Derivation of quantum hydrodynamic equations with Fermi-Dirac and Bose-Einstein statistics Luigi Barletti (Università di Firenze) Carlo Cintolesi (Università di Trieste) 6th MMKT Porto Ercole, june 9th 2012
Madelung equations The theory of quantum fluid equations dates back to 1926, when E. Madelung discovered the hydrodynamic form of Schrödinger equation: t n + div(nu) = 0, t u + 1 2 u 2 + (V 2 2 where ψ = n e is/ and u = S (and m = 1). n n ) = 0, E. Madelung, Quantentheorie in hydrodynamischer Form. Zeitschr. f. Phys. 40, 322 326 (1926)
Bohm potential Madelung equations look like an irrotational compressible Euler system with an additional term containing the Bohm potential 2 2 n n, named after David Bohm, who based on it his famous interpretation of quantum mechanics. D. Bohm, A suggested interpretation of the quantum theory in terms of hidden variables. Physical Review 85, 166 193 (1952)
Wigner functions An kinetic derivation of Madelung equations can be obtained by using the Wigner function 1 w(x, p, t) = (2π ) d/2 ψ (x + ξ2 ), t ψ (x ξ2 ), t e ip ξ/ dξ, R d and writing the equations for the moments n(x, t) = w(x, p, t) dp, nu(x, t) = p w(x, p, t) dp. E. Wigner, On the quantum correction for thermodynamic equilibrium. Physical Review 40, 749 759 (1932)
Mixed states Madelung equations hold for a pure state, described by a wave function ψ, but for a mixed (statistical) state the system is not formally closed. However, the derivation of quantum fluid models for collisional systems, necessarily requires a statistical description and, therefore, the problem arises of generalizing Madelung equations to such a situation. The question is not merely academic, because quantum fluid models can be of great interest for nanoelectronics.
Quantum hydrodynamics Indeed, a renewed interest for the subject dates back to the half of nineties with the work of C. Gardner, who proposed a quantum hydrodynamic model based on a local quantum Maxwellian obtained from Wigner s O( 2 ) corrections to thermal equilibrium. C.L. Gardner, The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54(2), 409 427 (1994)
Quantum entropy principle But it is only with the work of Degond and Ringhofer that the problem was set on a solid theoretical basis with the elaboration of the quantum version of the maximum entropy principle (QMEP). P. Degond, C. Ringhofer, Quantum moment hydrodynamics and the entropy principle. J. Stat. Phys. 112(3-4), 587 628 (2003) The QMEP has been exploited to generate several quantum fluid models of various kind.
Quantum fluid models from the QMEP (I) P. Degond, F. Méhats, C. Ringhofer, Quantum energy-transport and drift-diffusion models, J. Stat. Phys., 2005. A. Jüngel, D. Matthes, A derivation of the isothermal quantum hydrodynamic equations using entropy minimization, Z. Angew. Math. Mech., 2005. N. Ben Abdallah, F. Méhats, C. Negulescu. Adiabatic quantum-fluid transport models, Commun. Math. Sci., 2006. A. Jüngel, D. Matthes, J. P. Milisic, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 2006. P. Degond, S. Gallego, F. Méhats. Isothermal quantum hydrodynamics: derivation, asymptotic analysis, and simulation, Multiscale Model. Simul., 2007. P. Degond, S. Gallego, F. Méhats, An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes, J. Comput. Phys., 2007.
Quantum fluid models from the QMEP (II) S. Brull, F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 2010. L. B., F. Méhats, Quantum drift-diffusion modeling of spin transport in nanostructures. J. Math. Phys., 2010. L. B., G. Frosali, Diffusive limit of the two-band k p model for semiconductors. J. Stat. Phys., 2010. N. Zamponi, L. B., Quantum electronic trasport in graphene: a kinetic and fluid-dynamical approach. M2AS, 2011. N. Zamponi, Some fluid-dynamic models for quantum electron transport in graphene via entropy minimization. KRM, 2012. A. Jüngel. Transport Equations for Semiconductors. Springer, 2009. A. Jüngel. Dissipative quantum fluid models. Riv. Mat. Univ. Parma, 2012.
Different statistics Although the QMEP was originally stated for a general (convex) entropy functional, nevertheless, it has been applied only to Boltzmann entropy. Partial exceptions: P. Degond, S. Gallego, F. Méhats, An entropic quantum drift-diffusion model for electron transport in resonant tunneling diodes, J. Comput. Phys., 2007. M. Trovato, L. Reggiani, Quantum maximum entropy principle for a system of identical particles. Phys. Rev. E, 2010. A. Jüngel, S. Krause, P. Pietra, Diffusive semiconductor moment equations using Fermi-Dirac statistics. ZAMP, 2011. The work we are going to expose is exactly intended to fill this gap.
The kinetic model The starting point is the Wigner equation with BGK collisional term ( t + p m x + Θ [V ] ) w = g[w] w, τ where: w(x, p, t), is the Wigner function of the system; Θ [V ] = i [ ( V x + i 2 ) ( p V x i 2 p)] ; τ is a typical collisional time; g[w] is the local equilibrium state given by the QMEP.
Scaling the Wigner-BGK equation After a suitable scaling, we can write the Wigner-BGK equation as follows: ( t + p x + Θ ɛ [V ]) w = 1 (g[w] w) α α = τ t 0 (hydrodynamic parameter); ɛ = x 0 p 0 (semiclassical parameter).
QMEP We assume that g[w] is given by the QMEP: g[w] is the most probable state compatible with the information we have about it: g[w] has a constant temperature T g[w] has the same density and current as w: g[w] = n = w, pg[w] = J = pw. Most probable means that g[w] minimizes an entropy functional H. Then, g[w] is chosen as a minimizer of H among all Wigner functions that share with w the same moments n and J.
Entropy functional In terms of density operators, the suitable entropy functional for the isothermal case is the free-energy { H(ϱ) = Tr T [ϱ log ϱ + 1λ ] } (1 λϱ) log(1 λϱ) + Hϱ, which also incorporates the information on particle statistics: 1, Fermi-Dirac λ = 0, Maxwell-Boltzmann 1, Bose-Einstein
Constrained minimization problem The Wigner function g[w] satisfies the QMEP if: ϱ = Op ɛ (g[w]) is a minimizer of H under the constraints g[w] = n = w, pg[w] = J = pw. Here Op ɛ denotes the Weyl quantization, mapping 1-1 Wigner functions and density operators.
The minimizer Theorem A necessary condition for g = g[w] to be solution of the constrained minimization problem is that Lagrange multipliers A and B = (B 1,..., B d ) exist such that Op ɛ (g) = (e H(A,B) T + λ) 1, where ( ) H(A, B) = Op p B 2 ɛ 2 A, and A, B have to be determined as functions of n, J from the moment constraints.
Hydrodynamic limit Theorem In the hydrodynamic limit α 0, the solution w α of the Wigner-BGK equation tends to the local equilibrium state g[w 0 ], whose moments n and J satisfy the equations t n + J j = 0, x j t J i + x j pi p j g[w 0 ] + n x i V = 0.
Formal closure The unknown moment p i p j g[w] can be expressed in terms of the Lagrange multipliers as follows: x j pi p j g[w] = (nu i B j ) + ( ) B j nuj B j + n A. x j x i x i This provides a formal (and rather implicit) closure of the quantum hydrodynamic equations because A, B are depend on n, J through the moment constraints.
Difficulties with g[w] From now on, we shall simply denote by g the local equilibrium state g[w]. The phase-space function g is a very complicated object, involving back and forth Weyl quantization: g = Op ɛ 1 { exp [Op ɛ ( p B 2 2T A T )] + λ } 1 The only hope we have to get something explicit is expanding g semiclassically, i.e. in powers of ɛ.
Semiclassical expansion of g Skipping all technical details, we find that g has the following expansion g = g (0) + ɛ 2 g (2) + O(ɛ 4 ) where g (0) is the classical distribution g (0) = 1 e (p B)2 2T A T + λ and g (2) is a complicated expression involving A, B and their derivatives.
Semiclassical expansion of A and B What we really need is the expansion of A, B as functions of n, J, as it results from the constraints g = n and pg = J. Using the expansion of g, it turns out that A = A (0) + ɛ 2 A (2) + O(ɛ 4 ), B = B (0) + ɛ 2 B (2) + O(ɛ 4 ). At leading order we obtain ( A (0) = T φ 1 d 2 n (2πT ) d/2 ), B (0) = u = J/n, where φ s (z) = 1 λ Li s ( λe z ) = 1 + t (s 1) Γ(s) 0 e t z + λ dt.
Condition for invertibility 8 7 6 5 4 3 2 1 FD, d=3 MB, d=3 BE, d=3 BE, d=2 BE, d=1 Plots of φ d for some 2 values of λ and d. For λ < 0 and d 3, ranges from 0 to φ d 2 ζ ( ) d 2 / λ. 0 2 1.5 1 0.5 0 0.5 1 1.5 2 In the BE case, we have to assume n < (2πT ) d 2 ζ ( d 2 ) / λ. Exceeding particles fall in the condensate phase, not described by our model.
A and B: second-order expressions A (2) = 1 [ i u j ( i u j j u i ) 2 i i A (0)] φ 0 d 2 2 1 24T φ 0 d 24 2 1 B (2) i = n [ ] d 12Tn j ( i u j j u i )φ 0 d 2 1 ( i A (0) T ) 2 φ 0 d 2 3 φ 0 d 2 1 where and i := x i ( A φ 0 (0) ) ( s := φ s = φ T s φ 1 d 2 ( n n d ))
Semiclassical hydrodynamic equations (I) We can now substitute the expansions A A (0) + ɛ 2 A (2) and B B (0) + ɛ 2 B (2) in the quantum hydrodynamic equations t n + x j (nu j ) = 0, t (nu i) + x j (nu i B j ) + x i B j ( nuj B j ) + n x i (A + V ) = 0.
Semiclassical hydrodynamic equations (II) We obtain therefore our main result: t n + (nu j ) = 0, x j t (nu i) + (nu i u j ) + Tn ( n ) φ 1 d + n ( ) V + ɛ 2 Q(n) x j x i 2 n d x i = ɛ2 n 24T R jkr kj φ x i 2 φ 0 d 0 d 2 2 + 2 1 ɛ 2 n d 12T R ( ij R jk φ 0 d x k 2 1 ), where n d = (2πT ) d 2, Rij := x j u i x i u j
Modified Bohm potential The term ɛ 2 Q(n) can be identified as a modified Bohm potential: Q(n) = 1 2 φ 1 d 24 2 ( n n d )φ 0 d 2 2 φ 0 + φ 1 d d 2 1 2 ( n n d ) 2 φ0 d 2 3 φ 0 d 2 1, since lim Q(n) = 1 n. λ 0 6 n
Limit properties of φ s From known asymptotic properties of the polylogarithms we obtain: lim φ s(z) = e z, λ 0 φ s (z) e z, for z R and s R for z, λ R and s R φ s (z) z s, for z +, λ = 1 and s 1, 2,... Γ(s + 1) φ s (z) Γ(1 s)( z) s 1, for z 0, λ = 1 and s < 1
The Maxwell-Boltzmann limit In the M-B limit λ 0 the semiclassical equations become t n + (nu j ) = 0, x j t (nu i) + (nu i u j ) + T n + n ( V ɛ2 x j x i x i 6 = ɛ2 ( ) nrik R kj, 12T x j ) n n (corresponding to the equations found by Jüngel-Matthes ( 05) and Degond-Gallego-Méhats ( 07)).
Vanishing-temperature limit The behavior of the semiclassical hydrodynamic equations as T 0 depends dramatically on the sign of λ. Let us consider separately the three representative cases λ = 1 (FD), λ = 0 (MB) and λ = 1 (BE).
T 0 limit: Fermi-Dirac case The FD case is the richest: the limit T 0 is non-singular and yields completely degenerate fluid equations: t n + (nu j ) = 0, x j t (nu i) + (nu i u j ) + x j + n x i = ( d 2 ) 2 d d Γ ( ) 2 d d 2 2π ( V d 2 ɛ 2 d 6 ɛ 2 [ dπ (d 2) d 12 Γ ( d 2 + 1) 2 d ) n n n x i n 2 d n x i R jk R kj 4n 2 d ( ) ] + R ij R jk n d 2 d x k
T 0 limit: Maxwell-Boltzmann case This is the most singular case, because the limits λ 0 and T 0 are somehow incompatible. The formal limit of the semiclassical hydrodynamic equations with BE statistics is only compatible with an irrotational fluid (R = 0) and depends on how (0, 0) is approached in the parameter plane (λ, T ) Degond-Gallego-Méhats ( 07) show that the fully-quantum fluid equations admit a limit, which is given by Madelung equations.
T 0 limit: Bose-Einstein case, d 3 For λ = 1 and d 3, in the limit T 0 the fluid is completely condensate and, then, this case cannot be considered within our description.
T 0 limit: Bose-Einstein case, d = 2 In this case the limit T 0 is only compatible with an irrotational fluid. Moreover, we have to rescale the density as N = The resulting limit equations are t N + (Nu j ) = 0 x j t u i + u j u j + x i x i n 2πT. ) (V ɛ2 12 N = 0
T 0 limit: Bose-Einstein case, d = 1 In this, last, case we obtain t n + x (nu) = 0 t u + 1 2 x u2 + x ( V ɛ2 2 1 2 ) n n x 2 = 0
Conclusions We derived semiclassical isothermal hydrodynamic equations for Fermions or Bosons. The method exploits Degond and Ringhofer s Quantum Maximum Entropy Principle and the semiclassical expansion of the maximizer Wigner function. We obtained an Euler-like system with quantum corrections, of order 2, involving a modified Bohm potential and the velocity curl tensor. The Maxwell-Boltzmann limit and the T 0 limit have been investigated. L.B., C. Cintolesi, Derivation of isothermal quantum fluid equations with Fermi-Dirac and Bose-Einstein statistics. J. Stat. Phys. (to appear).
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