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Mathematical Models and Methods in Applied Sciences
cfWorld Scientific Publishing Company
DISSIPATIVITY AND IRREVERSIBILITY
OF ELECTROMAGNETIC SYSTEMS
MAURO FABRIZIO
Dipartimento di Matematica, Università di Bologna
Piazza di Porta S. Donato 5, 40126 Bologna, Italy
ANGELO MORRO
DIBE, Università di Genova
Via Opera Pia 11a, 16145 Genova, Italy
(..............)
A new approach is developed for the thermodynamics of electromagnetic materials with
memory by viewing them as dynamical systems. Though results are derived also for
non-linear materials with instantaneous response, an exhaustive set of thermodynamic
restrictions is derived for the case where the response is determined by linear functionals
of the histories of the electric field and the magnetic field. The kernels of the functionals
are shown to be subject to thermodynamic restrictions which are of basic importance in
wave propagation and initial boundary-value problems. The non-uniqueness of the free
enthalpy is examined and exemplified. The maximal free enthalpy is determined thus
showing that the Clausius-Duhem inequality may hold as an equality also for systems
with memory.
1. Introduction
Electromagnetic media usually show dispersive effects [1]. Only over a limited range
of frequencies, or in vacuum, can the velocity of propagation and wave attenuation
be regarded as constant in frequency. Dispersion is due essentially to the fact that
the polarization, the magnetization and the current density cannot follow the rapid
changes of the electromagnetic field. To describe these effects, simple models have
been elaborated in terms of a harmonic force and a damping force on the electrons
[2]. A more general and systematic theory hinges on the view that the body has
memory-dependent properties [3, 4].
The thermodynamic theory of electromagnetic solids with memory was first set
up in works by Coleman & Dill [5],[6] and then developed in a number of papers
and textbooks (cf. [7]). Lately a general theory of the electromagnetic field has
been set up by Gurtin [8] on the basis of gauge invariance properties. Yet some
conceptual reasons motivate a further investigation of the subject.
As with other models of material behaviour (cf. [9, 10]), simple electromagnetic
systems may be viewed as dynamical systems characterized by input-output relations. Here we consider polarizable and magnetizable bodies which allow also for
electric conduction. The memory of the material is described through functionals of
1
2
Dissipative electromagnetic systems
states and processes. Roughly, the state is a set of histories. In terms of states and
processes, we are able to establish the definition of cyclic processes and reversible
processes and hence to state the second law of thermodynamics for cycles. In materials with memory, cycles are quite a few. The thermodynamic scheme is then
completed by expressing the second law as a principle of dissipation of electromagnetic energy: the maximum work performed by the system in passing from a state
to any controllable state is bounded [11].
The conceptual novelty of the present approach to the constitutive properties of
materials with memory is that no recourse is made to norms and history spaces so
that the characterization of material properties is not affected by the choice of the
norm. Rather, a systematic scheme of electromagnetic systems is developed in terms
of dynamical systems. Though results are derived also for non-linear materials with
instantaneous response, an exhaustive set of thermodynamic restrictions is derived
for the case when the electric displacement D, the magnetic induction B, and the
current density J are determined by linear functionals of the histories of the electric
field E and the magnetic field H. For instance,
Z ∞
Z ∞
D(t) = 0E E(t) +
0E (u)E(t − u)du + 0H H(t) +
0H (u)H(t − u)du
0
0
where t is the time and 0E , 0E (u), 0H , 0H (u) are second-order tensors. Similar
0
relations are considered for B and J with tensors µ0E , µE
(u), µ0H , µ0H (u), and
0
0
κE (u), κH (u). As a consequence of thermodynamics it follows that
µH s (ω) > 0,
Es (ω) + κ̆Es (ω) + ω −1 κ∞E > 0,
ω > 0,
where the subscript s denotes the sine transform, κ is the integral of κ0 and κ̆(u) =
κ(u) − κ(∞). These inequalities are of basic importance in wave propagation and
initial boundary-value problems.
For electromagnetic materials with memory the free enthalpy, as well as the
free energy, is non-unique. Here we restrict attention to linear systems and hence
determine the maximal free enthalpy ψM . Physically, ψM exemplifies a remarkable
feature: the Clausius-Duhem inequality may hold as an equality though the system
is dissipative. We next exhibit other free enthalpies and prove the existence of
the minimal free enthalpy. Finally, we examine the dissipative properties at the
common boundary of two media thus deriving the thermodynamic restriction to be
imposed on some boundary conditions in electromagnetism.
2. Electromagnetic Systems
We denote by E the three-dimensional Euclidean point space and by V the associated translational space. We consider a body B occupying the region R ⊂ E.
Any point of B is labelled by the position vector x ∈ R. Henceforth it is understood that the statements are relative to any fixed point x ∈ R. The dependence
of E, H, D, B, J on x and t is constrained by the Maxwell equations. If the fields
Dissipative electromagnetic systems
3
E, H, D, B are smooth enough, the Maxwell equations can be written in the form
∇×E + Ḃ = 0,
∇ · B = 0,
∇×H − Ḋ = J,
∇ · D = ρ,
where ρ is the charge density, ∇ is the gradient operator and a superposed dot
denotes the time derivative.
Definition 1 An electromagnetic process P , of duration dP , is a piecewise continuous function on [0, dP ), with values in V × V , given by
P (t) = (ĖP (t), ḢP (t)),
t ∈ [0, dP );
(2.1)
ĖP , ḢP are the time derivative of the electric field EP and the magnetic field HP
(at a point) in [0, dP ).
The notation P[t1 ,t2 ) denotes the restriction of P to [t1 , t2 ) ⊂ [0, dP ). For ease
in writing, Pt stands for P[0,t) . Given two processes P1 , P2 of duration dP1 , dP2 , the
composition P1 ∗ P2 of P2 with P1 is defined as
š
P1 (t),
t ∈ [0, dP ),
(2.2)
(P1 ∗ P2 )(t) =
P2 (t − dP1 ), t ∈ [dP1 , dP1 + dP2 ).
The condition (2.1) characterizes simple materials in that the process involves
the time evolution of E and H at the pertinent point of B. Whenever the process P ,
at a point, is determined by the time evolution of the fields E and H in the whole
body B or in a neighbourhood of the point, the material is regarded as non-simple
and the associated phenomena as non-local. The materials considered in this paper
are simple.
A response of the material is associated with every process P and is characterized
by a function R : [0, dP ) → V × V × V given by
R(t) = (D(t), B(t), J(t)),
t ∈ [0, dP ),
(2.3)
namely the values of the displacement vector D, the magnetic induction B, and the
current density J.
The set Π of electromagnetic processes P satisfies the following properties.
i) If P ∈ Π then P[t1 ,t2 ) ∈ Π for every [t1 , t2 ) ⊂ [0, dP ).
ii) If P1 , P2 ∈ Π then P1 ∗ P2 ∈ Π.
We now introduce the notion of state (cf. [9]) and hence characterize the electromagnetic system.
Definition 2 A simple electromagnetic system is a set {Π, Σ, %, R} such that
1) Π is the set of processes,
2) Σ is an abstract set called the state space whose elements σ are called states,
3) % : Σ … Π → Σ is a function, called evolution function, subject to
%(σ, P1 ∗ P2 ) = %(%(σ, P1 ), P2 ),
∀P1 , P2 ∈ Π,
∀σ ∈ Σ,
(2.4)
4
Dissipative electromagnetic systems
4) R : Σ × (V × V ) → V × V × V is the response function which to each state and
value of a process, at a time t, assigns the response of the system at the same time
t.
We write the content of the fourth property as
D(t) = D̂(σ(t), P (t)),
B(t) = B̂(σ(t), P (t)),
J(t) = Ĵ(σ(t), P (t)).
(2.5)
The simplest example of electromagnetic system is the non-conducting dielectric
whose response (or constitutive model) is expressed by the continuous functions
D(t) = D̂(E(t)),
B(t) = B̂(H(t)),
J(t) = 0,
(2.6)
where
E · D̂(E) ≥ 0,
∀E ∈ V ;
H · B̂(H) ≥ 0,
∀H ∈ V.
In such a case the state σ is the pair (E, H). For, given a state σ0 = (E0 , H0 ) and
a process P = (ĖP , ḢP ), the transition function % is expressed by
E(t) =
Z
t
ĖP (ξ)dξ + E0 ,
H(t) =
0
Z
t
ḢP (ξ)dξ + H0 .
(2.7)
0
The response is given by (2.6). If, further, D̂ and B̂ are taken to be linear we have
D = E,
B = µH,
where , µ are positive definite tensors.
Another example is the electromagnetic system with magnetic hysteresis (cf.
[12]) which is described by the equations
D(t) = D̂(E(t)),
Ḃ(t) = B̂(H(t), Ḣ(t), B(t)),
J(t) = Ĵ(E(t)).
(2.8)
Here the state of the system is a triple,
σ = (E, H, B),
the transition function % is expressed by
Z t
E(t) =
ĖP (ξ)dξ + E0 ,
0
H(t) =
Z
t
ḢP (ξ)dξ + H0 ,
0
and the general integral of the differential equation
Ḃ(t) = B̂(H(t), ḢP (t), B(t))
where H(t) stands for the value determined through the process ḢP . Such integral,
along with the functions D̂, Ĵ, constitutes the response function.
For ease of notation, henceforth we omit the superscript P to denote the functions characterizing the process.
Dissipative electromagnetic systems
5
3. Materials with Memory
Any history is viewed as the set of the present value and the past history, e. g.,
Et = (E(t), Etr ), where Etr (s) = Et (s), s > 0. In the same way we write Ht =
(H(t), Htr ). For any pair of histories (Et , Ht ) we define the static continuation, of
duration a, to be the pair of histories ( a Et , a Ht ) such that
( a Et (s), a Ht (s)) =
n
(Et (s − a), Ht (s − a)),
(E(t), H(t)),
s>a
0≤s≤a
and the static relaxation, before a time b,
( b Et (s), b Ht (s)) =
n
(Et (s), Ht (s)), 0 ≤ s < b
(Et (b), Ht (b)), s ≥ b.
3.1. Dielectrics with memory
Dielectrics are modelled as non-conducting materials as follows.
Definition 3 A dielectric with memory is characterized by the constitutive equations
J=0
(3.9)
D(t) = D̂(Et , Ht ),
B(t) = B̂(Et , Ht ),
where (Et , Ht ) ' (E(t), Etr , H(t), Htr ) belong to the domain D = IR3 × IR3 × Dr
where Dr is a set of past histories (Etr , Htr ) : IR+ × IR+ 7→ IR3 × IR3 such that
1) Dr ⊃ L∞ (IR+ ) × L∞ (IR+ );
2) if (Et , Ht ) ∈ D then the static continuation ( a Et , a Ht ) is in D for every a > 0;
3) there are two functions D̃, B̃ on V × V such that
lim D̂( a Et , a Ht ) = D̃(E(t), H(t)),
a→∞
lim B̂( a Et , a Ht ) = B̃(E(t), H(t)), (3.10)
a→∞
the functions D̃, B̃ representing the constitutive functions of a dielectric without
memory;
4) if Et , Ht ∈ D then b Et , b Ht ∈ D for every b ≥ 0 and
lim D̂( b Et , b Ht ) = D̂(Et , Ht ),
b→∞
lim B̂( b Et , b Ht ) = B̂(Et , Ht ).
b→∞
By arguing as in the previous section, the hypotheses (3.9), and (3.10) allow us
to prove that dielectrics with memory are simple materials. Formally, we need only
replace the pair (E(t), H(t)), as the state, with (Et , Ht ). In the linear model, by
Definition 3 the fading memory property induces a precise condition on the memory
kernels.
In this regard we start from the standard linear representations of the constitutive functionals [1, 2],
Z ∞
Z ∞
B(t) = µ0 H(t) +
D(t) = 0 E(t) +
0 (s)Et (s)ds,
µ0 (s)Ht (s)ds (3.11)
0
0
6
Dissipative electromagnetic systems
where 0 , µ0 ∈ Sym and 0 , µ0 : IR+ 7→ Sym. Throughout the tensors 0 , µ0 and
0 , µ0 are allowed to depend on the position x ∈ R. Such dependence is understood
and not written.
Letting Et , Ht ∈ L∞ (IR+ ), by the property (1) we conclude that necessarily
0
 , µ0 ∈ L1 (IR+ ). Meanwhile, the properties (2), (3) are satisfied and
lim D̂( a Et ) = ∞ E(t),
a→∞
where
∞ = 0 +
Z
∞
0 (s)ds,
0
lim B̂( a Ht ) = µ∞ H(t),
a→∞
µ∞ = µ0 +
Z
∞
µ0 (s)ds.
0
This means that, as a → ∞, the functionals of the dielectric with memory are
required to become response functions of a dielectric (without memory). Hence the
equilibrium values ∞ , µ∞ are positive definite tensors. Finally, the property (4)
means that the domain D consists of the histories Et , Ht such that
Z ∞
Z ∞
lim
µ0 (s)ds Ht (b) = 0.
lim
0 (s)ds Et (b) = 0,
b→∞
b→∞
b
b
3.2. Conductors with memory
The metals and the ionized atmosphere are examples of conductors with a
complex-valued conductivity in time-harmonic fields (cf. [7], [13-15]). This is a
physical motivation for a model of conductor with memory where J and E are
related by a memory functional,
J(t) = Ĵ(Et ).
(3.12)
Meanwhile the electric displacement D and the magnetic induction B are given by
functions of E and H. A quite more general scheme is allowed by considering the
integrated history Èt of the electric field E up to time t, viz.
Z t
Èt (s) =
s ∈ IR+ ,
E(r)dr,
t−s
Since Èt (0) = 0, the static continuation Èat of Èt is defined to be
Èta (s) =
n
Èt (s − a), s > a,
s ≤ a.
0,
The correspondence between histories Et and integrated histories Èt is one-one and
hence the constitutive functional (3.12) for J may be written also as J(t) = J̃(Èt ).
Definition 4 A conductor with memory is characterized by the constitutive equations
D(t) = D̂(E(t), H(t)),
B(t) = B̂(E(t), H(t)),
J(t) = J̃(Èta )
Dissipative electromagnetic systems
7
such that
1) the domain D of the functional J̃ consists of the integrated histories Èt of the
electric field and D ⊃ L∞ (IR+ ),
2) if Èt ∈ D then Èat ∈ D,
3) lim J̃(Èta ) = 0, as a → ∞, for every Èt ∈ D,
4) if Et ∈ D then b Et ∈ D for every b ≥ 0 and
lim Ĵ( b Et ) = Ĵ(Et ).
b→∞
Remark. We might also define conductors with memory in terms of a memory
functional Ĵ(Et ), instead of J̃(Èt ), by replacing the asymptotic condition with
lim Ĵ( a Et ) = J(E(t)),
a→∞
where J(E) is the constitutive equation for the current density in a conductor. This
condition is more restrictive than the property (3) without being mathematically
more convenient.
For definiteness, let the functional on the history Et be linear in the form
Z ∞
κ(s)Et (s)ds.
J(t) =
0
Letting κ ∈ L1 (IR+ ) we integrate by parts to obtain
Z ∞
J(t) = −
κ0 (s)Èt (s)ds,
0
where κ0 (s) = dκ(s)/ds. Since D ⊃ L∞ (IR+ ) the function κ0 has to be an element
of L1 (IR+ ). Moreover by the property (3) it follows that the kernel κ must satisfy
Z ∞
κ(s + a)Et (s)ds = 0
lim
a→∞
0
for every Èt ∈ D.
4. Principles of Thermodynamics for electromagnetic systems
Later on we develop a thermodynamic analysis of the electromagnetic field through
the principle of electromagnetic energy dissipation. In this regard here we show
how this property follows from the second law of thermodynamics when isothermal
processes are involved.
Consider an electromagnetic system, with a uniform mass density ρ, occupying
the region R endowed with a smooth boundary ∂R. The first law of thermodynamics is stated with reference to cyclic processes namely pairs (σ, P ) such that
%(σ, P ) = σ. Let h be the rate at which heat is absorbed per unit mass.
8
Dissipative electromagnetic systems
First Law of Thermodynamics. At every cyclic process on [0, d)
Z dZ
[h(t) + Ḋ(t) · E(t) + Ḃ(t) · H(t) + J(t) · E(t)]dv dt = 0.
0
(4.13)
R
Denote by θ the absolute temperature, g = ∇θ the temperature gradient and q
the heat flux.
Second Law of Thermodynamics. At every cyclic process on [0, d)
Z dZ
‚ h(t)
ƒ
1
(4.14)
+ 2 q(t) · g(t) dv dt ≤ 0.
θ (t)
R θ(t)
0
Let now the temperature θ be constant, and hence g = 0. The inequality (4.14)
simplifies to
Z dZ
h(t)dv dt ≤ 0.
0
R
The first law (4.13) then implies that
Z dZ
[Ḋ(t) · E(t) + Ḃ(t) · H(t) + J(t) · E(t)]dv dt ≥ 0.
0
(4.15)
R
The validity of (4.15) for every cyclic process, on [0, d), is the global form of the
principle of the electromagnetic energy dissipation.
Here we restrict attention to simple materials for which the inequality holds for
every sub-body A of R. In such a case, the arbitrariness of A and the assumed
continuity of the integrand yield the local form of the principle, namely
Z d
[Ḋ(t) · E(t) + Ḃ(t) · H(t) + J(t) · E(t)]dt ≥ 0.
(4.16)
0
for every cyclic process on [0, d).
5. Reversibility
Also with reference to the formulations of the second law of thermodynamics in
standard physical frameworks, the notion of reversibility [16] still calls for a proper
role in a, possibly more precise, statement of the dissipation principle. In this regard
we begin by defining the reverse process of a given one.
Definition 5 For every process P of duration d, the reverse process P̃ is characterized by
˙
˙
P̃ (t) = (Ẽ(t),
H̃(t))
= lim+ (−Ė(d − τ ), −Ḣ(d − τ )),
τ →t
∀t ∈ [0, d).
If P is continuous, P̃ (t) = −P (d − t).
For convenience we denote by γ on Σ × V × V the electromagnetic power. The
value at any time t is given by
γ(σ(t), P (t)) = Ḋ(σ(t), P (t)) · E(t) + Ḃ(σ(t), P (t)) · H(t) + J(σ(t), P (t)) · E(t).
Dissipative electromagnetic systems
9
Definition 6 The process of states (σ, P ) is called reversible if
r1 )
%(σ, Pt ) = %(%(σ, P ), P̃d−t ),
∀t ∈ [0, d),
r2 )
γ(σ(t), P (t)) = −γ(σ(t), P̃ (d − t)),
∀t ∈ [0, d).
If at least one of these conditions is not true then the process of states is called
irreversible.
The definition of a reversible process of states, or simply reversible process, shows
that the reversibility is strictly related to the form of the constitutive equations.
Also reversibility is ascertained without having recourse to the thermodynamic principles. Two propositions are useful for later calculations.
Proposition 1 Every restriction of a reversible process is a reversible process.
To prove the assertion observe that if (σ, P ) is reversible then the condition (r2 )
holds for all t ∈ [0, d) and hence for every process (σ(t), P[t,t0 ) ) where σ(t) = %(σ, Pt )
and [t, t0 ) ⊂ [0, d). A similar argument proves that the restriction of a reversible
process satisfies (r1 ).
Proposition 2 If (σ, P ) and (%(σ, P ), P̂ ) are reversible processes then also (σ, P ∗
P̂ ) is reversible.
For, (σ, P ) and (%(σ, P ), P̂ ) are reversible processes and hence satisfy the properties (r1 ), (r2 ). By means of the semigroup properties (2.4) the properties (r1 ),
(r2 ) are seen to hold for the continuation (σ, P ∗ P̂ ).
Theorem 1 If the electromagnetic work
W (σ, P ) =
Z
d
γ(σ(t), P (t))dt
0
is non-negative for every process of states (σ, P ) in a class P then W vanishes at
every reversible process (σ, P ) ∈ P such that %(σ, P ), P̃ ) ∈ P.
Proof. Let (σ, P ) ∈ P be any reversible process. By (r1 ) and (r2 ) we have
Z
d
γ(%(%(σ, P ), P̃t ), P̃ (t))dt =
0
=−
Z
0
d
Z
d
0
γ(σ(d − t), P̃ (t))dt
γ(σ(d − t), P (d − t))dt =
Z
∞
γ(σ(t), P (t))dt.
0
A change of variable yields
W (%(σ, P ), P̃ ) = −W (σ, P ).
The hypothesis that W is non-negative for any process (σ, P ) ∈ P gives the desired
conclusion.
2
By way of example, we determine the reversible processes of a conducting body.
Still we consider the constitutive equations (2.5) and take the state σ as the pair
(E, H). We let
%(σ, P ) = σ(d).
%(σ, Pt ) = σ(t),
10
Dissipative electromagnetic systems
The property (r1 ) holds. For,
%(σ(d), P̃[d−t) ) = σ(d) +
Z
0
d−t
−Ṗ (d − ξ)dξ = σ(d) +
Z
t
Ṗ (τ )dτ = σ(t).
0
In connection with (r2 ), observe that D, B and J, at time t, are to be independent
of P (t). Moreover (r2 ) ultimately requires that
J(σ(t)) · E(t) = −J(σ(t)) · E(t)
for every time t ∈ [0, d). Hence the condition characterizing reversible processes is
that
J(E(t), H(t)) · E(t) = 0,
∀t ∈ [0, d).
Remark. A sufficient condition for a process of a conducting body to be reversible
is that E = 0 on [0, d). Now, physically sound constitutive equations for J are such
that J · E = 0 only if E = 0. In such a case E = 0 on [0, d) is also necessary for a
process to be reversible.
6. Principle of electromagnetic energy dissipation
The inequality (4.16) and the notion of reversibility allow us to write the following
statement.
Second Law (for isothermal processes). At any cycle (σ, P ) ∈ Σ × Π, the
electromagnetic work satisfies the inequality
Z
0
dP
‚
ƒ
Ḋ(σ(t), P (t)) · E(t) + Ḃ(σ(t), P (t)) · H(t) + J(σ(t), P (t)) · E(t) dt ≥ 0, (6.17)
for every simple electromagnetic system and equality holds for reversible cycles only.
It is worth emphasizing that cycles correspond to a restricted class of processes.
For materials with memory cycles consist of periodic processes. That is why, later
on, a dissipation principle is given for the general case of non-cyclic processes. In
this regard we need a more detailed characterization of non-cyclic processes.
Definition 7 The state σ ∈ Σ of an electromagnetic system is said to be controllable (or attainable) from all of Σ if, for any initial state σ i ∈ Σ, there is a process
P such that
%(σ i , P ) = σ.
The system is said to be controllable if any state σ ∈ Σ is controllable from all of
Σ.
Definition 8 The state space Σ of an electromagnetic system is said to be reachable
from a state σ if, whatever any final state σ f ∈ Σ, there is a process P ∈ Π such
that
%(σ, P ) = σ f .
Dissipative electromagnetic systems
11
If a state σ ∈ Σ is controllable, the space Σ need not be reachable from the state
σ. However, a controllable system makes Σ reachable from every state.
The dielectric is an example of a completely controllable system. For, if σ0 =
(E0 , H0 ) is any initial state and σ(t) = (E(t), H(t)) is any final state, it is always
possible to find a process P = (ĖP , ḢP ) such that
E(t) − E0 =
Z
dP
P
Ė (ξ)dξ,
0
H(t) − H0 =
Z
dP
ḢP (ξ)dξ.
(6.18)
0
Usually, instead, the state space of systems with memory are neither controllable
nor reachable.
Definition 9 Let D̃ ⊂ Σ be invariant under % in that, for every σ1 ∈ D̃ and P ∈ Π,
σ2 = %(σ1 , P ) ∈ D̃. A function e : D̃ → IR is called free energy if
e(σ2 )−e(σ1 ) ≤
Z
dP
0
‚
ƒ
Ḋ(σ(t), P (t))·E(t)+ Ḃ(σ(t), P (t))·H(t)+J(σ(t), P (t))·E(t) dt
(6.19)
Henceforth we let Σ be invariant under %.
Theorem 2 If the system is controllable, then by the Second Law there is a free
energy function e on the state space Σ.
Proof. For any state σ0 ∈ Σ and process P ∈ Π, the electromagnetic work W is
given
W (σ0 , P ) =
Z
0
dP
‚
ƒ
Ḋ(σ(t), P (t)) · E(t) + Ḃ(σ(t), P (t)) · H(t) + Ĵ(σ(t), P (t)) · E(t) dt
(6.20)
where σ(t) = %(σ0 , Pt ). Consider the state σ ∈ Σ and the set of real numbers
Q(σ0 , σ) := {W (σ0 , P ), ∀P : %(σ0 , P ) = σ}.
Fixed the states σ0 , σ, the set Q(σ0 , σ) is bounded below. For, since the system is
controllable there is a process P̃ ∈ Σ such that
%(σ, P̃ ) = σ0 .
(6.21)
The pair (σ0 , P ∗ P̃ ) is a cycle if %(σ0 , P ) = σ. Hence, by the Second Law we have
W (σ0 , P ∗ P̃ ) ≥ 0.
(6.22)
Accordingly,
W (σ0 , P ) + W (%(σ0 , P ), P̃ ) ≥ 0.
Given the state σ = %(σ0 , P ) and the process P̃ , let M = W (σ, P̃ ). Hence
W (σ0 , P ) ≥ −M
(6.23)
12
Dissipative electromagnetic systems
for every P ∈ Π such that %(σ0 , P ) = σ. Now, for every σ0 ∈ Σ, consider the set
Q(σ0 , σ) and, owing to (6.23), let
e(σ) = inf{Q(σ0 , σ),
∀P : %(σ0 , P ) = σ}.
(6.24)
The function e is a free energy on Σ in that
(e1 ) e : Σ 7→ IR,
(e2 ) if σ1 , σ2 ∈ Σ and P ∈ Π is such that %(σ1 , P ) = σ2 then (6.19) holds.
The property that e is a function on Σ follows from the controllability of the
system. To prove the property (e1 ), we argue that by (6.24), for every  > 0 there
is a process P such that %(σ0 , P ) = σ1 with
W (σ0 , P ) ≥ e(σ1 ) + .
Moreover, by (6.24),
W (σ0 , P ∗ P ) > e(σ2 ).
Accordingly, we have
e(σ2 ) − e(σ1 ) < W (σ0 , P ∗ P ) − W (σ0 , P ) +  < W (σ1 , P ) + .
Hence, by the arbitrariness of , the inequality (6.19) follows.
2
Remark. By analogy with [17] we can show that if there exists a state σ0 that is
controllable from all Σ then the Second Law implies the existence of a free energy
defined on the set Dσ0 of all states that are reachable from σ0 , namely
Dσ0 = {σ ∈ Σ : ∃P ∈ Π, %(σ0 , P ) = σ}.
Remark. As a consequence of Theorem 2, at any time t when the functional
e(σ(t)) is differentiable, the inequality
ė(σ(t)) ≤ Ḋ(σ(t), P (t)) · E(t) + Ḃ(σ(t), P (t)) · H(t) + J(σ(t), P (t)) · E(t) (6.25)
holds.
The controllability property might be much too severe thus making the Second
Law scarcely useful. A more general principle is then convenient. For any pair
σ ∈ Σ, σ 0 ∈ Dσ consider the scalar
M (σ, σ 0 ) := inf{W (σ, P ); P ∈ Π, %(σ, P ) = σ 0 }.
(6.26)
Hence, by analogy with [11] we can state the
Dissipation Principle. For every simple electromagnetic system the work done
from a state σ to every state σ0 ∈ Dσ is bounded below, viz. for any σ ∈ Σ there is
a constant Mσ such that
M (σ, σ 0 ) ≥ Mσ > −∞,
∀σ 0 ∈ Σ.
(6.27)
Dissipative electromagnetic systems
13
There exists a state σ † , called zero state, such that M (σ, σ) ≥ 0 for all σ ∈ Dσ† .
If there is a process P̄ ∈ Π such that W (σ, P̄ ) = M (σ, σ 0 ) and M (σ, σ 0 ) =
−M (σ 0 , σ) then (σ, P̄ ) is reversible.
The following theorem establishes a relation among the Second Law (cf. (6.17))
and the dissipation principle (cf. 6.27).
Theorem 3 The Second Law of Thermodynamics follows from the Dissipation
Principle. If, further, the system is controllable then the Second Law is also sufficient for the validity of the Dissipation Principle.
Proof. If (σ, P ) ∈ Σ × Π is a cycle, then so is the pair (σ, P ∗ P ). Letting P n be
the composition of n times the process P , we observe that (σ, P n ) is a cycle for any
integer n. Then, by the Dissipation Principle,
W (σ, P n ) = nW (σ, P ) ≥ M (σ, σ),
∀n ∈ IN.
(6.28)
Because W (σ, P ) is non-negative, this implies that (6.17) holds; otherwise, by (6.28),
the bound M (σ, σ) > −∞ would not be satisfied. Moreover, M (σ, σ) must vanish
because it is the infimum over a set of non-negative numbers which contains zero.
In fact, the trivial process P0 satisfies %(σ, P0 ) = σ and W (σ, P0 ) = 0. To prove
that equality in (6.17) holds only for reversible processes, observe that if there is a
non-trivial cycle (σ, P ) such that W (σ, P ) = 0, then the vanishing of M (σ, σ) and
the Dissipation Principle ensure that (σ, P ) is reversible.
Assume now that the system is controllable and that (6.17) holds for any cycle.
For any two states σ, σ 0 ∈ Σ, there is a process P 0 such that %(σ 0 , P 0 ) = σ. Then,
for each process P ∈ Π such that %(σ, P ) = σ 0 , the pair (σ, P ∗ P 0 ) is a cycle, and
(6.17) yields
(6.29)
W (σ, P ∗ P 0 ) = W (σ, P ) + W (σ 0 , P 0 ) ≥ 0.
Hence the work W (σ, P ) is necessarily bounded at all processes P with %(σ, P ) =
σ 0 and, moreover, M (σ, σ 0 ) + M (σ 0 , σ) ≥ M (σ, σ) = 0. Further, if M (σ, σ 0 ) =
−M (σ 0 , σ), and if there are processes P, P 0 satisfying %(σ, P ) = σ 0 , %(σ 0 , P 0 ) = σ,
and
W (σ, P ) = M (σ, σ 0 ),
W (σ 0 , P 0 ) = M (σ 0 , σ),
then, by the Second Law, the pair (σ, P ∗ P 0 ) is a reversible cycle in that
W (σ, P ∗ P 0 ) = W (σ, P ) + W (σ 0 , P 0 ) = 0.
Since P is obviously a restriction of P ∗ P 0 , it follows that (σ, P ) is also reversible.
Similarly, (σ 0 , P 0 ) is reversible. .
Theorem 4 The Dissipation Principle implies the existence of a free-energy functional for the system, eM : Dσ0 → IR for every state σ0 ∈ Σ.
Proof. Consider a reference state σ0 and any state σ ∈ Dσ0 that is reachable from
σ0 . Then we can define the functional eM (σ) by
eM (σ) := M (σ0 , σ).
(6.30)
14
Dissipative electromagnetic systems
In view of the definition of M (σ, σ 0 ), for every pair of states σ1 , σ2 ∈ Dσ0 and any
process P ∈ Π such that %(σ1 , P ) = σ2 , we have
eM (σ2 ) − eM (σ1 ) ≤ inf W (σ0 , P1 ∗ P ) − inf W (σ0 , P1 )
P1 ∈Π1
P1 ∈Π1
where Π1 = {P ∈ Π; %(σ0 , P1 ) = σ1 }. Hence, for any  > 0, there is a process P1
such that
eM (σ2 ) − eM (σ1 ) ≤ W (σ0 , P1 ∗ P ) +  − W (σ0 , P1 ) = W (σ1 , P ) + .
Since  is arbitrary, we have the characteristic inequality
eM (σ2 ) − eM (σ1 ) ≤ W (σ1 , P ).
.
Consider the functional ẽM relative to the zero state, ẽM (σ) = M (σ † , σ).
Proposition 3 The free energy functional ẽM on Dσ† is related to any other free
energy functional e defined on D ⊃ Dσ† by the inequality
ẽM (σ) ≥ e(σ),
∀σ ∈ Φσ† .
(6.31)
Proof. Since e is a free energy and e(σ † ) = 0, by (6.24) we have
e(σ) ≤ W (σ † , P )
(6.32)
for all P ∈ Π such that %(σ † , P ) = σ. By applying (6.30) it follows that for every
 > 0 there is a process P such that %(σ † , P ) = σ and
W (σ † , P ) < ẽM (σ) + .
(6.33)
Comparison of (6.32) and (6.33) and the arbitrariness of P and  yield (6.31). .
When the work of a given system, in passing from a state σ to any other state
reachable from σ, is bounded above then we can provide the definition of a new free
energy.
Theorem 5 The Dissipation Principle implies that the quantity
N (σ) = sup {−W (σ, P ) : P ∈ Π}
(6.34)
is bounded above for all σ ∈ Σ, then
em (σ) := N (σ)
is a free energy such that em (σ † ) = 0.
(6.35)
Dissipative electromagnetic systems
15
Proof. The requirement (e1 ) for a free energy is satisfied in that em is defined
on Σ. To examine (e2 ) we consider two states σ1 , σ2 and a process P 0 such that
%(σ1 , P 0 ) = σ2 . By (6.34) it follows that, for any P ∈ Π with %(σ2 , P ) = σ ∗ ,
−W (σ1 , P 0 ∗ P ) ≤ N (σ1 , σ ∗ )
(6.36)
and that, for any  > 0 there is a process P such that
−W (σ2 , P ) > N (σ2 , σ ∗ ) − .
(6.37)
Comparison of (6.35), (6.36) and (6.37) yields
em (σ2 ) − em (σ1 ) −  < −W (σ2 , P ) + W (σ1 , P 0 ∗ P ).
Letting P = P , by the arbitrariness of  we have the desired result in the form
em (σ2 ) − em (σ1 ) ≤ W (σ1 , P 0 ).
Moreover, N (σ † ) = 0 because it is the supremum over a set of non-positive numbers
containing the zero element. .
Proposition 4 For all systems such that the free energy functional em exists, any
other free energy e on Σσ† with e(σ † ) = 0 is related to em by
em (σ) ≤ e(σ),
∀σ ∈ Σσ† ,
(6.38)
that is, em is the minimal free energy of the system.
Proof. By (6.35), if σ ∈ Σσ† for any  > 0 there is a process P such that %(σ, P ) =
σ and
−W (σ, P ) > N (σ) −  = em (σ) − .
If e is a free energy then
e(σ ) − e(σ) ≤ W (σ, P ).
Since e(σ † ) = 0, it follows that e(σ ) ≥ 0 and hence
e(σ) ≥ −W (σ, P ).
By the arbitrariness of , (6.38) follows.
.
7. Dielectrics and Rate-Type Electromagnetic Materials
By analogy with [10] we now establish the restrictions placed by the dissipation
principle on conducting media that are described by the constitutive equations
D = D̂(E, H),
B = B̂(E, H),
J = Ĵ(E, H).
(7.39)
To simplify the notation we use the same symbol for the function and the value.
16
Dissipative electromagnetic systems
On the basis of (7.39), the state σ is defined to be the pair E, H, i.e. σ = (E, H),
the state transition is given by (6.18), and D, B, J are the response functions. At
every cyclic process (σ, P ) ∈ Σ × Π the inequality
0 ≤ W (σ, P ) =
Z
d
0
‚
ƒ
Ḋ(E(t), H(t))·E(t)+Ḃ(E(t), H(t))·H(t)+J(E(t), H(t))·E(t) dt
holds. Two integrations by parts yield
Z
0
d
‚
ƒ
D(E(t), H(t)) · Ė(t) + B(E(t), H(t)) · Ḣ(t) − J(E(t), H(t)) · E(t) dt ≤ 0. (7.40)
Let P0 be the trivial process given by Ė(t) = 0, Ḣ(t) = 0, ∀t ∈ [0, dP0 ). Hence
(σ, P0 ) is cyclic for every (initial) state σ. The dissipation inequality reduces to
J(E, H) · E ≥ 0,
∀E, H ∈ V.
(7.41)
By (7.40), for every cycle,
Z
d
0
‚
ƒ
D(σ(t)) · Ė(t) + B(σ(t)) · Ḣ(t) dt ≤
Z
0
d
‚
ƒ
J(σ(t)) · E(t) dt.
(7.42)
With any smooth, oriented, closed curve C + in the state space Σ ⊂ V × V we
associate a cyclic process (σ, P ) such that the sequence of states σ(t) = %(σ, Pt )
covers the curve C + in the given orientation. Let
Z
D(E, H) · dE + B(E, H) · dH,
I(C + ) =
C+
whence
+
I(C ) =
Z
0
d
[D(E(t), H(t)) · Ė(t) + B(E(t), H(t)) · Ḣ(t)]dt,
(7.43)
Hence (7.42) and (7.43) imply that
I(C + ) ≤
Z
d
0
J(E(t), H(t)) · E(t)dt.
(7.44)
By the continuity of the vector functions E(t), H(t) on [0, d) it follows that
M (C + ) = sup{J(E, H) · E; (E, H) ∈ C + }
exists and is finite. Hence (7.44) implies that
I(C + ) ≤ d M (C + ).
(7.45)
I(C + ) ≤ 0.
(7.46)
The arbitrariness of d yields
Dissipative electromagnetic systems
17
Denote by C − the curve obtained by C + by reversing the orientation. By repeating
step by step the previous argument we obtain the analogous result
I(C − ) ≤ 0.
(7.47)
Since
I(C + ) = −I(C − ),
it follows that
I(C) = 0
(7.48)
for every closed curve C in Σ.
The result (7.48), along with the assumed continuity of D(E, H) and B(E, H),
implies that there exists a state function ψ(E, H) such that
ψ̇(E, H) = −D(E, H) · Ė − B(E, H) · Ḣ
whence
D(E, H) = −∂E ψ(E, H),
B(E, H) = −∂H ψ(E, H)
(7.49)
where ∂E = ∂/∂E, ∂H = ∂/∂H. Hence, for any process (σ1 , P ) the work W takes
the form
Z d
ƒd
‚
W (σ1 , P ) = ψ(%(σ1 , P )) − ψ(σ1 ) +
Ĵ(E(t), H(t)) · E(t)dt + D · E + B · H 0 .
0
By means of the function e = ψ − D · E − B · H, the inequality (7.41) yields the
bound (6.19), namely
e(σ2 ) − e(σ1 ) ≤ W (σ1 , P )
(7.50)
where σ2 = %(σ1 , P ). Hence e is a free energy.
Remark. As a consequence of (7.49), for C 1 functions D̂, B̂, the symmetry properties
∂E D = (∂E D)T ,
∂H B = (∂H B)T ,
∂H D = ∂E B,
hold. Also eqs. (7.49) imply that the potential ψ and the free energy e are unique to
within an additive constant. For, if e1 , e2 are two free energies then ψ1 −ψ2 = e1 −e2 .
Because
∂E (ψ1 − ψ2 ) = 0,
∂H (ψ1 − ψ2 ) = 0,
it follows that ψ1 − ψ2 is constant.
A second model of electromagnetic medium describes the electric conduction
through a rate-type model. For isotropic materials the constitutive equations are
taken in the form
D = E,
B = µH,
αJ̇ = κE − J,
18
Dissipative electromagnetic systems
where α, , µ, κ are suitable scalars possibly dependent on the position. Such equations are linear and the third one is a rate-type equation for the current density J.
The state of the body is then the triple (E, H, J).
The electromagnetic work is expressed as
W (σ, P )
=
=
Z
d
€
α
1 
J̇ · J + J2 dt
κ
κ
0
Z d
‚ d €  2 µ 2 α 2 1 2ƒ
E + H + J + J dt.
dt 2
2
κ
κ
0
E · Ė + µH · Ḣ +
The dissipation principle results in
Z
d
0
1 2
J dt ≥ 0,
κ
which implies that the conductivity κ is positive.
It is a routine calculation to check that e = (1/2)(E2 + µH2 + (α/κ)J2 ) is the
free energy, to within an additive constant.
8. Thermodynamic Restrictions on Linear Electromagnetic Systems
The known linear models of simple electromagnetic systems are in fact particular
cases of the response functions
Z ∞
Z ∞
0
t
0H (u)Ht (u)du, (8.51)
D(t) = 0E E(t) +
E (u)E (u)du + 0H H(t) +
0
0
B(t) = µ0E E(t) +
Z
∞
0
J(t) =
0
(u)Et (u)du + µ0H H(t) +
µE
Z
0
∞
0
κE
(u)Et (u)du +
Z
∞
0
Z
∞
0
µ0H (u)Ht (u)du, (8.52)
κ0H (u)Ht (u)du.
(8.53)
Of course 0E , 0E (u), ... have values in Lin(V ) and the dependence on x ∈ R is
understood and not written. The state σ is the pair of histories Et , Ht . Hence a
pair (σ, P ) is cyclic if the histories Et , Ht are periodic.
Concerning the dependence on the argument u, as usual we let 0E , 0H , ..., κ0H ∈
1
L (IR+ ). Let
Z
∞
ˆE = 0E +
0
0E (u) exp(iωu)du,
and so on for ˆH , ..., κ̂H . For time-harmonic fields (∝ exp(−iωt)) we have
D = ˆE E + ˆH H,
B = µ̂E E + µ̂H H,
J = κ̂E E + κ̂H H.
By definition, ˆE , ..., κ̂H are elements of Lin(V ) with complex values. Moreover,
Z ∞
lim ˆE = 0E ,
lim ˆE = 0E +
0E (u)du =: ∞E ,
ω→∞
ω→0
0
Dissipative electromagnetic systems
19
and so on.
We determine the thermodynamic restrictions by considering the electromagnetic field
E(x, t) = E1 (x) sin ωt + E2 (x) cos ωt,
H(x, t) = H1 (x) sin ωt + H2 (x) cos ωt,
(8.54)
where ω > 0. Correspondingly we obtain
B = B1 sin ωt + B2 cos ωt,
D = D1 sin ωt + D2 cos ωt,
J = J1 sin ωt + J2 cos ωt,
where B1 , B2 , D1 , D2 , J1 , J2 are appropriate linear vector functions of E1 , E2 , H1 ,
H2 , parameterized by the frequency ω. Preliminarily we have to ascertain that the
field (8.54) satisfy the Maxwell equations. Substitution yields
∇×E1 − ωB2 = 0,
∇×E2 + ωB1 = 0,
∇ · B1 = 0,
∇ · B2 = 0,
(8.55)
∇×H1 + ωD2 = J1 , ∇×H2 − ωD1 = J2 , ω∇ · D1 = −∇ · J2 , ω∇ · D2 = ∇ · J1 .
(8.56)
If the values E1 , E2 , H1 , H2 at a point x are fixed so are B1 , B2 , D1 , D2 , J1 , J2 .
Hence (8.54) satisfies the Maxwell equations for arbitrary values of E1 , E2 , H1 , H2
at any point of the body provided only that the gradients ∇E1 , ∇E2 , ∇H1 , ∇H2
satisfy (8.55) and (8.56).
The field (8.54) is admissible for arbitrary vectors E1 , E2 , H1 , H2 , provided only
that the space dependence is appropriate. We can then determine some consequences of the inequality (6.17) on the constitutive equations for D, B, J. Substitution of (8.54) in (6.17) and integration over the period [0, d] yields
E2 · D1 − E1 · D2 + H2 · B1 − H1 · B2 + ω −1 (E1 · J1 + E2 · J2 ) > 0.
(8.57)
As ω → ∞, by Riemann-Lebesgue’s lemma we have
D1 → 0E E1 + 0H H1 ,
D2 → 0E E2 + 0H H2 ,
and similarly for B and J. Hence, as ω → ∞, (8.57) simplifies to
E2 · [0E − (0E )T ]E1 + H2 · [µ0H − (µ0H )T ]H1
+E2 · [0H − (µ0E )T ]H1 + H2 · [µ0E − (0H )T ]E1 ≥ 0.
(8.58)
The arbitrariness of E1 , E2 , H1 , H2 implies that
0E = (0E )T ,
µ0H = (µ0H )T ,
0H = (µ0E )T .
(8.59)
Also, letting ω → 0 we obtain (8.58) with 0E,H , µ0E,H replaced by ∞E,H , µ∞E,H .
Using again the arbitrariness of E1 , E2 , H1 , H2 gives
∞E = (∞E )T ,
µ∞H = (µ∞H )T ,
∞H = (µ∞E )T .
(8.60)
20
Dissipative electromagnetic systems
By regarding formally the 4-tuples of vectors (E1 , E2 , H1 , H2 ) as elements of the
twelve-dimensional Euclidean vector space V12 , eq. (8.57) results in the positive
definiteness of a proper tensor Λ in V12 for every ω ∈ IR++ . In particular it follows
that
0
µ0Hs > 0,
∀ω ∈ IR++ ,
(8.61)
ωEs
+ κ0Ec > 0,
where the subscripts c and s denote half-range cosine and sine Fourier transform.
The inequalities (8.61) generalize a result by Gentili on isotropic dielectrics with
memory [19]. Also, letting ω be small enough and using the continuity of the
quantities involved we conclude that
κ0Ec (0) > 0,
in V . Since, e.g.,
κ0Ec (0) =
Z
∞
0
κ0Hc (0) = 0,
κ0E (u)du = κ∞E ,
we can write
κ∞E > 0,
κ∞H = 0.
(8.62)
Theorem 6 The constitutive equations (8.51) to (8.53) meet the strong principle of
electromagnetic energy dissipation if and only if (8.59) and the positive definiteness
of Λ hold.
Proof. The state and the process are the pairs σ(t) = (Et , Ht ), P = (ĖP , ḢP ). The
“only if” part has just been proved. To prove the “if” part we consider a periodic
state function σ of period d. Accordingly we represent σ, namely the histories Et
and Ht , by Fourier series as
Et (x, u) =
∞
X
k=0
Ek1 (x) sin kω(t − u) + Ek2 (x) cos kω(t − u),
and similarly for Ht , where ω = 2π/d. Upon suitable dependences on x, any 4-tuple
of values Ek1 , Ek2 , Hk1 , Hk2 is compatible with Maxwell’s equations. The function
D(t) is then given by
D(t) =
∞
X
[Dh1 sin kωt + Dh2 cos kωt]
h=0
where
Dh1 = [0E + 0Ec (hω)]Eh1 + 0Es (hω)Eh2 + [0H + 0Hc (hω)]Hh1 − 0Hs (hω)Hh2 ,
Dh2 = [0E − 0Es (hω)]Eh1 + 0Ec (hω)Eh2 + [0H − 0Hs (hω)]Hh1 + 0Hc (hω)Hh2 .
The functions B(t) and J(t) are obtained by simply replacing  with µ and κ,
respectively.
Dissipative electromagnetic systems
21
The process P is taken as the corresponding pair (Ė, Ḣ) on [0, d]. The work
W (σ, P ) along the cycle, is given by
W (σ, P ) = ω
Z
∞ X
∞
dX
0 h=0 k=1
+
Z
h{[Dh1 cos hωt − Dh2 sin hωt] · [Ek1 sin kωt + Ek2 cos kωt]
+[Bh1 cos hωt − Bh2 sin hωt] · [Hk1 sin kωt + Hk2 cos kωt]}dt
∞ X
∞
dX
[Jh1 sin hωt + Jh2 sin hωt] · [Ek1 sin kωt + Ek2 cos kωt]dt.
0 h=0 k=0
Term by term integration, as t ∈ [0, d], shows that the only non-zero terms are those
with h = k. Further, integration yields a common factor d/2 = π/ω. Hence, letting
Ξk be the 4-tuple of vectors (Ek1 , Ek2 , Hk1 , Hk2 ), in view of (8.59), we have
W (σ, P ) = π
∞
X
k=1
kΞk · Λ(kω)Ξk +
π
{E01 · κ0Ec (0)E01 + E02 · κ0Ec (0)E02 }.
ω
The positive definiteness of Λ and κ0Ec (0) yields the desired conclusion W (σ, P ) > 0
for every non-trivial cycle, which is the content of the strong principle of electromagnetic energy dissipation. .
Consider the relaxation functions
Z u
Z u
Z u
0
0
(ξ)dξ, µH (u) = µ0H +
µ0H (ξ)dξ.
(ξ)dξ, κE (u) =
E (u) = 0E +
κE
E
0
0
0
0
, µ0H ∈ L1 (IR+ ), the limit values ∞E , κ∞E , µ∞H of E (u), κE (u),
Since 0E , κE
µH (u) as u → ∞, hold finite. It is convenient to consider also
˘E (u) = E (u) − ∞E ,
κ̆E (u) = κE (u) − κ∞E ,
µ̆H (u) = µH (u) − µ∞H .
Lemma 1 If µ0H ∈ L1 (IR+ ) then
µ0H − µH (u) < 0
and
Z
∞
0
µ̆Hc (ω)dω = µ0H − µ∞H .
Proof. By the Fourier inverse transform we have
Z
2 ∞ 0
µ0H (u) =
µHs (ω) sin ωu dω
π 0
whence, by integration,
µH (u) − µ0H =
2
π
Z
0
∞
1 − cos ωu 0
µHs (ω)dω.
ω
22
Dissipative electromagnetic systems
Since µ0Hs > 0 the first result follows. Now, an integration by parts yields
1
µ̆Hc (ω) = − µ0Hs (ω)
ω
and hence
Z
2 ∞
(1 − cos ωu)µ̆0Hc (ω)dω.
π 0
The limit as u → ∞ and Riemann-Lebesgue’s lemma yield the second result.
As a consequence of Lemma 1 we have
µH (u) − µ0H = −
.
µ∞H > µ0H ,
namely, the equilibrium magnetic permeability µ∞H is greater than the instantaneous magnetic permeability µ0H .
An integration by parts yields
0
κEc
(ω) = κ∞E + ω κ̆Es
and hence (8.61) becomes
0
Es
+ κ̆Es +
1
κ∞E > 0.
ω
Application of the Fourier inverse transform and integration with respect to u yields
Z

2 ∞ 1 − cos ωu €
1
κ̆Es + κ∞E dω.
E (u) − 0E > −
π 0
ω
ω
If, further, the solid is non-conducting - and hence κE = 0 - then
E (u) − 0E > 0.
Taking the limit as u → ∞ yields
∞E > 0E .
For non-conducting solids, the equilibrium electric permittivity ∞E is greater than
the instantaneous electric permittivity 0E .
If H , µE , κH are zero then simpler relations follow. Letting , µ, κ stand for
E , µH , κE we have
µ0s > 0,
s0 + κ̆s + ω −1 κ∞ > 0.
9. Free Enthalpy of Linear Systems with Memory
Among the potentials for dielectrics with memory, the free enthalpy ψ : D 7→ IR,
defined by ψ = e − B · H − D · E, proves very convenient (cf. [5, 6]). By (6.19), for
any pair (σ1 , P ), such that σ1 ∈ D and %(σ1 , P ) = σ2 ∈ D, the inequality
Z d
ψ(σ2 ) − ψ(σ1 ) ≤
[−B(S(τ )) · Ḣ(τ ) − D(S(τ )) · Ė(τ ) + J(S(τ )) · E(τ )]dτ (9.63)
0
Dissipative electromagnetic systems
23
holds where S(τ ) = (σ(τ ), P (τ )). A function ψ : D 7→ IR is a free enthalpy if (9.63)
holds for every σ1 ∈ D̃ and P ∈ Π such that σ2 = %(σ1 , P ) while e = ψ +B·H+D·E
is positive definite, namely e(σ(t)) > 0 when E(t), H(t), Etr , Htr are not all zero.
For, (9.63) and the definition of ψ yield (6.19).
As usual, we denote by E† (t) and H† (t) the histories constantly equal to E(t)
and H(t). We now remind a theorem of Coleman and Dill [5, 6].
Theorem 7 Every free enthalpy of a dielectric with memory satisfies
B = −∂H ψ,
D = −∂E ψ,
(9.64)
wherever ψ is differentiable with respect to E(t) and H(t), and is minimal at constant histories in that, if (Et , Ht ) ∈ D then
ψ(Et , Ht ) ≥ ψ(E† (t), H† (t)).
(9.65)
Proof. By (9.63) we have
lim
h→0+
ψ(σ(t + h)) − ψ(σ(t))
≤ −B(σ(t)) · Ḣ(t) − D(σ(t)) · Ė(t) + J(σ(t)) · E(t)
h
where σ(t) = (Et , Ht ) and Ė and Ḣ stand for the right time derivative of E and H.
Indicate separately the dependence on the present values E(t), H(t) and the past
histories Ert , Hrt . Hence ψ = ψ(E(t), Ert , H(t), Htr ) and we have
∂E ψ · Ė + ∂H ψ · Ḣ + δψ ≤ −B · Ḣ − D · Ė + J · E
where δψ is the differential of ψ as the present values E(t), H(t) are kept fixed. Since
δψ(Et , Ht ) depends on the state (Et , Ht ), as in [6] we can choose the derivatives
Ė, Ḣ arbitrarily while leaving ∂E ψ, ∂H ψ, B, D unchanged. It then follows that (9.64)
holds. To prove the minimum property we need only choose σ1 = (Et , Ht ) and
σ2 = (E† (t), H† (t)); σ2 can be obtained by σ1 through a constant trivial process of
infinite duration. Hence (9.63) implies (9.65). .
As a by-product we have the following
Proposition 5 A function ψ : D 7→ IR is a free enthalpy if and only if, for every
smooth process P ,
ψ̇ ≤ −B · Ḣ − D · Ė + J · E,
(9.66)
Definition 10 If (9.63), or (9.66), holds as an equality, the corresponding free
enthalpy is called maximal.
For later convenience we recall a result given in [20].
Proposition 6 If K ∈ L1 (IR+ , Sym) and w ∈ L2 (IR+ ) then
Z ∞ Z u1
w(u1 ) · K(u1 − u2 )w(u2 )du2 du1 > 0
0
0
24
Dissipative electromagnetic systems
for any non-zero w if and only if
Kc (ω) > 0,
ω ∈ IR+ .
We now restrict attention to linear systems with H , µE , κH = 0 and let , µ, κ
stand for E , µH , κE . For definiteness we first consider non-conducting (J = 0),
linear, dielectrics for which (9.66) reduces to
ψ̇ ≤ −B · Ḣ − D · Ė.
(9.67)
The tensor functions 0 , µ0 , which are taken to have values in Sym, are required to
satisfy the thermodynamic restrictions (8.61), namely
0s > 0,
µ0s > 0,
∀ω ∈ IR++ .
For the sake of simplicity some examples of free enthalpy functional are given by
assuming that
0 (u) = 0 (u)T ,
µ0 (u) = µ0 (u)T ,
∀u ∈ IR+ .
Let Ĕt , H̆t denote the difference histories of E, H relative to the present values
E(t), H(t); e.g. Ĕt (u) = Et (u) − E(t). Also observe that if  = (|u1 − u2 |) then
12 :=
∂ 2 (|u1 − u2 |)
= −00 (|u1 − u2 |) − 2δ(u1 − u2 )0 (|u1 − u2 |)
∂u1 ∂u2
where a prime denotes the derivative with respect to the argument. The same
relation holds for µ. Consider the functional
Z ∞Z ∞
t
t
1
1
ψM (E , H ) = − 2 E(t) · ∞ E(t) − 2
[Ĕt (u1 ) · 12 (|u1 − u2 |)Ĕt (u2 )]du1 du2
0
0
Z ∞Z ∞
1
1
[H̆t (u1 ) · µ12 (|u1 − u2 |)H̆t (u2 )]du1 du2 .
− 2 H(t) · µ∞ H(t) − 2
0
0
Time differentiation and integrations by parts yield
Z ∞Z ∞
Ėt (u1 ) · 1 (|u1 − u2 |)Ėt (u2 )du1 du2
ψ̇M (Et , Ht ) = −D(t) · Ė(t) − 12
0
0
Z ∞Z ∞
1
−B(t) · Ḣ(t) − 2
Ḣt (u1 ) · µ1 (|u1 − u2 |)Ḣt (u2 )du1 du2 ,
0
0
where, e.g., 1 = ∂(|u1 − u2 |)/∂u1 . Now, for  (and µ) we have
1 = sgn(u1 − u2 )0 (|u1 − u2 |),
and hence 1 , µ1 are odd by interchange of u1 and u2 . Accordingly the two integrals
vanish and (9.65) holds with the equality sign. This in turn shows that the functional
Dissipative electromagnetic systems
25
ψM is the maximal free enthalpy. Of course (9.64) and (9.65) hold. Correspondingly,
the free energy e = ψ + B · H + D · E takes the form
Z ∞Z ∞
eM (Et , Ht ) = 12 E(t) · 0 E(t) − 12
Et (u1 ) · 12 (|u1 − u2 |)Et (u2 )du1 du2
Z ∞0 Z ∞0
Ht (u1 ) · µ12 (|u1 − u2 |)Ht (u2 )du1 du2 .
+ 12 H(t) · µ0 H(t) − 21
0
0
Incidentally, for any vector function w on IR+ , the integral of w(u1 ) · 12 (|u1 −
u2 |)w(u2 ) on IR+ × IR+ equals
Z ∞ Z u1
−2
w(u1 ) · K(u1 − u2 )w(u2 )du1 du2
0
0
where K(u1 − u2 ) =  (|u1 − u2 |) + 00 δ(u1 − u2 ) and 00 is the value of 0 (u) at
u = 0. Moreover,
Kc (ω) = c00 (ω) + 00 = ω0s (ω).
00
The analogous result holds for µ. Hence, because ω0s (ω) > 0 and ωµ0s (ω) > 0 for
every non-zero ω, via obvious identifications and Proposition 6 we obtain that eM
is positive definite.
Another expression for the free enthalpy is easily found if
0 > 0,
µ0 > 0;
µ00 ≤ 0.
00 ≤ 0,
Examine the functional
ψG (Et , Ht ) =
− 12 E(t) · ∞ E(t) +
1
2
− 21 H(t) · µ∞ H(t) +
Z
1
2
∞
0
Z
0
[Ĕt (u) · 0 (u)Ĕt (u) du
∞
H̆t (u) · µ0 (u)H̆t (u) du.
Time differentiation and an integration by parts yield
Z ∞
t
t
1
ψ̇G (E , H ) = −D(t) · Ė(t) + 2
[Ĕt (u) · 00 (u)Ĕt (u) du
0
Z ∞
1
−B(t) · Ḣ(t) + 2
H̆t (u) · µ00 (u)H̆t (u) du.
0
Because 00 and µ00 are non-positive the validity of (9.67) follows and hence ψG is a
free enthalpy. The corresponding free energy takes the form
Z ∞
t
t
1
1
Et (u) · 0 (u)Et (u)du
eG (E , H ) = 2 E(t) · 0 E(t) + 2
0
Z ∞
+ 12 H(t) · µ0 H(t) + 21
Ht (u) · µ0 (u)Ht (u)du
0
0
0
The assumption  > 0, µ > 0 yields the positive definiteness of eG .
26
Dissipative electromagnetic systems
A further free enthalpy functional holds if the functions 0 , µ0 are proportional
to decreasing exponentials,
0 (u) = 00 exp(−γu),
µ0 (u) = µ00 exp(−νu),
γ, ν > 0.
Because ∞ > 0 and µ∞ > µ0 , both 00 and µ00 are required to be positive definite.
Let
Z ∞
i2
h
0 (u)Ĕt (u)du
ψD (Et , Ht ) = − 12 E(t) · ∞ E(t) + 21 (∞ − 0 )−1/2
0
Z ∞
h
i2
−1/2
1
1
− 2 H(t) · µ∞ H(t) + 2 (µ∞ − µ0 )
µ0 (u)H̆t (u)du .
0
Time differentiation and an integration by parts yield
Z ∞
Z
h
i h
−1/2
0
t
−1/2
 (u)Ĕ (u)du · (∞ − 0 )
ψ̇D = (∞ − 0 )
0
∞
00 (u)Ĕt (u)du
0
−D(t) · Ė(t) + ...
i
the dots denoting the analogous terms where D → B, E → H,  → µ. Because
00 (u) = −γ0 (u), µ00 (u) = −νµ0 (u), it follows that (9.67) holds. Correspondingly,
the free energy eD takes the form
Z ∞
h
i2
t
t
−1/2
1
1
eD (E , H ) = 2 E(t) · 0 E(t) + 2 (∞ − 0 )
0 (u)Et (u)du
0
Z ∞
i2
h
−1/2
µ0 (u)Et (u)du .
+ 21 H(t) · µ0 H(t) + 21 (µ∞ − µ0 )
0
The positive definiteness of eD is apparent.
Finally, consider the functional
Z ∞Z ∞
t
t
1
1
Ĕt (u1 ) · 00 (u1 + u2 )Ĕt (u2 )du1 du2
ψEH (E , H ) = − 2 E(t) · ∞ E(t) − 2
0
0
Z ∞Z ∞
1
1
− 2 H(t) · µ∞ H(t) − 2
H̆t (u1 ) · µ00 (u1 + u2 )H̆t (u2 )du1 du2 .
0
0
Time differentiation and an integration by parts yield
Z ∞Z ∞
ψ̇EH = −D(t) · Ė(t) +
Ėt (u1 ) · 0 (u1 + u2 )Ėt (u2 )du1 du2
Z ∞0 Z ∞0
−B(t) · Ḣ(t) +
Ḣt (u1 ) · µ0 (u1 + u2 )Ḣt (u2 )du1 du2 .
0
0
In addition, the corresponding free energy eEH can be written as
Z ∞Z ∞
eEH (Et , Ht ) = 21 E(t) · 0 E(t) − 12
Et (u1 ) · 00 (u1 + u2 )Et (u2 )du1 du2
Z ∞0 Z ∞0
1
1
Ht (u1 ) · µ00 (u1 + u2 )Ht (u2 )du1 du2 .
+ 2 H(t) · µ0 H(t) − 2
0
0
Dissipative electromagnetic systems
27
If 0 and µ0 are superpositions of exponentials of the form
X
X
µ0 (u) = M
αk µk exp(−αk u),
αk k exp(−αk u),
0 (u) = Ξ
k
k
where Ξ, M are positive definite and αk > 0 while k , µk ≥ 0, then (9.67) holds and
eEH is positive definite. There are cases where the representation in terms of Fourier
transforms proves convenient. Paralleling the procedure of [18], the occurrence of
a convolution and an application of the Parseval-Plancherel theorem allow the free
enthalpy functional ψM to be written in the frequency domain as
ψM (Et , Ht ) = − 12 E(t) · ∞ E(t) − 12 H(t) · µ∞ H(t)
€
€
1
E(t) 
E(t)  0
· s (ω) Ets (ω) −
+ (Etc (ω) · 0s (ω)Etc (ω))]dω
+
[ω Est (ω) −
π 0
ω
ω
Z
€
1 ∞ € t
H(t)  0
H(t) 
+
[ω Hs (ω) −
· µs (ω) Hst (ω) −
+ (Htc (ω) · µ0s (ω)Htc (ω))]dω.
π 0
ω
ω
Z
∞
To write ψG in the frequency domain we need the following result.
Lemma 2 If w, K ∈ L1 (IR) and K(u) = K(−u), K(u) = KT (u), ∀u ∈ IR, then
Z ∞
w(u) · K(u)w(u)du
−∞
Z ∞Z ∞
1
[wC (ω) · KC (ω − ξ)wC (ξ) + wS (ω) · KC (ω − ξ)wS (ξ)]dωdξ.
=
(2π)2 −∞ −∞
Proof. The inversion formula allows w to be expressed in terms of the Fourier
transform wF as
Z ∞
1
wF (u) exp(iωu)dω.
w(u) =
2π −∞
Substitution and a change of the order of integration yields
Z ∞
Z ∞Z ∞
1
w(u) · K(u)w(u)du =
wF (ω) · KF (−(ω + ξ))wF (ξ)dωdξ.
(2π)2 −∞ −∞
−∞
Since K is even the full-range sine transform KS vanishes. Let the subscript C
denote the full-range cosine transform. Hence wF = wC + iwS . A change of
variable, ω → −ω and the symmetry of K yield the result. .
As a consequence, if w and K are defined on IR+ we carry w and K over to IR−
by letting w(u) = 0 and K(u) = K(−u) as u ∈ IR− . Hence we have KC = 2Kc
and wC = wc , wS = ws . Substitution gives
Z ∞
w(u) · K(u)w(u)du
0
Z ∞Z ∞
1
=
[wc (ω) · Kc (ω − ξ)wc (ξ) + ws (ω) · Kc (ω − ξ)ws (ξ)]dωdξ.
2π 2 −∞ −∞
28
Dissipative electromagnetic systems
Alternatively, we let w(u) = ±w(−u) as u ∈ IR− and observe that wC = 2wc , wS =
0 or wC = 0, wS = 2ws . Hence we have
Z ∞
Z ∞Z ∞
1
wc (ω) · Kc (ω − ξ)wc (ξ)dωdξ
w(u) · K(u)w(u)du =
π 2 −∞ −∞
0
Z ∞Z ∞
1
ws (ω) · Kc (ω − ξ)ws (ξ)dωdξ.
=
π 2 −∞ −∞
Also, by applying the Parseval-Plancherel theorem and observing that the integrands Kc (ω)wc (ω) and Ks (ω)ws (ω) are even we have
Z
Z
Z ∞
2 ∞
2 ∞
Kc (ω)wc (ω)dω =
Ks (ω)ws (ω)dω.
K(u)w(u)du =
π 0
π 0
0
The identification of K, w with 0 , Et or µ0 , Ht allows ψG to be written, e.g., as
Z ∞
2
t
t
1
ψG (E , H ) = − 2 E(t) · ∞ E(t) + E(t) ·
c0 (ω)Etc (ω)dω
π
0
Z ∞
Z ∞
1
t
− 2
c0 (ω − ξ)Ect (ω)dω dξ
E (ξ) ·
2π −∞ c
−∞
Z ∞
2
1
µ0c (ω)Htc (ω)dω
− 2 H(t) · µ∞ H(t) + H(t) ·
π
0
Z ∞
Z ∞
1
− 2
Htc (ξ) ·
µc0 (ω − ξ)Htc (ω)dω dξ.
2π −∞
−∞
Finally, the application of the Parseval-Plancherel theorem yields
Z ∞
€
2‚
E(t)  ƒ2
ψD (Et , Ht ) = − 21 E(t) · ∞ E(t) + 2 (∞ − 0 )−1/2
dω
s0 (ω) Ets (ω) −
π
ω
0
Z ∞
€
2‚
H(t)  ƒ2
− 12 H(t) · µ∞ H(t) + 2 (µ∞ − µ0 )−1/2
dω .
µ0s (ω) Hst (ω) −
π
ω
0
Conducting materials are now considered. A simple but realistic model of the
ionosphere is given by the constitutive relations
Z ∞
J(t) =
κ0 (u)Et (u)du,
(9.68)
B(t) = µ H(t),
D(t) =  E(t),
0
R∞
where κ0 ∈ L1 (IR+ ). The equilibrium conductivity κ∞ = 0 κ0 (u)du is positive
definite. Again, the possible dependence of , µ and κ0 (u) on the position x is
understood and not written. In such a case the property (9.66) for a functional ψ
to be a free enthalpy simplifies to
ψ̇(t) ≤ −B(H(t)) · Ḣ(t) − D(E(t)) · Ė(t) + E(t) · J(Et ).
(9.69)
By (8.59) the tensors  and µ are required to be symmetric. For technical convenience we also let κ0 : IR+ 7→ Sym. Hence the functional
ψM (Et , H(t)) = − 12 E(t) · E(t) − 12 H(t) · µH(t)
Z ∞Z ∞
Et (u1 ) · κ0 (|u1 − u2 |)Et (u2 )du1 du2
+ 12
0
0
(9.70)
Dissipative electromagnetic systems
29
is a free enthalpy. For, time differentiation yields
ψ̇M (Et , H(t)) = −Ė(t) · E(t) − Ḣ(t) · µH(t)
Z ∞Z ∞
+
Ėt (u1 ) · κ0 (|u1 − u2 |)Et (u2 )du1 du2 .
0
0
Since Ėt (u1 ) = −dEt (u1 )/du1 , integrate by parts with respect to u1 and observe
that
∂κ0 (|u1 − u2 |)
= sgn(u1 − u2 )κ00 (|u1 − u2 |)
∂u1
is odd by interchange of u1 and u2 . Hence, because κ0 ∈ L1 (IR+ ), we obtain
ψ̇M = −D · Ė − B · Ḣ + J · E.
The functional ψM is a free enthalpy, indeed the maximal one.
If κ0 > 0, κ00 < 0 and κ000 > 0 then also
Z ∞
t
1
1
1
ψG (E(t), H(t), È ) = − 2 E(t) · E(t) − 2 H(t) · µH(t) − 2
Èt (u) · κ00 (u)Èt (u)du
0
is a free enthalpy. Time differentiate and observe that
Z ∞
Z ∞
κ00 (u)Èt (u)du = −
κ0 (u)Et (u)du = −J(t).
0
0
Because E(t − u) = dÈt (u)/du, an integration by parts and the condition κ00 ∈
L1 (IR+ ) yield
Z ∞
Èt (u) · κ000 (u)Èt (u)du
ψ̇G = −D · Ė − B · Ḣ + E · J −
0
whence the inequality (9.69) follows.
If κ0 is a decreasing exponential, κ0 (u) = κ00 exp(−γu), γ > 0, then
Z ∞

‘2
ψD (E(t), H(t), Èt ) = − 21 E(t)·E(t)− 21 H(t)·µH(t)+ 21 (γκ∞ )−1/2
κ00 (u)Èt (u)du
0
is a free enthalpy. Time differentiation, integration by parts and the observation
that
κ000 = −γκ00 = γ 2 κ0
yield
Z
h
−1/2
ψ̇D = −D · Ė − B · Ḣ + E · J − κ∞
∞
0
i2
κ00 (u)Èt (u)du .
Hence the inequality (9.69) holds.
The free enthalpies ψM , ψG , ψD are also written in the frequency domain. Preliminarily we observe that, because 0 = 0, by (8.61) we have κ0c (ω) > 0, ∀ω ∈ IR+ .
Now, because
Z ∞Z u
Z ∞Z ∞
Et (u) · κ0 (u − τ )Et (τ )dτ du
Et (u1 ) · κ0 (|u1 − u2 |)Et (u2 )du1 du2 = 2
0
0
0
0
30
Dissipative electromagnetic systems
and
Z
∞
0
Z
u
0
Et (u) · κ0 (u − τ )Et (τ )dτ du
Z ∞
1
[Et (ω) · κ0c (ω)Ect (ω) + Est (ω) · κ0c (ω)Ets (ω)dω)]dω
=
2π −∞ c
we have
ψM (Et , H(t))
=
− 12 E(t) · E(t) − 12 H(t) · µH(t)
Z ∞
1
+
[Ets (ω) · κc0 (ω)Ets (ω) + Etc (ω) · κ0c (ω)Ect (ω)]dω.
2π 0
The positive definiteness of κc0 emphasizes the minimum property of the free enthalpy at constant histories.
A direct application of Lemma 2 shows that
−
1
(2π)2
Z
∞
−∞
Z
∞
−∞
ψG = − 12 E(t) · E(t) − 12 H(t) · µH(t)
[Ètc (ω) · κ00c (ω − ξ)Èct (ξ) + Èst (ω) · κ00c (ω − ξ)Èts (ξ)]dωdξ.
Finally, the Parseval-Plancherel theorem allows ψD to be written as
Z ∞
i2
2h
−1/2
1
1
κ00s (ω)Èts (ω)dω .
ψD = − 2 E(t) · E(t) − 2 H(t) · µH(t) − 2 (γκ∞ )
π
0
10. Dissipativity at Interfaces
Interfaces model regions where the material properties change sharply in a direction.
Hence we are led to considering surfaces, across which the electromagnetic field is
discontinuous, endowed with appropriate thermodynamic properties. Let S be the
common boundary between two regions, say 1 and 2. Denote by n the unit normal
to S, drawn from region 2 to region 1, and, for any function f on the spatial domain,
let [[f ]] = f1 − f2 where f1 and f2 are the limiting values of f (y) as y approaches
the point x on S from the region 1 and 2, respectively. If ρτ is the surface charge
density and Jτ is the surface current density, the boundary conditions read
[[B]] · n = 0,
[[E]]×n = 0,
[[D]] · n = ρτ ,
(10.71)
[[H]]×n = −Jτ .
(10.72)
Consider a region Ω comprising a part of S. The inequality (4.15) and Poynting’s
theorem allow us to write
Z dZ
E×H · n da dt ≤ 0.
0
∂Ω
Dissipative electromagnetic systems
31
Let now Ω be a Gaussian pillbox, half in one medium and half in the other, with
bases orthogonal to the local normal n. We obtain
Z dZ
[[Eτ ×H · n]]da dt ≤ 0
0
A
where Eτ is the tangential electric field and A is the projection of Ω on S. Since
[[Eτ ]] = 0, the arbitrariness of A and the second relation in (10.72) imply that
Z d
(10.73)
Eτ (x, t)J̇τ (x, t)dt ≥ 0
0
for every point x ∈ S and every cycle on [0, d).
The condition (10.73) is a general constraint on the behaviour of electromagnetic
interfaces. A description of the constitutive properties of an interface may be given
in the form
(10.74)
Jτ (t) = Ĵτ (σ(t), P (t)),
the dependence of σ, P , and hence Jτ , on x ∈ S being understood. The response R
in Definition 2 is then generalized by letting
R = (D̂, B̂, Ĵ, Ĵτ ).
Some simple, but significant, examples show the operative role of (10.73). If the
external medium (1) is a perfect conductor we let
Eτ = 0
at
S
and hence (10.73) holds identically. If, rather, the external medium is a conductor,
but not a perfect conductor, we can assume that
H1 ×n = 0.
Hence (10.72) gives
H2 ×n = Jτ .
The constitutive equation for Jτ is taken as
Jτ (x, t) = δ0 (x)Eτ (x, t).
The positivity of the surface conductivity δ0 is a necessary and sufficient condition
for (10.73) to hold. A more general constitutive equation incorporates memory
effects. By analogy with [22] we let
Z ∞
Jτ (x, t) = δ0 (x)Eτ (x, t) +
δ 0 (x, ξ)Etτ (x, ξ)dξ,
0
0
where δ0 is taken to be non-negative and δ (x, ·) ∈ L1 (IR+ ) for every x ∈ S. The
thermodynamic condition (10.73) becomes
Z ∞
Z d
ƒ
‚
δ 0 (ξ)Eτt (ξ)dξ dt ≥ 0
Eτ (t) · δ0 Eτ (t) +
0
0
for every cycle on [0, d), the dependence on x being understood.
32
Dissipative electromagnetic systems
Theorem 8 The constitutive relation (10.74) satisfies the inequality (10.73) for
every cycle if and only if
δ0 + δc0 (ω) > 0,
∀ω ∈ IR++ .
(10.75)
Proof. Let Eτ (x, t) = E0 (x) cos ωt and Hτ (x, t) = H0 (x) cos ωt, t ∈ IR. Hence the
state (Et , Ht ) on [0, 2π/ω) is a cycle. Upon substitution and integration on t we
have
Z 2π/ω
π
0<
Eτ (t)×Jτ (t)dt = E20 [δ0 + δc0 (ω)]
ω
0
whence (10.75) follows. Conversely, let (Et , Ht ) constitute a cycle on [0, d). Hence
E and H are periodic with period d. Letting ω = 2π/d we represent Eτ (t) through
the Fourier series
∞
X
Eτ (t) =
Ak cos kωt + Bk sin kωt.
k=0
Substitution in (10.73) and integration with respect to t yields
Z 2π/ω
∞
πX 2
Eτ (t) · Jτ (t)dt =
(Ak + B2k )[δ0 + δc (kω)],
ω
0
k=1
which completes the proof. .
Application of the Riemann-Lebesgue lemma to (10.75) yields
δ0 ≥ 0.
By the second condition in (10.72) we consider the constitutive relation
Z ∞
Eτ (t) = η0 Hτ (t)×n +
η(u)Hτt (u)×ndu
(10.76)
0
1
1
and let η ∈ L (IR ) ∩ H (IR ). Hence we show that there are functionals such that,
to each history Htτ , associates a function of t, say φ(t), that satisfies
+
+
φ̇(t) ≤ Eτ (t)×Hτ (t) · n.
Any such functional plays a role similar to that of the free energy and hence we call
it boundary free energy. As we see in a moment also the boundary free energy is
non-unique.
Let Hτ (·) ∈ L1 (IR+ ) and consider the integrated history H̀t .
Z t
t
H (u) =
Hτ (ξ)dξ.
t−u
The function H̀t (u) is differentiable with respect to t and u;
dH̀t (u)
= Hτ (t) − Htτ (u),
dt
dH̀t (u)
= Htτ (u).
du
Dissipative electromagnetic systems
33
The functionals
Ψ1 (H̀t ) =
1
2
Z
∞
0
Z
∞
0
η12 (|u1 − u2 |)[H̀t (u1 )×n] · [H̀t (u2 )×n]du1 du2 ,
and
t
Ψ2 (H̀ ) = −
1
2
Z
∞
η 0 (u)[H̀t (u)×n]2 du
0
are now considered. Time differentiation and an integration by parts yield
Z ∞Z ∞
Ψ̇1 = −[Hτ (t)×n] ·
η1 (u1 − u2 |)Htτ (u2 )du2 du1
0
0
Z ∞Z ∞
η1 (|u1 − u2 |)[Htτ (u1 )×n] · [Htτ (u2 )×n]du2 du1 .
+
0
0
The last integral vanishes. Hence, an integration by parts and use of (10.76) yield
Ψ̇1 = Eτ ×Hτ · n − η0 [Hτ ×n]2 .
Since η0 > 0 the desired inequality follows. Similarly, time differentiation yields
Z ∞
Z ∞
0
t
Ψ̇2 = −[Hτ ×n] ·
η (u)H̀ (u)×ndu +
η 0 (u)[Hτt (u)×n] · [H̀t (u)×n]du.
0
0
Two integrations by parts and use (10.76) yield
Z
Ψ̇2 = Eτ ×Hτ · n − η0 [Hτ ×n]2 − 21
∞
η 00 (u)[H̀t (u)×n]2 du.
0
Provided only that η 00 > 0 the inequality for Ψ2 follows.
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