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−µc j D c,n + c
E × H + E D × H 0
+〈ρvn − j D n 〉∆µ − c
E 0 2 × H D 2
· n
· n , (124)
where 2 denotes the dielectric, ie n points into the dielectric. The entropy
production is now
R sf = fn ∆T + 〈ρc vn − j D c,n〉∆µc
+
〈vn gj〉−Π D
jn ∆vt,j
+〈ρvn − j D n 〉 ∆µ eff
+
n ×
E D + E
· c ∆H 0 t
+
H D
2 + v2/c × D2 · c
E 0 2 × n
. (125)
This Rsf yields 9 boundary conditions. Ignoring cross terms, we have especially
n ×
E D + E
= βsc ∆H 0 t, βs > 0 . (126)
Here we have altogether 18 boundary conditions. They suffice to determine
the 7 collective modes of the conductor and the 9 collective modes of the
dielectric, to fix the normal component of B + B D and to determine the lab
velocity of the interface.
5 Two Illustrative Examples
The hydrodynamics is not simply a coarse-grained theory; it has a scaleinvariance
built in: Varying the resolution, the hydrodynamics remains valid.
This feature makes the theory especially general and widely applicable. The
same applies to the boundary conditions derived above, which are valid where
ever the bulk theory is. Below are two examples for the vacuum-conductor
interface, where we shall calculate some coefficients from two simple models.
In both cases, we shall employ a higher resolution hydrodynamic theory to
obtain the coefficient of the coarser-grained one.
More specifically, we shall in the first example calculate the coefficient in the
boundary condition Eq(126), and understand the discontinuity of the coarsegrained
magnetic field H as given by a highly conducting surface layer (and
surface current) that is not explicitly accounted for in the given resolution.
This is of course a trivial case, but it does serve to draw one scenario in which
the discontinuity of the magnetic field is not surprising. The crucial point
here is that the coarse-grained boundary conditions as presented above are
applicable also for situations where a higher resolution theory is not easily
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