An isothermal theory of free energies and free enthalpies, corresponding to linear constitutive relations with memory, is presented for isotropic non-magnetic materials. This is a second paper, following recent work on a general tensor theory of isothermal dielectrics and on the form of the minimum free energy. Both papers are based on continuum thermodynamics. For a standard choice of relaxation function, the minimum and maximum free energies are given explicitly, using a method previously developed in a mechanics context. Also, a new family of intermediate free energy functionals is derived for dielectrics. All these are solutions of a constrained optimization problem.

A general theory of free energies for dielectric materials under isothermal conditions was given recently [1]. The present work presents more detailed formulae and results for free energies in non-magnetic isotropic materials. We shall derive expressions for the minimum, maximum and other intermediate free energy functionals which also have an extremum property. Two approaches are possible: (1) To present the developments as in mechanics [2], based explicitly on thermodynamics; and (2) To use the method given independently [3-5] for dielectric materials with memory, but with quite diﬀerent methods, notation and terminology. In this work, as in [1], the fist approach will be adopted but with reference to the connection between the two methods. The intermediate free energies are analogous to quantities known in mechanics [2]. They are new and particularly physically relevant in the context of dielectrics, in that they apply to memory models that are standard for such materials, but not usually applied to viscoelastic behaviour.

Applications of the results of this work, from a physical point of view, include the establishment of bounds on the level of dissipation in the material, using the fact that the total dissipation associated with the minimum(maximum) free energy is an upper(lower) bound on the actual physical dissipation.

On the matter of notation, vectors and tensors are denoted by lowercase and uppercase boldface characters respectively and scalars by ordinary script. The real line is denoted by R, the non-negative reals by R+ and the strictly positive reals by R++. Similarly, R− is the set of non-positive reals and R−− the strictly negative reals. Complex quantities arise in the frequency domain so we have complex vector spaces for which the dot product involves using the complex conjugate of objects in the dual space. The magnitude squared, denoted by |.|2 refers to the dot product of objects with their complex conjugates.

Consider a rigid non-magnetic isotropic dielectric material subject to a varying electric field. Let the body under consideration occupy a volume B⊂R3. A typical point in B is x while t is a given time. The electric field on this region is E(x,t), with electric displacement denoted by D(x,t). The magnetic field and induction contributions are neglected. The space variables are generally omitted.

Let ψ(t) be any free energy of the material, for the isothermal case, and D(t) the rate of dissipation. Then the first and second laws of thermodynamics can be written as [1]

ψ˙+D=D..E, D≥0 (3.1)

Let Et:R+↦V= R3 be defined bya

Et(s) = E(t−s), s∈R+ (3.2)

aThis quantity would be denoted as in [3-5], while the future continuation, defined by (3.3), would be written as . The future continuation is referred to in [3] as the recovery field.

Equation (3.2) gives the history and also the current value E(t) of the electric field. A given future continuation is denoted by

Et(s) = E(t−s), s∈ R−− (3.3)

Let us define the free enthalpy [1] as

F= ψ−D.E (3.4)

In terms of this quantity, (3.1) becomes

F.+D = -D.E., D≥0 (3.5)

The relative history Etr is defined by

Etr(s) = Et(s)-E(t) (3.6)

A relative future continuation is also defined by (3.6) for s∈R−−

We define the equilibrium free enthalpy Fe(t) to be that given for the static history Et(s) = E(t), s∈R+.

Therefore,

Fe(t) = F˜e(E(t)) (3.7)

emphasizing that the function Fe(t) depends only on E(t).

A. Required properties of a free enthalpy

Let us state the characteristic properties of a free enthalpy, provable within a general framework [1,6-8]:

P1: ∂F(t)∂Λ(t) = −D(t) (3.8)

P2: For any history Eta and current value Ea(t)

F˜(Eta,Ea(t))≥F˜e(Ea(t)) (3.9)

P3: Condition (3.5) holds.

These will be referred to as the Graffi conditions by analogy with those for a free energy in mechanics ([2] for example).

A special case of the linear model introduced in [1] is described in this section. This is the material considered in [3-5], namely a passive, homogeneous isotropic non-magnetic dielectric. All kernels are scalar quantities. We have

F(t) = Fe(t)−12∫0∞∫0∞Etr(s).G12(s,u)Etr(u)dsdu (4.1a)

= Fe(t)−12∫0∞∫0∞E˙t(s).G˜(s,u)E˙t(u)dsdu (4.1b)

Fe(t) = -12G∞E(t).E(t) (4.1c)

where (4.1b) follows from (4.1a) by partial integrations. Also

E˙t(u) = ∂∂tEt(u) = -∂∂tEt(u) = -∂∂tEtr(u) (4.2)

G˜(s,u) = G(s,u)-G∞

and

∂2∂s∂uG(s,u) ≡ G12(s,u) (4.3a)

lims→∞G(s,u) = G∞, u∈R+ (4.3b)

lims→∞∂∂uG(s,u) = 0, u∈R+ (4.3c)

lims→∞∂∂sG(s,u) = 0, u∈R+ (4.3d)

G(s,u) = G(u,s) (4.3e)

with similar limits at large u holding for fixed s. The parameter G∞ in (4.1a) is defined by (4.3b). Relation (3.8) gives the linear constitutive relation

D(t) = G∞E(t)+∫0∞G′(u)Etr(u)du (4.4a)

= G0E(t)+∫0∞G′(u)Et(u)du (4.4b)

= G∞E(t)+∫0∞G˜(u)E˙t(u)du (4.4c)

= ∫0∞G(u)E˙t(u)du (4.4d)

where

G(u) = G(0,u) (4.5a)

G0 = G(0,0) = G(0) (4.5b)

G˜(u) = G(u)-G∞ (4.5c)

G′(u) = dduG(u) (4.5d)

The quantity G(s) is the relaxation function of the material. In writing the final form of (4.4), we are assuming that E(−∞) = Et(∞) vanishes; we furthermore assume that it goes to zero sufficiently strongly so that various integrals, introduced below, exist. The quantity G∞ is related to the relaxation function through

G∞ = G(s,∞) = G(0,∞) = G(∞) (4.6)

We deduce from (3.5), (4.1a) and the time derivative of (4.1b) that

D(t) = 12∫0∞∫0∞E˙t(s).[G1(s,u)+G2(s,u)]E˙t(u)dsdu (4.7a)

= 12∫0∞∫0∞Etr(s).[G112(s,u)+G212(s,u)]Etr(u)dsdu (4.7b)

where subscripts refer to partial diﬀerentiation with respect to the first or second argument. Equation (4.7b) follows from (4.7a) by partial integrations, using (4.1a), (4.3), and the fact that Etr(0) vanishes. Relation (4.1a), expressed in terms of histories, becomes

F(t) = 12G0E(t).E(t)-D(t).E(t) (4.8)

−12∫0∞∫0∞Et(s).G12(s,u)Et(u)dsdu

From (3.4), (4.2) and (4.8), we deduce that

ψ(t) = ϕ0(t)−∫0∞∫0∞Et(s).G12(s,u)Et(u)dsdu (4.9a)

= U(t)−12∫0∞∫0∞Etr(s).G12(s,u)Etr(u)dsdu (4.9b)

= U(t)−12∫0∞∫0∞E˙t(s).G˜(s,u)E˙t(u)dsdu (4.9c)

Where

ϕ0(t) = 12G0E(t).E(t), U(t) = D(t).E(t)−12G∞E(t).E(t) (4.10)

The integral terms, with the negative signs included, are non-negative [1]. The total work done by the electric field up to time t is

W(t) = ∫−∞tD˙(u).E(u)du (4.11a)

= D(t).E(t)−∫−∞tD(u).E˙(u)du (4.11b)

Under the assumption that ψ(−∞) = 0, an integration of (3.1) over (−∞,t] yields

ψ(t)+D(t) = W(t) (4.12)

Where

D(t) = ∫−∞tD(u)du≥0 (4.13)

is the total dissipation up to time t. We take G0 to be equal to the vacuum permittivity ∈0 [3-5], giving

D(t) = ∈0E(t)+P(t) (4.14)

where, using (4.4b), one sees that the polarization P(t) is given by the causal relationship

P(t) = ∫0∞G′(u)Et(u)du = ∫−∞tG′(t−s)E(s)ds (4.15)

Then, (4.11a) becomes

W(t)ϕ0(t)+Wint(t), ϕ0(t) = 12∈0|E(t)|2 (4.16)

where ϕ0(t) is the quantity introduced in (4.10) and Wint(t) is given by

Wint(t) = ∫−∞tP˙(s).E(s)ds (4.17)

is the accumulation of energy (density) transferred to the medium at the point under consideration, from the beginning of the pulse-medium interaction until time t. We can write (4.12) as

ψint(t)+D(t) = Wint(t) (4.18)

where ψint(t) is given by the integral term in (4.9a).

A. Minimal states

The fundamental definition of the state of a material with memory at time t is the history of the independent field variable and its current value (Et, E(t)). Also, diﬀerent histories may be members of the same minimal state [1]. Two states (Et1,E1(t)), (Et2,E2(t)) are equivalent, or in the same minimal state if from a time t onwards, we have

D1(t+s) = D2(t+s) if E1(t+s) = E2(t+s), so s≥0 (4.19)

Where D1, D2 are defined by (4.4) for these states. Then it follows that

∫0∞G′(s+u)(Et1(u)−Et2(u))du = 0, s≥0 (4.20)

A fundamental distinction between materials is that for certain relaxation functions, namely those with only isolated singularities (in the frequency domain), the set of minimal states is non-singleton, while if some branch cuts are present in the relaxation function, the material has only singleton minimal states [2,9]. For relaxation functions with only isolated singularities, there is a maximum free energy that is less than the work function W(t) and also a range of related intermediate free energies, which are discussed later. On the other hand, if branch cuts are present, the maximum free energy is W(t).

In this work, we will deal exclusively with the case where the relaxation function has only isolated singularities, so that the minimal states will be non-singleton.

For such materials, the free energy functional is positive semi-definite ([2], page 152).

Note that the statement that (Et1,E1(t)) and (Et2,E2(t)) are equivalent is the same as the assertion that (Etd,0) is equivalent to the zero state (0,0), where 0 is the zero in R3 (and also the zero history), while

Etd(s) = Et1(s)−Et2(s) (4.21)

A functional of (Et, E(t)) which yields the same value for all members of the same minimal state will be referred to as a functional of the minimal state (abbreviated to FMS) or as a minimal state variable.

Let (Et1,E1(t)), (Et2,E2(t)) be any equivalent states. Then, a free energy is a functional of the minimal state if

ψ(Et1,E1(t)) = ψ(Et2,E2(t)) It is not necessary that a free energy have this property, though it holds for the minimum and all related free energies introduced later.

For any f∈L2(R), we denote its Fourier transform fF∈L2(R) byb fF(ω) = ∫−∞∞f(ξ)e−iωξdξ (5.1a) = f+(ω)+f−(ω) (5.1b) f+(ω) = ∫0∞f(ξ)e−iωξdξ (5.1c) f−(ω) = ∫−∞∞f(ξ)e−iωξdξ (5.1d) bThe quantity fF(ω), defined by (5.1), would be denoted in [3-5] by 2π−−√fˆ(−ω) or 2π−−√fˆ¯¯¯(ω) for real ω. Only real valued functions will be considered so that f¯F(ω) = fF(−ω), ω∈R (5.2)

Where the bar denotes the complex conjugate. For complex values of ω, (5.2) becomes fF(ω)¯¯¯¯¯¯¯¯¯ = fF(−ω¯¯) (5.3) Functions defined on R+ are identified with functions on R which vanish identically on R−−. For such functions, fF(ω) = f+(ω) = fc(ω)−ifs(ω) (5.4) Where fc(ω), fs(ω) are respectively the Fourier cosine and sine transforms. A property of Fourier transforms which will be used later is f+(ω)−→−ω→∞f(0)iω, so that fs(ω)−→−ω→∞f(0)ω (5.5) If f(0) is non-zero. A. The kernel G(u) We can write the Fourier transforms of G′(u) and G˜(u) in (4.4) as G′+(ω) = G′c(ω)−iG′s(ω) = χ+(ω) = χc(ω)−iχs(ω) (5.6) G˜+(ω) = G˜c(ω)−iG˜s(ω)

The quantity χ+(ω) is the susceptibility, denoted by χ(ω) in [3- 5]. By partial integration, one can show that G′+(ω) = −(G0−G∞)+iωG˜+(ω) (5.7) giving, in particular, that G˜s(ω) = χs(ω) = −ωG˜c(ω) (5.8) Observe that G′+(0) = ∫0∞G′(s)ds = G∞−G0 (5.9) Using (5.2), we see that G+¯¯¯¯¯′(ω) = χ+¯¯¯¯(ω) = G′+(−ω) = χ+(−ω) (5.10) which will be useful later. By considering periodic behavior in Et(s), we can show that a consequence of the second law is that ([2], page 140) G′s(ω) = χs(ω) = −ωG˜c(ω) > 0, 0 < ω < ∞ (5.11) It follows from (5.11) that G∞ > G0 [1]. We have G0 = ∈0>0, so that G∞>0

B. The complex frequency plane and the function H(ω)

We will be considering frequency domain quantities, defined by analytic continuation from integral definitions, as functions on the complex ω plane, denoted by Ω, where

Ω+ = {ω∈Ω|Im[ω]∈R+} Ω(+) = {ω∈Ω|Im[ω]∈R++}

Similarly, Ω− and Ω(−) are the lower half-planes including and excluding the real axis, respectively.

The quantities f±(ω), defined by (5.1), are analytic in Ω(∓) respectively ([2], page 547). Thus, the quantity G′+(ω) is analytic on Ω(−). It will be assumed that G′+(ω) is analytic on R and thus on Ω−, or more precisely, on an open set containing Ω−. Also, it will be taken to be analytic at infinity. This function is defined by analytic continuation in regions of Ω+ where the Fourier integral does not converge. The quantity G′s(ω) has singularities in both Ω(+) and Ω(−) that are mirror images of each other. It goes to zero at the origin and must also be analytic there. A quantity central to our considerations is defined by

H(ω) = ωG′s(ω) = ωχs(ω) = −ω2G˜c(ω) (5.13)

It is a non-negative, even function of the frequency and goes to zero quadratically at the origin. The relation (see (5.5))

ilimω→∞ωG′+(ω) = limω→∞ωG′s(ω) = G′(0) (5.14)

yields

G′(0) = H(∞) = H∞ (5.15)

For the model considered later, H(ω) goes to zero at large ω so that H∞ = 0. The Fourier transforms of the history and continuation are denoted by Et+(ω) and Et−(ω) respectivelyc. These are particular examples of (5.1c) and (5.1d). The quantity Et+(ω) is analytic on Ω(−) and Et−(ω) is analytic on Ω(+). Both are assumed to be analytic on an open set including R. It is further assumed that they are analytic at infinity.

cReferring to footnote a, we see that Etr±(ω) would be denoted in [3-5] by 2π−−√Eˆtr∓(−ω).

The Fourier transforms of the relative history and continuation, defined by (3.6), have the form

Etr±(ω) = Et±(ω)∓E(t)iω∓ (5.16)

where the notation ω± is defined as ω ± iα, α > 0. The parameter α is assumed to tend to zero after any integrations have been carried out (see for example [2], page 551). We have

E˙t+(ω) = ddtEt+(ω) = -iωEt+(ω)+E(t) = -iωEtr+(ω) (5.17)

The second relation follows from (3.2) and an integration by parts in the Fourier integral defining Et+(ω). Also, based on arguments from [1] and [2], page 146, for example, we can express the constitutive equation (4.4) in the form

D(t) = G∞E(t)+iπ∫−∞∞H(ω)ωEtr+(ω)dω = G∞E(t)−1π∫−∞∞H(ω)ω2E˙tr+(ω)dω (5.18) = G0E(t)+iπ∫−∞∞H(ω)ωEt+(ω)dω

Various forms for the work function, defined by (4.11), are given for the general tensor case in [1]. Those required in this work for scalar kernels are summarized here. The analogous mechanics version of these relations may be found in [2], page 153, for example. It can be shown that

W(t) = U(t)−12∫0∞∫0∞G12(|u−s|)Etr(u).Etr(s)dsdu = ϕ0(t)−12∫0∞∫0∞G12(|u−s|)Et(u).Et(s)dsdu (6.1) where U(t) and ϕ0(t) are defined by (4.10). We see from (6.1) that W(t) can be cast in the form (4.9) by putting G12(s,u) = G12(|u−s|) (6.2) In terms of frequency domain quantities, we find that [1] W(t) = U(t)−12π∫−∞∞E¯¯¯tr+(ω).H(ω)Etr+(ω)dω = ϕ0(t)+12π∫−∞∞E¯¯¯t+(ω).H(ω)Et+(ω)dω (6.3)

The quantity W(t)-D(t)•E(t) can be shown to obey the properties specified in subsection ⅢA of a free enthalpy, with zero dissipation. Because of the vanishing dissipation, it must be the maximum free energy associated with the material or greater than this quantity, an observation which follows from (4.12). For relaxation functions with only isolated singu-larities, as introduced later, there is a maximum free energy which is a functional of the minimum state and is less that W(t).

Consider the scalar product on the function space of electric fields, defined by [2,5,9-12] (Et1,Et2) = (Et1F,Et2F) = ∫−∞∞∫−∞∞Et1(s).G(|s−u|)Et2(u)dsdu (6.4a) = 12π∫−∞∞E¯¯¯t1F(ω).H(ω)Et2F(ω)dω = (Et2,Et1) = (Et2F,Et1F) (6.4b)

Recalling (5.1), we see that the quantity EtF(ω) is the Fourier transform of Et(s),s∈R in the time domain. Also, the subscript F in the bracket notation indicates that the frequency domain version is being used. The related norm is

(Et,Et) = ∥Et∥2 = ∥EtF∥2 (6.5) In the notation of (6.5), the work function is given by W(t) = U(t)+∥Etr∥2 (6.6a) = ϕ0(t)+∥Et∥2 (6.6b) where Etr(s) = 0, s∈R−− in (6.6a) and Et(s) = 0, s∈R−− in (6.6b).

We consider materials such that G′+(ω) (or G˜(ω)+) has only a finite number of isolated singularities in Ω(+). Thus, H(ω) has only isolated singularities in Ω(±), which are mirror images of each other in the real axis, as ascribed to G′s(ω) before (5.13)). This means that it can be put in the form of a ratio of polynomials. We will take the singularities to be a finite number of simple poles. The quantity H(ω) has a finite number of zeros in Ω±(ω), also mirror images of each other. It is real and non-negative on R, an even function of ω and therefore a function of ω2, in view of its analyticity about the origin. It vanishes quadratically at the origin. This non-negative quantity can be factorized in general as outlined in [2,13]. Thus, we have that

H+(ω)H−(ω) = H(ω) (7.1)

where

H±(ω) = H∓(−ω) = H∓¯¯¯¯¯(ω) (7.2a) H(ω) = |H±(ω)|2, ω∈R (7.2b)

The factorization (7.1) is unique up to multiplication by a phase factor eiα, where α is a constant. Relation (7.2a) reduces this arbitrariness to multiplication by a factor ±1.

The factorization (7.1) is equivalent to a homogeneous Riemann-Hilbert problem on the half-plane [1,3-5].

The quantity H+(ω) (H−(ω)) has all its singularities in Ω(+) (Ω(−)) and all its zeros in Ω+ (Ω−) ([2], page 239). There are many other factorizations, obtained by interchanging some or all of the zeros of H+(ω) and H−(ω), while retaining the same singularity structure. The diﬀerent factorizations are labeled by the subscript or superscript f. We have

H(ω) = Hf+(ω)Hf−(ω) (7.3a) Hf±(ω) = Hf±(−ω) = Hf∓¯¯¯¯¯¯(ω) (7.3b)

Certain types of exchanges must be excluded to ensure that (7.3b) are true [11], ([2], page 338). Each factorization generally yields a diﬀerent free energy, though there may be exceptions. All these free energies are FMSs [2]. The factorization with no exchange of zeros, which is that given by (7.1), yields the minimum free energy Ψm(t).

Remark Ⅶ.1.Each exchange of zeros, starting from these factors, can be shown to yield a free energy which is greater than or equal to the previous quantity ([2], page 363).

Note that the zeros of Hf±(ω) at the origin play no part in these exchanges.

Remark Ⅶ.2. A particularly interesting choice of ψf(t), which we denote by ψM(t), is obtained by interchanging all the zeros. This can be identified as the maximum free energy among all those that are FMSsd. It is less than the work function, which is not a FMS for materials with only isolated singularities [2].

dThe minimum and maximum free energies are given also in [5]. The intermediate free energies, introduced in this work, have not been discussed previously in the context of dielectrics, but are analogous to those derived in mechanics in for example [2],10] -12]. If there are N diﬀerent factorizations of H(ω), then f = N is chosen to denote the maximum free energy.

Note that there are several (indeed many, for a large number of isolated singularities) diﬀerent zero exchange pathways leading from the minimum to the maximum free energy.

The most general free energy and rate of dissipation arising from these factorizations is given by

ψ(t) = ∑f=1Nλfψf(t), D(t) = ∑f=1NλfDf(t), ∑f=1Nλf = 1, λf≥0 (7.4)

A particular case of this linear form is the free energy proposed in [11].

Remark Ⅶ.3. The set of all free energies at time t associated with a given material is denoted by Φ(t). The boundary of this set is where one or more of the fundamental properties listed in subsection ⅢA is breaking down; an example, which will arise later, would be where, for a given non-zero history, the rate of dissipation is zero, so that a small shift in its kernel parameters results in it becoming negative.

All the ψf(t) emerge from extremum arguments. This is apparent in the case of the minimum free energy for a given state, which is obtained by finding an optimal continuation Etm(s), s∈R− yielding the maximum recoverable work from this state. Also, the maximum free energy is determined by finding the optimal history EtM(s), s∈R+ which minimizes the work done to achieve the given state. It must be an equivalent state to the given history. In other words, the two histories must be in the same minimal state. The intermediate free energies are also obtained from an extremum principle, but involving an optimization of the history/continuation Et(s), s∈R, as we shall see in section Ⅷ.

The relevant theoretical developments motivating the results of section Ⅷ are presented in [2], chapter 15 for mechanics; they also apply to dielectrics. We will present an abbreviated version of these arguments, and also demonstrate that the functionals which emerge are in fact free energies, with the required properties. It can be shown that all these quantities, including the intermediate functionals, are on the boundary of Φ(t) for the material ([2], page 365), which is consistent with the fact that they emerge from an extremum principle. This property is discussed briefly below.

Let us define the quantities EtR(s) and Etd(s) by the relation

EtR(s) = Et(s)+Etd(s), s∈R (8.1)

where Et(s) is the given history, which is zero for s∈R−−. The quantity EtR is typically non-zero on R and Etd follows from (8.1). In the frequency domain, the relation becomes

EtRF(ω) ≡ EtR+(ω)+EtR−(ω) = Et+(ω)+EtdF(ω), ω∈R (8.2)

where the quantities EtR±(ω) are particular cases of (5.1c) and (5.1d). Consider the norm ∥ER∥2 = ∥ERF∥2, defined by (6.5). This can be written in a form similar to the quantity W(∞) in [1], from which the minimum free energy was deduced. We have, from (6.4b) and (6.5), that ∥∥EtRF∥∥2 = 12π∫−∞∞[Et+¯¯¯¯¯(ω)+EtdF¯¯¯¯¯¯(ω)].H(ω)[Et+(ω)+EtdF(ω)]dω (8.3) Let us choose a particular factorization of H(ω), as given by (7.3). Also, we define [1] Hf−(ω)Et+(ω)−qft−(ω)−qft+(ω) (8.4a) qft±(ω) = 12πi∫−∞∞Hf−(ω′)Et+(ω′)ω′−ω∓dω′ (8.4b)

where the qft±(ω) are analytic in Ω∓, respectively. The singularities of qft−(ω) are the same as those of Hf−(ω), as may be perceived by closing the contour in (8.4b) on Ω(−). Singularities on the real axis are excluded by assumption. Using the factorization in (8.4), we can express ∥ERF∥2 in the form

∥∥EtRF∥∥2 = 12π∫−∞∞∣∣qft−(ω)−qft+(ω)+Hf−(ω)EtdF(ω)∣∣2dω (8.5)

Let us further define

Hf−(ω)EtdF(ω) = uft−(ω)−uft+(ω) uft±(ω) = 12πi∫−∞∞Hf−(ω′)EtdF(ω′)ω′−ω∓dω′ (8.6) Note that, by virtue of Cauchy's integral formula, uft−(ω) = qftd−(ω)+Hf−(ω)Etd−(ω) (8.7a) = qftR−(ω)−qft−(ω)+Hf−(ω)Etd−(ω) (8.7b) uft+(ω) = qftd+(ω) = qftR+(ω)−qft+(ω) (8.7c) Where qftd±(ω) are the quantities defined by (8.4b) but with Et+(ω′) replaced by Etd+(ω′) = EtR+(ω′)−Et+(ω′) Also, qftR±(ω) = 12πi∫−∞∞Hf−(ω′)EtR+(ω′)ω′−ω∓dω′ (8.8)

The optimization problem which yields the free energy functional associated with a given factorization of H(ω) is given as follows ([2], pages 346, 350). We minimize ∥∥EtRF∥∥2, subject to the constraint

uft−(ω) = 12πi∫−∞∞Hf−(ω′)EtdF(ω′)ω′−ω+dω′ = 0, (8.9) The resultant free energy is ψf(t) = ϕ0(t)+∥∥Etf∥∥2, (8.10)

where Etf is the history/continuation EtRF that is the solution of the constrained minimization problem. The formulation of this problem is discussed at length in [2], chapter 15 and earlier papers. In particular, it is shown that each ψf(t) has all the required properties of a free energy, which will in any case be shown later for the present context. Using (8.6), (8.7c) and (8.9) in (8.5) together with proposition 1 of [1], we find that Hf−(ω)EtdF(ω) = qft+(ω)−qftR+(ω), (8.11a) ∥∥EtRF∥∥2 = 12π∫−∞∞∣∣qft−(ω)−qftR+(ω)∣∣2dω, (8.11b) = 12π∫−∞∞[∣∣qft−(ω)∣∣2+∣∣qftR+(ω)∣∣2]dω, (8.11c)

where qftR+(ω) is given by (8.8). Since qftR−(ω) depends only on the given history Et+(ω), the solution to the minimization problem is obtained by choosing EtRF(ω) = Etf(ω) to give that

qftR+(ω) = 0, (8.12) yielding ∥∥Etf∥∥2 = 12π∫−∞∞∣∣qft−(ω)∣∣2dω, (8.13) From (8.11a) and (8.12), we obtain that

Hf−(ω)EtdF(ω) = Hf−(ω)EtRF(ω)−Hf−(ω)Et+(ω) = qft+(ω).(8.14)

Then, (8.4a) gives

Hf−(ω)EtRF(ω) = Hf−(ω)Etf(ω) = qft−(ω), (8.15)

and

Etf(ω) = qft−(ω)Hf−(ω) = 12πiHf−(ω)∫−∞∞Hf−(ω′)Et+(ω′)ω′−ω+dω′. (8.16)

The observation after (8.4) about the singularities of qft−(ω) and Hf−(ω) yield that the singularity structure of Etf(ω) is determined by the zeros of Hf−(ω).

Remark Ⅷ.1. Note that for f≠1, N, the quantity Hf−(ω) has some of its zeros in Ω− and some in Ω+. For f = 1, all the zeros of Hf−(ω) are in Ω−, so that Etf(ω) has singularities only in Ω− and its inverse Fourier transform is thus non-zero only on R−. This latter quantity is the negative of the optimal continuation, yielding the minimum free energy, which is discussed for the general tensor case in [1]. It can be shown that both the constraint (8.9) and the minimization of ∥∥EtRF∥∥2 result in the same relation, namely (8.16) for f = 1.

For f = N, all the zeros of Hf−(ω) are in Ω+, which gives that Etf(ω) has singularities only in Ω+. Therefore, its inverse Fourier transform is non-zero only on R+ and is the optimal history yielding the maximum free energy. In this case, the quantity ∥∥EtRF∥∥2 to be minimized is the work function and (8.9) is the condition that the optimum history and the given history are in the same minimal state.

For intermediate cases, Etf(ω) is non-zero on R and minimizes ∥∥EtRF∥∥2, subject to the constraints (8.9), for various factorizations. These constraints are transitional between the two extremes discussed above.

We will see that all of these cases result in free energies that are on the boundary of Φ(t), defined by remark Ⅶ.3. This property is apparent for the minimum and maximum free energies. For these and all intermediate cases, we will see that the corresponding rates of dissipation are positive semi-definite rather than positive definite quadratic forms. They are zero for certain non-zero histories so that small variations in their kernels may lead to negative rates of dissipation, for these histories, in contradiction to the second law, as given by (3.1).

Thus, the fundamental physical content of the constrained optimizations is that they lead to free energies on the boundary of Φ(t).

The free energy (8.10) has the form ψf(t) = ϕ0(t)+12π∫−∞∞∣∣qft−(ω)∣∣2dω (8.17a) = ϕ0(t)+12π∫−∞∞Etf¯¯¯¯(ω).H(ω)Etf(ω)dω,(8.17b) by virtue of (8.13) and (8.15). We have the following relations ([1,10,13,14] for example) ddtqft+(ω) = −iωqft+(ω)−Kf(t), (8.18a) ddtqft−(ω) = −iωqft−(ω)−Kf(t)+Hf−(ω)E(t), (8.18b) Kf(t) = 12π∫−∞∞Hf−(ω)Etr+(ω)dω,(8.18c) = −12πi∫−∞∞Hf−(ω)ωE˙t+(ω)dω (8.18d) and lim|ω|→∞ωqft+(ω) = iKf(t), (8.19a) lim|ω|→∞ωqft−(ω) = i(Kf(t)−HsrE(t)), (8.19b) 12π∫−∞∞qft+(±ω)dω = −12Kf(t), (8.19c) 12π∫−∞∞qft+(±ω)dω= −12(Kf(t)−HsrE(t)). (8.19d)

Remark Ⅷ.2. For the material considered in section Ⅸ, the quantity H(ω) vanishes for large values of ω. Thus, Hsr in (8.18) and (8.19) is zero. Also, , (8.20) ∫−∞∞Hf−(ω)ω = 0, (8.20) as can be seen by closing the contour on Ω(+), so that, recalling (5.16), it is clear that (8.18c) is replaced by Kf(t) = 12π∫−∞∞Hf−(ω)Et+(ω)dω (8.21) The work function, given by (6.3), takes the form [1] W(t) = ϕ0(t)+12π∫−∞∞[∣∣qft−(ω)∣∣2+∣∣qft+(ω)∣∣2]dω (8.22) From (4.12), (8.17) and (8.22), we deduce that the total dissipation corresponding to the minimum free energy is given by Df(t) = ∫−∞tDf(u)du = 12π∫−∞∞∣∣qft+(ω)∣∣2dω (8.23) Differentiating this relation with respect to t and using (8.18a), (8.19c), gives Df(t) = |Kf(t)|2 (8.24)

Since Kf(t), given by (8.21) can vanish for non-zero histories, Df(t) is a positive semi-definite rather than a positive definite quadratic form, as noted in remark Ⅷ.1.

We can re-express these results in terms of relative histories [1]. Let us put Pft(ω) = Hf−(ω)Eftr+(ω) = pft−(ω)−pft+(ω) pft±(ω) = 12πi∫−∞∞Pft(ω′)ω′−ω∓dω′ = qft±(ω)+12π∫−∞∞Hf−(ω′)ω′(ω′−ω∓)dω′E(t) (8.25) By closing the contour on Ω+, it emerges that pft−(ω) = qft−(ω)−Hf−(ω)iωE(t) (8.26a) pft+(ω) = qft+(ω) (8.26b) Also, the work function, given by (8.22), becomes W(t) = U(t)+12π∫−∞∞[∣∣pft−(ω)∣∣2+∣∣pft+(ω)∣∣2] dω (8.27) while (8.17b) and (8.23) are given by ψf(t) = U(t)+12π∫−∞∞∣∣pft−(ω)∣∣2dω (8.28a) Df(t) = 12π∫−∞∞∣∣pft+(ω)∣∣2dω (8.28b) Equation (8.24) is unchanged.

We now briefly present double integral forms of Ψf(t), Df(t) and D(t), which are analogous to (though simpler than) the corresponding formulae in [1] for the minimum free energy in the full tensor case. Using (8.17b) and (8.28a), we can write Ψf(t) in the forms

ψf(t) = ϕ0(t)+i4π2∫−∞∞∫−∞∞Et+¯¯¯¯¯(ω).Mf(ω1,ω2)Et+(ω2)ω+1−ω−2dω1dω2 (8.29a) = U(t)+i4π2∫−∞∞∫−∞∞Etr+¯¯¯¯¯¯(ω1).Mf(ω1,ω2)Etr+(ω2)ω+1−ω−2dω1dω2 (8.29b) Mf(ω1,ω2) = Hf+(ω1)Hf−(ω2) (8.29c) Also, Df(t), given by (8.24), can be expressed as Df(t) = i4π2∫−∞∞∫−∞∞Etr+¯¯¯¯¯¯(ω1).Mf(ω1,ω2)Etr+(ω2)dω1dω2 (8.30) From (8.23) and (8.28b), we deduce that Df(t) = i4π2∫−∞∞∫−∞∞Et+¯¯¯¯¯(ω1).Mf(ω1,ω2)Et+(ω2)ω−1−ω+2dω1dω2 = −i4π2∫−∞∞∫−∞∞Etr+¯¯¯¯¯¯(ω1).Mf(ω1,ω2)Etr+(ω2)ω−1−ω+2dω1dω2 (8.31) One finds that ∫−∞∞∫−∞∞Et+¯¯¯¯¯(ω1).Mf(ω1,ω2)Et+(ω2)ω+1−ω−2dω1dω2 = 0 (8.32)

by integrating over ω2 for example and closing the contour on Ω−, since Hf+(ω) and Ef+(ω) have no singularity in the lower half plane. This allows us to write (8.29a) in the explicitly convergent form

ψf(t) = ϕ0(t)+i4π2∫−∞∞∫−∞∞Et+¯¯¯¯¯(ω1).[Mf(ω1,ω2)−Mf(ω2,ω1)]Et+(ω2)ω1−ω2dω1dω2 (8.33) The fact that the integral term is non-negative implies that [1] △f(ω) = i([Hf+(ω)]′Hf−(ω)−Hf+(ω)[Hf−(ω)]′)≥0, ω∈R (8.34) By invoking (5.17), we see that the quantity Etr+(ω) may be replaced by iE˙t+(ω)ω/ in the above double integral expressions. The free enthalpy Ff(t) corresponding to Ψf(t) may be deduced from (3.4) to be Ff(t) = ψf(t)−D(t).E(t) (8.35a) = −S(t)+12π∫−∞∞∣∣qft−(ω)∣∣2dω (8.35b) = Fe(t)+12π∫−∞∞∣∣Pft−(ω)∣∣2dω (8.35c) where

S(t) = D(t).E(t)−ϕ0(t) (8.36)

and Fe(t) is given by (4.1c). It is easy to show that Ff(t) obeys the Graﬃ conditions listed in subsection ⅢA. Property P2 is immediately apparent. The relation (4.12) holds by virtue of (8.17), (8.22) and (8.23). The time derivative of (4.12) yields (3.1) with Df(t) given by (8.24), which is equivalent to P3. Property P1 can be proved with the aid of (8.35b), by showing that

∂S(t)∂E(t) = D(t) (8.37)

Remark Ⅷ.3. It was shown in [14] (see also [2], page 340) that qft−(ω), defined by (8.4), is a function of the minimal state in the sense defined after (4.21). This result transfers to the present context without alteration. From (8.17), we deduce that Ψf(t) is a function of the minimal state as defined by (4.22).

For the minimum free energy Ψm(t), we denote Ψint(t), introduced in (4.18), as Ψrec(t), given by

ψrec(t) = 12π∫−∞∞|qt−(ω)|2dω (8.38)

where qt−(ω) is defined by (8.4) for f = 1. This is the recoverable energy defined in [3]. The quantity Dm(t), given by (8.23) for f = 1, is that referred to as the irrecoverable energy in [3]. Corresponding quantities ψf(t)−ϕ0(t) and Df(t), for each f, may be similarly labeled.

The optimal history/continuation, given by the inverse transform of (8.1), can be shown to have a discontinuity at the origin which is infinite if Hsr = 0, while at t = ±∞, it is non-zero ([1,2], page 354).

We have H(ω) = ωχs(ω) = −ω2i(χ+(ω)−χ+¯¯¯¯(ω))≥0, ω∈R (9.1) by virtue of (5.11). As in [3-5], we take the susceptibility to be modeled by a sum of Lorentz oscillatorse eThis behavior, while often used to describe memory behavior of dielectrics, is not usually applied to viscoelastic materials. The singularity structure given by (9.4) corresponds to exponential decay with sinusoidal behavior in the time domain, while viscoelastic materials are generally modeled by simple exponential decay, which in the frequency domain yields simple poles on the positive imaginary axis. χ+(ω) = ∑n=1Nfnω2pnω2n−iγnω−ω2 (9.2)

in terms of the oscillator strength fn, the plasma frequency ωpn, the resonant frequency ωn and the damping rate γn of each oscillator. All of these are positive parameters. It is assumed that ω2n>γ2n4 (9.3) The singularities of χ+(ω) are of course isolated. They are simple poles at ζn=12[iγn+σn] , ζ′n = 12[iγn−σn] = −ζ¯n (9.4) σn = 4ω2n−γ2n−−−−−−−√, n=1,2.....,N.

Thus, they occur in pairs, equidistant from the positive imaginary axis, as required by (5.3). Separating the poles in (9.2), we can write χ+(ω) = ∑n = 1Nfnω2pnσn{1ω−ζn−1ω−ζ′n} (9.5) In the time domain, this translates into G′(s) = i∑n = 1Nfnω2pnσn(eiζns−eiζ′s), s∈R+ (9.6)

which are decaying exponentials multiplying sine functions [15], so that there will be oscillatory behavior superimposed on the exponential decay. From (9.2) and (9.1), we have

H(ω) = ∑n = 1Nfnω2pnγnω2(ω2n−ω2)2+γ2nω2 (9.7) which can be written as a ratio of two (factored) polynomials: H(ω) = H1ω2∏N−1l = 1[(ω−ηl)(ω−η′l)(ω−η¯l)(ω−η¯′l)]∏Nl = 1[(ω−ζl)(ω−ζ′l)(ω−ζ¯l)(ω−ζ¯′l)] (9.8a) H1 = ∑n=1Nfnω2pnγn , ω∈R (9.8b)

where the denominator of (9.8a) uses the notation of (9.4) while the numerator is factored to yield the zeros of H(ω); these must occur also in pairs, as in (9.4), so that η′l=−η¯l for each l. We have explicitly included the fact that H(ω) vanishes quadratically at the origin. Note that the smaller number of zeros reflects the fact that H(ω) behaves as ω−2 for large ω.

The factorization can be carried out by inspection. We obtain H−(ω) = h1∏N−1l = 1(ω−η¯l)(ω−η¯′l)∏Nl = 1(ω−ζ¯l)(ω−ζ¯′l) ζ¯l∈Ω(−), l = 1, 2, ..., η¯l∈Ω(−),l = 1, 2, ...N-1 (9.9) H+(ω) = H¯¯¯−(ω) = H−(−ω)

Where h21 = H1

The most general case of a rational function, which is considered in [2,11], can be obtained from (9.8) by allowing singularities to coalesce. As a result of this, some zeros may also coalesce. Indeed, we cannot exclude the possibility that some of the zeros in (9.8) have a power higher than unity, even if all the singularities are simple poles. For simplicity, it is assumed that this does not happen for our choice of parameters.

Let us define for l = 1, 2, . . ., 2N −2, αl = ⎧⎩⎨ζl+1¯¯¯¯¯¯¯2,ζ′l2,¯¯¯¯¯¯l oddl even, βl = ⎧⎩⎨⎪⎪⎪⎪ηl+1¯¯¯¯¯¯¯η′l2¯¯¯¯¯¯,η′l2¯¯¯¯¯,l oddl even (9.10)

and α2N−1 = ζN¯¯¯¯, α2N =ζ′N¯¯¯¯¯ (9.11) We can write (9.5) as χ+(ω) = ∑l = 12Nglω−α1¯¯¯¯ = −∑l = 12Nglω+α1, αl¯¯¯∈Ω(+) (9.12) where (5.10) has been used and gl = ⎧⎩⎨⎪⎪−fiω2piσi,fiω2piσi,i = l+12, l odd,i = l2, l even. (9.13) In this notation, (9.6) becomes G′(s) = i∑l = 12Ngleiαl¯¯¯s (9.14a) = −i∑l = 12Ngle−iαls (9.14b) Using (9.14b) in (4.20), we see that two electric field histories are in the same minimal state if and only if

Et1+(α1) = Et2+(α1), l=1,2,...,2N (9.15) Thus, a minimal state is defined by the quantities (e1,e1,...e2N), where el(t) = Et+(α1), l = 1,2,...,2N (9.16) Relation (5.3) gives el¯¯¯(t) = Et+(αl)¯¯¯¯¯¯¯¯¯¯¯ = Et+(−αl¯¯¯), l = 1,2,...,2N (9.17) Formula (5.17) for ω = α1 yields e˙l(t) = −iαlel(t)+E(t) (9.18) Equation (4.4b) can be written as D(t) = G0E(t)−i∑l = 12Nglel(t) = G0E(t)+i∑l = 12Nglel¯¯¯(t) (9.19) with the aid of (9.14), so that

D˙(t) = G0E˙(t)−i∑l = 12Ngle˙l(t) (9.20)

If a free energy or free enthalpy is to depend only on the el(t), l = 1,2,...,2N, then the most general forms of G˜(s,u) and G12(s,u) in (4.2) and (4.9) are

G˜(s,u) = −∑i,j = 12NCijeiαi¯¯¯¯s−iαju (9.21) G12(s,u) = −∑i,j = 12NC′ijeiαi¯¯¯¯s−iαju, C′ij = Cijαi¯¯¯αj where C and C' are non-negative hermitian matrices. Then, (4.9) gives that ψ(t) = ϕ0(t)+12∑i,j = 12NC′ijei¯¯¯ . ej = U(t)+12∑i,j = 12NCije¯˙i . e˙j (9.22) Relation (4.5a) gives ∑i = 12NCij = −gjαj, j = 1, 2,..., 2N (9.23)

Taking the complex conjugate of this relation yields

∑j = 12NCij = −giαi¯¯¯¯, i=1, 2,..., 2N (9.24) Equation (3.1) is satisfied if D(t) = 12∑i,j = 12NΓije¯˙i.e˙j (9.25) Γij = −i(α¯¯i−αj)Cij, i, j= 1, 2, ..., 2N The matrix with components Γij must also be hermitian and non-negative. Let us now consider the free energies Ψf(t). We can put (9.9) in the form H−(ω) = h1ω∏2N−2l = 1(ω−βl)∏2Nl = 1(ω−αl) (9.26a) = h1ω∑i = 12NRiω−αi (9.26b) αl∈Ω(−), l = 1, 2, ..., 2N,βl∈Ω(−), l = 1, 2, ..., 2N-2 where Ri = (ω−αi)H−(ω)∣∣∣∣∣ω = αi = h1αi∏2N−2l = 1(αi−βl)∏2Nl = 1l ≠ i(αi−αl) (9.27) Also,

H+(ω) = h1ω∏2N−2l = 1(ω−βl¯¯¯)∏2Nl = 1(ω−αl¯¯¯) (9.28a) = −h1ω∏2N−2l = 1(ω+γl)∏2Nl = 1(ω−αl) (9.28b) = h1ω∑i = 12NR¯¯¯iω−αi¯¯¯ (9.28c) = h1ω∑i = 12NRiω+αi (9.28d) where (7.3) has been used. The relationship ∑i = 12NRi = 0 (9.29) must hold since H−(ω) tends to h1ω−1 at large ω. Therefore, we can also write (9.26b), (9.28c) and (9.28d) as H−(ω) = h1∑i = 12NαiRiω−αi H+(ω) = h1∑i = 12Nα¯¯iR¯¯¯iω−α¯¯i = h1∑i = 12NαiRiω+αi (9.30)

We identify also a much larger class of factorizations of H(ω), determined by interchanging particular βl in (9.26a) with βl¯¯¯ in (9.28a). These diﬀerent factorizations are labeled by the subscript or superscript f. Thus,

Hf−(ω) = h1ω∏2N−2l = 1(ω−ζfl)∏2Nl = 1(ω−αl) (9.31a) = h1ω∑i = 12NRfiω−αl (9.31b) = h1∑i = 12NαiRfiω−αi (9.31c) αl∈Ω(−1), l = 1, 2, ..., 2Nβl∈Ω(−1), l = 1, 2, ..., 2N−2 ζfl = λflβl+(1−λfl)β¯l, λfl = 0 or 1 (9.31d) Rfi = (ω−αi)Hf−(ω)∣∣∣∣∣ω= αi = h1αi∏2N−2l = 1(αi−ζfl)∏2Nl = 1l ≠ i(αi−αl) (9.31e)

Observe that (9.29) also holds for the Rf, a property which has been used in writing (9.31c). There are 22N−2 diﬀerent factorizations. Referring to the discussion in sections Ⅶ and Ⅷ, we note that if all the λfl are equal to one, then Ψf(t) is the minimum free energy Ψm(t), while if all are zero, we obtain the maximum free energy ΨM(t). All other possibilities yield functionals that are intermediate between these two extremes. These observations follow from remark Ⅶ.1. Observe that

H(ω) = Hf+(ω)Hf−(ω) = H1(ω)2 ∑i,j = 12NRfi¯¯¯¯Rfj(ω−α¯¯i)(ω−αj) = H1ω2∑i,j = 12NRfi¯¯¯¯Rfjα¯¯i−αj{1(ω−α¯¯i)−1(ω−αj)} (9.32) Now, we have the relation χ+(ω) = 1π∫−∞∞H(ω′)ω′(ω′−ω−)dω′ (9.33)

which follows by applying an integration over R to (9.1) and completing the contour over Ω(−). The integral over χ+¯¯¯¯(ω) vanishes. Thus, we have, on integrating over Ω+,

χ+(ω) = H1π∫−∞∞ω′ω′−ω−∑i,j = 12NRfi¯¯¯¯Rfjα¯¯i−αj{1ω′−α¯¯i−1ω′−αj}dω′ (9.34a) = -2iHi∑i,j = 12Nα¯¯iRfi¯¯¯¯Rfj(α¯¯i−αj)(ω−α¯¯i) (9.34b) The quantity qft−(ω), defined by (8.4), may be evaluated by closing the contour on Ω(−), giving qft+(ω) = 12πi∫−∞∞Hf−(ω′)Et+(ω′)ω′−ω+dω′ = h1∑i=12NαiRfiei(ω)ω−αi (9.35) From (8.4a), we have qft+(ω) = qft−(ω)−Hf−(ω)Et+(ω) = h1∑i = 12NRfi[αiEt+(αi)−ωEt+(ω)]ω−αi (9.36)

which has singularities at those of Et+(ω) in Ω(+) but none in Ω(−). These explicit relations allow analytic continuation of qt±(ω) to the whole complex plane, excluding singular points. From (8.26a), (9.18), (9.31) and (9.35), we see that the quantity Pft−(ω) is given by

Pft−(ω) = 12πi∫−∞∞Hf−(ω')Etr+(ω')ω'−ω+dω' = ih1∑i = 12NRfie˙i(t)ω−αi (9.37)

The optimal history/continuation in the frequency domain, which is a special case of (8.16), is given by formulae which are generalizations of those in [2], page 366. We deduce from (8.17b) and (9.35) that

ψf(t) = ϕ0(t)+iH1∑i,j = 12Nαi¯¯¯αjRfi¯¯¯¯Rfjαi¯¯¯−αjei¯¯¯(t).ej(t) (9.38a) = ϕ0(t)−12∫0∞∫0∞Et(s1).Gf12(s1,s2)Et(s2)ds1ds2 (9.38b) Where Gf12(s1,s2) = -2iH1∑i,j = 12Nαi¯¯¯αjRfi¯¯¯¯Rfjαi¯¯¯−αjeiαi¯¯¯¯s1−iαjs2 (9.39)

Also, from (8.28a) and (9.37), ψf(t) = U(t)+iH1∑i,j = 12NRfi¯¯¯¯Rfjαi¯¯¯−αje˙¯i(t).e˙j(t) (9.40a) = U(t)−12∫0∞∫0∞E˙t(s1).G˜f(s1,s2)E˙t(s2)ds1ds2 (9.40b)

. Where

G˜f(s1,s2) = -2iH1∑i,j = 12NRfi¯¯¯¯Rfjαi¯¯¯−αjeiαi¯¯¯¯s1−iαjs2 (9.41)

We write (9.38a) and (9.40a) in the form (9.22), giving

ψf(t) = ϕ0(t)+12∑i,j = 12NCfijei¯¯¯(t).ej(t)

= U(t)+12∑i,j = 12NCfije˙¯(t).e˙j(t), Cfij = 2iH1Rfi¯¯¯¯Rfjαi¯¯¯¯−αj (9.42)

Comparing (9.34b) with (9.12), we see that

gi = -2iH1∑j = 12Nαi¯¯¯Rfi¯¯¯¯Rfjαi¯¯¯−αj = -2iH1∑j = 12NαjRfi¯¯¯¯Rfjαi¯¯¯−αj (9.43)

where (9.29), for the Rfi, has been used. The relation (4.5a), which yields that Gf2(0,s) = G′(s), where the latter quantity is given by (9.14), can be confirmed with the aid of (9.41) and (9.43). Relation (9.23), which is equivalent to (4.5a), can be confirmed for Cf.

Recall from (9.16) that the quantities e1, i = 1,2,...,2N represent a particular minimal state and all the Ψf(t) are manifestly FMSs.

From (8.18d), (8.21), (9.31b), and (9.31c), we have

Kf(t) = -ih1[∑i = 12NαiRfiei(t)] = h1[∑i = 12NRfie¯i(t)] (9.44)

so that

Df(t) = H1∣∣∣∣∑i = 12NαiRfiei(t)∣∣∣∣2 = H1∣∣∣∣∑i = 12NRfie˙j(t)∣∣∣∣2 (9.45a)

= H1∑i,i = 12NRfi¯¯¯¯e˙¯(t).e˙j(t) (9.45b)

= H1∑i,i = 12Nαi¯¯¯αjRfi¯¯¯¯Rfjei¯¯¯(t).ej(t) (9.45c)

This is in agreement with (9.25). Relations (9.38) and (9.45) can be derived also from (8.29) and (8.30). Note that Df(t) vanishes if

∑i = 12NαiRfiei(t) = 0 (9.46)

Solutions to this equation will exist for non-zero values of ei(t). Therefore, (9.45b) is a positive semi-definite rather than a positive definite quadratic form, so that the associated matrix will have some zero eigenvalues. If one of these zero eigenvalues were to become slightly negative, then the second law would no longer hold. Thus, the free energy Ψf(t) is on the boundary of Φ(t), defined in remark Ⅶ.3. This observation is of course a special case of that after (8.24), and relates to remark Ⅷ.1.

A. Sinusoidal histories for non-magnetic materials

Let us consider the formulae (9.44) and (9.38) for histories

Et(s) = Aeiω0(t−s)+A¯¯¯e−iω0(t−s), s∈R+ (9.47)

and

Et+(ω) = Aeiω0ti(ω+ω0)+A¯¯¯e−iω0ti(ω+ω0) (9.48)

giving that el(t), defined by (9.16), has the form

el(t) = Aeiω0ti(ω0+αl)+A¯¯¯e−iω0ti(ω0+αl) (9.49)

The relation (9.38a) becomes

ψf(t) = ϕ0(t)+iH1∑i,j = 12Nαi¯¯¯αjRfi¯¯¯¯Rfjαi¯¯¯−αj⎧⎩⎨⎪⎪−A.A(ω0+αi¯¯¯¯)(ω0+αj)e2iω0t−A¯¯¯.A¯¯¯(ω0+αi¯¯¯¯)(ω0+αj)e−2iω0t+A.A¯¯¯(ω0+αi¯¯¯¯)(ω0+αj)+A¯¯¯.A(ω0+αi¯¯¯¯)(ω0+αj)⎫⎭⎬⎪⎪ (9.50)

Also, from (9.44), Kf(t), which determines the rate of dissipation, has the form

Kf(t) = -h1∑i = 12NαiRfi{Aeiω0t(αi+ω0)+A¯¯¯e−iω0t(αi−ω0)} (9.51)

Remark Ⅸ.1. General formulae for the minimum free energy and the associated rate of dissipation were given for sinusoidal histories in [1]. These generalize immediately to apply to Ψf(t) and Df(t) by simply replacing H±(ω) (or the tensor version of these quantities used in [1]) with Hf±(ω)

Thus, the formulae for Ψf(t) and Kf(t) are as follows

ψf(t) = ϕ0(t)+Bf1(ω0)e2iω0tA.A++B¯¯¯f1(ω0)e−2iω0tA¯¯¯.A¯¯¯+Bf2(ω0)|A|2 Bf1(ω0) = 12{X′+(ω0)+iω0[Hf−(−ω0)]2} (9.52) Bf2(ω0) = ddω0[ω0Xc(ω0)]+Δf(ω0) Df(t) = |Kf(t)|2, Kf(t) = Hf+(ω0)Aeiω0t+Hf−(ω0)A¯¯¯e−iω0t

where ∆f(ω) is given by (8.34). It will now be shown that relations (9.50) and (9.51) can also be obtained from these, but after somewhat difficult manipulations. We see from (9.26), (9.28) and remark Ⅸ.1 that

[Hf−(ω0)]2 = Hf+(ω0)Hf−(ω0) = H1ω20∑i,j = 12NRfi¯¯¯¯Rfj(ω0−αi¯¯¯)(ω0−αj) (9.53) Referring to (9.34), we have X¯¯¯+(−ω0) = X+(ω0) = 2iH1∑i,j = 12NαjR¯¯¯fjRfj(α¯¯i−αj)(ω0−αj) (9.54) so that

X+(ω0) = -iH1∑i,j = 12NRfi¯¯¯¯Rfj(α¯¯i−αj){α¯¯iω0−α¯¯i−α¯¯jω0−αj} (9.55a) = -iH1∑i,j = 12NRfi¯¯¯¯Rfj[(α¯¯i−αj)ω0+2α¯¯iαj](α¯¯i−αj)(ω0−α¯¯i)(ω0+α¯¯i) (9.55b)

We obtain

12[X+(ω0)+iω0[Hf−(ω0)]2] = -iH1∑i,j = 12Nα¯¯iαjRfi¯¯¯¯Rfj(α¯¯i−αj)(ω0−α¯¯i)(ω0+αj) (9.56)

which agrees with the coefficient of A.Ae2iω0t in (9.50). From (9.55a), we see that

ddω0[ω0X+(ω0)] = iH1∑i,j = 12NRfi¯¯¯¯Rfj(α¯¯i−αj){α¯¯i2(ω0−α¯¯i)2+α2j(ω0+αj)2} (9.57)

so that

ddω0[ω0Xc(ω0)] = iH12∑i,j = 12NRfi¯¯¯¯Rfj(α¯¯i−αj) (9.58) {α¯¯i2(ω0−α¯¯i)2+α2j(ω0+αj)2+α¯¯i2(ω0+α¯¯i)2+α2j(ω0−αj)2} Also, using (9.30), we see that the quantity ∆f(ω0), defined by (8.34), is given by Δf(ω0) = -iH1∑i,j = 12Nα¯¯iαjRfi¯¯¯¯Rfj{1(ω0−α¯¯i)2(ω0−αj)−1(ω0−α¯¯i)(ω0−αj)2} = -iH1∑i,j = 12Nα¯¯iαjRfi¯¯¯¯Rfjα¯¯i−αj{1ω0−α¯¯i(1ω0−α¯¯i−1ω0−αj)−1(ω0−αj)(1ω0−α¯¯−1ω0−αj)} (9.59)

Now, by virtue of (9.29), for the Rfi, it follows that

∑j =12NαjRfjα¯¯i−αj = ∑j =12Nα¯¯iRfjα¯¯i−αj, ∑i =12Nα¯¯iRfjα¯¯i−αj = ∑j =12NαjRfjα¯¯i−αj (9.60)

We deduce the relation

Δf(ω0) = -iH1⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪∑2Ni,j = 1Rfi¯¯¯¯Rfjα¯¯i−αj[α¯¯i2(ω0−α¯¯i)2+α2j(ω0−αj)2]− 2∑2Ni,j = 1α¯¯iαjRfi¯¯¯¯Rfj(α¯¯i−αj)(ω0−α¯¯i)(ω0−αj)⎫⎭⎬⎪⎪⎪⎪⎪⎪⎪⎪ (9.61) Using the symmetry of ∆f(ω0) with respect to ω0, we find that Δf(ω0) = -ddω[ω0Xc(ω0)] +iH1∑i,j = 12Nα¯¯iαjRfi¯¯¯¯Rfjα¯¯i−αj{1(ω0−α¯¯i)(ω0−αj)+1(ω0+α¯¯i)(ω0+αj)} (9.62)

and therefore

Bf2(ω0) = iH1∑i,j = 12Nα¯¯iαjRfi¯¯¯¯Rfjα¯¯i−αj {1(ω0−α¯¯i)(ω0−αj)+1(ω0+α¯¯i)(ω0+αj)} (9.63)

which agrees with the constant term (proportional to |A|2) in (9.50).

The main results are listed below.

1. An isothermal theory of free energies, based on continuum thermodynamics and corresponding to linear constitutive relations with memory, is presented for isotropic non-magnetic materials. For a standard choice of relaxation function, the minimum and maximum free energies are given explicitly.

2. The central new result of this work is the determination of a family of intermediate free energy functionals, for the same relaxation function. These are analogous, though not identical, to those known within a mechanics framework. However, the assumed singularity structure of the frequency domain version of the relaxation function (the susceptibility, for dielectrics) is more relevant, in a physical sense, to dielectrics than to mechanics, as discussed in footnote 5, and this singularity structure is the basis of the new results.

3. An important aspect of these developments, from a physical point of view, is that all of these quantities are solutions of a constrained minimization problem, leading to free energy functionals that are on the boundary of the set of all free energies. Going outside of this boundary would yield functionals that do not obey properties imposed by the laws of thermodynamics.