinear form (equivalent for smooth positive solutions) tn = 6 xn + 3 xf 1[n] + 2 xf 2[n] on T (0, ), n(, 0) = n 0 on T. (QD6) with the nonlinear operators F 1[n] = 4 x n ( 4( x 4 n) x n ), F2[n] = 8 ( 2 x n 4( x 4 n) 2)2. 3

8 Basic analysis of the equation Existence of solutions Semilinear form (equivalent for smooth positive solutions) tn = 6 xn + 3 xf 1[n] + 2 xf 2[n] on T (0, ), n(, 0) = n 0 on T. (QD6) with the nonlinear operators F 1[n] = 4 x n ( 4( x 4 n) x n ), F2[n] = 8 ( 2 x n 4( x 4 n) 2)2. Theorem (Existence and uniqueness of a classical solution, [B., Jüngel, Matthes 12.]) n 0 H 2 (T) strictly positive. There exist T > 0 and unique smooth and strictly positive classical solution n C ((0, T ); C (T)) to (QD6) with n(t) n 0 in H 2 (T) as t 0. 3

9 Basic analysis of the equation Existence of solutions Semilinear form (equivalent for smooth positive solutions) tn = 6 xn + 3 xf 1[n] + 2 xf 2[n] on T (0, ), n(, 0) = n 0 on T. (QD6) with the nonlinear operators F 1[n] = 4 x n ( 4( x 4 n) x n ), F2[n] = 8 ( 2 x n 4( x 4 n) 2)2. Theorem (Existence and uniqueness of a classical solution, [B., Jüngel, Matthes 12.]) n 0 H 2 (T) strictly positive. There exist T > 0 and unique smooth and strictly positive classical solution n C ((0, T ); C (T)) to (QD6) with n(t) n 0 in H 2 (T) as t 0. Theorem (Global existence of weak nonnegative solutions, [B., Jüngel, Matthes 12.]) n 0 L 1 (T) of finite entropy H[n 0] <. There exists a nonnegative function n W 1,4/3 loc (0, ; H 3 (T)), satisfying n L 2 loc(0, ; H 3 (T)) and n(0) = n 0, that is a solution to (QD6) in the following weak sense: ( tn, ϕ dt + 3 x n xϕ 3 + F 1[n] xϕ 3 F 2[n] xϕ 2 ) dx dt = T for all test functions ϕ L 4 (0, T ; H 3 (T)) and given terminal time T > 0. 3

10 Analysis of the equation Global nonnegative weak solutions Functional inequality (entropy dissipation bound) d dt H[n(t)] κ ( 3 x n x n 6) dx, κ > 0, (t > 0), (EDB) T 4

11 Analysis of the equation Global nonnegative weak solutions Functional inequality (entropy dissipation bound) d dt H[n(t)] κ ( 3 x n x n 6) dx, κ > 0, (t > 0), (EDB) Sketch of the existence proof. semi-discrete problem T 1 τ (n n0) = 6 xn + 3 xf 1[n] + 2 xf 2[n] + ε( 6 x log n log n) change of variables n = e y and regularization in y linearization by means of the fixed point formulation, Leray-Schauder: n ε = e yε > 0 deregularization (ε 0): n ε n [ (EDB), Sobolev embeddings. ] passage to the continous limit (τ 0) [ (EDB), Gagliardo-Nirenberg inequality, Aubin s lemma. ] 4

12 Analysis of the equation Global nonnegative weak solutions Functional inequality (entropy dissipation bound) d dt H[n(t)] κ ( 3 x n x n 6) dx, κ > 0, (t > 0), (EDB) Sketch of the existence proof. semi-discrete problem T 1 τ (n n0) = 6 xn + 3 xf 1[n] + 2 xf 2[n] + ε( 6 x log n log n) change of variables n = e y and regularization in y linearization by means of the fixed point formulation, Leray-Schauder: n ε = e yε > 0 deregularization (ε 0): n ε n [ (EDB), Sobolev embeddings. ] passage to the continous limit (τ 0) [ (EDB), Gagliardo-Nirenberg inequality, Aubin s lemma. ] Exponential convergence in time. n(t; ) 1 L 1 (T) 2H[n 0]e 16π6 κt for all t > 0. [ log-sobolev inequality n log n dx 1 T 32π x n dx, Csiszár-Kullback-Pinsker inequality. ] T 4

13 Other (interesting) properties Lyapunov functionals entropies dissipation of α-functionals (along smooth positive solutions) 1 E α[n] = (n α αn + α 1) dx, α 0, 1, α(α 1) T d dt Eα[n(t)] ( κα 3 x n x n 6) dx, t > 0 T for < α < dissipation of the Fisher information (along smooth positive solutions) F[n] = 2 x n 2 d dx; F[n(t)] 0, t > 0. dt T 5

14 Other (interesting) properties Lyapunov functionals entropies dissipation of α-functionals (along smooth positive solutions) 1 E α[n] = (n α αn + α 1) dx, α 0, 1, α(α 1) T d dt Eα[n(t)] ( κα 3 x n x n 6) dx, t > 0 T for < α < dissipation of the Fisher information (along smooth positive solutions) F[n] = 2 x n 2 d dx; F[n(t)] 0, t > 0. dt formal representation tn = x ( T δe[n] n x δn ) ; E[n] = 1 n x 2 log n 2 dx. 2 T 5

15 Other (interesting) properties Lyapunov functionals entropies 5 dissipation of α-functionals (along smooth positive solutions) 1 E α[n] = (n α αn + α 1) dx, α 0, 1, α(α 1) T d dt Eα[n(t)] ( κα 3 x n x n 6) dx, t > 0 T for < α < dissipation of the Fisher information (along smooth positive solutions) F[n] = 2 x n 2 d dx; F[n(t)] 0, t > 0. dt formal representation tn = x ( T δe[n] n x δn ) ; E[n] = 1 n x 2 log n 2 dx. 2 T gradient flows with respect to the L 2 -Wasserstein distance [Ambrosio, Gigli, Savaré 05.] tn + x(n xv) = 0, v = xψ, Ψ = δe[n] δn

16 Other (interesting) properties Gradient flow representation Recall the approximated quantum diffusion model ( ( tn = xn 2 ε2 2 6 x n x n )) ( ( x + ε4 1 n 360 x n x 2 ( 2 x log n) )) n 2 x(n x 2 log n) Formal representation ( δh[n] ) ( tn = x n x + ε2 δf[n] ) ( δn 12 x n x + ε4 δe[n] ) δn 360 x n x δn heat eq. [JKO 98.] DLSS eq. [GST 06.] sixth order eq. [?] 6

17 Other (interesting) properties Gradient flow representation Recall the approximated quantum diffusion model ( ( tn = xn 2 ε2 2 6 x n x n )) ( ( x + ε4 1 n 360 x n x 2 ( 2 x log n) )) n 2 x(n x 2 log n) Formal representation ( δh[n] ) ( tn = x n x + ε2 δf[n] ) ( δn 12 x n x + ε4 δe[n] ) δn 360 x n x δn Dissipation identities heat eq. [JKO 98.] DLSS eq. [GST 06.] sixth order eq. [?] d dt H[n] = 1 2 F[n] along the heat flow d dt d2 F[n] = 2 H[n] = 8E[n] dt2 d H[n] = 8E[n] dt along the DLSS flow along the heat flow 6

18 Thank you for your attention! Danke für Ihre Aufmerksamkeit!

19 QDD Model - Outline of the derivation starting point - collisional quantum Liouville-von Neumann equation: iε tϱ = [H, ϱ] + iεq(ϱ), H = ε2 V - Hamiltonian, [H, ϱ] = Hϱ ϱh, V - electric potential 2 ϱ - density operator (trace class hermitian operator on L 2 (R d )), ε = λ db L - scaled Planck constant, λ db - de Broglie wavelength, L - reference length Q - collision operator Wigner Transform: W : L 2 = {ϱ Tr{ϱϱ } < } L 2 (R 2d ) W (ϱ)(x, p) = ϱ (x + ε 2 ξ, x ε ) 2 ξ e iξ p dξ, (ϱϕ)(x) = ϱ(x, x )ϕ(x )dx, ϕ L 2 (R d ), ϱ - integral kernel of ϱ. 7

20 QDD Model - Outline of the derivation Wigner Transform of the Liouville-von Neumann eq. iε tϱ = [H, ϱ] + iεq(ϱ) Wigner-Boltzmann equation: tw + p xw + Θ[V ]w = Q(w), x, p R d, t > 0, with w = W (ϱ), (Θ[V ]w)(x, p, t) = ( i (2π) d ε R V (x + ε 2d 2 ξ, t) V (x ε 2 ξ, t)) w(x, p, t)e i(p p ) ξ dp dξ. specify the collision operator Q(w): quantum entropy of the quantum mechanical state is given by: 1 H(w) = (2πε) d w(x, p, ) ((Log w)(x, p, ) 1 + p 2 ) 2 V (x, ) dxdp, with Log w = W (log W 1 (w)) quantum logarithm 8

21 QDD Model - Outline of the derivation Wigner Transform of the Liouville-von Neumann eq. iε tϱ = [H, ϱ] + iεq(ϱ) Wigner-Boltzmann equation: tw + p xw + Θ[V ]w = Q(w), x, p R d, t > 0, with w = W (ϱ), (Θ[V ]w)(x, p, t) = ( i (2π) d ε R V (x + ε 2d 2 ξ, t) V (x ε 2 ξ, t)) w(x, p, t)e i(p p ) ξ dp dξ. specify the collision operator Q(w): quantum entropy of the quantum mechanical state is given by: 1 H(w) = (2πε) d w(x, p, ) ((Log w)(x, p, ) 1 + p 2 ) 2 V (x, ) dxdp, with Log w = W (log W 1 (w)) quantum logarithm define quantum Maxwellian as minimizer of H(w) under constraint that particle density u(x, t) = 1 (2πε) d w(x, p, t)dp is given 8

22 QDD Model - Outline of the derivation formal solution of the constrained minimization problem: M[w](x, p, t) = Exp ( A(x, t) p 2 ), 2 A Lagrange multiplier, Exp w = W (exp W 1 (w)) quantum exponential define Q(w) = M[w] w properties of Q: 1. collisional invariant: Q(w)dp = 0 2. null-space: Q(w) = 0 w = M[w] 3. quantum entropy decay: ( Q(w) Log w + p 2 ) 2 V (x, ) dxdp 0 9

23 QDD Model - Outline of the derivation consider diffusion scaling, i.e. replace the time t and Q(w) by t/α and Q(w)/α, respectively Wigner-Boltzmann-Poisson system: α 2 tw α + α(p xw α + Θ[V α]w α) = M[w α] w α, x, p R d, t > 0, w α(,, 0) = w 0, λ 2 D V α = 1 (2πε) d w α(x, p, t)dp C(x) 10

24 QDD Model - Outline of the derivation Theorem (Nonlocal quantum drift-diffusion equations) Let (w α, V α) be a solution of the Wigner-Boltzmann-Poisson system. Then, formally, w α w and V α V as α 0, where w(x, p, t) = Exp(A(x, t) p 2 /2) and (A, V ) is a solution of the quantum drift-diffusion equations u t div J u = 0, J u = u (A V ), λ 2 D V = u C(x), t > 0. u(, 0) = u 0 in R d, The particle density u and the Lagrange multiplier A are related through 1 ) u(x, t) = Exp (A(x, t) p 2 dp. (2πε) d 2 11

25 QDD Model - Outline of the derivation Sketch of the proof. perform the limit α 0 for the moment equation: α t w αdp + div x pw αdp + Θ[V α]w αdp = α 1 Q(w α)dp = 0 use certain properties of the moments of Θ[V α] employ the Chapman-Enskog method 12

26 QDD Model - Entropy dissipation property recall quantum kinetic entropy is given by 1 ) H(w) = w(x, p, ) ((Log w)(x, p, ) 1 + p 2 (2πε) d R 2d 2 V (x, ) dxdp quantum fluid entropy is obtained inserting w 0 = Exp(A p 2 /2) Log w 0 = A p 2 /2: 1 H(w 0) = w 0(x, p, )(A 1 V )dxdp (2πε) d R 2d = u(a 1 V )dx =: S(u) R d 13

27 QDD Model - Entropy dissipation property Theorem (Degond, Ringhofer, 2004) If u is a solution of the nonlocal quantum drift-diffusion equations, for given potential V, then d dt S(u) u tv dx. R d Proof. since u(x, t) = 1 (2πε) d Exp ) (A(x, t) p 2 dp u t = ua t 2 straightforward computation showes d dt S(u) + u (A V ) 2 dx = u tv dx 14

28 QDD Model - O(ε 6 )-approximation expansion of quantum exponential in powers of ε O(ε 6 )-approximation of the Lagrange multiplier A: A = log u + d log( 2π) ε2 6 + ε4 360 u u ( log u u 2 : (u 2 log u)) + O(ε 6 ) local quantum (drift-)diffusion equation with zero electric field (V = 0): ( ( u t = u ε2 u )) 6 div u u + ε4 360 div u(, 0) = u 0 ( u ( log u u 2 : (u 2 log u)) ) + O(ε 6 ) in R d 15

29 QDD Model - O(ε 6 )-approximation entropy dissipation property of the local quantum diffusion equation: recall the quantum fluid entropy: S(u) = u(a 1)dx inserting O(ε 6 )-approximation of A O(ε 6 )-approximation of the quantum fluid entropy: Rd ( ε 26 S 4 (u) = u(log u 1) + u 2 + ε4 720 u 2 log u 2) dx S 4 is a Lyapunov functional of the local quantum diffusion equation, i.e. d ( dt S4(u) + log u ε2 u 6 u R d u + ε4 360 ( log u )) u 2 : (u 2 2 log u) = 0 16

30 Related PDEs recall the local quantum diffusion equation: (u ( u t = u ε2 6 div u )) (u ( + ε4 u 360 div 12 2 log u 2 + u 1 2 :(u 2 log u) O(ε 2 )-approximation - semiclassical limit, ε 0: u t = u O(ε 4 )-approximation - setting ε 2 = 6 and neglecting other terms the Derrida-Lebowitz-Speer-Spohn equation: ( ( u )) u t + div u = 0 u O(ε 6 )-approximation - setting ε 4 = 360 and neglecting other terms: ( 1 u t div u ( 2 2 log u u 2 : (u 2 log u)) ) = 0 )) 17

31 Related PDEs - particular structure ( u t = div u δφ ) δu (u), Φ flow the heat equation, u t = u: Φ(u) = u(log u 1)dx = E 1(u), δφ (u) = log u δu the Derrida-Lebowitz-Speer-Spohn equation, u t + div ( u ( u u )) = 0: Φ(u) = 1 u log u 2 dx = u 2 dx = F 1(u), 4 δφ δu (u) = u u the sixth-order equation, u t div(u ( log u 2 + u 1 2 :(u 2 log u)))=0: Φ(u) = 1 u 2 log u 2 δφ dx, 2 δu (u) = log u u 2 : (u 2 log u) 18

32 Related PDEs - dissipation of the physical entropy the heat equation, u t = u: de 1(u) = u log u 2 dx de1(u) dt dt + 4F 1(u) = 0 the Derrida-Lebowitz-Speer-Spohn equation, u t + div ( u ( u u )) = 0: de 1(u) dt = 1 2 u 2 log u 2 dx the sixth-order equation, u t div(u ( log u u 2 :(u 2 log u)))=0: de 1(u) dt = [u 3 log u 2 2u( 2 log u) 2 : 2 log u ] dx 19

33 Related PDEs - the sixth-order equation ( 1 u t div u ( 2 2 log u u 2 : (u 2 log u)) ) = 0 u(, 0) = u D case - periodic boundary conditions: 1 dissipation of generalized entropies E α(u) = α(α 1) u α dx [Jüngel, Matthes]: de for α , E α are entropies, i.e. α(u) 0. dt global-in-time existence of weak nonnegative solutions [Jüngel, Milišić]: result based on an exponential variable transformation (u = exp(y)) and the entropy-entropy production inequality: de 1 (u) dt + c (( u) 2 xxx + ( 4 u) 2 x ( 4 u) 2 xx + ( 6 u) 6 x) dx 0, c > 0. 20

34 Related PDEs - the sixth-order equation radially symmetric case: 1 dissipation of generalized entropies E α(u) = α(α 1) r d 1 u α dr: E α are entropies for: d = 2, α , d = 3, α , d = 4, α in particular, the following entropy-entropy production inequality holds: de 1 (u) + c r d 1 ( r u) 2 dt r dr 0, c > 0. multi-dimensional case - periodic boundary conditions (expectations): dissipation of variety of generalized entropies E α(u) = u α dx 1 α(α 1) 21

35 Related PDEs - the sixth-order equation radially symmetric case: 1 dissipation of generalized entropies E α(u) = α(α 1) r d 1 u α dr: E α are entropies for: d = 2, α , d = 3, α , d = 4, α in particular, the following entropy-entropy production inequality holds: de 1 (u) + c r d 1 ( r u) 2 dt r dr 0, c > 0. multi-dimensional case - periodic boundary conditions (expectations): 1 dissipation of variety of generalized entropies E α(u) = α(α 1) u α dx the entropy-entropy production inequality de 1 (u) dt + c ( 3 u u u u 6) dx 0, c > 0. 21