Boundary Conditions of the

Hydrodynamic Theory of Electromagnetism

Sven Symalla and Mario Liu 1

Institut fur Theoretische Physik, Universitat Hannover,30167 Hannover, Germany

Abstract

The complete set of boundary conditions for the hydrodynamic theory of polarizableand magnetizable media is obtained from the differential equations of the theory.The qualitative distinction of different numbers of variables that exists betweenthe dielectrics and conductors is stressed. The theory and the boundary conditionspresented here are covariant to linear order in v/c.

Key words: hydrodynamic Maxwell equations, boundary conditions, covariance

1 Introduction

It was not until a few years ago that the Maxwell equations were incorporatedinto the hydrodynamic theory in a rigorous and consistent way, complete withdissipative terms and valid also in the nonlinear regime [1]. The resultanttheory accounts for the low frequency dynamics of macroscopic systems thatare exposed to external fields, or contain electric charges and currents. It washelpful in understanding the curious spin-up behavior of ferrofluids [2], andhas produced a number of interesting predictions [3,4].

The four following paragraphs are a quick summary, addressing those who aresurprised by these statements, yet do not intend to read the references [1–4]:The macroscopic Maxwell equations certainly belong to the best studied andverified differential equations in physics, especially in the optical range of fre-quency. Yet they are usually taken as a set of stand-alone equations, decoupledfrom other macroscopic variables. So the evolution of the electromagnetic fieldis generally obtained with given and set values of density, temperature and

1 email liu@itp.uni-hannover.de

Preprint submitted to Elsevier Preprint 28 May 1998

local velocity, among others. In a few notable exceptions, these parameters aretaken as functions of time. A rigorous and trustworthy theory, on the otherhand, needs to consider these macroscopic, hydrodynamic quantities also asvariables, and include their dynamics with that of the field. For instance, giventhe continuity equation for the momentum and its modification through thepresence of electromagnetic field, we will find an unambiguous answer to thenotoriously difficult question of what is the ponderomotive force for a non-stationary, dissipative system of nonlinear constitutive relations [2,3]. This isa typical feed-back effect of how fields influence the hydrodynamic variables.And it is simply neglected if the velocity is given as a parameter, even if afunction of time.

Starting from the thermodynamic approach to electromagnetism as given inthe justifiably famous Vol 8 of Landau and Lifshitz [5], a theory that accountsfor the dynamics of both the field and the hydrodynamic variables simulta-neously is essentially what has been achieved in [1–4]. Most of the papers areconfined to the true hydrodynamic regime of local equilibrium, though a firstattempt of generalizing to the dispersive, optical frequency range has provenrather successful [3].

One striking result of this theory is the appearance of dissipative fields, HD

and ED that are gradients of the original fields. They account for dissipativephenomena such as the restoration of equilibrium – even if the constitutiverelations are nonlinear. (In contrast, the identification of the imaginary partsof ε and µ with dissipation holds only for strictly linear constitutive relations.)Despite the k-dependence, or the appearance of spatial dispersion, HD and ED

include temporal dispersion. (In fact, even in the linear case, these dissipativefields are more general that the accounts of temporal dispersion. The differenceis large in select systems, especially dielectric ferrofluids [4].)

The higher order gradient terms in the dissipative fields necessitate additionalboundary conditions for the Maxwell equations, although no additional vari-ables are being considered. This is the principle difference to the many workson the subject of “Additional Boundary Conditions”, where either higher fre-quency dynamic variables such as the exitonic degrees of freedom, or lowfrequency, surface variables are being considered [6].

One of the main foundations of this new, hydrodynamic theory of electro-magnetism is a pragmatic combination of the Galilean and the Lorentz trans-formation to first order in v/c: The former is employed for all hydrodynamicvariables, including the temporal and spatial derivative, and the latter used forthe field variables. As a result, the equations of motion are not covariant. Thisis cause for worry, because although the difference between the Galilean andthe first-order Lorentz transformation is known to be irrelevant in usual hy-drodynamic theories, circumstances are not as clear-cut when electromagnetic

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fields are involved.

This is the reason a fully relativistic version of the same theory was derived [7],employing only the Lorentz transformation, and demonstrating unambigu-ously the consistency of the hydrodynamic theory of electromagnetism. Whilethis is clearly the correct theory to apply for relativistic systems such as quicklyspinning astrophysical objects, it falls short when used to decide whether theoriginal theory is indeed the appropriate and rigorously valid one for non-relativistic systems. For this purpose, one needs to go one step further, derivethe complete set of boundary conditions, and thoroughly check whether anyterm not contained in the original theory could conceivably be important. Thefirst part of this work is the content of the present paper.

The hydrodynamic theory and the boundary conditions, covariant to firstorder in v/c, are presented here. We distinguish clearly between dielectrics andconductors, because the theory for conductors has less independent variables,and hence also less boundary conditions. And because the transition betweenthe two theories is subtle and prone to errors.

The boundary conditions are derived from the bulk equations themselves,with the help of irreversible thermodynamics. This deviates from the strictlymathematical concept of differential equations, where boundary conditionsare extrinsic information, used to select a special solution out of the manifoldsatisfying the equation.

The most well known example of the approach in physics is the derivationof boundary conditions from the Maxwell equations, yielding results such asthe continuity of the normal component of the magnetic field B [5]. Andthere are many more recent examples: Boundary conditions have been derivedespecially for broken symmetry systems, (such as liquid crystals [8], superfluid4He [9] and 3He[10], including the A→B transition [11],) but also for the moremundane shear flows of isotropic liquids [12].

The boundary conditions to be derived below are being applied to understandthe viscosity of ferrofluids, and to dynamo theory. These results will be pre-sented elsewhere. (In fact, one or two of the more unconventional boundaryconditions have already been used [2,4], though without proper derivation orpresentation.)

The paper has four more chapters, the first two to present the hydrodynamictheory, for dielectrics and conductors, respectively, and the third to set upthe boundary conditions. A fourth chapter considers two specific situations toclarify the physics contained in some of the boundary conditions.

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2 The Hydrodynamic Theory for Dielectrics

2.1 Lorentz Transformation and Thermodynamics

The starting point of every hydrodynamic theory is the thermodynamic theory.The energy density εtot is a function of all the other conserved quantities, theentropy density s and the field variables B and D. For a two-component liquidwe have, in the rest frame,

dεtot = dε+ c2 dρ = T ds+ (µ+ c2) dρ

+µc dρc + H · dB + E · dD , (1)

where the constraints

∇ ·B = 0 , ∇ ·D = ρel (2)

are satisfied in equilibrium. (ρel is the charge density.) Note that εtot is the totalenergy including the rest mass. The mass densities, ρ and ρc, are connectedto the particle numbers n1 and n2 by

ρ = m1 n1 +m2 n2 , ρc = m2 n2, (3)

where m1 and m2 denote the respective masses. The choice of µ + c2 as theconjugate variable to ρ renders the expansion in the small parameter ε/ρ c2

simple. (At most densities, the rest mass is certainly the by far dominatingcontribution.)

The equilibrium fluxes of energy and momentum in the rest frame are [5],

Q = cE×H , (4)

Πij = (T s+ µ ρ+ µc ρc + E ·D + H ·B− ε) δij−1

2[HiBj + EiDj + (i↔ j))] . (5)

These variables and their fluxes constitute the energy-momentum 4-tensorΠµν ,

Π00 = εtot , Π0k = Πk0 = Qk/c = c gtotk ,

Πik = Πik = Πki . (6)

The local conservation laws ensure that Πµν satisfies

4

∂νΠµν = 0 (7)

and is symmetric. (The greek indices go from 0 to 3, the latin ones from 1 to3.)

The Lorentz transformation will now yield these thermodynamic expressionsfor an arbitrary inertial frame. Denoting the rest frame quantities with thesuperscript 0, the Lorentz transformation

Πµν = Λ µα

[Παβ

]0Λ νβ , (8)

for v = v ex employs with the matrix

Λ νµ =

γ γ β 0 0

γ β γ 0 0

0 0 1 0

0 0 0 1

, (9)

where

β = v/c , γ = 1/√

1− β2 . (10)

Up to order v2

c2, we have

εtot = (1 + v2/c2)εtot,0

+2v [1 + v2/ c2] · gtot,0

+vi Π0ij vj/c

2 , (11)

c gtoti =Qi/c = vi/c (1 + v2/c2) εtot,0

+[1 + v2/(2 c2)] c gtot,0i

+3 vi c gtot,0l vl/(2 c

2)

+[1 + v2/(2 c2)] vl Π0li/c

+vi vl (Π0lk/c) vk/(2 c

2), (12)

Πik = vi vk εtot,0/c2 + [1 + v2/(2 c2)] vi g

tot,0k

+[1 + v2/(2 c2)] vk gtot,0i + vi vk g

tot,0l vl/(v c

2)

+Π0ik + Π0

il vl vk/(2 c2)

+Π0lk vl vi/(2 c

2) . (13)

As ε¿ ρ c2, we shall only include terms of linear order in vc, except for terms

connected to the rest mass, ρ c2, where quadratic order will be included. (In the

5

energy flux, this leads to a third order term.) Then the lab-frame expressionsfor the energy, momentum and the fluxes are

gtot = (T s+ µ ρ+ µc ρc + ρ c2)v/c2 + E×H/c , (14)

Q = c2 gtot + vl gl v , (15)

Πij = (T s+ µ ρ+ µc ρc + v · g+E ·D + H ·B− ε) δij+1

2[gi vj − EiDj −HiBj + (i↔ j))] , (16)

where

g= gtot −D×B/c . (17)

The other thermodynamic variables also need to be transformed. First, thefour fields:

B0 = B− v × E/c , D0 = D + v ×H/c , (18)

H0 = H− v ×D/c , E0 = E + v ×B/c . (19)

Both chemical potentials, µ+ c2 and µc, obey analogous formulas:

µ+ c2 = [1− v2/(2 c2)] (µ0 + c2) , (20)

µc = [1− v2/(2 c2)]µ0c . (21)

However, because of c2, only µ is altered

µ = µ0 − v2/2 , (22)

while µc = µ0c remains invariant in linear order of v

c. The quantities s, ρc

and T are also invariant to this order, s = s0, ρc = ρ0c and T = T 0. In the

combination ρ c2, we have

ρ= [1 + v2/(2 c2)] ρ0 ; (23)

otherwise, it is ρ = ρ0.

Because ε¿ ρ c2, the nonrelativistic expression for gtot is

gtot = ρv + E×H/c . (24)

6

Putting all this together, the Π00-component yields

dε= dεtot − c2 dρ = T ds+ µ dρ+ µc dρc + v · dg+H · dB + E · dD . (25)

The isotropy of the system, ie the invariance of the energy density under aninfinitesimal rotation implies:

v × g + H×B + E×D = 0 . (26)

2.2 The Equilibrium Conditions

Maximizing the integrated entropy∫

d3r s under the constrains of variousconservation laws and Eqs(2), the resulting Euler-Lagrange equations, or theequilibrium conditions are [7]

∂tT = 0 , ∂tµ = 0 , ∂tµc = 0 ,

∂tH0 = 0 , ∂tE

0 = 0 , vij = 0 ,

∇µ+ ∂tv + µ ∂tv/c2 = 0 ,

∇0T − 1

µ+ c2T ∇0µ = 0 ,

∇0µc −1

µ+ c2µc∇0µ = 0 ,

∇0 ×H0 +1

µ+ c2H0 ×∇0µ = 0 ,

∇0 × E0 +1

µ+ c2E0 ×∇0µ = 0 (27)

where

∇0 = ∇+v

c2∂t , vij = 1

2(∇ivj +∇jvi) . (28)

Note the preponderance of rest frame quantities. Except for the chemical po-tential, all spatial derivatives are small. If the charges are eventually able tomove around, the total entropy may be further maximized if

E0 = 0 . (29)

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2.3 The Hydrodynamic Equations

In local equilibrium, the conditions of Eqs(27) are not met, and the left handsides (that we shall call thermodynamic forces) are not zero. This leads todissipative terms (denoted below by the superscript D) and entropy productionas functions of the thermodynamic forces, which parameterize the deviationfrom global equilibrium. The variables of course still obey continuity equations,

0 = ∂t(B + BD) + c∇× (E + ED) , (30)

0 = ∂t(D + DD) + jDel + (ρel + ρD

el)v

−c∇× (H + HD) , (31)

0 = ∂t(ρ+ ρD) +∇ · (ρv − jD) , (32)

0 = ∂t(ρc + ρDc ) +∇ · (ρc v − jD

c ) , (33)

R/T = ∂t(s+ sD) +∇ · (sv − fD) , (34)

0 = ∂t (gtoti + gtot,D

i ) +∇j (Πij − ΠDij) , (35)

0 = ∂t(εtot + εD) +∇ · (Q + QD) (36)

with

∇ · (B + BD) = 0 , (37)

∇ · (D + DD) = ρel + ρDel . (38)

We may subtract the equation of motion for mass from that of the total energyεtot, to arrive at the continuity equation for the non-relativistic form of energyconservation, more usual in hydrodynamic theories,

0 = ∂t(ε+ εD − ρD c2) +∇ · (Q + QD

−ρ c2 v + c2 jD) . (39)

The covariance of the Eqs(30 - 38) has a number of consequences:

• The equilibrium contributions of energy, momentum and their fluxes con-stitute the equilibrium energy-momentum 4-tensor Πµν ; the same appliesto the nonequilibrium contributions, εD, QD, gtot,D and −ΠD

ij constitute thenonequilibrium 4-tensor ΠD,µν .• Analogously, all fields, the equilibrium and nonequilibrium ones, (E,B),

(ED,BD), (D,H) and (DD, HD) constitute 4-tensors of the form

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(E,B)µν =

0 −Ex −Ey −EzEx 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

. (40)

• The equilibrium quantities (s, sv/c), (ρ, ρv/c), (ρc, ρc v/c) and (ρel c, ρel v)are 4-vectors. The same applies to the nonequilibrium quantities (sD c,−fD),(ρD c,−jD), (ρD

c c,−jDc ) and (ρD

el c, jDel).

The dissipative contributions of the variables sD, ρD, ρDc , gtot,D, BD, DD and

ρDel vanish in the local rest frame, v = 0, because this is where the nonrela-

tivistic and relativistic physics overlap. Here, according to the concept of localequilibrium, the variables only contain equilibrium information. As a result,their form in an arbitrary frame is given as:

BD = v × ED/c , DD = −v ×HD/c , (41)

ρDel = v · jD

el/c2 , sD = −v · fD/c2 , (42)

ρD = −v · jD/c2 , ρDc = −v · jD

c /c2 , (43)

gtot,Di = −ΠD

ij vj/c2 , (44)

and because of the symmetry of the energy-momentum 4-tensor,

QDi = c2 gtot,D

i = −vj ΠDij, (45)

with

εD = O(v2/c2

)(46)

yet an order higher.

To actually obtain the dissipative currents, we substitute ∂tεtot in Eq(36) for

the expressions of Eq(25),

T ∂ts+ (µ+ c2) ∂tρ+ µc ∂tρc + v · ∂tg + H · ∂tB+E · ∂tD + ∂tε

D = −∇ · (Q + QD) , (47)

insert the appropriate equations of motion, Eqs(30 - 35), excluding terms oforder higher than v

c, and arrive at two expressions, the energy flux QD

i andthe entropy production R. The energy flux is

9

QDi =−(µ+ c2) jD

i − µc jDc,i − T fD

i

+c[ED ×H0 + E0 ×HD

]i− vj ΠD

ij , (48)

which in conjunction with Eq(45) leads to

(µ+ c2) jDi =−µc j

Dc,i − T fD

i

+c[ED ×H0 + E0 ×HD

]i. (49)

(It is now plain that jDi ∼ c−2 is relativistically small, and it is well justified

to ignore it in non-relativistic theories.)

The entropy production is

R= fD · ∇0T + ΠDij vij + jD

c · ∇0µc + jD · ∇0µ+ jDel E

0

+ED · c(∇0 ×H0

)−HD · c

(∇0 × E0

). (50)

Inserting Eq (49), the number of independent thermodynamic forces is reducedby one,

R= fD ·(∇0T − 1

µ+ c2T ∇0µ

)+ ΠD

ij vij

+jDc ·(∇0µc −

1

µ+ c2µc∇0µ

)+ jD

el E0

+ED · c(∇0 ×H0 +

1

µ+ c2H0 ×∇0µ

)

−HD · c(∇0 × E0 +

1

µ+ c2E0 ×∇0µ

). (51)

It is instructive to compare this equation with Eq(27). Clearly, the thermo-dynamic forces there appear here again. If they vanish, we have equilibrium,and the entropy production R is zero. If they do not, the leading terms in R,being a positive definite function, must be quadratic in these forces. In thisorder, the dissipative currents, fD, ΠD

ij, jDc , ED, HD, jD

el, are proportional tothe forces, with the coefficients obeying the Onsager symmetry relations.

3 The Hydrodynamic Theory for Conductors

If the electrical conductivity is large enough, such that the displacement cur-rent ∂tD in

10

0 = D + jDel + ρel v − c∇× (H + HD) (52)

is negligible with respect to jDel = σE0, it may then be eliminated. This is the

quasi-stationary approximation[5], always justified in the low-frequency limit:Consider (for v ≡ 0)

|D| ≈ω εE ¿ jD = σ E , (53)

(where ε is the dielectric constant,) and find

ω ε/σ ¿ 1. (54)

We shall in this work refer to a system as a conductor if the quasi-stationaryapproximation is valid. (If not, the displacement current ∂tD needs to beincluded and, despite some residual conductivity, the equations of the lastsection apply.)

In the Heaviside-Lorentz system of units, the conductivity of metals is of theorder of σ = 1018/s, eight to ten orders of magnitude above the frequenciesat which local equilibrium reigns and the hydrodynamic theory is valid. Sothe quasi-stationary approximations always apply in these systems, which cantherefore be considered as conductors without any qualification. If the con-ductivity is below 1010/s, circumstances are more complicated and depend onthe given frequency.

The elimination of the displacement current ∂tD is a qualitatively importantstep, as this is equivalent to the statement that D is no longer an independentvariable of the hydrodynamic theory. Indeed, the relaxation time of the electricfield is τ = ε

σ, as

|E| ≈ E/τ ≈ D/ε ≈ σ E/ε . (55)

and the electric field has ample time to relax for frequencies ω τ ¿ 1. So,taking E0 as zero in equilibrium, the rest frame thermodynamics is now,

dεtot =T ds+ (µ+ c2) dρ+ µc dρc + H · dB . (56)

Note that the electric field need not vanish in the lab-frame, v 6= 0, though ofcourse only via the Lorentz transformation

E = −v ×B/c , D = −v ×H/c , (57)

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without changing the fact that they are not independent. Note also that wehave only implemented the fact that the equilibrium electric field vanishes.Off equilibrium, it does not, and as we know may drive a current through awire. It remains dependent, however, and is in this case given by

c∇× (H + HD) = σE. (58)

There are two ways to arrive at the hydrodynamic theory for conductors, onemay either set to zero all terms ∼ D0 and E0 in the previous theory, or onemay start from scratch with a reduced set of variables. Both methods lead tothe same results, which are presented below.

The basic thermodynamic identity for an arbitrary inertial system is,

dε= dεtot − c2 dρ = T ds+ µ dρ+ µc dρc

+v · dgtot + H · dB (59)

where

gtot = (T s+ µ ρ+ µc ρc + ρ c2)v

c2− 1

c

(v

c×B

)×H

≈ ρv − 1

c

(v

c×B

)×H . (60)

In equilibrium, the magnetic field satisfies

∇ ·B = 0 . (61)

To linear order in vc, the fields

H0 = H , B0 = B (62)

are Lorentz invariant.

The change in the equilibrium conditions are in the Euler-Lagrange equationsfor the field. Only

∂tH = 0 , ∇0 ×H +1

µ+ c2H×∇0µ = 0 , (63)

remain, with E0 ≡ 0 implied. The rotational identity Eq(26) reduces to

H×B = 0 . (64)

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The hydrodynamic equations are

0 = ∂t(B + BD) + c∇× (−v ×B/c+ ED) , (65)

0 = ∂t(ρ+ ρD) +∇ · (ρv − jD) , (66)

0 = ∂t(ρc + ρDc ) +∇ · (ρc v − jD

c ) , (67)

R

T= ∂t(s+ sD) +∇ · (sv − fD) , (68)

0 = ∂t (gtoti + gtot,D

i ) +∇j (Πij − ΠDij) , (69)

0 = ∂t(ε+ εD − ρD c2)

+∇ · (Q + QD − ρ c2 v + jD) (70)

with

0 =∇ · (B + BD) , BD = v × ED/c , (71)

Q = (T s+ µ ρ+ µc ρc + ρ c2 + v · g)v

− (v ×B)×H , (72)

g= gtot + (v ×H)×B/c2 = ρv , (73)

Πij = [T s+ µ ρ+ µc ρc + v · g + H ·B− ε] δij+1

2[gi vj −HiBj + (i↔ j))] , (74)

(µ+ c2) jDi = −T fD

i − µc jDc,i + c

[ED ×H

]i, (75)

R= fD ·(∇0T − 1

µ+ c2T ∇0µ

)+ ΠD

ij vij

+jDc ·(∇0µc −

1

µ+ c2µc∇0µ

)

+ED · c(∇0 ×H +

1

µ+ c2H×∇0µ

). (76)

The quantities sD, ρD, ρDc , QD

i , gtot,D, εD satisfy the same relations as before,see Eq(42 - 46). The dissipative fluxes are again obtained in an expansion ofthe 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 jD

el,

jDel + ρel v := c∇×H + ∂t (v ×H) /c ,

ρ+ ρDel := −∇ · (v ×H) /c . (77)

Note that HD ∼ ∇× E0 has been set to zero, while ρD is given by Eq(42).

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4 Boundary Conditions

A solution of the hydrodynamic equations is possible only if in addition to theequations we also have the appropriate initial and boundary conditions. Thelatter are obtained from the bulk equations themselves. Therefore, numberand type of the boundary conditions depend on the two systems comprisingthe interface. The boundary conditions are best derived in the rest frame of theinterface, (or more generally in the rest frame of the portion of the interfaceunder consideration).

4.1 The Dielectric-Dielectric Interface, without Phase Transition

We shall first consider the simpler case in which no mass current may crossthe interface, ie in the absence of the possibility of a phase transition, such asgiven at an interface made from two different substances.

The first boundary condition is the continuity of the normal component of theenergy flux,

∆(Qn +QDn ) = 0 , (78)

where the subscript n denotes the component normal to the interface. Thiscondition is obtained by integrating the energy conservation,

εtot +∇ · (Q + QD) = 0,

over an infinitesimally thick slab around the stationary interface. Physically, itsimply implies that the same amount of energy enters and leaves the interface.

The notation, here and below, is given as

∆A≡A(r− rsf → −0)

−A(r− rsf → +0) ≡ A1 − A2 , (79)

A≡〈A〉 ≡ 12

(A1 + A2) , (80)

where rsf is a given point on the surface. The normal vector n hence pointsinto region 2. Also,

∆(AB)≡A1 B1 − A2 B2 = A∆B +B∆A . (81)

All vectors are divided into an tangential and a normal component, say

14

vt =v − (v · n) n , (82)

vn = (v · n) n . (83)

The lack of mass transfer implies

ρ1 v1,n = jD1,n , ρc,1 v1,n = jD

c,1,n ,

ρ2 v2,n = jD2,n , ρc,2 v2,n = jD

c,2,n . (84)

Because jD1,n and jD

2,n are relativistically small quantities, so are v1,n and v2,n,hence also jD

c,1,n and jDc,2,n.

Three more boundary conditions follow from the conservation of momentum,one per component. In comparison to the conservation of energy, however,there is a complication arising from the possibility of an isotropic surfacepressure[13],

Πtotij ≡ Πij − ΠD

ij − αsf (δij − ninj) δ(|r− rsf |), (85)

where αsf is the surface energy density of [13]. After some algebra, we obtain

∆(Πnn − ΠDnn) = αsf

(1

R1

+1

R2

), (86)

∆(Πt,i − ΠDt,i) t1,i = −t1 · ∇αsf , (87)

∆(Πt,i − ΠDt,i) t2,i = −t2 · ∇αsf (88)

where

Πt,i = Πlj nj (δil − ni nl) ,ΠD

t,i = ΠDlj nj (δil − ni nl) , (89)

R1, R2 denote the curvature radii, and t1, t2 the attendant principle directions.The radii are positive if they point into region 1, and the bulge is toward region2.

From the Maxwell equations, we deduce the boundary conditions,

∆(Bn +BDn ) = 0 , (90)

∆(Et + EDt ) = 0 , (91)

∆(Dn +DDn ) = −σsf , (92)

∆(Ht + HDt ) = n× jel,sf/c , (93)

15

where σsf denotes the surface charge density and jel,sf the surface current, asyet undetermined.

Combining Eqs(78) and (84), we obtain

0 = ∆(Qn +QDn + (µ+ c2) jD

n − (µ+ c2) ρ vn) . (94)

which in conjunction with Eqs(49, 15, 45) implies

0 = ∆[(T s+ v · g) vn − ΠD

nj vj − T fDn

+c(E×H + ED ×H0 + E0 ×HD

)· n]. (95)

If the interface is curved, we shall from here on choose the inertial systemvt = 0 (which gets rid of a term ∼ vtαsf in Rsf), otherwise, vt is arbitrary.As vn is negligibly small in the absence of phase transition, Eq(95) is quicklyconverted into the surface entropy production,

Rsf ≡−T ∆fn = fn ∆T − ΠDjn ∆vt,j

+c∆(E×H + ED ×H0 + E0 ×HD

)· n (96)

with fn ≡ (svn − fDn ), here and below. The surface entropy production Rsf ≡

−T ∆fn is the difference between the entropy current exiting and entering thesurface. Just as its bulk counter part, it is positive definite, invariant undertime inversion and vanishes in equilibrium. Some more algebra then yields

Rsf = fn ∆T + c(n× ED

)·∆H + c

(HD × n

)·∆E

+(−ΠDjn +HD

j Bn + EDj Dn + 1

4σsf∆Ej) ∆vt,j

+ [n× jel,sf ] · (n× E0) , (97)

a nice sum of independent pairs of force and fluxes. Taking them to be propor-tional to each other, they represent another seven boundary conditions, anddetermine the surface current jel,sf .

One symmetry element peculiar to the interface is the invariance of Rsf underthe simultaneous operation n → −n and ∆ → −∆. Taking this and theisotropy into account, we find

fn =κs∆T (98) n× ED

n× jel,sf

=

βs γs

γs σs

c∆Ht

n× E0

, (99)

16

−ΠD,efft,i(

HD × n)i

=

ηs ζs

ζs αs

∆vt,i

c∆Et,i

(100)

where −ΠD,efft,i ≡ −ΠD

t,i +HDt,iBn + ED

t,iDn + 14σsf∆Et,j, and

γs = −γs , ζs = −ζs . (101)

αs, βs, ηs, κs, σs > 0 . (102)

These are altogether 21 boundary conditions. They suffice to determine (i) all(outgoing) hydrodynamic modes of the dielectric medium, 9 for each side; (ii)the normal components of B + BD and D + DD, (iii) the lab velocity. See [14]for a more detailed consideration (in single-component liquids).

4.2 The Dielectric-Dielectric Interface, with Phase Transition

Mass and concentration currents across the interface renders the considerationslightly more complicated.

The 12 continuity conditions remain:

∆(Qn +QDn ) = 0 , (103)

∆(ρ vn − jDn ) = 0 , (104)

∆(ρc vn − jDc,n) = 0 (105)

∆(Πnn − ΠDnn) = Psf , (106)

∆(Πt,i − ΠDt,i) t1,i = t1 · ∇αsf , (107)

∆(Πt,i − ΠDt,i) t2,i = t2 · ∇αsf (108)

∆(BDn +Bn) = 0 , (109)

∆(Et + EDt ) = 0 , (110)

∆(DDn +Dn) = −σsf , (111)

∆(Ht + HDt ) = n× jel,sf/c . (112)

The rest of 9 boundary conditions must now be deduced from Rsf . CombiningEq(103) and (104), we obtain

0 = ∆(Qn +QDn )− (µ+ c2) ∆(ρ vn − jD

n )

= ∆(Qn +QDn + (µ+ c2) jD

n − (µ+ c2) ρ vn)

+〈ρ vn − jDn 〉∆µ , (113)

17

which in conjunction with Eqs(49, 15, 45) leads to

0 = ∆[(T s+ µc ρc + v · g) vn − ΠD

nj vj − T fDn

−µc jDc,n + c

(E×H + ED ×H0

+E0 ×HD)· n]

+ 〈ρ vn − jDn 〉∆µ . (114)

Employing Eq(105), we have

Rsf =−T ∆fn = fn ∆T + 〈ρ vn − jDn 〉∆µ

+〈ρc vn − jDc,n〉∆µc −∆(vj ΠD

jn)

+∆ (v · g vn) + c∆(E×H + ED ×H0

+E0 ×HD)· n, (115)

which is more suitably written as

Rsf = fn ∆T + 〈ρc vn − jDc,n〉∆µc

+[〈vn gj〉 − ΠD

jn + 14σsf∆Ej

]∆vt,j

+〈ρ vn − jDn 〉∆µeff

+c(n× 〈ED − vn

c×B〉

)·∆H0

+c(〈HD +

vn

c×D〉 × n

)·∆E0

+ [n× jel,sf ] · (n× E0) , (116)

where

µeff ≡(µ+

v2n gn

(ρ vn − jDn )− ΠD

nn vn(ρ vn − jD

n )

). (117)

This Rsf yields 9 boundary conditions and the value of the surface currentjel,sf .

4.3 Interfaces involving Conductors

The essential difference to the boundary conditions considered until now isthe fact that the electric field D is no longer an independent variable. As adirect result, neither are the two boundary conditions,

18

∆(DDn +Dn) = −σsf , (118)

∆(Ht + HDt ) = n× jel,sf/c. (119)

This also affects the surface entropy production. Starting from

0 = ∆(Qn +QDn )− (µ+ c2) ∆(ρ vn − jD

n )

= ∆(Qn +QDn + (µ+ c2) jD

n − (µ+ c2) ρ vn)

+〈ρ vn − jDn 〉∆µ (120)

we have for the conductor-conductor interface

0 = ∆[(T s+ µc ρc + v · g) vn − ΠD

nj vj

−T fDn − µc j

Dc,n + c

(E×H + ED ×H

)· n]

+〈ρ vn − jDn 〉∆µ, (121)

and hence

Rsf = −T ∆fn = fn ∆T + 〈ρ vn − jDn 〉∆µ

+〈ρc vn − jDc,n〉∆µc −∆(vj ΠD

jn) + ∆ (v · g vn)

+c∆(E×H + ED ×H

)· n (122)

or

Rsf = fn ∆T + 〈ρc vn − jDc,n〉∆µc

+[〈vn gj〉 − ΠD

jn

]∆vt,j

+〈ρ vn − jDn 〉∆µeff

+c[n×

(ED + E

)]·∆Ht , (123)

where µeff is the same as before, see Eq(117). This expression yields 7 connect-ing conditions. So we have a total of 16 boundary conditions for the conductor-conductor interface. They suffice to determine all outgoing collective modes,7 for each side, the normal component of B + BD, and the lab velocity ofthe interface. (Note that the number of the collective modes is reduced in aconductor, because there are no sq-Modes [4]. Also, the electromagnetic wavesare reduced to magnetic, diffusive modes.)

For the conductor-dielectric interface, Eq(120) implies

0 = ∆[(T s+ µc ρc + v · g) vn − ΠD

nj vj − T fDn

19

−µc jDc,n + c

(E×H + ED ×H0

)· n]

+〈ρ vn − jDn 〉∆µ− c

(E0

2 ×HD2

)· n , (124)

where 2 denotes the dielectric, ie n points into the dielectric. The entropyproduction is now

Rsf = fn ∆T + 〈ρc vn − jDc,n〉∆µc

+[〈vn gj〉 − ΠD

jn

]∆vt,j

+〈ρ vn − jDn 〉∆µeff

+[n×

(ED + E

)]· c∆H0

t

+(HD

2 + v2/c×D2

)· c(E0

2 × n). (125)

This Rsf yields 9 boundary conditions. Ignoring cross terms, we have especially

n×(ED + E

)= βsc∆H0

t , βs > 0 . (126)

Here we have altogether 18 boundary conditions. They suffice to determinethe 7 collective modes of the conductor and the 9 collective modes of thedielectric, to fix the normal component of B + BD and to determine the labvelocity of the interface.

5 Two Illustrative Examples

The hydrodynamics is not simply a coarse-grained theory; it has a scale-invariance built in: Varying the resolution, the hydrodynamics remains valid.This feature makes the theory especially general and widely applicable. Thesame applies to the boundary conditions derived above, which are valid whereever the bulk theory is. Below are two examples for the vacuum-conductorinterface, where we shall calculate some coefficients from two simple models.In both cases, we shall employ a higher resolution hydrodynamic theory toobtain the coefficient of the coarser-grained one.

More specifically, we shall in the first example calculate the coefficient in theboundary condition Eq(126), and understand the discontinuity of the coarse-grained magnetic field H as given by a highly conducting surface layer (andsurface 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 whichthe discontinuity of the magnetic field is not surprising. The crucial pointhere is that the coarse-grained boundary conditions as presented above areapplicable also for situations where a higher resolution theory is not easily

20

available, such as for a turbulent boundary layer. If we check our ambitionswith respect to resolving the turbulence in details, this layer and its turbu-lence enhanced magnetic diffusivity should be well accounted for by the sameboundary condition.

The second example is less straight forward. Here, we employ the hydrody-namic theory for a dielectric as the higher resolution description of the con-ductor. This sounds somewhat surprising at first and should be explained.The dielectric theory (including a finite conductivity) contains a stationary,collective mode (the sq-mode) that is given by an exponential decay of theelectromagnetic field from the system’s surface [4]. Because the decay lengthshrinks with the conductivity, the hydrodynamic theory for conductors doesnot contain this mode, (though this may be amended by including higher or-der gradient terms). Nevertheless, the derived boundary conditions are suchthat the long ranged effects of the sq-mode are well accounted for, albeit ona grain size on which the sq-mode is no longer resolved. This statement isexplicitly proven by comparing the dielectric with the conductor theory.

5.1 The Highly Conducting Surface Layer

We consider an infinitely long and conducting wire (region 1), of r0, whichis located in a vacuum (region 2), and subjected to a parallel, constant andhomogeneous electrical field E = E0 ez.

coarse-grained theory

Within the wire we have

ED = const ez = E0 ez, (127)

and from Eq(76)

∇×H =1

β cED , (128)

in vacuum we have

∇×H = 0 . (129)

The connecting condition, Eq(126), is

21

H1,t −H2,t =− E0

βs ceϕ, (130)

(ϕ, r, z: cylinder coordinates). So in the wire

H(r) =1

2 β cE0 r eϕ, (131)

and outside

H(r) =

(E0 r0

2 β c+E0

βs c

)r0

reϕ . (132)

5.1.1 high resolution theory

We have, within the wire,

∇× E = 0 , c∇×H = σE, (133)

and outside,

∇× E = 0 , ∇×H = 0, (134)

connected by

∆Ht = 0 , ∆Et = 0. (135)

The solutions are, within the wire,

E = E0 ez , H(r) =σ

2 cE0 r eϕ, (136)

and outside

H(r) =σ

2 cE0

r20

reϕ . (137)

Clearly, both theories agree if

β = 1/σ , 1/βs = 0. (138)

22

To obtain a finite βs, we add a thin layer (say of thickness d = r0/1000) ofhighly conducting substance (say σL = 1018 s) to the wire (say σ = 1015 s).

The electric field remains constant throughout,

E =E0 ez , (139)

so ∆Et = 0 and ∇×E = 0 are satisfied. As a result of employing c∇×H = jel

twice, we have within the wire,

H(r) =σ

2 cE0 r eϕ , (140)

within the layer,

H(r) =

(σL

2 cE0 r +

(σ − σL)E0

2 c

r20

r

)eϕ (141)

and in vacuum,

H(r) =

(σL

2 cE0 (r0 + d)2 +

(σ − σL)

2 cE0 r

20

)1

reϕ

=

(σ

2 cE0 r0 +

σL d

cE0

)r0

reϕ +O

(d

r0

). (142)

For the given numbers the second parenthesis has the same magnitude as thefirst.

5.1.2 comparison

Choosing not to resolve the added layer, a comparison of Eq(132) with Eq(142)yields

1/βs =σL d . (143)

5.2 sq-Mode

5.2.1 the dielectric theory

The half space x < 0 is vacuum, and the half space x > 0 is a conductor,labeled 1 and 2, respectively. Region 1 has

23

H ≡ 0 , E =E0 ey . (144)

The stationary Maxwell equations, in region 2,

0 = c∇× (E + ED) , (145)

0 = jDel − c∇× (H + HD), (146)

with

HD =−α c∇× E, ED = β c∇×H, jDel = σE (147)

are satisfied by the general solution

Ey =Ec + A e−x/λ , (148)

Hz =−σcEc x+Hc −

λ

β cA e−x/λ,

where

λ=

√αβ c2

1 + β σ; (149)

The constant amplitudes Ec, Hc and A are to be determined from the con-necting conditions at x = 0. These are (91), (93) and (97) (no surface current)

Ht + HDt = 0 , Et + ED

t = E0 ey , (150)

cHDt = ζ1 n× ED , (151)

where the last equation is a result of ED ≡ 0 and HD ≡ 0 in vacuum, leadingto

Rsf = ...cHDt · (n× ED). (152)

So the special solution is

Ey =1

1 + β σE0 + A e−x/λ , (153)

Hz =− σ

c (1 + β σ)E0 x−

λ

β cA e−x/λ − σ λ

cA, (154)

EDy =

β σ

1 + β σE0 − A e−x/λ , (155)

24

HDz =

α c

λA e−x/λ (156)

with

A=σ β ζ1

(α c2/λ+ ζ1) (1 + β σ)E0 . (157)

5.2.2 the conducting theory

The stationary Maxwell equation (65) is reduced to

∇× ED = 0 , (158)

with

ED = βc c∇×H , (159)

cf Eq (76). The connecting conditions are

EDt =E0 ey , (160)

n× ED =−βs cHt ; (161)

the first being a result of putting E2 ≡ 0 and ED1 ≡ 0 in Eq(110), the second

follows from Eq(126). The solutions are

EDy ≡E0 , (162)

Hz =− 1

βc cE0 x−

1

βs cE0 . (163)

Now, comparing both results, we have finally

βc =1 + β σ

σ, (164)

1/βs =λσ2 β ζ1

(α c2/λ+ ζ1) (1 + β σ). (165)

References

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25

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