Semiclassical Hydrodynamic Equations

The semiclassical hydrodynamic equations are obtained by taking moments ψ = [1, mξ, mξ²/2]on the semiclassical Boltzmann equation of (2.9) with the collision term (2.10), then integrating the resulting equations over all ξ.

∂_t The integrals of the collision terms in all three cases should preserve the conservation property. That means the conservation of mass, momentum and energy need to be satised all the time, which is called the compatibility condition,

where dΞ = m³dξ/h³ is the innitesimal volume in momentum space. We note that the collision in BGK model should also preserve the compatibility

2.4. Semiclassical Hydrodynamic Equations 15

The denitions of the number density, number density ux, and energy density are given, respectively, by

n(x, t) =

Other denitions of higher order moments such as internal energy density ε(x, t), stress tensor Pij(x, t), heat ux vector qi(x, t) are also given,

We can have the familiar form of hydrodynamic equations,

∂n

i. We have briey described the equilibrium distribu-tion including MB, FD, and BE distribudistribu-tions. The next step is to derive the corresponding hydrodynamic equations under these statistics. Although the

hydrodynamic equations are the same in both classical and semiclassical ver-sion, the transport coecients are dierent due to the dierences between f^eq,Q and f^eq,C. After some straightforward algebraic manipulations, the equation of state of pressure p, the number density n and total energies E in classical and quantum statistics as:

g_ν is the generalized FD/BE function which is dened as

g_ν(z)≡ 1 wherein Γ(ν) is the Gamma function. Notice that in classical limit, gν(z) will approach z no matter what the number ν is. We can check that the pressure pq, number density nqand total energies Eqwill approach the classical counterpart in classical limit. We can also check the specic heat γ = Cp/C_v, in ideal classical gas,γc = 5/3 for monatomic gas, but γq = ⁵₃^g5/2_g^(z)g2 ^1/2(z)

3/2(z)

in ideal quantum gas, which depends on the fugacity z and not constant.

Obviously, γqapproaches γcin classical limit. Moreover, more hydrodynamical

2.4. Semiclassical Hydrodynamic Equations 17

coecients like viscosity and thermal conductivity are given by

µq = τ nkBTg_5/2(z) g_3/2(z) = τ P κ_q = τ5k_B

2m[7g_7/2(z)

2g_3/2(z)− 5g_5/2(z)

2g_3/2² (z)]nk_BT, (2.20) and the classical coecients are also listed for comparing:

µ_c = τ nk_BT = τ P κ_c = τ5k_B

2mnk_BT = τ5k_B

2mP. (2.21)

All the semiclassical coecients will approach their classical counterparts and Prandtl number equals 1 in classical coecients. However, in semiclassical model, the Prandtl number depends on the fugacity z and is no more a con-stant number. Similar results could be found in linearized semiclassical Boltz-mann equations [23]. Until now, the semiclassical BoltzBoltz-mann Equation and the semiclassical hydrodynamical equations have been introduced and com-pared with their classical counterparts. In those descriptions there appears an important parameter z which is not shown in classical Boltzmann equation or MB statistics. The physical meaning of the fugacity z is described below:

Recall number density in semiclassical representation nq = ^g3/2_Λ3^(z), if we con-sider 0 < z < 1, then g3/2(z) ≃ z, we have z = nq(h²/2πmk_BT )^3/2 = n_qΛ³. From this equation, z can be interpreted as the ratio of Λ³ to average volume occupied by particles. In other words, it is the ratio of occupied length to particles thermal wavelength. When z is small, it means the order of spa-tial dimension is larger than the thermal wavelength and we can neglect the degeneracy eect of particles (highly non-degenerate gas). For large z, the degeneracy eect is important (highly degenerate gas) because the order of thermal wavelength is comparable to the spatial dimension. Moreover, the thermal wavelength is proportional to T^−1/2, that means when the number

density is in normal condition this eect will be obvious especially in the low temperature. In summary, when z → 0 the quantum distribution will coin-cide with the classical one, and the physical explanation is that the length dimension of particle is larger than the particle de Broglie wavelength. The wave property will not be important. When z is considerably large, two length scales become comparable and one cannot omit the quantum eect anymore.

So, we can think the fugacity z is the index of the degree of degeneracy. The fugacity z has some restrictions in two dierent quantum distributions. In the case of Boson, z should not exceed 1 because of the non-negative density, and in the Fermion case there are no such restrictions on z.

Finally, the actual correction values of the generalized Fermi function is shown in Fig. 2.1. One nds the BE and FD curves overlap MB curve in z → 0 limit, that means the classical statistics could only work in the classical limit (z → 0) of quantum statistics. Moreover, z is restricted in 0 ≤ z ≤ 1 for Bose gas, the function g3/2(z) increases monotonically with z and is bounded, its largest value being g3/2(1) = 2.612.

2.4. Semiclassical Hydrodynamic Equations 19

Figure 2.1: Bose and Fermi functions in dierent fugacity z. (a) Bose Func-tion g1.5^Bose(z), g2.5^Bose(z), and ^g_g^{2.5BoseBose(z)}

Chapter 3

Semiclassical Lattice Boltzmann Method

Contents

3.1 Overview . . . 21 3.2 The Derivations of Conventional LB method . . . 22 3.2.1 Time Discretization . . . 22 3.2.2 Space and Velocity Discretization . . . 23

3.3 The Derivations of Single Relaxation Time Semiclas-sical LB method . . . 28

3.3.1 Summary of Single Relaxation Time Semiclassical LB Method . . . 32 3.4 Generalized Semiclassical LB method . . . 35

3.1 Overview

In this chapter, the semiclassical LB method will be derived from semiclas-sical Boltzmann equation with BGK approximation (2.11). In general, there are two steps for deriving LB method from the continuous Boltzmann equa-tion. First, the time discretization is achieved by integrating the Boltzmann

equation along characteristic line. Second, the discretization of space is done by low Mach expansion or Hermite polynomials expansion. The derivations of semiclassical LB method is following the Hermite expansion procedures and the results are extended to multiple relaxation time version. The derived equations could be validated by reducing the SLB equation to classical one in classical limit.