Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 29

This is the classical definition of the density of states. If the quantum effect is taken
into account for the electron, we have

m √ 2 m

ε

D ( ε )= , (1.60)

h 3 π

2

where h is the Planck constant divided by 2 π .By D ( ε ) ,Eq. (1.58) can be written as

(Xu and Wang 2005)

q ( r , t )= v ( r , t ) f ( r , ε , t ) ε D ( ε ) d ε . (1.61)

ε

By Eqs. (1.47) and (1.51), we have

( + , ε ( + ) , + ) − ( , ε ( ) , ) −

f r d r v d v t d t f r v t f 0 f

= , (1.62)

d t τ

where d r and d v are the incremental of the position and velocity vectors, respec-
tively. Note that it takes approximately time period

to decay to
its equilibrium state f 0 . Under the assumption that no external forces act on the heat
transfer medium, the acceleration of this decaying process is zero (Xu and Wang
2005). Therefore Eq. (1.62) can be rewritten as

τ 0 for f ( r , ε ( v ) , t )

( + , ε ( ) , + ) − ( , ε ( ) , + )

f r d r v t d t f r v t d t

d t

(1.63)

f ( r , ε ( v ) , t + d t ) − f ( r , ε ( v ) , t ) f 0 − f

+ = .

d t τ

By applying a Taylor expansion, we have

f ( r + d r , ε ( v ) , t + d t ) − f ( r , ε ( v ) , t + d t )

d t

(1.64)

d r · ∇ f ( r , ε ( v ) , t + d t )+ higher order terms

= .

d t

Therefore, there exists a value τ T such that v · ∇ f ( r , ε ( v ) , t + τ T ) is the best approx-

imation of the first term on the left side of Eq. (1.63). As d t ≈ τ 0 (Tien et al. 1998,

Xu and Wang 2005), Eq. (1.63) becomes

τ v · ∇ f ( r , ε ( v ) , t + τ T )+ f ( r , ε ( v ) , t + τ )= f 0 . (1.65)

For the electron, f 0 is the Fermi-Dirac equilibrium distribution

1

exp ε − μ
k B T

,

f 0 ( ε )= (1.66)

1 +