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The hydrodynamic equations are
with
0=∂t(B + B D )+c ∇×(−v × B/c + E D ) , (65)
0=∂t(ρ + ρ D )+∇·(ρ v − j D ) , (66)
0=∂t(ρc + ρ D c )+∇·(ρc v − j D c ) , (67)
R
T = ∂t(s + s D )+∇·(s v − f D ) , (68)
0=∂t (g tot
i + g tot,D
i )+∇j (Πij − Π D ij) ,
0=∂t(ε + ε
(69)
D − ρ D c 2 )
+∇·(Q + Q D − ρc 2 v + j D ) (70)
0=∇·(B + B D ) , B D = v × E D /c , (71)
Q =(Ts+ µρ+ µc ρc + ρc 2 + v · g) v
− (v × B) × H , (72)
g = g tot +(v × H) × B/c 2 = ρ v , (73)
Πij =[Ts+ µρ+ µc ρc + v · g + H · B − ε] δij
+ 1
2 [gi vj − Hi Bj +(i ↔ j))] , (74)
(µ + c 2 ) j D i = −T f D i − µc j D c,i + c
E D × H
, (75)
i
R = f D
· ∇ 0 T − 1
µ + c2 T ∇0
µ +Π D ij vij
+j D
c · ∇ 0 µc − 1
µ + c2 µc ∇ 0
µ
+E D
· c ∇ 0 × H + 1
µ + c2 H ×∇0
µ . (76)
The quantities s D , ρ D , ρ D c , Q D i , g tot,D , ε D satisfy the same relations as before,
see Eq(42 - 46). The dissipative fluxes are again obtained in an expansion of
the entropy production R, Eq(76).
The two Maxwell equations (31, 38) are not part of the hydrodynamic theory;
rather, they simply define the quantities ρel and j D el,
j D el + ρel v := c ∇×H + ∂t (v × H) /c ,
ρ + ρ D el := −∇ · (v × H) /c . (77)
Note that H D ∼∇×E 0 has been set to zero, while ρ D is given by Eq(42).
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