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2.3 The Hydrodynamic Equations
In local equilibrium, the conditions of Eqs(27) are not met, and the left hand
sides (that we shall call thermodynamic forces) are not zero. This leads to
dissipative terms (denoted below by the superscript D ) and entropy production
as functions of the thermodynamic forces, which parameterize the deviation
from global equilibrium. The variables of course still obey continuity equations,
with
0=∂t(B + B D )+c ∇×(E + E D ) ,
0=∂t(D + D
(30)
D )+j D el +(ρel + ρ D el) v
−c ∇×(H + H D ) , (31)
0=∂t(ρ + ρ D )+∇·(ρ v − j D ) , (32)
0=∂t(ρc + ρ D c )+∇·(ρc v − j D c ) , (33)
R/T = ∂t(s + s D )+∇·(s v − f D ) , (34)
0=∂t (g tot
i + g tot,D
i )+∇j (Πij − Π D ij) , (35)
0=∂t(ɛ tot + ε D )+∇·(Q + Q D ) (36)
∇·(B + B D )=0, (37)
∇·(D + D D )=ρel + ρ D el . (38)
We may subtract the equation of motion for mass from that of the total energy
ɛ tot , to arrive at the continuity equation for the non-relativistic form of energy
conservation, more usual in hydrodynamic theories,
0=∂t(ε + ε D − ρ D c 2 )+∇·(Q + Q D
−ρc 2 v + c 2 j D ) . (39)
The covariance of the Eqs(30 - 38) has a number of consequences:
• The equilibrium contributions of energy, momentum and their fluxes constitute
the equilibrium energy-momentum 4-tensor Π µν ; the same applies
to the nonequilibrium contributions, ε D , Q D , g tot,D and −Π D ij constitute the
nonequilibrium 4-tensor Π D,µν .
• Analogously, all fields, the equilibrium and nonequilibrium ones, (E, B),
(E D , B D ), (D, H) and (D D , H D ) constitute 4-tensors of the form
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