Maximum recoverable work in linear thermoelectromagnetism
We give a general closed expression for the minimum free energy in terms of Fourier-transformed quantities for a thermoelectromagnetic conductor with memory effects for the electric current density and the heat flux, when the integrated histories of the electric field and of the temperature gradien...
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| Цитувати: | Maximum recoverable work in linear thermoelectromagnetism / G. Amendola, A. Manes // Нелінійні коливання. — 2006. — Т. 9, № 3. — С. 287-319. — Бібліогр.: 18 назв. — англ. |
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| citation_txt | Maximum recoverable work in linear thermoelectromagnetism / G. Amendola, A. Manes // Нелінійні коливання. — 2006. — Т. 9, № 3. — С. 287-319. — Бібліогр.: 18 назв. — англ. |
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| description | We give a general closed expression for the minimum free energy in terms of Fourier-transformed quantities for a thermoelectromagnetic conductor with memory effects for the electric current density and the
heat flux, when the integrated histories of the electric field and of the temperature gradient are chosen to
characterize the states of the material. An equivalent formulation is derived and applied to the discrete
spectrum model material response.
Наведено загальний замкнений вираз для мiнiмальної вiльної енергiї в термiнах перетворення
Фур’є для провiдника з ефектом запам’ятовування густини електричного струму та теплового потоку у випадку, коли для характеризацiї стану матерiалу розглядається iнтегральна
iсторiя електричного струму та градiєнта температури. Отримано i застосовано еквiвалентне формулювання до моделi реакцiї матерiалу, що має дискретний спектр.
|
| first_indexed | 2025-12-07T15:26:10Z |
| format | Article |
| fulltext |
UDC 517 . 9
MAXIMUM RECOVERABLE WORK
IN LINEAR THERMOELECTROMAGNETISM*
МАКСИМАЛЬНА ЕНЕРГIЯ , ЯКУ МОЖНА ПОВЕРНУТИ
В ЛIНIЙНОМУ ТЕРМОЕЛЕКТРОМАГНЕТИЗМI
G. Amendola
Dipartimento di Matematica Applicata ”U.Dini”
via Diotisalvi, 2, 56126, Pisa, Italy
A. Manes
Dipartimento di Matematica ”L.Tonelli”
via F. Buonarroti, 2, 56127, Pisa, Italy
We give a general closed expression for the minimum free energy in terms of Fourier-transformed quanti-
ties for a thermoelectromagnetic conductor with memory effects for the electric current density and the
heat flux, when the integrated histories of the electric field and of the temperature gradient are chosen to
characterize the states of the material. An equivalent formulation is derived and applied to the discrete
spectrum model material response.
Наведено загальний замкнений вираз для мiнiмальної вiльної енергiї в термiнах перетворення
Фур’є для провiдника з ефектом запам’ятовування густини електричного струму та тепло-
вого потоку у випадку, коли для характеризацiї стану матерiалу розглядається iнтегральна
iсторiя електричного струму та градiєнта температури. Отримано i застосовано еквiвалент-
не формулювання до моделi реакцiї матерiалу, що має дискретний спектр.
1. Introduction. In a recent work [1] we have studied the problem of finding an explicit form
for the minimum free energy of a linear thermoelectromagnetic conductor characterized, in
particular, by two constitutive equations for the electric current density and for the heat flux
expressed by means of two local functionals of the histories of the electric field and the tempera-
ture gradient, respectively [2 – 4]. The investigation of such a problem is very important since
the minimum free energy is related to the maximum recoverable work, that is, the maximum
quantity of work we can obtain from the material at a given state.
The presence of thermal effects in electromagnetism was considered, in particular, in [5,
6] by Coleman & Dill, who also derived the restrictions imposed on the constitutive equati-
ons by the laws of thermodynamics. Also in [7] these problems were studied for the materials
considered later on in [1], giving a theorem of uniqueness, existence, and asymptotic stability
too.
In this work we follow Golden’s lines of [8], where the minimum free energy is determined
for a linear viscoelastic material in a scalar case, together with the procedure used in [9] for
an analogous problem always in viscoelasticity, see also [10 – 12]. While in [1] the states of the
thermoelectromagnetic material are characterized by the actual values of the electric and the
magnetic fields, of the temperature and the histories of the electric field and of the temperature
*The work is performed under the auspices of CNR and MIUR.
c© G. Amendola, A. Manes, 2006
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3 287
288 G. AMENDOLA, A. MANES
gradient, in the present work we consider the integrated histories of the electric field and of
the temperature gradient instead of their histories. Thus, new appropriate definitions of the
continuation of histories and processes are required.
The layout of this article is the following. After introducing the constitutive equations we
recall the thermodynamic restrictions on them in Section 2, writing some fundamental relati-
onships. In Section 3, we consider the definitions of states and processes and in the following
Section 4 we give a first definition of equivalence between integrated histories. After consi-
dering the thermoelectromagnetic work in Section 5, we give an equivalent definition of equi-
valence by means of the boundedness of the work in Section 6. Then, in Section 7 we derive
the expression of the maximum recoverable work, of which another equivalent form, derived
in Section 8, is used to study the particular case of discrete spectrum materials in the last Sec-
tion 9.
2. Preliminaries and basic equations. A rigid electromagnetic B occupies a region Ω, which
is a bounded and simply-connected domain of the three-dimensional Euclidean space R3, with
a smooth boundary ∂Ω, whose outward normal unit is denoted by n.
We are concerned with the linear theory of thermoelectromagnetism, which is characterized
by the following constitutive equations:
D(x, t) = εE(x, t) + ϑ(x, t)a, B(x, t) = µH(x, t), (2.1)
J(x, t) =
+∞∫
0
α(s)Et(x, s)ds, q(x, t) = −
+∞∫
0
k(s)gt(x, s)ds, (2.2)
h(x, t) = cϑ̇(x, t) + Θ0
[
A1 · Ḋ(x, t)/ε+ A2 · Ḃ(x, t)/µ
]
, (2.3)
where D and B are the electric displacement and the magnetic induction, E and H denote the
electric and magnetic fields, ϑ is the temperature relative to the absolute reference temperature
Θ0, uniform in Ω; moreover, the current density J and the heat flux q are expressed by two
functionals of the histories, up to time t, of the electric field, Et(x, s) = E(x, t− s) ∀s ∈ R+ =
= [0,+∞), and of the temperature gradient g = ∇ϑ, gt(x, s) = g(x, t− s) ∀s ∈ R+, respecti-
vely; finally, h denotes the rate at which heat is absorbed per unit volume.
On supposing that the body is homogeneous and isotropic, all the parameters, that is, the
dielectric constant ε > 0, the magnetic permeability µ > 0 and the specific heat c > 0, the
vectors a, A1 and A2 as well as the memory kernels α and k are constant for any point x ∈ Ω̄.
In particular, the kernels α and k, called the electric and thermal conductivities, are two
relaxation functions α : R+ → R and k : R+ → R given by [2, 3]
α(t) = α0 +
t∫
0
α′(τ)dτ, k(t) = k0 +
t∫
0
k′(τ)dτ ∀t ∈ R+, (2.4)
where α0 and k0 denote the initial values, at time t = 0, of the two functions α and k, with
lim
t→+∞
limα(t) = 0 and lim
t→+∞
limk(t) = 0 under the hypothesis that they belong to H1(R+).
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 289
It is useful to introduce the integrated histories of E and g, which are two functions Ēt(x, ·) :
R+ → R3 and gt(x, ·) : R+ → R3 defined as follows:
Ēt(x, s) =
s∫
0
Et(x, λ)dλ, ḡt(x, s) =
s∫
0
gt(x, λ)dλ; (2.5)
these quantities, taking account of the hypotheses assumed for α and k, allow us to rewrite the
constitutive equations (2.2) in an equivalent form,
J(x, t) = −
+∞∫
0
α′(s)Ēt(x, s)ds, q(x, t) =
+∞∫
0
k′(s)ḡt(x, s)ds. (2.6)
The constitutive equations (2.1) – (2.3) characterize the behaviour of a simple material [6,
13], for which we have deduced the restrictions that the thermodynamic principles place on
the assumed constitutive equations in [7]. These principles state [14, 15] that the following two
relations: ∮
[h(t) + Ḋ(t) ·E(t) + Ḃ(t) ·H(t) + J(t) ·E(t)]dt = 0, (2.7)
∮ {
h(t)/[Θ0 + ϑ(t)] + q(t) · g(t)/[Θ0 + ϑ(t)]2
}
dt ≤ 0 (2.8)
must hold for any cyclic process. In the last of these, the equality sign is referred only to reversi-
ble processes.
Here we have understood the dependence on the position x ∈ Ω of all the functions; this
will be done later on because the statements will be relative to any fixed x ∈ Ω.
Since we are concerned with a linear theory, only the linearized expression of (2.8) must be
considered; thus, (2.8) becomes [16]
1
Θ2
0
∮
{h(t)[Θ0 − ϑ(t)] + q(t) · g(t)} dt ≤ 0 (2.9)
and, hence, eliminating Θ0h(t) by means of (2.7), it follows that
1
Θ2
0
∮ {
h(t)ϑ(t) + Θ0[Ḋ(t) ·E(t) + Ḃ(t) ·H(t) + E(t) · J(t)] −q(t) · g(t)} dt ≥ 0. (2.10)
This inequality must be satisfied by the constitutive equations; therefore, substituting (2.1)
and (2.3) into it, we obtain an inequality, which, because of the arbitrarinesses of ϑ, E, g and of
the same ϑ with respect to Ė and Ḣ, gives [7]
A1 = a, A2 = 0 (2.11)
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3
290 G. AMENDOLA, A. MANES
and other two inequalities, concerning with J and q, which yield
+∞∫
0
α(s) cos(ωs)ds > 0,
+∞∫
0
k(s) cos(ωs)ds > 0 ∀ω 6= 0. (2.12)
These results change the given form to the constitutive equation (2.3), which, taking account
of (2.1)1, reduces to
h(x, t) =
(
c+ Θ0a2/ε
)
ϑ̇(x, t) + Θ0a · Ė(x, t) (2.13)
so that (2.10) assumes the following form:∮ [
c
Θ0
ϑ̇(t)ϑ(t) +
1
ε
Ḋ(t) ·D(t) +
1
µ
Ḃ(t) ·B(t) + J(t) ·E(t)− 1
Θ0
q(t) · g(t)
]
dt ≥ 0. (2.14)
We recall that in thermoelectromagnetism the equations which are to be considered are
Maxwell’s ones besides the energy equation expressed by h(x, t) = −∇·q(x, t)+r(x, t), where
h is given by (2.13) and thus can be eliminated while r denotes the heat sources.
We now introduce the Fourier transform of any function f : R → Rn, denoted by fF and
given by
fF (ω) =
+∞∫
−∞
e−iωξf(ξ)dξ = f+(ω) + f−(ω), (2.15)
where
f+(ω) =
+∞∫
0
e−iωξf(ξ)dξ, f−(ω) =
0∫
−∞
e−iωξf(ξ)dξ; (2.16)
moreover, we consider the half-range Fourier cosine and sine transforms
fc(ω) =
+∞∫
0
cos(ωξ)f(ξ)dξ, fs(ω) =
+∞∫
0
sin(ωξ)f(ξ)dξ, (2.17)
which hold if the function f is defined only on R+ as well as f+. In particular, we remember that
if f is any function defined on R+, it can be extended on R in several ways. For this purpose
we can consider its usual extension to (−∞, 0), where it vanishes identically, or its extensions
made with an even (f(ξ) = f(−ξ) ∀ξ < 0) or an odd (f(ξ) = −f(−ξ) ∀ξ < 0) function thus
obtaining the following relations:
fF (ω) = f+(ω) = fc(ω)− ifs(ω), fF (ω) = 2fc(ω), fF (ω) = −2ifs(ω), (2.18)
which hold in the three cases, respectively.
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 291
We observe that the functions f±, given by (2.16), can be considered as functions of z ∈ C,
that is defined in the complex plane C; these new functions f±(z) are analytic for z ∈ C(∓),
where we
C(−) =
{
z ∈ C : Im z ∈ R−−} , C(+) =
{
z ∈ C : Im z ∈ R++
}
, (2.19)
R−−(R++) denoting the strictly negative (positive) reals. We can extend, by hypothesis [8], the
analyticity on C∓, where
C− =
{
z ∈ C : Im z ∈ R−} , C+ =
{
z ∈ C : Im z ∈ R+
}
. (2.20)
We shall use the notation f(±)(z) to denote that the function has zeros and singularities in C±.
Therefore we can rewrite (2.12) in terms of the definition (2.17)1, that is, the thermodynamic
restrictions are expressed by
αc(ω) > 0, kc(ω) > 0 ∀ω ∈ R, (2.21)
with the new hypotheses that αc(0) > 0 and kc(0) > 0. Moreover, we recall the following
results [1, 17]:
α′s(ω) = −ωαc(ω), k′s(ω) = −ωkc(ω), (2.22)
whence, if α′′, k′′ ∈ L2(R+) and | α′(0) |< +∞, | k′(0) |< +∞, we have
lim
ω→∞
limωα′s(ω) = − lim
ω→∞
limω2αc(ω) = α′(0) ≤ 0, (2.23)
lim
ω→∞
limωk′s(ω) = − lim
ω→∞
limω2kc(ω) = k′(0) ≤ 0. (2.24)
Still now we assume
α′(0) < 0, k′(0) < 0. (2.25)
Let us introduce the electric and thermal conductivities
ν(α)(t) =
t∫
0
α(ξ)dξ, ν(k)(t) =
t∫
0
k(ξ)dξ; (2.26)
in particular, we have
ν(α)
∞ =
+∞∫
0
α(ξ)dξ = αc(0) > 0, ν(k)
∞ =
+∞∫
0
k(ξ)dξ = kc(0) > 0. (2.27)
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3
292 G. AMENDOLA, A. MANES
The static continuation, with duration τ ∈ R++, of two given histories Et(s) and gt(s)
∀s ∈ R+ is defined by
Et(τ) =
{
E(t), s ∈ [0, τ ],
Et(s− τ), s > τ,
gt(τ) =
{
g(t), s ∈ [0, τ ],
gt(s− τ), s > τ.
(2.28)
The integrated histories, which correspond to these continuations, are
Ēt+τ (s) =
s∫
0
E(t)dη = sE(t), s ∈ [0, τ ],
τE(t) +
s−τ∫
0
Et(ρ)dρ, s > τ,
ḡt+τ (s) =
s∫
0
g(t)dη = sg(t), s ∈ [0, τ ],
τg(t) +
s−τ∫
0
gt(ρ)dρ, s > τ,
(2.29)
and must be considered in the expressions (2.6) to evaluate the current density and the heat
flux yielded after the static continuations (2.28); thus, we obtain
J(t+ τ) = ν(α)(τ)E(t)−
+∞∫
0
α′(τ + ρ)Ēt(ρ)dρ, (2.30)
q(t+ τ) = −ν(k)(τ)g(t) +
+∞∫
0
k′(τ + ρ)ḡt(ρ)dρ. (2.31)
We observe that in the constitutive equations the present value of the temperature gradient
has not the same role, which, on the contrary, that one of the electric field has, since this appears
explicitly in the constitutive equation (2.1)1 for the electric displacement; the presence of g(t)
in (2.31) is analogous to the one of E(t) in (2.30), but this is due only to the static continuations
of both the values E(t) and g(t).
Finally, we note that the asymptotic values (2.27) allow us to obtain the current density and
the heat flux at time t when the constant histories E† = Et(s) = E and g† = gt(s) = g
∀s ∈ R+ are considered. In fact, from (2.6) or directly from (2.2) it follows that
J(t) = ν(α)
∞ E, q(t) = −ν(k)
∞ g,
which, on account of (2.27), express the physical results of a constant current density with the
same versus of the electric field E and of a constant heat flux whose versus is opposite to the
one of the temperature gradient g.
3. States and processes for the thermoelectromagnetic body. The behaviour of our thermo-
electromagnetic solid is characterized by the assumed constitutive equations (2.1), (2.3) and
(2.6), which, as we have already noted, allows us to consider B as a simple material. Thus, B can
be described in terms of states and processes.
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 293
Taking into account the relations (2.1) and (2.3), the function
σ(t) = (E(t),H(t), ϑ(t), Ēt, ḡt) (3.1)
can be assumed to express the thermodynamic state at time t at any fixed x ∈ Ω. We note
that, contrary to the choice made in [1], the integrated histories of the electric field and of
the temperature gradient are now chosen to express the memory effects on the instantaneous
values of the current density and of the heat flux.
A map P : [0, d) → R3×R3×R×R3, piecewise continuous on the time interval [0, d) ⊂ R,
defined as
P (τ) = (ĖP (τ), ḢP (τ), ϑ̇P (τ),gP (τ)) ∀τ ∈ [0, d) (3.2)
is said to be a kinetic process of duration d ∈ R+. Here we have the time derivatives of
the electric and magnetic fields EP and HP and of the temperature ϑP together with the
temperature gradient at any instant τ of the time interval [0, d). We shall denote by P[t1,t2) the
restriction of the process P to the time interval [t1, t2) ⊂ [0, d); moreover, the set of the admi-
ssible states will be denoted by Σ, while Π will denote the set of all admissible processes. Thus,
we can introduce the function ρ : Σ × Π → Σ, defined by σf = ρ(σi, P ) ∈ Σ, which maps an
initial state σi ∈ Σ and a process P ∈ Π into the final state σf and it is said to be the evolution
function; in particular, we call cycle the pair (σ, P ) such that σ(d) = ρ(σ(0), P ) = σ(0).
The response of the material is given by the function
U(t) = (D(t),B(t),J(t),q(t)), (3.3)
where the instantaneous values of D, B, J and q are given by (2.1) and (2.6), and therefore it
depends on the pair (σ, P ), that is, the output function
U = Ũ(σ, P ) (3.4)
where Ũ : Σ×Π → R3 ×R3 ×R3 ×R3.
We can introduce the linear functionals J̃ : Γα → R3 and q̃ : Γk → R3, on account of
(2.6), such that
J̃(Ēt) = −
+∞∫
0
α′(s)Ēt(s)ds, q̃(ḡt) =
+∞∫
0
k′(s)ḡt(s)ds, (3.5)
where, taking in mind (2.30), (2.31), the function spaces Γα and Γk are defined by
Γα =
Ēt : (0,+∞) → R3;
∣∣∣∣∣
+∞∫
0
α′(s+ τ)Ēt (s) ds
∣∣∣∣∣ < +∞ ∀τ ≥ 0
, (3.6)
Γk =
ḡt : (0,+∞) → R3;
∣∣∣∣∣
+∞∫
0
k′(s+ τ)ḡt (s) ds
∣∣∣∣∣ < +∞ ∀τ ≥ 0
. (3.7)
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294 G. AMENDOLA, A. MANES
Any process P = P (τ), of duration d, is defined in the time interval [0, d) ⊂ R+ by means
of (3.2); it may be applied at any instant t ∈ R+.
First, we suppose that P is applied at time t = 0, when the state is σ(0) = (E∗(0),H∗(0),
ϑ∗(0), Ē0
∗, ḡ
0
∗) ∈ Σ. In this case τ ≡ t and therefore (3.2) becomes P (t) = (ĖP (t), ḢP (t),
ϑ̇P (t),gP (t)) ∈ Π; the evolution function yields a set of states σ(t) = (E(t),H(t), ϑ(t), Ēt, ḡt)
for any t ∈ (0, d], characterized by
E(t) = E∗(0) +
t∫
0
ĖP (ξ)dξ, H(t) = H∗(0) +
t∫
0
ḢP (ξ)dξ, (3.8)
ϑ(t) = ϑ∗(0) +
t∫
0
ϑ̇P (ξ)dξ, (3.9)
where E(t) ≡ EP (τ) and similarly for H and ϑ, and
Ēt(s) =
t∫
t−s
E(η)dη, 0 ≤ s < t,
Ē0
∗(s− t) +
t∫
0
E(η)dη, s ≥ t,
(3.10)
ḡt(s) =
t∫
t−s
gP (η)dη, 0 ≤ s < t,
ḡ0
∗(s− t) +
t∫
0
gP (η)dη, s ≥ t,
(3.11)
where E(η) is given by (3.8)1 and gP (η) is assigned in the process.
Now, we apply the process P at time t > 0, when the initial state is σ(t) = (E(t),H(t),
ϑ(t), Ēt, ḡt). The process P (τ) = (ĖP (τ), ḢP (τ), ϑ̇P (τ),gP (τ)), defined on [0, d), is related to
EP : (0, d] → R3, EP (τ) = E(t) +
τ∫
0
ĖP (η)dη, (3.12)
HP : (0, d] → R3, HP (τ) = H(t) +
τ∫
0
ḢP (η)dη, (3.13)
ϑP : (0, d] → R, ϑP (τ) = ϑ(t) +
τ∫
0
ϑ̇P (η)dη (3.14)
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 295
for any τ ∈ (0, d]. In particular, the process P assigns the temperature gradient
gP : [0, d) → R3, gP (τ) = g(t+ τ). (3.15)
Moreover such a process, starting at time t, to which corresponds the initial integrated histories
Ēt and ḡt that appear in σ(t), induces the continuation of the initial Ēt and ḡt by means of
EP (τ) ≡ E(t+ τ) and (3.15)2, with t+ τ ≤ t+ d, and expressed by
Ēt+d(s) = (EP ∗ Ē)t+d(s) =
Ēd
P (s) =
s∫
0
Ed
P (η)dη, 0 ≤ s < d,
Ēd
P (d) + Ēt(s− d), s ≥ d,
(3.16)
ḡt+d(s) = (gP ∗ ḡ)t+d(s) =
ḡd
P (s) =
s∫
0
gd
P (η)dη, 0 ≤ s < d,
ḡd
P (d) + ḡt(s− d), s ≥ d,
(3.17)
where
Ēd
P (d) =
d∫
0
EP (η)dη, Ēt(s− d) =
t∫
t−(s−d)
E(ξ)dξ ∀s ≥ d, (3.18)
ḡd
P (d) =
d∫
0
gP (η)dη, ḡt(s− d) =
t∫
t−(s−d)
g(ξ)dξ ∀s ≥ d, (3.19)
are the integrated histories corresponding to P in [0, d) and the initial integrated histories,
respectively for E and g.
The current density and the heat flux, yielded by the application of a process P , of duration
d, to a given initial state σ(t), are J̃((EP ∗ Ē)t+d) and q̃((gP ∗ ḡ)t+d) and can be evaluated by
using (3.5) with (3.16) and (3.17). We derive these results in the case where a restriction of the
process P[0,τ), applied at time t when the state is σ(t) = (E(t),H(t), ϑ(t), Ēt, ḡt), is considered;
in this case d must be substituted by τ in (3.16), (3.17) to evaluate (3.5), which thus give
J̃(Ēt+τ ) = −
+∞∫
0
α′(s)(EP ∗ Ē)(t+ τ − s)ds =
= α(τ)Ēτ
P (τ)−
τ∫
0
α′(η)Ēτ
P (η)dη −
+∞∫
τ
α′(ρ)Ēt(ρ− τ)dρ, (3.20)
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296 G. AMENDOLA, A. MANES
q̃(ḡt+τ ) =
+∞∫
0
k′(s)(gP ∗ ḡ)(t+ τ − s)ds =
= −k(τ)ḡτ
P (τ) +
τ∫
0
k′(η)ḡτ
P (η)dη +
+∞∫
τ
k′(ρ)ḡt(ρ− τ)dρ. (3.21)
4. Equivalence of states and of integrated histories. We can now introduce an equivalence
relation in the state space Σ by means of the following definition [11].
Definition 4.1. Two states σj ∈ Σ, j = 1, 2, are equivalent if
Ũ(σ1, P ) = Ũ(σ2, P ) ∀P ∈ Π. (4.1)
Thus, whatever may be the admissible process, two equivalent states yield the same response
of the material and therefore they are indistinguishable. This definition satisfies the requi-
rements of an equivalence relation, denoted by R. Therefore, if Σ contains equivalent states,
then we can introduce the corresponding quotient space ΣR, whose elements σR are the classes
of equivalent states.
Definition 4.2. We say that a state of the material is minimal if it is characterized by a minimum
set of data.
The introduction of the functionals in (3.5) allows us to give a new definition of equivalence
relative to the integrated histories of the electric field and of the heat flux.
Definition 4.3. Given two states σj(t) = (Ej(t),Hj(t), ϑj(t), Ēt
j , ḡ
t
j), j = 1, 2, corresponding
to the same values of the magnetic field, Hj(t) = H(t), j = 1, 2, and of the temperature, ϑj(t) =
= ϑ(t), j = 1, 2, the integrated histories of the electric field, Ēt
j , j = 1, 2, and of the temperature
gradient, ḡt
j , j = 1, 2, are said equivalent if for every EP : (0, τ ] → R3, gP : [0, τ) → R3 and
for every τ > 0 the relations
E1(t) = E2(t), J̃((EP ∗ Ē1)t+τ ) = J̃((EP ∗ Ē2)t+τ ), (4.2)
q̃((gP ∗ ḡ1)t+τ ) = q̃((gP ∗ ḡ2)t+τ ) (4.3)
hold, whatever HP : (0, τ ] → R3 and ϑP : (0, τ ] → R may be.
With this definition the integrated histories of E and g, which yield the same current density
and the same heat flux, respectively, are identified. Moreover, the conditions required in Defi-
nition 4.1 are now satisfied. In particular, we observe that for any process the values of HP (τ)
and ϑP (τ) do not depend on the two couples of the integrated histories Ēt
j and ḡt
j , j = 1, 2;
furthermore, we note that for both the integrated histories we must consider the same process
P and hence the same value of the temperature gradient, gP (ρ) ∀ρ ∈ [0, τ), which yields, in
particular, gP (0) = g(t) by virtue of (3.15). Therefore, it follows that the equivalence of two
integrated histories of E and g yield the same response of the material.
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 297
If we consider the continuation of the zero integrated histories of E and g, Ēt(s) = 0̄†(s) =
= 0 and ḡt(s) = 0̄†(s) = 0 ∀s ∈ R+, by means of a process P[0,τ) applied at time t when the
initial state is σ(t) = (E(t),H(t), ϑ(t), 0̄†, 0̄†), (3.16) and (3.17) yield
(EP ∗ 0̄†)t+τ (s) =
{
Ēτ
P (s), 0 ≤ s < τ,
Ēτ
P (τ), s ≥ τ,
(gP ∗ 0̄†)t+d(s) =
{
ḡτ
P (s), 0 ≤ s < τ,
ḡτ
P (τ), s ≥ τ.
(4.4)
As a consequence of Definition 4.3, from (3.20), (3.21) it is easy to see that two integrated
histories Ēt and ḡt are equivalent to their relative zero integrated histories if for every τ > 0
+∞∫
τ
α′(s)Ēt(s− τ)ds =
+∞∫
0
α′(τ + ξ)Ēt(ξ)dξ = 0, (4.5)
+∞∫
τ
k′(s)ḡt(s− τ)ds =
+∞∫
0
k′(τ + ξ)ḡt(ξ)dξ = 0. (4.6)
Thus we see that both the relations (4.2), (4.3) and these last relations (4.5), (4.6) express
the same equivalence between two couples of integrated histories. In fact, given the integrated
histories Ēt
j and ḡt
j , j = 1, 2, equivalent in the sense of Definition 4.3, obviously they must
satisfy (4.2), (4.3), from which we deduce that
+∞∫
τ
α′(s)Ēt
1(s− τ)ds =
+∞∫
τ
α′(s)Ēt
2(s− τ)ds, (4.7)
+∞∫
τ
k′(s)ḡt
1(s− τ)ds =
+∞∫
τ
k′(s)ḡt
2(s− τ)ds (4.8)
must hold for all τ > 0 too. Finally, it is enough to define Ēt(s − τ) = Ēt
1(s − τ) − Ēt
2(s − τ)
and ḡt(s − τ) = ḡt
1(s − τ) − ḡt
2(s − τ) to verify that (4.7), (4.8) coincide with (4.5), (4.6) and
conclude that the integrated histories obtained by the two differences are equivalent to their
zero integrated histories.
5. Thermoelectromagnetic work. The local form of the second law of thermodynamics
is expressed by (2.10), to which we have given the form (2.14) taking account of the consti-
tutive equations of B. These expressions allow us to deduce the work done on any process P
[6, 13 – 16]. With reference to the second form (2.14), the work done on a process P (τ) =
= (ĖP (τ), ḢP (τ), ϑ̇P (τ),gP (τ)), defined for any τ ∈ [0, d) and applied at time t when the state
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298 G. AMENDOLA, A. MANES
is σi(t) = (Ei(t),Hi(t), ϑi(t), Ēt
i, ḡ
t
i), can be written as follows:
W (σi(t), P ) = W̃ (Ei(t),Hi(t), ϑi(t), Ēt
i, ḡ
t
i; ĖP , ḢP , ϑ̇P ,gP ) =
=
d∫
0
[
c
Θ0
ϑ̇P (τ)ϑP (τ) +
1
ε
Ḋ(EP (τ), ϑP (τ)) ·D(EP (τ), ϑP (τ)) +
1
µ
Ḃ(HP (τ)) ·
·B(HP (τ)) + J̃((EP ∗ Ēi)t+τ ) ·EP (τ)− 1
Θ0
q̃((gP ∗ ḡi)t+τ ) · gP (τ)
]
dτ. (5.1)
Here we have considered the variable τ ∈ [0, d) and therefore EP (τ), HP (τ), ϑP (τ) are expres-
sed by (3.12) – (3.14), gP (τ) is given by P and has the form (3.15) and finally EP ∗ Ēi, gP ∗ ḡi
are the continuations (3.16), (3.17). If we consider the variable ξ ∈ [t, t + d) we can transform
(5.1) as follows:
W (σi(t), P ) =
t+d∫
t
[
c
Θ0
ϑ̇(ξ)ϑ(ξ) +
1
ε
Ḋ(E(ξ), ϑ(ξ)) ·D(E(ξ), ϑ(ξ)) +
+
1
µ
Ḃ(H(ξ)) ·B(H(ξ)) + J̃((EP ∗ Ēi)ξ) ·E(ξ)− 1
Θ0
q̃((gP ∗ ḡi)ξ) · g(ξ)
]
dξ.
(5.2)
For the sake of simplicity we eliminate i in σi(t) and from (5.2) we derive
W (σ(t), P ) =
1
2
[
c
Θ0
ϑ2(t+ d) +
1
ε
D2(t+ d) +
1
µ
B2(t+ d)
]
− 1
2
[
c
Θ0
ϑ2(t) +
+
1
ε
D2(t) +
1
µ
B2(t)
]
−
t+d∫
t
+∞∫
0
α′(s)(EP ∗ Ē)ξ(s)ds ·E(ξ)dξ−
− 1
Θ0
t+d∫
t
+∞∫
0
k′(s)(gP ∗ ḡ)ξ(s)ds · g(ξ)dξ, (5.3)
where we have to use (3.16), (3.17) for the continuations of the integrated histories.
In Section 3 we have considered the particular case where the process is applied at time
t = 0. Now we suppose at this initial instant the state to be
σ0(0) = σ(0) = (0,0, 0, 0̄†, 0̄†), (5.4)
that is, E(0) = 0, H(0) = 0, ϑ(0) = 0, Ē0(s) = 0̄†(s) = 0 and ḡ0(s) = 0̄†(s) = 0 ∀s ∈ R+;
therefore, the ensuing fields, now denoted by E0, H0, ϑ0 to distinguish this particular case,
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 299
taking into account (3.8) – (3.11), assume the form
E0(t) =
t∫
0
ĖP (s)ds, H0(t) =
t∫
0
ḢP (s)ds, ϑ0(t) =
t∫
0
ϑ̇P (s)ds (5.5)
and
(EP ∗ 0̄†)t(s) =
{
Ēt
0(s), 0 ≤ s < t,
Ēt
0(t), s ≥ t,
(gP ∗ 0̄†)t(s) =
{
ḡt
P (s), 0 ≤ s < t,
ḡt
P (t), s ≥ t.
(5.6)
With these hypotheses we are able to distinguish the work due only to the process and give the
following definition.
Definition 5.1. Let P (t) = (ĖP (t), ḢP (t), ϑ̇P (t),gP (t)) be a process of duration d applied at
time t = 0 and related to E0(t), H0(t), ϑ0(t), (EP ∗ 0̄†)t, and (gP ∗ 0̄†)t, given by (5.5), (5.6), if
the work done on P is finite, then P is said to be a finite work process.
Such a work is given by (5.3), which, using (5.5), (5.6), assumes the form
W (σ0(0), P ) = W̃ (0,0, 0, 0̄†, 0̄†; ĖP , ḢP , ϑ̇P ,gP ) =
=
d∫
0
[
c
Θ0
ϑ̇0(t)ϑ0(t) + +
1
ε
Ḋ(E0(t), ϑ0(t)) ·D(E0(t), ϑ0(t))+
+
1
µ
Ḃ(H0(t)) ·B(H0(t)) + J̃((Ē ∗ 0̄†)t) ·E0(t)−
− 1
Θ0
q̃((gP ∗ 0̄†)t) · gP (t)
]
dt, (5.7)
which, taking account of (2.6) with (5.6), can be written as follows:
W (σ0(0), P ) =
1
2
[
c
Θ0
ϑ2
0(d) +
1
ε
D2
0(d) +
1
µ
B2
0(d)
]
−
−
d∫
0
t∫
0
α′(s)Ēt
0(s)ds+
+∞∫
t
α′(s)Ēt
0(t)ds
·E0(t)dt−
− 1
Θ0
d∫
0
t∫
0
k′(s)ḡt
P (s)ds+
+∞∫
t
k′(s)ḡt
P (t)ds
· gP (t)dt. (5.8)
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300 G. AMENDOLA, A. MANES
Lemma 5.1. The work obtained by any process, which is a finite work process in the sense of
Definition 5.1, is positive.
Proof. Let P be a finite work process of duration d; such a process, starting from the state
σ0(0) at time t = 0, yields the work (5.8), which, integrating by parts the integrals over [0, t]
and evaluating the other two, becomes
W (σ0(0), P ) =
1
2
[
c
Θ0
ϑ2
0(d) +
1
ε
D2
0(d) +
1
µ
B2
0(d)
]
+
+
d∫
0
t∫
0
α(s)Et
0(s)ds ·E0(t) +
1
Θ0
t∫
0
k(s)gt
P (s)ds · gP (t)
dt. (5.9)
Assuming that the functions in (5.9) are equal to zero for any t > d, the integral in (5.9)
being extended on R+, and using ∗ to denote the complex conjugate, by means of Plancherel’s
theorem assumes the form
+∞∫
0
t∫
0
α(s)Et
0(s)ds ·E0(t) +
1
Θ0
t∫
0
k(s)gt
P (s)ds · gP (t)
dt =
=
1
2π
+∞∫
−∞
[
αF (ω)E0F (ω) ·E∗0F
(ω) +
1
Θ0
kF (ω)gPF
(ω) · g∗PF
(ω)
]
dω =
=
1
2π
+∞∫
−∞
{
αc(ω)[E2
0c
(ω) + E2
0s
(ω)] +
1
Θ0
kc(ω)[g2
Pc
(ω) + g2
Ps
(ω)]
}
dω > 0. (5.10)
The last result follows easily from (2.21) and from the consideration that the Fourier transforms
of the functions, defined on R+ and equal to zero on R−−, by means of (2.18) can be expressed
in terms of their cosine and sine transforms, which are even and odd functions, respectively.
The lemma is proved.
Usually, the duration of a process P is finite, d < +∞; however, as we have already done
in the proof of the previous lemma, we can define P on R+ by putting P (τ) = (ĖP (τ), ḢP (τ),
ϑ̇P (τ),gP (τ)) = (0,0, 0,0) for any τ ≥ d. If we assume that EP (τ) = 0, HP (τ) = 0, ϑP (τ) =
= 0 for any τ > d too, (5.7) can be written as follows:
W (σ0(0), P ) =
d∫
0
[
c
Θ0
ϑ̇0(ξ)ϑ0(ξ) +
1
ε
Ḋ0(ξ) ·D0(ξ) +
1
µ
Ḃ0(ξ) ·B0(ξ)
]
dξ−
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 301
−
+∞∫
0
η∫
0
α′(s)Ēη
P (s)ds+
∫ +∞
η
α′(s)Ēη
P (η)ds
·EP (η)dη−
− 1
Θ0
+∞∫
0
η∫
0
k′(s)ḡη
P (s)ds+
∫ +∞
η
k′(s)ḡη
P (η)ds
· gP (η)dη, (5.11)
which, taking into account that the initial state is σ0(0), given by (5.4), and integrating by parts,
reduces to
W (σ0(0), P ) =
1
2
[
c
Θ0
ϑ2(d) +
1
ε
D2(d) +
1
µ
B2(d)
]
+
+
+∞∫
0
η∫
0
α(s)Eη
P (s)ds ·EP (η) +
1
Θ0
η∫
0
k(s)gη
P (s)ds · gP (η)
dη =
=
1
2
[
c
Θ0
ϑ2(d) +
1
ε
D2(d) +
1
µ
B2(d)
]
+
+
1
2
+∞∫
0
+∞∫
0
α(| η − ρ |)EP (ρ)dρ ·EP (η)+
+
1
Θ0
+∞∫
0
k(| η − ρ |)gP (ρ)dρ · gP (η)
dη. (5.12)
In the last relation we have the even functions α(| η − ρ |) and k(| η − ρ |), whose Fourier
transforms can be written in terms of their Fourier cosine transforms; therefore, using (2.18)2,
(5.12) becomes
W (σ0(0), P ) =
1
2
[
c
Θ0
ϑ2(d) +
1
ε
D2(d) +
1
µ
B2(d)
]
+
+
1
2π
+∞∫
−∞
[
αc(ω)EP+(ω) ·E∗P+(ω) +
1
Θ0
kc(ω)gP+(ω) · g∗P+(ω)
]
dω.
(5.13)
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302 G. AMENDOLA, A. MANES
This result allows us to introduce the functional spaces
H̃α(R+,R3) =
E : R+ → R3;
+∞∫
−∞
αc(ω)E+(ω) ·E∗+(ω)dω < +∞
, (5.14)
H̃k(R+,R3) =
g : R+ → R3;
+∞∫
−∞
kc(ω)g+(ω) · g∗+(ω)dω < +∞
, (5.15)
which characterize the finite work processes. The completions of these spaces, with respect to
the norms corresponding to the inner products defined by (E1,E2)α =
∫ +∞
−∞
αc(ω)E1+(ω) ·
·E∗2+(ω)dω and (g1,g2)k =
∫ +∞
−∞
kc(ω)g1+(ω) · g∗2+(ω)dω, respectively, yield two Hilbert’s
spaces, which are denoted by Hα(R+,R3) and Hk(R+,R3) and characterize the spaces of the
processes.
Let us now assume σ(t) = (E(t),H(t), ϑ(t), Ēt, ḡt) as the initial state of B such that its
integrated histories Ēt ∈ Γα and ḡt ∈ Γk, see (3.6), (3.7), are admissible histories which yield
a finite work during any process, characterized by gP ∈ Hk(R+,R3) and related to EP ∈
∈ Hα(R+,R3). If P = (ĖP , ḢP , ϑ̇P ,gP ) is one of these processes with duration d < +∞, it
may be extended on R+ on supposing that P (τ) = (0,0, 0,0) for every τ ≥ d and that it is
related to EP (τ) = 0, HP (τ) = 0, ϑP (τ) = 0 ∀τ > d. The work done on such a process is
given by (5.1), which, using (3.20), (3.21), now assumes the following form:
W (σ(t), P ) =
d∫
0
[
c
Θ0
ϑ̇P (τ)ϑP (τ) +
1
ε
Ḋ(EP (τ), ϑP (τ)) ·D(EP (τ), ϑP (τ)) +
+
1
µ
Ḃ(HP (τ)) ·B(HP (τ))
]
dτ +
+∞∫
0
α(τ)Ēτ
P (τ)−
τ∫
0
α′(s)Ēτ
P (s)ds −
−
+∞∫
0
α′(τ + ξ)Ēt(ξ)dξ
·EP (τ)dτ +
1
Θ0
+∞∫
0
k(τ)ḡτ
P (τ)−
τ∫
0
k′(s)ḡτ
P (s)ds −
−
+∞∫
0
k′(τ + ξ)ḡt(ξ)dξ
· gP (τ)dτ. (5.16)
This expression, by evaluating the first integral and integrating by parts in the other two,
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 303
can be changed as follows:
W (σ(t), P ) =
1
2
[
c
Θ0
ϑ2
P (d) +
1
ε
D2(EP (d), ϑP (d)) +
1
µ
B2(HP (d)
]
−
− 1
2
[
c
Θ0
ϑ2
P (0) +
1
ε
D2(EP (0), ϑP (0)) +
1
µ
B2(HP (0))
]
+
+
+∞∫
0
1
2
+∞∫
0
α(| τ − η |)EP (η)dη − I(α)(τ, Ē
t)
·EP (τ)dτ+
+
1
Θ0
+∞∫
0
1
2
+∞∫
0
k(| τ − η |)gP (η)dη − I(k)(τ, ḡ
t)
· gP (τ)dτ, (5.17)
where we have put
I(α)(τ, Ē
t) =
+∞∫
0
α′(τ + ξ)Ēt(ξ)dξ, I(k)(τ, ḡ
t) =
+∞∫
0
k′(τ + ξ)ḡt(ξ)dξ, τ ≥ 0. (5.18)
These two quantities, defined on R+, are present in (2.30), (2.31) and hence are related to the
static continuations having the duration τ ; their induced regularities allow us to evaluate the
Fourier transforms,
I(α)+(ω, Ēt) =
+∞∫
0
e−iωτI(α)(τ, Ē
t)dτ, I(k)+(ω, ḡt) =
+∞∫
0
e−iωτI(k)(τ, ḡ
t)dτ.
Thus, by means of Plancherel’s theorem, (5.17) becomes
W (σ(t), P ) =
1
2
[
c
Θ0
ϑ2
P (d) +
1
ε
D2
P (EP (d), ϑP (d)) +
1
µ
B2
P (HP (d))
]
−
− 1
2
[
c
Θ0
ϑ2
P (0) +
1
ε
D2
P (EP (0), ϑP (0)) +
1
µ
B2
P (HP (0))
]
+
+
1
2π
+∞∫
−∞
[
αc(ω)EP+(ω) ·E∗P+(ω)dω +
1
Θ0
kc(ω)gP+(ω) · g∗P+(ω)
]
dω−
− 1
2π
+∞∫
−∞
[
I(α)+(ω, Ēt) ·E∗P+(ω) +
1
Θ0
I(k)+(ω, ḡt) · g∗P+(ω)
]
dω.
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304 G. AMENDOLA, A. MANES
6. The equivalence relation done in terms of the thermoelectromagnetic work. In Defini-
tion 4.3 we have called equivalent two couples of integrated histories, Ēt
j and ḡt
j , j = 1, 2, if
they give the same electric current density and the same heat flux when the body is subjected
to the same process; moreover, as we have already observed, this equivalence relation yields
the same response of the material because of the equality of the values of E, H and ϑ at the
initial instant of the process. An analogous equivalence relation may be done by means of the
termolectromagnetic work as follows.
Definition 6.1. Let σj(t) = (Ej(t),Hj(t), ϑj(t), Ēt
j , ḡ
t
j), j = 1, 2, be two states of B. Two
couples of integrated histories Ēt
1, ḡ1 and Ēt
2, ḡt
2 are said to be w-equivalent if and only if the
equality
W̃ (E1(t),H1(t), ϑ1(t), Ēt
1, ḡ
t
1; ĖP , ḢP , ϑ̇P ,gP ) =
= W̃ (E2(t),H2(t), ϑ2(t), Ēt
2, ḡ
t
2; ĖP , ḢP , ϑ̇P ,gP ) (6.1)
is satisfied for every ĖP : [0, τ) → R3, ḢP : [0, τ) → R3, ϑ̇P : [0, τ) → R, gP : [0, τ) → R3
and for every τ > 0.
The two definitions of equivalence are equivalent as the following theorem states.
Theorem 6.1. For the thermoelectromagnetic body B, two couples of integrated histories of
the electric field and of the temperature gradient are w-equivalent if and only if they are equivalent
in the sense of Definition 4.3.
Proof. It is obvious that if two couples of integrated histories, Ēt
1, ḡ1 and Ēt
2, ḡt
2, are equi-
valent in the sense of Definition 4.3, then for every EP : (0, τ ] → R3, HP : (0, τ ] → R3,
ϑP : (0, τ ] → R, gP : [0, τ) → R3 and for every τ > 0 we have
d∫
0
[
c
Θ0
ϑ̇P (τ)ϑP (τ) +
1
ε
Ḋ(EP (τ), ϑP (τ)) ·D(EP (τ), ϑP (τ)) +
1
µ
Ḃ(HP (τ)) ·
·B(HP (τ)) + J̃((EP ∗ Ē1)t+τ ) ·EP (τ)− 1
Θ0
q̃((gP ∗ ḡ1)t+τ ) · gP (τ)
]
dτ =
=
d∫
0
[
c
Θ0
ϑ̇P (τ)ϑP (τ) +
1
ε
Ḋ(EP (τ), ϑP (τ)) ·D(EP (τ), ϑP (τ)) +
1
µ
Ḃ(HP (τ)) ·
· B(HP (τ)) + J̃((EP ∗ Ē2)t+τ ) ·EP (τ)− 1
Θ0
q̃((gP ∗ ḡ2)t+τ ) · gP (τ)
]
dτ, (6.2)
because of (3.12) – (3.15) and of the conditions imposed by Definition 4.3. This relation yields
the equality of the two works done on the same process of duration d, applied to the states
(Ej(t),Hj(t), ϑj(t), Ēt
j , ḡ
t
j), j = 1, 2, whose instantaneous values coincide.
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 305
On the other hand, on supposing that the two couples of integrated histories, Ēt
1, ḡ1 and Ēt
2,
ḡt
2, are w-equivalent, (6.1) holds for any P with an arbitrary duration d. Using the expression
(5.17) of the work, (6.1) yields
c
2Θ0
{
ϑ2
P1
(d)− ϑ2
P2
(d)− [ϑ2
P1
(0)− ϑ2
P2
(0)]
}
+
1
2ε
{
D2
P1
(d)−D2
P2
(d) −
−[D2
P1
(0)−D2
P2
(0)]
}
+
1
2µ
{
B2
P1
(d)−B2
P2
(d)− [B2
P1
(0)−B2
P2
(0)]
}
+
+
1
2
+∞∫
0
+∞∫
0
α(| τ − η |)[EP1(η) ·EP1(τ)−EP2(η) ·EP2(τ)]dηdτ−
−
+∞∫
0
[I(α)(τ, Ē
t
1) ·EP1(τ)− I(α)(τ, Ē
t
2) ·EP2(τ)]dτ−
− 1
Θ0
+∞∫
0
[I(k)(τ, ḡ
t
1)− I(k)(τ, ḡ
t
2)] · gP (τ)dτ = 0, (6.3)
where the quantities evaluated in d and 0 are given by (3.14) and (2.1) taking into account
(3.12) – (3.14). In particular, we have ϑPj (d) = ϑj(t)+
∫ d
0
ϑ̇P (s)ds and ϑPj (0) = ϑj(t), j = 1, 2,
and analogous relations for EPj and HPj , j = 1, 2. Obviously, since the same gP appears on
both sides of (6.1) the integral with k(| τ − ξ |) has been eliminated. Substituting all these
relations into (6.3), we get
{(
c
Θ0
+
a2
ε
)
[ϑ1(t)− ϑ2(t)] + a · [E1(t)−E2(t)]
} d∫
0
ϑ̇P (s)ds+ {a[ϑ1(t) −
−ϑ2(t)] + ε[E1(t)−E2(t)]} ·
d∫
0
ĖP (s)ds+ µ[H1(t)−H2(t)] ·
d∫
0
ḢP (s)ds+
+
1
2
[E2
1(t)−E2
2(t)]
+∞∫
0
+∞∫
0
α(| τ − η |)dηdτ +
1
2
[E1(t)−E2(t)]·
·
+∞∫
0
+∞∫
0
α(| τ − η |)
τ∫
0
ĖP (s)ds+
η∫
0
ĖP (s)ds
dηdτ − +∞∫
0
[I(α)(τ, Ē
t
1)·
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306 G. AMENDOLA, A. MANES
·E1(t)− I(α)(τ, Ē
t
2) ·E2(t)]dτ −
+∞∫
0
[I(α)(τ, Ē
t
1)− I(α)(τ, Ē
t
2)]·
·
τ∫
0
ĖP (s)dsdτ − 1
Θ0
+∞∫
0
[I(k)(τ, ḡ
t
1)− I(k)(τ, ḡ
t
2]) · gP (τ)dτ = 0, (6.4)
which must hold for any P and any d > 0. The arbitrariness of ϑ̇P and the one of ĖP yield the
following system: (
c
Θ0
+
a2
ε
)
[ϑ1(t)− ϑ2(t)] + a · [E1(t)−E2(t)] = 0,
(6.5)
a[ϑ1(t)− ϑ2(t)] + ε[E1(t)−E2(t)] = 0,
whence it follows that
ϑ1(t) = ϑ2(t), E1(t) = E2(t); (6.6)
moreover, since ḢP and gP are also arbitrary, we get
H1(t) = H2(t), I(α)(τ, Ē
t
1) = I(α)(τ, Ē
t
2), I(k)(τ, ḡ
t
1) = I(k)(τ, ḡ
t
2). (6.7)
From (6.7)2,3 and (5.18) we obtain
+∞∫
0
α′(τ + ξ)[Ēt
1(ξ)− Ēt
2(ξ)])dξ = 0,
+∞∫
0
k′(τ + ξ)[ḡt
1(ξ)− ḡt
2(ξ)]dξ = 0, (6.8)
which, together with the conditions (6.6) and (6.7)1, expresses the equivalence of the two
couples of integrated histories Ēt
j , ḡt
j , j = 1, 2, since the differences Ēt = Ēt
1− Ēt
2, ḡt = ḡt
1−ḡt
2
satisfy (4.5)2 and (4.6)2.
7. Maximum recoverable work. The maximum recoverable work is the maximum work
obtainable from the material at a given state. It is defined as follows.
Definition 7.1. Given a state σ of B, the maximum work obtained by starting from σ is
WR(σ) = sup {−W (σ, P ) : P ∈ Π} , (7.1)
where Π denotes the set of finite work processes.
We observe that from thermodynamic considerations we have WR(σ) < +∞; moreover,
WR(σ) is a nonnegative function of the state, since the null process, which belongs to Π, yields
a null work. In many works [2, 8, 12] it has been shown that such a work (7.1) coincides with the
minimum free energy, that is,
ψm(σ) = WR(σ). (7.2)
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 307
In order to derive an expression for these quantities we consider an initial state σ(t) =
= (E(t),H(t), ϑ(t), Ēt, ḡt) at a fixed time t when we apply a process P ∈ Π with a finite durati-
on d but extended on [d,+∞), where P = 0 and we assume the following values: EP (d) = 0,
HP (d) = 0, ϑP (d) = 0. Then, we consider the corresponding work expressed by (5.17), which
now reduces to
W (σ, P ) = −1
2
{
c
Θ0
ϑ2(t) +
1
ε
[εE(t) + ϑ(t)a]2 + µH2(t)
}
+
+
1
2
+∞∫
0
+∞∫
0
[
α(| τ − ξ |)EP (η) ·EP (τ) +
1
Θ0
k(| τ − η |)gP (η) · gP (τ)
]
dηdτ−
−
+∞∫
0
[
I(α)(τ, Ē
t) ·EP (τ) +
1
Θ0
I(k)(τ, ḡ
t) · gP (τ)
]
dτ. (7.3)
The required maximum recoverable work will be obtained by an opportune process P (m)
related to E(m) and g(m); therefore, we consider the set of processes related to
EP (τ) = E(m)(τ) + γe(τ), gP (τ) = g(m)(τ) + δv(τ) τ ∈ R+, (7.4)
with γ and δ real parameters, e and v arbitrary smooth functions with e(0) = 0 and v(0) = 0,
and we study the maximum of −W (σ, P ) by substituting (7.4) into (7.3) and evaluating
∂
∂γ
[−W (σ, P )] |γ=0= −
+∞∫
0
+∞∫
0
α(| τ − η |)E(m)(η)dη − I(α)(τ, Ē
t)
·E(τ)dτ = 0,
(7.5)
∂
∂δ
[−W (σ, P )] |δ=0= − 1
Θ0
+∞∫
0
+∞∫
0
k(| τ − η |)g(m)(η)dη − I(k)(τ, ḡ
t)
· v(τ)dτ = 0,
whence it follows that
+∞∫
0
α(| τ − η |)E(m)(η)dη = I(α)(τ, Ē
t),
(7.6)
+∞∫
0
k(| τ − η |)g(m)(η)dη = I(k)(τ, ḡ
t)
for all τ ∈ R+. These relations turn out to be two integral equations of the Wiener – Hopf type
and of the first kind, which are not solvable in the general case. Nevertheless, the thermodynamic
properties of the kernels α and k and some theorems on factorization allow us to determine the
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308 G. AMENDOLA, A. MANES
solutions E(m) and g(m) of (7.6), which corresponds to the maximum recoverable work, whose
expression, derived from (7.1), (7.3) with (7.6), is
WR(σ) =
1
2
{
c
Θ0
ϑ2(t) +
1
ε
[εE(t) + ϑ(t)a]2 + µH2(t)
}
+
+
1
2
+∞∫
0
+∞∫
0
[
α(| τ − ξ |)E(m)(η) ·E(m)(τ) +
+
1
Θ0
k(| τ − η |)g(m)(η) · g(m)(τ)
]
dηdτ (7.7)
which, applying Plancherel’s theorem, becomes
WR(σ) =
1
2
{
c
Θ0
ϑ2(t) +
1
ε
[εE(t) + ϑ(t)a]2 + µH2(t)
}
+
+
1
2π
+∞∫
−∞
[
αc(ω)E(m)
+ (ω) · (E(m)
+ (ω))∗ +
1
Θ0
kc(ω)g(m)
+ (ω) · (g(m)
+ (ω))∗
]
dω. (7.8)
It remains to solve the Wiener – Hopf equations (7.6). For this purpose let
r(α)(τ) =
+∞∫
−∞
α(| τ − s |)E(m)(s)ds, r(k)(τ) =
+∞∫
−∞
k(| τ − s |)g(m)(s)ds ∀τ ∈ R−, (7.9)
be equal to zero on R++. We observe that supp(r(α)) ⊆ R−, supp(r(k)) ⊆ R−, supp(E(m)) ⊆
⊆ R+, supp(g(m)) ⊆ R+, supp(I(α)(·, Ēt)) ⊆ R+, supp(I(k)(·, ḡt)) ⊆ R+; therefore, (7.6) can
be rewritten as follows:
+∞∫
0
α(| τ − η |)E(m)(η)dη = I(α)(τ, Ē
t) + r(α)(τ),
(7.10)
+∞∫
0
k(| τ − η |)g(m)(η)dη = I(k)(τ, ḡ
t) + r(k)(τ)
for all τ ∈ R, whence Fourier’s transform yields
2αc(ω)E(m)
+ (ω) = I(α)+(ω, Ēt) + r(α)
− (ω), 2kc(ω)g(m)
+ (ω) = I(k)+(ω, ḡ) + r(k)
− (ω). (7.11)
Let us introduce
K(α)(ω) = (1 + ω2)αc(ω), K(k)(ω) = (1 + ω2)kc(ω), (7.12)
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 309
which are two functions without zeros for any real ω ∈ R, and at infinity, because of the
properties (2.22) – (2.25). They can be factorized as well as αc(ω) and kc(ω) and, therefore,
we have
K(α)(ω) = K
(α)
(+)(ω)K(α)
(−)(ω), K(k)(ω) = K
(k)
(+)(ω)K(k)
(−)(ω), (7.13)
αc(ω) = α(+)(ω)α(−)(ω), kc(ω) = k(+)(ω)k(−)(ω), (7.14)
whence, from (7.12) it follows that
α(±)(ω) =
1
1± iω
K
(α)
(±)(ω), k(±)(ω) =
1
1± iω
K
(k)
(±)(ω). (7.15)
Thus, from (7.11) we obtain
α(+)(ω)E(m)
+ (ω) =
1
2α(−)(ω)
[I(α)+(ω, Ēt) + r(α)
− (ω)], (7.16)
k(+)(ω)g(m)
+ (ω) =
1
2k(−)(ω)
[I(k)+(ω, ḡt) + r(k)
− (ω)]. (7.17)
Let us consider
Pt
(α)(z) =
1
4πi
+∞∫
−∞
I(α)+(ω, Ēt)/α(−)(ω)
ω − z
dω, Pt
(α)(±)(ω) = lim
β→0∓
limPt
(α)(ω + iβ), (7.18)
Pt
(k)(z) =
1
4πi
+∞∫
−∞
I(k)+(ω, ḡt)/k(−)(ω)
ω − z
dω, Pt
(k)(±)(ω) = lim
β→0∓
limPt
(k)(ω + iβ), (7.19)
which, by using the Plemelj formulae [18], yield
I(α)+(ω, Ēt)
2α(−)(ω)
= Pt
(α)(−)(ω)−Pt
(α)(+)(ω), (7.20)
I(k)+(ω, ḡt)
2k(−)(ω)
= Pt
(k)(−)(ω)−Pt
(k)(+)(ω). (7.21)
We observe that both Pt
(α)(±)(z) and Pt
(k)(±)(z) have zeros and singularities in z ∈ C±, and
hence they are analytic in C(∓) and, by the hypothesis on the Fourier transforms [8], also on R.
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310 G. AMENDOLA, A. MANES
From (7.16), (7.17), taking into account (7.20), (7.21), we have
α(+)(ω)E(m)
+ (ω) + Pt
(α)(+)(ω) = Pt
(α)(−)(ω) +
r(α)
− (ω)
2α(−)(ω)
, (7.22)
k(+)(ω)g(m)
+ (ω) + Pt
(k)(+)(ω) = Pt
(k)(−)(ω) +
r(k)
− (ω)
2k(−)(ω)
, (7.23)
where the quantities in the left-hand sides, considered as functions of z, are analytic on C−,
while the others in the right-hand sides are analytic on C+; consequently, the functions in the
left-hand sides have analytic extensions on C and vanish at infinity, therefore are equal to zero
and hence we get
E(m)
+ (ω) = −
Pt
(α)(+)(ω)
α(+)(ω)
, g(m)
+ (ω) = −
Pt
(k)(+)(ω)
k(+)(ω)
; (7.24)
analogous relations may be derived by putting the right-hand sides of (7.22), (7.23) equal to zero.
These last relations (7.24) substituted into (7.8) yield the required expression of the minimum
free energy
ψm(σ(t)) =
1
2
{
c
Θ0
ϑ2(t) +
1
ε
[εE(t) + ϑ(t)a]2 + µH2(t)
}
+
+
1
2π
+∞∫
−∞
[
| Pt
(α)(+)(ω) |2 +
1
Θ0
| Pt
(k)(+)(ω) |2
]
dω. (7.25)
8. An equivalent formulation of ψm. The minimum free energy ψm, we have now derived,
is expressed in terms of Pt
(α)(+)(ω) and Pt
(k)(+)(ω), which are related to Ēt and ḡt. In order
to obtain new relations in function of these last quantities, we extend the kernels α′(s) and
k′(s) on R−− with odd functions, denoted by α′(o)(s) and k′(o)(s), such that α′(o)(s) = α′(s),
k′(o)(s) = k′(s) ∀s ≥ 0; moreover, we take the usual extensions on R−− for Ēt and ḡt, i.e.,
Ēt(s) = 0, ḡt(s) = 0 ∀s < 0. Identifying these functions with their extensions, (5.18) become
I(α)(τ, Ē
t) =
+∞∫
−∞
α′(o)(τ + ξ)Ēt(ξ)dξ, I(k)(τ, ḡ
t) =
+∞∫
−∞
k′(o)(τ + ξ)ḡt(ξ)dξ, τ ≥ 0, (8.1)
which, by defining
I(N)
(α) (τ, Ēt) =
+∞∫
−∞
α′(o)(τ + ξ)Ēt(ξ)dξ, I(N)
(k) (τ, ḡt) =
+∞∫
−∞
k′(o)(τ + ξ)ḡt(ξ)dξ, τ < 0, (8.2)
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 311
can be extended on R as follows:
I(R)
(α) (τ, Ēt) =
+∞∫
−∞
α′(o)(τ + ξ)Ēt(ξ)dξ =
I(α)(τ, Ēt) ∀τ ≥ 0,
I(N)
(α) (τ, Ēt) ∀τ < 0,
(8.3)
I(R)
(k) (τ, ḡt) =
+∞∫
−∞
k′(o)(τ + ξ)ḡt(ξ)dξ =
I(k)(τ, ḡt) ∀τ ≥ 0,
I(N)
(k) (τ, ḡt) ∀τ < 0.
(8.4)
Introducing Ēt
N (s) = Ēt(−s), ḡt
N (s) = ḡt(−s) ∀s ≤ 0 with their extensions Ēt
N (s) = 0,
ḡt
N (s) = 0 ∀s > 0, their Fourier’s transforms are
Ēt
NF
(ω) = Ēt
N−(ω) =
(
Ēt
+(ω)
)∗
, ḡt
NF
(ω) = ḡt
N−(ω) =
(
ḡt
+(ω)
)∗
. (8.5)
Thus, (8.3) and (8.4) can be written as
I(R)
(α) (τ, Ēt) =
+∞∫
−∞
α′(o)(τ − s)Ēt
N (s)ds, I(R)
(k) (τ, ḡt) =
+∞∫
−∞
k′(o)(τ − s)ḡt
N (s)ds, (8.6)
whose Fourier’s transforms, taking into account (2.18)3, are given by
I(R)
(α)F
(ω, Ēt) = −2iα′s(ω)
(
Ēt
+(ω)
)∗
, I(R)
(k)F
(ω, ḡt) = −2ik′s(ω)
(
ḡt
+(ω)
)∗
. (8.7)
Hence, using (2.22) and (7.14), it follows that
1
2α(−)(ω)
I(R)
(α)F
(ω, Ēt) = iωα(+)(ω)
(
Ēt
+(ω)
)∗
, (8.8)
1
2k(−)(ω)
I(R)
(k)F
(ω, ḡt) = iωk(+)(ω)
(
ḡt
+(ω)
)∗
. (8.9)
Directly from the definitions (8.3), (8.4) we obtain
I(R)
(α)F
(ω, Ēt) = I(N)
(α)−(ω, Ēt) + I(α)+(ω, Ēt), (8.10)
I(R)
(k)F
(ω, ḡt) = I(N)
(k)−(ω, ḡt) + I(k)+(ω, ḡt), (8.11)
whence, taking into account (7.20), (7.21), we get
1
2α(−)(ω)
I(R)
(α)F
(ω, Ēt) =
1
2α(−)(ω)
I(N)
(α)−(ω, Ēt) + Pt
(α)(−)(ω)−Pt
(α)(+)(ω), (8.12)
1
2k(−)(ω)
I(R)
(k)F
(ω, ḡt) =
1
2k(−)(ω)
I(N)
(k)−(ω, ḡt) + Pt
(k)(−)(ω)−Pt
(k)(+)(ω), (8.13)
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312 G. AMENDOLA, A. MANES
where, using the Plemelj formulae, we also have
1
2α(−)(ω)
I(R)
(α)F
(ω, Ēt) = P(1)t
(α)(−)(ω)−P(1)t
(α)(+)(ω), (8.14)
1
2k(−)(ω)
I(R)
(k)F
(ω, ḡt) = P(1)t
(k)(−)(ω)−P(1)t
(k)(+)(ω), (8.15)
P(1)t
(α)(±)(ω) and P(1)t
(k)(±)(ω) being defined as in (7.18) and (7.19).
Thus, (8.12) – (8.15) yield two relations, which define the functions
V(α)(ω) ≡ Pt
(α)(+)(ω)−P(1)t
(α)(+)(ω) = Pt
(α)(−)(ω)−P(1)t
(α)(−)(ω) +
1
2α(−)(ω)
I(N)
(α)−(ω, Ēt),
(8.16)
V(k)(ω) ≡ Pt
(k)(+)(ω)−P(1)t
(k)(+)(ω) = Pt
(k)(−)(ω)−P(1)t
(k)(−)(ω) +
1
2k(−)(ω)
I(N)
(k)−(ω, ḡt), (8.17)
with two different expressions, which assure the analyticity on C+ and on C−, respectively, and
vanish at infinity; therefore, we have V(α)(ω) = 0 and V(k)(ω) = 0. Hence, it follows that
Pt
(α)(+)(ω) = P(1)t
(α)(+)(ω), Pt
(α)(−)(ω) = P(1)t
(α)(−)(ω)− 1
2α(−)(ω)
I(N)
(α)−(ω, Ēt), (8.18)
Pt
(k)(+)(ω) = P(1)t
(k)(+)(ω), Pt
(k)(−)(ω) = P(1)t
(k)(−)(ω)− 1
2k(−)(ω)
I(N)
(k)−(ω, ḡt). (8.19)
Hence, taking account of (7.18), (7.19), (8.18), (8.19) and (8.8), (8.9), we get
Pt
(α)(+)(ω) = P(1)t
(α)(+)(ω) = lim
z→ω−
lim
1
2πi
+∞∫
−∞
iω′α(+)(ω′)
(
Ēt
+(ω′)
)∗
ω′ − z
dω′, (8.20)
Pt
(k)(+)(ω) = P(1)t
(k)(+)(ω) = lim
z→ω−
lim
1
2πi
+∞∫
−∞
iω′k(+)(ω′)
(
ḡt
+(ω′)
)∗
ω′ − z
dω′, (8.21)
from which we have
(
Pt
(α)(+)(ω)
)∗
= i lim
η→ω+
lim
1
2πi
+∞∫
−∞
ω′α(−)(ω′)Ēt
+(ω′)
ω′ − η
dω′, (8.22)
(
Pt
(k)(+)(ω)
)∗
= i lim
η→ω+
lim
1
2πi
+∞∫
−∞
ω′k(−)(ω′)ḡt
+(ω′)
ω′ − η
dω′. (8.23)
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 313
Application of the Plemelj formulae to the last two relations yields
ωα(−)(ω)Ēt
+(ω) = Qt
(α)(−)(ω)−Qt
(α)(+)(ω), (8.24)
ωk(−)(ω)ḡt
+(ω) = Qt
(k)(−)(ω)−Qt
(k)(+)(ω), (8.25)
where Qt
(α)(±)(z), Q
t
(k)(±)(z) have zeros and singularities for z ∈ C±, being
Qt
(α)(±)(ω) = lim
z→ω∓
lim
1
2πi
+∞∫
−∞
ω′α(−)(ω′)Ēt
+(ω′)
ω′ − z
dω′, (8.26)
Qt
(k)(±)(ω) = lim
z→ω∓
lim
1
2πi
+∞∫
−∞
ω′k(−)(ω′)ḡt
+(ω′)
ω′ − z
dω′. (8.27)
Comparison of (8.22), (8.23) with (8.26), (8.27) yields(
Pt
(α)(+)(ω)
)∗
= iQt
(α)(−)(ω),
(
Pt
(k)(+)(ω)
)∗
= iQt
(k)(−)(ω), (8.28)
which, substituting into (7.25), gives the required new expression
ψm(t) =
1
2
{
c
Θ0
ϑ2(t) +
1
ε
[εE(t) + ϑ(t)a]2 + µH2(t)
}
+
+
1
2π
+∞∫
−∞
[
| Qt
(α)(−)(ω) |2 +
1
Θ0
| Qt
(k)(−)(ω) |2
]
dω. (8.29)
We note that both the current density and the heat flux can be expressed in terms of the
last quantities we have derived. For this purpose we apply the Plancherel theorem to (3.5);
using (2.18)3, since α′ and k′ are considered as two odd functions, (2.22), where αc and kc are
factorized by means of (7.14), and (8.24), (8.25), we get
J̃(Ēt) =
i
π
+∞∫
−∞
α(+)(ω)
[
Qt
(α)(−)(ω)−Qt
(α)(+)(ω)
]
dω,
q̃(ḡt) = − i
π
+∞∫
−∞
k(+)(ω)
[
Qt
(k)(−)(ω)−Qt
(k)(+)(ω)
]
dω.
These expressions, for the analyticity ofα(+)(ω)Qt
(α)(+)(ω) and k(+)(ω)Qt
(k)(+)(ω) in C−, reduce
to
J̃(Ēt) =
i
π
+∞∫
−∞
α(+)(ω)Qt
(α)(−)(ω)dω, q̃(ḡt) = − i
π
+∞∫
−∞
k(+)(ω)Qt
(k)(−)(ω)dω,
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314 G. AMENDOLA, A. MANES
which must be real and give the required relations.
9. A discrete spectrum model. We now apply the results of Section 8 to the particular class
of response functions that characterize the discrete spectrum model. The relaxation functions
α and k have the form
α(t) =
n∑
i=1
gie
−αit, t ≥ 0,
0, t < 0,
k(t) =
n∑
i=1
hie
−kit, t ≥ 0,
0, t < 0,
(9.1)
where gi, αi, hi, ki ∈ R++, i = 1, 2, . . . , n, n ∈ N and αj < αj+1, kj < kj+1, j = 1, 2, . . . , n−
−1. These hypotheses assure that α(0) =
∑n
i=1 gi > 0, k(0) =
∑n
i=1 hi > 0, which are two
conditions derived in [1] on account of (2.21) by using the inverse Fourier transforms of αc
and kc.
The Fourier transforms of (9.1) are
αF (ω) =
n∑
i=1
gi
αi + iω
, kF (ω) =
n∑
i=1
hi
ki + iω
, ω ∈ R,
whence, taking account of (2.18)1, it follows that
αc(ω) =
n∑
i=1
αigi
ω2 + α2
i
, kc(ω) =
n∑
i=1
kihi
ω2 + k2
i
, ω ∈ R,
and we write (7.12) as
K(α)(ω) =
n∑
i=1
αigi
ω2 + 1
ω2 + α2
i
, K(k)(ω) =
n∑
i=1
kihi
ω2 + 1
ω2 + k2
i
, ω ∈ R. (9.2)
The last two expressions coincide with the ones derived in [1]. Thus, we recall the results of the
study of the function f (α)(z) = K(α)(z) and f (k)(z) = K(k)(z), where z = −ω2.
Let n 6= 1. If we suppose that α2
i , k2
i 6= 1’, i = 1, 2, ..., n, the functions f (α)(z) and f (k)(z)
have n simple poles at α2
i and k2
i , i = 1, 2, ..., n. The number of simple zeros is n if 1 < α2
1 and
1 < k2
1 or α2
n < 1 and k2
n < 1, which are denoted by γ2
1 = 1, γ2
j , j = 2, 3, ..., n, and δ21 = 1, δ2j ,
j = 2, 3, ..., n; if α2
p < 1 < α2
p+1 and k2
p′ < 1 < k2
p′+1, where p and p′ are two integer numbers,
which may also coincide but in any case they must assume only one of the values 1, 2, ..., n− 1,
then the zeros γ2
p+1 and δ2p′+1 are such that γ2
p+1lim
≥
<1 = γ2
1 and δ2p′+1lim
≥
<1 = δ21 and hence
they can be equal to 1, therefore, they have multiplicity 2, and the number of the distinct zeros
reduces to n− 1. In any case the zeros different from 1 are such that
α2
1 < γ2
2 < α2
2 < ... < α2
p < γ2
p+1 < α2
p+1 < ... < α2
n−1 < γ2
n < α2
n,
(9.3)
k2
1 < δ22 < k2
2 < ... < k2
p′ < δ2p′+1 < k2
p′+1 < ... < k2
n−1 < δ2n < k2
n,
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 315
moreover, (9.2) can be written as
K(α)(ω) = K(α)
∞
n∏
i=1
{
ω2 + γ2
i
ω2 + α2
i
}
, K(k)(ω) = K(k)
∞
n∏
i=1
{
ω2 + δ2i
ω2 + k2
i
}
, (9.4)
where
K(α)
∞ = lim
ω→±∞
limK(α)(ω) =
n∑
i=1
αigi > 0, K(k)
∞ = lim
ω→±∞
limK(k)(ω) =
n∑
i=1
kihi > 0
and γ1 = 1, δ1 = 1, and only one of the other zeros, say γ2
p+1 and δ2p′+1, can be equal to 1, which
thus becomes a zero of multiplicity 2.
The factorizations (7.13), taking account of (9.4), yield, in particular,
K
(α)
(−)(ω) = k(α)
∞
n∏
i=1
{
ω + iγi
ω + iαi
}
, K
(k)
(−)(ω) = k(k)
∞
n∏
i=1
{
ω + iδi
ω + iki
}
, (9.5)
where
k(α)
∞ =
√
K
(α)
∞ , k(k)
∞ =
√
K
(k)
∞ .
We must consider (8.26), (8.27), which give Qt
(α)(−)(ω) and Qt
(k)(−)(ω) present in (8.29); in them
α(−)(ω) and k(−)(ω), contrary to what occurs in [1], are multiplied by ω, therefore the new zeros
γ0 = 0 and δ0 = 0 are introduced and (7.15), taking account of (9.5), yield
ωα(−)(ω) = ik(α)
∞
ω
ω + i
n∏
i=1
{
ω + iγi
ω + iαi
}
, ωk(−)(ω) = ik(k)
∞
ω
ω + i
n∏
i=1
{
ω + iδi
ω + iki
}
. (9.6)
These expressions, taking into account that γ2
1 = 1, δ21 = 1 and putting ρ1 = γ0 = 0 and
ρj = γj , j = 2, 3, ..., n, φ1 = δ0 = 0 and φj = δj , j = 2, 3, ..., n, can be written as follows:
ωα(−)(ω) = ik(α)
∞
n∏
i=1
{
ω + iρi
ω + iαi
}
= ik(α)
∞
(
1 + i
n∑
r=1
Ar
ω + iαr
)
, (9.7)
ωk(−)(ω) = ik(k)
∞
n∏
i=1
{
ω + iφi
ω + iki
}
= ik(k)
∞
(
1 + i
n∑
r=1
Br
ω + ikr
)
, (9.8)
with
Ar = (ρr − αr)
n∏
i=1,i6=r
{
ρi − αr
αi − αr
}
, Br = (φr − kr)
n∏
i=1,i6=r
{
φi − kr)
ki − kr
}
, (9.9)
where we can have ρp+1 = γp+1 = 1 and φp′+1 = δp′+1 = 1.
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316 G. AMENDOLA, A. MANES
We now consider the case where one of the inverse decay times αi and of ki, i = 1, 2, ..., n,
is equal to 1. If α2
1 = k2
1 = 1 or α2
n = k2
n = 1 then f (α)(z) and f (k)(z) have n − 1 zeros γ2
j , δ2j ,
j = 2, 3, ..., n, and n−1 poles α2
j , k2
j , j = 2, 3, ..., n, or α2
i , k2
i , i = 1, 2, ..., n−1, in the two cases;
the presence of the factor
ω
ω + i
in (9.6) yields the zeros ρ1 = 0, φ1 = 0 and thus (9.7), (9.8)
and (9.9), where α2
1 = k2
1 = 1 or α2
n = k2
n = 1, still hold. It may be that α2
p = 1, 1 < p < n,
k2
p′ = 1, 1 < p′ < n, therefore we have n − 1 zeros γ2
j , δ2j , j = 2, 3, ..., n, and n − 1 poles α2
i ,
i = 1, 2, ..., p− 1, p+ 1, ..., n, k2
i , i = 1, 2, ..., p′ − 1, p′ + 1, ..., n, all ordered as in (9.3) and also
by α2
p−1 < γ2
p < 1 < γ2
p+1 < α2
p+1, k2
p′−1 < δ2p′ < 1 < δ2p′+1 < k2
p′+1; however, (9.7), (9.8) and
(9.9) hold with α2
p = k2
p′ = 1.
Let n = 1. This particular case must be considered separately by writing (7.15) by means of
(9.5) with n = 1; we get
ωα(−)(ω) =
ik
(α)
∞ ω
ω + iα1
= ik(α)
∞
(
1 +
iA1
ω + iα1
)
, A1 = −α1, k
(α)
∞ =
√
α1g1, (9.10)
ωk(−)(ω) =
ik
(k)
∞ ω
ω + ik1
= ik(k)
∞
(
1 +
iB1
ω + ik1
)
, B1 = −k1, k
(k)
∞ =
√
k1h1. (9.11)
When n 6= 1, that is in the general case, from (8.26), (8.27) with (9.7)2, (9.8)2 we have
Qt
(α)(−)(ω) = i
k
(α)
∞
2πi
+∞∫
−∞
Ēt
+(ω′)
ω′ − ω+
dω′ −
n∑
r=1
k
(α)
∞ Ar
2πi
+∞∫
−∞
Ēt
+(ω′)/(ω′ − ω+)
ω′ − (−iαr)
dω′,
Qt
(k)(−)(ω) = i
k
(k)
∞
2πi
+∞∫
−∞
ḡt
+(ω′)
ω′ − ω+
dω′ −
n∑
r=1
k
(k)
∞ Br
2πi
+∞∫
−∞
ḡt
+(ω′)/(ω′ − ω+)
ω′ − (−ikr)
dω′,
where the first integrals of the two expressions vanish because Ēt
+, ḡt
+, considered as functions
of z ∈ C, are analytic on C(−), and the same integrals can be extended on an infinite contour
on C(−) without changing their values, which are zero; then, it remains to evaluate the second
integrals by closing again on C(−) and taking account of the sense of the integrations, thus
obtaining
Qt
(α)(−)(ω) = −k(α)
∞
n∑
r=1
Ar
ω + iαr
Ēt
+(−iαr), (9.12)
Qt
(k)(−)(ω) = −k(k)
∞
n∑
r=1
Br
ω + ikr
ḡt
+(−ikr). (9.13)
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MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 317
We observe that (2.16)1 gives
Ēt
+(−iαr) =
+∞∫
0
e−αrsĒt(s)ds =
(
Ēt
+(−iαr)
)∗
, (9.14)
ḡt
+(−ikr) =
+∞∫
0
e−krsḡt(s)ds =
(
ḡt
+(−ikr)
)∗
, (9.15)
which allow us to obtain from (9.12), (9.13) that
(
Qt
(α)(−)(ω)
)∗
= −k(α)
∞
n∑
r=1
Ar
ω − iαr
Ēt
+(−iαr),
(
Qt
(k)(−)(ω)
)∗
= −k(k)
∞
n∑
r=1
Br
ω − ikr
ḡt
+(−iαr).
Finally, we can evaluate the integrals
1
2π
+∞∫
−∞
∣∣∣Qt
(α)(−)(ω)
∣∣∣2 dω =
=
(
k(α)
∞
)2
n∑
r,l=1
ArAlĒt
+(−iαr) · Ēt
+(−iαl)
1
2πi
+∞∫
−∞
i/(ω + iαr)
ω − iαl
dω =
= K(α)
∞
n∑
r,l=1
ArAl
αr + αl
Ēt
+(−iαr) · Ēt
+(−iαl),
1
2π
+∞∫
−∞
∣∣∣Qt
(k)(−)(ω)
∣∣∣2 dω =
=
(
k(k)
∞
)2
n∑
r,l=1
BrBlḡt
+(−ikr) · ḡt
+(−ikl)
1
2πi
+∞∫
−∞
i/(ω + ikr)
ω − ikl
dω =
= K(k)
∞
n∑
r,l=1
BrBl
kr + kl
ḡt
+(−ikr) · ḡt
+(−ikl),
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3
318 G. AMENDOLA, A. MANES
which, after using (9.14)2, (9.15)2, can be substituted into (8.29) and give
ψm(t) =
1
2
{
c
Θ0
ϑ2(t) +
1
ε
[εE(t) + ϑ(t)a]2 + µH2(t)
}
+
+
1
2
+∞∫
0
+∞∫
0
2
n∑
r,l=1
K(α)
∞
ArAl
αr + αl
e−(αrs1+αls2)Ēt(s1) · Ēt(s2) +
+
1
Θ0
n∑
r,l=1
K(k)
∞
BrBl
kr + kl
e−(krs1+kls2)ḡt(s1) · ḡt(s2)
ds1ds2.
This expression in the case where n = 1, taking account of (9.10)2,3, (9.11)2,3, assumes the
simpler form
ψm(t) =
1
2
{
c
Θ0
ϑ2(t) +
1
ε
[εE(t) + ϑ(t)a]2 + µH2(t)
}
+
+
1
2
α2
1g1
+∞∫
0
e−α1sĒt(s)ds
2
+
1
Θ0
k2
1h1
+∞∫
0
e−k1sḡt(s)ds
2
.
In this last case it is easy to derive by means of integrations by parts the results obtained in [1],
where the histories of E and g are considered instead of their integrated histories.
1. Amendola G., Manes A. Minimum free energy in linear thermoelectromagnetism // Quart. Appl. Math. —
2005. — 63, № 4. — P. 645 – 672.
2. Gentili G. Thermodynamic potentials for electromagnetic fields in the ionosphere // Int. J. Eng. Sci. — 1995.
— 33, № 11. — P. 1561 – 1575.
3. Gurtin M. E., Pipkin A. C. A general theory of heat conduction with finite wave speeds // Arch. Ration.
Mech. and Anal. — 1968. — 31. — P. 113 – 126.
4. Cattaneo C. Sulla conduzione del calore // Atti Semin. mat. e fis. Univ. Modena. — 1948. — 3. — P. 83 – 101.
5. Coleman B. D., Dill E. H. On the thermodynamics of electromagnetic fields in materials with memory //
Arch. Ration. Mech. and Anal. — 1971. — 41. — P. 132 – 162.
6. Coleman B. D., Dill E. H. Thermodynamic restrictions on the constitutive equations of electromagnetic
theory // ZAMP. — 1971. — 22. — P. 691 – 702.
7. Amendola G. Linear stability for a thermoelectromagnetic material with memory // Quart. Appl. Math. —
2002. — 59, № 1. — P. 67 – 84.
8. Golden J. M. Free energy in the frequency domain: the scalar case // Ibid. — 2000. — 58, № 1. — P. 127 – 150.
9. Gentili G. Maximum recoverable work, minimum free energy and state space in linear viscoelasticity // Ibid.
— 2002. — 60, № 1. — P. 153 – 182.
10. Deseri L., Gentili G., Golden J. M. On the minimal free energy and the Saint-Venant principle in linear
viscoelasticity // Math. Models and Methods for Smart Materials / Eds M. Fabrizio, B. Lazzari, A. Morro. —
Singapore: World Sci., 2002.
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3
MAXIMUM RECOVERABLE WORK IN LINEAR THERMOELECTROMAGNETISM 319
11. Fabrizio M., Golden J. M. Maximum and minimum free energies for a linear viscoelastic material // Quart.
Appl. Math. — 2002. — 60, № 2. — P. 341 – 381.
12. Fabrizio M., Giorgi C., Morro A. Free energies and dissipation properties for systems with memory // Arch.
Ration. Mech. and Anal. — 1994. — 125. — P. 341 – 373.
13. Coleman B. D., Owen D. R. A mathematical foundation of thermodynamics // Ibid. — 1974. — 54. — P. 1 –
104.
14. Fabrizio M., Morro A. Electromagnetism of continuous media. — Oxford Univ. Press, 2003.
15. Fabrizio M., Morro A. Thermodynamics of electromagnetic isothermal systems with memory // J. Nonequil.
Thermodyn. — 1997. — 22. — P. 110 – 128.
16. Fabrizio M., Morro A. Mathematical problems in linear viscoelasticity // SIAM Stud. Appl. Math. — 1992.
17. Amendola G. The minimum free energy for an electromagnetic conductor // Appl. Anal. — 2005. — 84, № 1.
— P. 67 – 87.
18. Muskhelishvili N. I. Singular integral equations. — Noordhoff: Groningen, 1953.
Received 25.05.2006
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 3
|
| id | nasplib_isofts_kiev_ua-123456789-178167 |
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| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T15:26:10Z |
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| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Amendola, G. Manes, A. 2021-02-18T07:46:50Z 2021-02-18T07:46:50Z 2006 Maximum recoverable work in linear thermoelectromagnetism / G. Amendola, A. Manes // Нелінійні коливання. — 2006. — Т. 9, № 3. — С. 287-319. — Бібліогр.: 18 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178167 517.9 We give a general closed expression for the minimum free energy in terms of Fourier-transformed quantities for a thermoelectromagnetic conductor with memory effects for the electric current density and the heat flux, when the integrated histories of the electric field and of the temperature gradient are chosen to characterize the states of the material. An equivalent formulation is derived and applied to the discrete spectrum model material response. Наведено загальний замкнений вираз для мiнiмальної вiльної енергiї в термiнах перетворення Фур’є для провiдника з ефектом запам’ятовування густини електричного струму та теплового потоку у випадку, коли для характеризацiї стану матерiалу розглядається iнтегральна iсторiя електричного струму та градiєнта температури. Отримано i застосовано еквiвалентне формулювання до моделi реакцiї матерiалу, що має дискретний спектр. The work is performed under the auspices of CNR and MIUR. en Інститут математики НАН України Нелінійні коливання Maximum recoverable work in linear thermoelectromagnetism Article published earlier |
| spellingShingle | Maximum recoverable work in linear thermoelectromagnetism Amendola, G. Manes, A. |
| title | Maximum recoverable work in linear thermoelectromagnetism |
| title_full | Maximum recoverable work in linear thermoelectromagnetism |
| title_fullStr | Maximum recoverable work in linear thermoelectromagnetism |
| title_full_unstemmed | Maximum recoverable work in linear thermoelectromagnetism |
| title_short | Maximum recoverable work in linear thermoelectromagnetism |
| title_sort | maximum recoverable work in linear thermoelectromagnetism |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/178167 |
| work_keys_str_mv | AT amendolag maximumrecoverableworkinlinearthermoelectromagnetism AT manesa maximumrecoverableworkinlinearthermoelectromagnetism |