Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials

Abstract

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.

Introduction

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 $\mathcal{R}$, the non-negative reals by ${\mathcal{R}}^{+}$ and the strictly positive reals by ${\mathcal{R}}^{++}$. Similarly, ${\mathcal{R}}^{-}$ is the set of non-positive reals and ${\mathcal{R}}^{--}$ 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.

General Relations

Consider a rigid non-magnetic isotropic dielectric material subject to a varying electric field. Let the body under consideration occupy a volume $\mathcal{B}\subset {\mathcal{R}}^3$. 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 $\psi \left(t\right)$ 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]

$\dot{\psi }\text{+}D\text{=}\stackrel{.}{D}.E,D\ge 0\text{ (3}\text{.1)}$

Let $E^t\text{:}{\mathcal{R}}^{+}\mapsto \mathcal{V}\text{= }{\mathcal{R}}^3$ be defined by^a

$E^t\left(s\right)\text{ = E}\left(t-s\right),s\in {\mathcal{R}}^{+}\text{ (3}\text{.2)}$

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

$E^t\left(s\right)\text{ = E}\left(t-s\right),s\in {\mathcal{R}}^{--}\text{ (3}\text{.3)}$

Let us define the free enthalpy [1] as

$\mathcal{F}\text{= }\psi -\text{D}\text{.E (3}\text{.4)}$

In terms of this quantity, (3.1) becomes

$\stackrel{.}{\mathcal{F}}+D\text{ = -D}\text{.}\stackrel{.}{\text{E}},D\ge \text{0 (3}\text{.5)}$

The relative history ${\text{E}}_r^t$ is defined by

${\text{E}}_r^t\left(s\right){\text{ = E}}^t\left(s\right)\text{-E}\left(t\right)\text{ (3}\text{.6)}$

A relative future continuation is also defined by (3.6) for $s\in {\mathcal{R}}^{--}$

We define the equilibrium free enthalpy ${\mathcal{F}}_e\left(t\right)$ to be that given for the static history ${\text{E}}^t\left(s\right)\text{ = E}\left(t\right),s\in {\mathcal{R}}^{+}$.

Therefore,

${\mathcal{F}}_e\left(t\right)\text{ = }{\tilde{\mathcal{F}}}_e\left(\text{E}\left(t\right)\right)\text{ (3}\text{.7)}$

emphasizing that the function ${\mathcal{F}}_e\left(t\right)$ 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: $\frac{\partial \mathcal{F}(t)}{\partial \Lambda (t)}\text{ = }-\text{D}\left(t\right)\text{ (3}\text{.8)}$

P2: For any history $E_a^t$ and current value $E_a\left(t\right)$

$\tilde{\mathcal{F}}\left(E_a^t,{\text{E}}_a\left(t\right)\right)\ge {\tilde{\mathcal{F}}}_e\left({\text{E}}_a\left(t\right)\right)\text{ (3}\text{.9)}$

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 Linear Memory Model

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

$\mathcal{F}\left(t\right)\text{ = }{\mathcal{F}}_e\left(t\right)-\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\text{E}}_r^t}}\left(s\right).G_{12}\left(s,u\right){\text{E}}_r^t\left(u\right)dsdu\text{ (4}\text{.1a)}$

$={\mathcal{F}}_e\left(t\right)-\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\dot{E}}^t\left(s\right)}}.\tilde{G}(s,u){\dot{E}}^t\left(u\right)dsdu\left(4.1b\right)$

${\mathcal{F}}_e\left(t\right)\text{ = -}\frac{1}{2}{G}_{\infty }\text{E}\left(t\right).\text{E}\left(t\right)\text{ (4}\text{.1c)}$

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

${\dot{E}}^t\left(u\right)\text{ = }\frac{\partial }{\partial t}{\text{E}}^t\left(u\right)\text{ = -}\frac{\partial }{\partial t}{\text{E}}^t\left(u\right)\text{ = -}\frac{\partial }{\partial t}{\text{E}}_r^t\left(u\right)\left(4.2\right)$

$\tilde{G}\left(s,u\right)\text{ = }G\left(s,u\right)\text{-}{G}_{\infty }$

and

$\frac{{\partial }^2}{\partial s\partial u}G\left(s,u\right)\equiv G_{12}\left(s,u\right)\text{ (4}\text{.3a)}$

$\underset{s\to \infty }{\lim }G\left(s,u\right)\text{ = }{G}_{\infty },u\in {\mathcal{R}}^{+}\text{ (4}\text{.3b)}$

$\underset{s\to \infty }{\lim }\frac{\partial }{\partial u}G\left(s,u\right)\text{ = 0},u\in {\mathcal{R}}^{+}\text{ (4}\text{.3c)}$

$\underset{s\to \infty }{\lim }\frac{\partial }{\partial s}G\left(s,u\right)\text{ = 0},u\in {\mathcal{R}}^{+}\text{ (4}\text{.3d)}$

$G\left(s,u\right)\text{ = }G\left(u,s\right)\text{ (4}\text{.3e)}$

with similar limits at large u holding for fixed s. The parameter $G_{\infty }$ in (4.1a) is defined by (4.3b). Relation (3.8) gives the linear constitutive relation

$\text{D}\left(t\right)\text{ = }{G}_{\infty }\text{E}\left(t\right)\text{+}{\displaystyle \underset{0}{\overset{\infty }{\int }}{G}^{\prime }}\left(u\right){\text{E}}_r^t\left(u\right)du\text{ (4}\text{.4a)}$

$=G_0\text{E}\left(t\right)\text{+}{\displaystyle \underset{0}{\overset{\infty }{\int }}{G}^{\prime }}\left(u\right){\text{E}}^t\left(u\right)du\text{ (4}\text{.4b)}$

$=G_{\infty }\text{E}\left(t\right)\text{+}{\displaystyle \underset{0}{\overset{\infty }{\int }}\tilde{G}\left(u\right)}{\dot{E}}^t\left(u\right)du\left(4.4c\right)$

$={\displaystyle \underset{0}{\overset{\infty }{\int }}G\left(u\right)}{\dot{E}}^t\left(u\right)du\left(4.4d\right)$

where

$G\left(u\right)\text{ = }G\left(0,u\right)\text{ (4}\text{.5a)}$

$G_0\text{ = }G\left(0,0\right)\text{ = }G\left(0\right)\text{ (4}\text{.5b)}$

$\tilde{G}\left(u\right)\text{ = }G\left(u\right)\text{-}{G}_{\infty }\text{ (4}\text{.5c)}$

$G^{\prime }\left(u\right)\text{ = }\frac{d}{du}G\left(u\right)\text{ (4}\text{.5d)}$

The quantity G(s) is the relaxation function of the material. In writing the final form of (4.4), we are assuming that $\text{E}\left(-\infty \right){\text{ = E}}^t\left(\infty \right)$ vanishes; we furthermore assume that it goes to zero sufficiently strongly so that various integrals, introduced below, exist. The quantity $G_{\infty }$ is related to the relaxation function through

$G_{\infty }\text{ = }G\left(s,\infty \right)\text{ = }G\left(0,\infty \right)\text{ = }G\left(\infty \right)\text{ (4}\text{.6)}$

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

$D\left(t\right)\text{ = }\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\dot{E}}^t}}\left(s\right)\text{.}\left[G_1\left(s,u\right)+G_2\left(s,u\right)\right]{\dot{E}}^t\left(u\right)dsdu\left(4.7a\right)$

$=\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\text{E}}_r^t}}\left(s\right)\text{.}\left[G_{112}\left(s,u\right)+G_{212}\left(s,u\right)\right]{\text{E}}_r^t\left(u\right)dsdu\text{ (4}\text{.7b)}$

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 $E_r^t\left(0\right)$ vanishes. Relation (4.1a), expressed in terms of histories, becomes

$\mathcal{F}\left(t\right)\text{ = }\frac{1}{2}{G}_0\text{E}\left(t\right)\text{.E}\left(t\right)\text{-D}\left(t\right)\text{.E}\left(t\right)\text{ (4}\text{.8)}$

$-\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\text{E}}^t}\left(s\right)}\text{.}{G}_{12}\left(s,u\right){\text{E}}^t\left(u\right)dsdu$

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

$\psi \left(t\right)={\varphi }_0\left(t\right)-{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\text{E}}^t}}\left(s\right).G_{12}\left(s,u\right){\text{E}}^t\left(u\right)dsdu\text{ (4}\text{.9a)}$

$=U\left(t\right)-\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\text{E}}_r^t}}\left(s\right).G_{12}\left(s,u\right){\text{E}}_r^t\left(u\right)dsdu\text{ (4}\text{.9b)}$

$=U\left(t\right)-\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\dot{E}}^t}}\left(s\right).\tilde{G}\left(s,u\right){\dot{E}}^t\left(u\right)dsdu\left(4.9c\right)$

Where

${\varphi }_0\left(t\right)=\frac{1}{2}{G}_0\text{E}\left(t\right).E\left(t\right),U\left(t\right)=\text{ D}\left(t\right).\text{E}\left(t\right)-\frac{1}{2}{G}_{\infty }\text{E}\left(t\right).\text{E}\left(t\right)\text{ (4}\text{.10)}$

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

$W\left(t\right)={\displaystyle \underset{-\infty }{\overset{t}{\int }}\dot{D}\left(u\right)}\text{.E}\left(u\right)du\left(4.11a\right)$

$=\text{ D}\left(t\right).E\left(t\right)-{\displaystyle \underset{-\infty }{\overset{t}{\int }}\text{D}}\left(u\right).\dot{E}\left(u\right)du\text{ (4}\text{.11b)}$

Under the assumption that $\psi \left(-\infty \right)\text{ = 0}$, an integration of (3.1) over $\left(-\infty ,t\right]$ yields

$\psi \left(t\right)+\mathcal{D}\left(t\right)=W\left(t\right)\text{ (4}\text{.12)}$

Where

$\mathcal{D}\left(t\right)={\displaystyle \underset{-\infty }{\overset{t}{\int }}D}\left(u\right)du\ge 0\text{ (4}\text{.13)}$

is the total dissipation up to time t. We take $G_0$ to be equal to the vacuum permittivity ${\in }_0$ [3-5], giving

$\text{D}\left(t\right)={\in }_0\text{E}\left(t\right)+\text{P}\left(t\right)\text{ (4}\text{.14)}$

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

$\text{P}\left(t\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{G}^{\prime }}\left(u\right){\text{E}}^t\left(u\right)du={\displaystyle \underset{-\infty }{\overset{t}{\int }}{G}^{\prime }\left(t-s\right)}\text{E}\left(s\right)ds\text{ (4}\text{.15)}$

Then, (4.11a) becomes

$W\left(t\right){\varphi }_0\left(t\right)+W_{int}\left(t\right),{\varphi }_0\left(t\right)=\frac{1}{2}{\in }_0{\left|\text{E}\left(t\right)\right|}^2\text{ (4}\text{.16)}$

where ${\varphi }_0\left(t\right)$ is the quantity introduced in (4.10) and $W_{int}\left(t\right)$ is given by

$W_{int}\left(t\right)={\displaystyle \underset{-\infty }{\overset{t}{\int }}\dot{P}\left(s\right)}.\text{E}\left(s\right)ds\left(4.17\right)$

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

${\psi }_{int}\left(t\right)+\mathcal{D}\left(t\right)=W_{int}\left(t\right)\text{ (4}\text{.18)}$

where ${\psi }_{int}\left(t\right)$ 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 (E^t, E(t)). Also, different histories may be members of the same minimal state [1]. Two states $\left({\text{E}}_1^t,{\text{E}}_1\left(t\right)\right)$, $\left({\text{E}}_2^t,{\text{E}}_2\left(t\right)\right)$ are equivalent, or in the same minimal state if from a time t onwards, we have

${\text{D}}_1\left(t+s\right)={\text{ D}}_2\left(t+s\right){\text{ if E}}_1\left(t+s\right)={\text{ E}}_2\left(t+s\right),\text{ so }s\ge 0\text{ (4}\text{.19)}$

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

${\displaystyle \underset{0}{\overset{\infty }{\int }}{G}^{\prime }\left(s+u\right)}\left({\text{E}}_1^t\left(u\right)-{\text{E}}_2^t\left(u\right)\right)du=0,s\ge 0\text{ (4}\text{.20)}$

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 $\left({\text{E}}_1^t,{\text{E}}_1\left(t\right)\right)$ and $\left({\text{E}}_2^t,{\text{E}}_2\left(t\right)\right)$ are equivalent is the same as the assertion that $\left(E_d^t,0\right)$ is equivalent to the zero state (0,0), where 0 is the zero in ${\mathcal{R}}^3$ (and also the zero history), while

$E_d^t\left(s\right)={\text{ E}}_1^t\left(s\right)-{\text{E}}_2^t\left(s\right)\text{ (4}\text{.21)}$

A functional of (E_t, 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 $\left({\text{E}}_1^t,{\text{E}}_1\left(t\right)\right)$, $\left({\text{E}}_2^t,{\text{E}}_2\left(t\right)\right)$ be any equivalent states. Then, a free energy is a functional of the minimal state if

$$\psi \left({\text{E}}_1^t,{\text{E}}_1\left(t\right)\right)=\psi \left({\text{E}}_2^t,{\text{E}}_2\left(t\right)\right)$$

It is not necessary that a free energy have this property, though it holds for the minimum and all related free energies introduced later.

Kernels and Field Variables in the Frequency Domain

For any $f\in L^2\left(\mathcal{R}\right)$, we denote its Fourier transform $f_F\in L^2\left(\mathcal{R}\right)$ by^b

$f_F\left(\omega \right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f}\left(\xi \right)e^{-i\omega \text{ξ}}\text{dξ (5}\text{.1a)}$

$\text{= }{f}_{+}\left(\omega \right)+f_{-}\left(\omega \right)\text{ (5}\text{.1b)}$

$f_{+}\left(\omega \right)={\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(\text{ξ}\right)}{e}^{-i\omega \text{ξ}}d\text{ξ (5}\text{.1c)}$

$f_{-}\left(\omega \right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}f\left(\text{ξ}\right)}{e}^{-i\omega \text{ξ}}d\text{ξ (5}\text{.1d)}$

Only real valued functions will be considered so that

${\overline{f}}_F\left(\omega \right)=f_F\left(-\omega \right),\omega \in \mathcal{R}\text{ (5}\text{.2)}$

Where the bar denotes the complex conjugate. For complex values of ω, (5.2) becomes

$\overline{f_F\left(\omega \right)}=f_F\left(-\overline{\omega }\right)\text{ (5}\text{.3)}$

Functions defined on ${\mathcal{R}}^{+}$ are identified with functions on R which vanish identically on ${\mathcal{R}}^{--}$. For such functions,

$f_F\left(\omega \right)=f_{+}\left(\omega \right)=f_c\left(\omega \right)-i{f}_s\left(\omega \right)\text{ (5}\text{.4)}$

Where $f_c\left(\omega \right)$, $f_s\left(\omega \right)$ are respectively the Fourier cosine and sine transforms. A property of Fourier transforms which will be used later is

$f_{+}\left(\omega \right)\underset{\omega \to \infty }{\to }\frac{f\left(0\right)}{i\omega },\text{ so that }{f}_s\left(\omega \right)\underset{\omega \to \infty }{\to }\frac{f\left(0\right)}{\omega }\text{ (5}\text{.5)}$

If $f\left(0\right)$ is non-zero.

A. The kernel G(u)

We can write the Fourier transforms of $G^{\prime }\left(u\right)$ and $\tilde{G}\left(u\right)$ in (4.4) as

${G^{\prime }}_{+}\left(\omega \right)={G^{\prime }}_c\left(\omega \right)-i{G^{\prime }}_s\left(\omega \right)={\chi }_{+}\left(\omega \right)={\chi }_c\left(\omega \right)-i{\chi }_s\left(\omega \right)\text{ (5}\text{.6)}$

${\tilde{G}}_{+}\left(\omega \right)={\tilde{G}}_c\left(\omega \right)-i{\tilde{G}}_s\left(\omega \right)$

The quantity ${\chi }_{+}\left(\omega \right)$ is the susceptibility, denoted by $\chi \left(\omega \right)$ in [3- 5]. By partial integration, one can show that

${G^{\prime }}_{+}\left(\omega \right)=-\left(G_0-G_{\infty }\right)+i\omega {\tilde{G}}_{+}\left(\omega \right)\text{ (5}\text{.7)}$

giving, in particular, that

${\tilde{G}}_s\left(\omega \right)={\chi }_s\left(\omega \right)=-\omega {\tilde{G}}_c\left(\omega \right)\text{ (5}\text{.8)}$

Observe that

${G^{\prime }}_{+}\left(0\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}{G}^{\prime }\left(s\right)}ds=G_{\infty }-G_0\text{ (5}\text{.9)}$

Using (5.2), we see that

${\overline{G_{+}}}^{\prime }\left(\omega \right)=\overline{{\chi }_{+}}\left(\omega \right)={G^{\prime }}_{+}\left(-\omega \right)={\chi }_{+}\left(-\omega \right)\text{ (5}\text{.10)}$

which will be useful later. By considering periodic behavior in $E^t\left(s\right)$, we can show that a consequence of the second law is that ([2], page 140)

${G^{\prime }}_s\left(\omega \right)={\chi }_s\left(\omega \right)=-\omega {\tilde{G}}_c\left(\omega \right)>0,0<\omega <\infty \text{ (5}\text{.11)}$

It follows from (5.11) that $G_{\infty }>G_0$ [1]. We have $G_0={\in }_0>0$, so that $G_{\infty }>0$

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

${\Omega }^{+}=\left\{\omega \in \Omega |Im\left[\omega \right]\in {\mathcal{R}}^{+}\right\}$

${\Omega }^{(+)}=\left\{\omega \in \Omega |Im\left[\omega \right]\in {\mathcal{R}}^{++}\right\}$

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

The quantities $f_{\pm }\left(\omega \right)$, defined by (5.1), are analytic in Ω^(∓) respectively ([2], page 547). Thus, the quantity ${G^{\prime }}_{+}\left(\omega \right)$ is analytic on Ω⁽⁻⁾. It will be assumed that ${G^{\prime }}_{+}\left(\omega \right)$ is analytic on $\mathcal{R}$ 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 ${G^{\prime }}_s\left(\omega \right)$ 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

$H\left(\omega \right)=\omega {G^{\prime }}_s\left(\omega \right)=\omega {\chi }_s\left(\omega \right)=-{\omega }^2{\tilde{G}}_c\left(\omega \right)\text{ (5}\text{.13)}$

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

$i\underset{\omega \to \infty }{\lim }\omega {G^{\prime }}_{+}\left(\omega \right)=\underset{\omega \to \infty }{\lim }\omega {G^{\prime }}_s\left(\omega \right)=G^{\prime }\left(0\right)\text{ (5}\text{.14)}$

yields

$G^{\prime }\left(0\right)=H\left(\infty \right)=H_{\infty }\text{ (5}\text{.15)}$

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 ${\text{E}}_{+}^t\left(\omega \right)$ and ${\text{E}}_{-}^t\left(\omega \right)$ respectively^c. These are particular examples of (5.1c) and (5.1d). The quantity ${\text{E}}_{+}^t\left(\omega \right)$ is analytic on Ω⁽⁻⁾ and ${\text{E}}_{-}^t\left(\omega \right)$ is analytic on Ω⁽⁺⁾. Both are assumed to be analytic on an open set including $\mathcal{R}$. It is further assumed that they are analytic at infinity.

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

${\text{E}}_{r\pm }^t\left(\omega \right)={\text{ E}}_{\pm }^t\left(\omega \right)\mp \frac{\text{E}\left(t\right)}{i{\omega }^{\mp }}\text{ (5}\text{.16)}$

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

${\dot{E}}_{+}^t\left(\omega \right)=\frac{d}{dt}{\text{E}}_{+}^t\left(\omega \right)\text{ = -}i\omega {\text{E}}_{+}^t\left(\omega \right)+\text{E}\left(t\right)\text{ = -}i\omega {\text{E}}_{r+}^t\left(\omega \right)\left(5.17\right)$

The second relation follows from (3.2) and an integration by parts in the Fourier integral defining ${\text{E}}_{+}^t\left(\omega \right)$. Also, based on arguments from [1] and [2], page 146, for example, we can express the constitutive equation (4.4) in the form

$\text{D}\left(t\right)\text{ = }{G}_{\infty }\text{E}\left(t\right)+\frac{i}{\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H\left(\omega \right)}{\omega }}{\text{E}}_{r+}^t\left(\omega \right)d\omega$

$=G_{\infty }\text{E}\left(t\right)-\frac{1}{\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H\left(\omega \right)}{{\omega }^2}}{\dot{E}}_{r+}^t\left(\omega \right)d\omega \left(5.18\right)$

$=G_0\text{E}\left(t\right)+\frac{i}{\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H\left(\omega \right)}{\omega }}{\text{E}}_{+}^t\left(\omega \right)d\omega$

The Work Function

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

$W\left(t\right)=U\left(t\right)-\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{G}_{12}}}\left(\left|u-s\right|\right){\text{E}}_r^t\left(u\right).{\text{E}}_r^t\left(s\right)dsdu$

$={\varphi }_0\left(t\right)-\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{G}_{12}}}\left(\left|u-s\right|\right){\text{E}}^t\left(u\right).{\text{E}}^t\left(s\right)dsdu\text{ (6}\text{.1)}$

where U(t) and ${\varphi }_0\left(t\right)$ are defined by (4.10). We see from (6.1) that W(t) can be cast in the form (4.9) by putting

$G_{12}\left(s,u\right)=G_{12}\left(\left|u-s\right|\right)\text{ (6}\text{.2)}$

In terms of frequency domain quantities, we find that [1]

$W\left(t\right)=U\left(t\right)-\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\overline{E}}_{r+}^t}\left(\omega \right).H\left(\omega \right){\text{E}}_{r+}^t\left(\omega \right)d\omega$

$={\varphi }_0\left(t\right)+\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\overline{E}}_{+}^t}\left(\omega \right).H\left(\omega \right){\text{E}}_{+}^t\left(\omega \right)d\omega \left(6.3\right)$

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]

$\left({\text{E}}_1^t,{\text{E}}_2^t\right)=\left({\text{E}}_{1F}^t,{\text{E}}_{2F}^t\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\text{E}}_1^t}}\left(s\right).G\left(\left|s-u\right|\right){\text{E}}_2^t\left(u\right)dsdu\text{ (6}\text{.4a)}$

$=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\overline{E}}_{1F}^t}\left(\omega \right).H\left(\omega \right){\text{E}}_{2F}^t\left(\omega \right)d\omega =\left({\text{E}}_2^t,{\text{E}}_1^t\right)\text{ = }\left({\text{E}}_{2F}^t,{\text{E}}_{1F}^t\right)\left(6.4b\right)$

Recalling (5.1), we see that the quantity ${\text{E}}_F^t\left(\omega \right)$ is the Fourier transform of ${\text{E}}^t\left(s\right)$,$s\in \mathcal{R}$ 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

$\left({\text{E}}^t{\text{,E}}^t\right)={\Vert {\text{E}}^t\Vert }^2={\Vert {\text{E}}_F^t\Vert }^2\text{ (6}\text{.5)}$

In the notation of (6.5), the work function is given by

$W\left(t\right)=U\left(t\right)+{\Vert {\text{E}}_r^t\Vert }^2\text{ (6}\text{.6a)}$

$={\varphi }_0\left(t\right)+{\Vert {\text{E}}^t\Vert }^2\text{ (6}\text{.6b)}$

where ${\text{E}}_r^t\left(s\right)=0$, $s\in {\mathcal{R}}^{--}$ in (6.6a) and ${\text{E}}^t\left(s\right)=0$, $s\in {\mathcal{R}}^{--}$ in (6.6b).

Factorization of H (ω)

We consider materials such that ${G^{\prime }}_{+}\left(\omega \right)$ (or $\tilde{G}{}_{+}\left(\omega \right)$) 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 ${G^{\prime }}_s\left(\omega \right)$ 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 $\mathcal{R}$, 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

$H_{+}\left(\omega \right)H_{-}\left(\omega \right)=H\left(\omega \right)\text{ (7}\text{.1)}$

where

$H_{\pm }\left(\omega \right)=H_{\mp }\left(-\omega \right)=\overline{H_{\mp }}\left(\omega \right)\text{ (7}\text{.2a)}$

$H\left(\omega \right)={\left|H_{\pm }\left(\omega \right)\right|}^2,\omega \in \mathcal{R}\text{ (7}\text{.2b)}$

The factorization (7.1) is unique up to multiplication by a phase factor e^iα, 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

$H\left(\omega \right)=H_{+}^f\left(\omega \right)H_{-}^f\left(\omega \right)\text{ (7}\text{.3a)}$

$H_{\pm }^f\left(\omega \right)=H_{\pm }^f\left(-\omega \right)=\overline{H_{\mp }^f}\left(\omega \right)\text{ (7}\text{.3b)}$

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 Ψ_m(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 $H_{\pm }^f\left(\omega \right)$ at the origin play no part in these exchanges.

Remark Ⅶ.2. A particularly interesting choice of ${\psi }_f\left(t\right)$, which we denote by ${\psi }_M\left(t\right)$, is obtained by interchanging all the zeros. This can be identified as the maximum free energy among all those that are FMSs^d. It is less than the work function, which is not a FMS for materials with only isolated singularities [2].

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

$\psi \left(t\right)={\displaystyle \sum _{f=1}^N{\text{λ}}_f{\psi }_f}\left(t\right),D\left(t\right)={\displaystyle \sum _{f=1}^N{\text{λ}}_f{D}_f}\left(t\right),{\displaystyle \sum _{f=1}^N{\text{λ}}_f=1},{\text{ λ}}_f\ge 0\text{ (7}\text{.4)}$

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 ${\psi }_f\left(t\right)$ 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 ${\text{E}}_m^t\left(s\right)$, $s\in {\mathcal{R}}^{-}$ yielding the maximum recoverable work from this state. Also, the maximum free energy is determined by finding the optimal history ${\text{E}}_M^t\left(s\right)$, $s\in {\mathcal{R}}^{+}$ 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 E^t(s), $s\in \mathcal{R}$, 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.

The Free Energy Associated with a Particular Factorization

Let us define the quantities ${\text{E}}_R^t\left(s\right)$ and ${\text{E}}_d^t\left(s\right)$ by the relation

${\text{E}}_R^t\left(s\right)={\text{ E}}^t\left(s\right)+{\text{E}}_d^t\left(s\right),s\in \mathcal{R}\text{ (8}\text{.1)}$

where ${\text{E}}^t\left(s\right)$ is the given history, which is zero for $s\in {\mathcal{R}}^{--}$. The quantity ${\text{E}}_R^t$ is typically non-zero on $\mathcal{R}$ and ${\text{E}}_d^t$ follows from (8.1). In the frequency domain, the relation becomes

${\text{E}}_{RF}^t\left(\omega \right)\equiv {\text{ E}}_{R+}^t\left(\omega \right)+{\text{E}}_{R-}^t\left(\omega \right)={\text{ E}}_{+}^t\left(\omega \right)+{\text{E}}_{dF}^t\left(\omega \right),\omega \in \mathcal{R}\text{ (8}\text{.2)}$

where the quantities ${\text{E}}_{R\pm }^t\left(\omega \right)$ are particular cases of (5.1c) and (5.1d). Consider the norm ${\Vert {\text{E}}_R\Vert }^2={\Vert {\text{E}}_{RF}\Vert }^2$, 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

${\Vert {\text{E}}_{RF}^t\Vert }^2=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left[\overline{{\text{E}}_{+}^t}\left(\omega \right)+\overline{{\text{E}}_{dF}^t}\left(\omega \right)\right]}.H\left(\omega \right)\left[{\text{E}}_{+}^t\left(\omega \right)+{\text{E}}_{dF}^t\left(\omega \right)\right]d\omega \text{ (8}\text{.3)}$

Let us choose a particular factorization of H(ω), as given by (7.3). Also, we define [1]

$H_{-}^f\left(\omega \right){\text{E}}_{+}^t\left(\omega \right)-{\text{q}}_{-}^{ft}\left(\omega \right)-{\text{q}}_{+}^{ft}\left(\omega \right)\text{ (8}\text{.4a)}$

${\text{q}}_{\pm }^{ft}\left(\omega \right)=\frac{1}{2\pi i}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H_{-}^f\left({\omega }^{\prime }\right){\text{E}}_{+}^t\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-{\omega }^{\mp }}}d{\omega }^{\prime }\text{ (8}\text{.4b)}$

where the ${\text{q}}_{\pm }^{ft}\left(\omega \right)$ are analytic in Ω^∓, respectively. The singularities of ${\text{q}}_{-}^{ft}\left(\omega \right)$ are the same as those of $H_{-}^f\left(\omega \right)$, 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 ${\Vert {\text{E}}_{RF}\Vert }^2$ in the form

${\Vert {\text{E}}_{RF}^t\Vert }^2=\frac{1}{2\pi }{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left|{\text{q}}_{-}^{ft}\left(\omega \right)-{\text{q}}_{+}^{ft}\left(\omega \right)+H_{-}^f\left(\omega \right){\text{E}}_{dF}^t\left(\omega \right)\right|}}^{2}d\omega \text{ (8}\text{.5)}$

Let us further define

$H_{-}^f\left(\omega \right){\text{E}}_{dF}^t\left(\omega \right)={\text{ u}}_{-}^{ft}\left(\omega \right)-{\text{u}}_{+}^{ft}\left(\omega \right)$

${\text{u}}_{\pm }^{ft}\left(\omega \right)=\frac{1}{2\pi i}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H_{-}^f\left({\omega }^{\prime }\right){\text{E}}_{dF}^t\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-{\omega }^{\mp }}}d{\omega }^{\prime }\text{ (8}\text{.6)}$

Note that, by virtue of Cauchy's integral formula,

${\text{u}}_{-}^{ft}\left(\omega \right)={\text{ q}}_{d-}^{ft}\left(\omega \right)+H_{-}^f\left(\omega \right){\text{E}}_{d-}^t\left(\omega \right)\text{ (8}\text{.7a)}$

$={\text{ q}}_{R-}^{ft}\left(\omega \right)-{\text{q}}_{-}^{ft}\left(\omega \right)+H_{-}^f\left(\omega \right){\text{E}}_{d-}^t\left(\omega \right)\text{ (8}\text{.7b)}$

${\text{u}}_{+}^{ft}\left(\omega \right)={\text{ q}}_{d+}^{ft}\left(\omega \right)={\text{ q}}_{R+}^{ft}\left(\omega \right)-{\text{q}}_{+}^{ft}\left(\omega \right)\text{ (8}\text{.7c)}$

Where ${\text{q}}_{d\pm }^{ft}\left(\omega \right)$ are the quantities defined by (8.4b) but with ${\text{E}}_{+}^t\left({\omega }^{\prime }\right)$ replaced by

${\text{E}}_{d+}^t\left({\omega }^{\prime }\right)={\text{ E}}_{R+}^t\left({\omega }^{\prime }\right)-{\text{E}}_{+}^t\left({\omega }^{\prime }\right)$

Also,

${\text{q}}_{R\pm }^{ft}\left(\omega \right)=\frac{1}{2\pi i}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H_{-}^f\left({\omega }^{\prime }\right){\text{E}}_{R+}^t\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-{\omega }^{\mp }}}d{\omega }^{\prime }\text{ (8}\text{.8)}$

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 ${\Vert {\text{E}}_{RF}^t\Vert }^2$, subject to the constraint

${\text{u}}_{-}^{ft}\left(\omega \right)=\frac{1}{2\pi i}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H_{-}^f\left({\omega }^{\prime }\right){\text{E}}_{dF}^t\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-{\omega }^{+}}}d{\omega }^{\prime }=0,(8.9)$

The resultant free energy is

${\psi }_f\left(t\right)={\varphi }_0\left(t\right)+{\Vert {\text{E}}_f^t\Vert }^2,(8.10)$

where ${\text{E}}_f^t$ is the history/continuation ${\text{E}}_{RF}^t$ 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 ${\psi }_f\left(t\right)$ 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

$H_{-}^f\left(\omega \right){\text{E}}_{dF}^t\left(\omega \right)={\text{ q}}_{+}^{ft}\left(\omega \right)-{\text{q}}_{R+}^{ft}\left(\omega \right),(8.11a)$

${\Vert {\text{E}}_{RF}^t\Vert }^2=\frac{1}{2\pi }{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left|{\text{q}}_{-}^{ft}\left(\omega \right)-{\text{q}}_{R+}^{ft}\left(\omega \right)\right|}}^{2}d\omega ,(8.11b)$

$=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left[{\left|{\text{q}}_{-}^{ft}\left(\omega \right)\right|}^2\text{+}{\left|{\text{q}}_{R+}^{ft}\left(\omega \right)\right|}^2\right]}d\omega ,(8.11c)$

where ${\text{q}}_{R+}^{ft}\left(\omega \right)$ is given by (8.8). Since ${\text{q}}_{R-}^{ft}\left(\omega \right)$ depends only on the given history ${\text{E}}_{+}^t\left(\omega \right)$, the solution to the minimization problem is obtained by choosing ${\text{E}}_{RF}^t\left(\omega \right)={\text{ E}}_f^t\left(\omega \right)$ to give that

${\text{q}}_{R\text{+}}^{ft}\left(\omega \right)=0,(8.12)$

yielding

${\Vert {\text{E}}_f^t\Vert }^2=\frac{1}{2\pi }{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left|{\text{q}}_{-}^{ft}\left(\omega \right)\right|}}^{2}d\omega ,(8.13)$

From (8.11a) and (8.12), we obtain that

$H_{-}^f\left(\omega \right){\text{E}}_{dF}^t\left(\omega \right)=H_{-}^f\left(\omega \right){\text{E}}_{RF}^t\left(\omega \right)-H_{-}^f\left(\omega \right){\text{E}}_{+}^t\left(\omega \right)={\text{ q}}_{+}^{ft}\left(\omega \right).(8.14)$

Then, (8.4a) gives

$H_{-}^f\left(\omega \right){\text{E}}_{RF}^t\left(\omega \right)=H_{-}^f\left(\omega \right){\text{E}}_f^t\left(\omega \right)={\text{ q}}_{-}^{ft}\left(\omega \right),(8.15)$

and

${\text{E}}_f^t\left(\omega \right)=\frac{{\text{q}}_{-}^{ft}\left(\omega \right)}{H_{-}^f\left(\omega \right)}=\frac{1}{2\pi i{H}_{-}^f\left(\omega \right)}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H_{-}^f\left({\omega }^{\prime }\right){\text{E}}_{+}^t\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-{\omega }^{+}}}d{\omega }^{\prime }.(8.16)$

The observation after (8.4) about the singularities of $q_{-}^{ft}\left(\omega \right)$ and $H_{-}^f\left(\omega \right)$ yield that the singularity structure of ${\text{E}}_f^t\left(\omega \right)$ is determined by the zeros of $H_{-}^f\left(\omega \right)$.

Remark Ⅷ.1. Note that for $f\ne 1$, N, the quantity $H_{-}^f\left(\omega \right)$ has some of its zeros in Ω⁻ and some in Ω⁺. For f = 1, all the zeros of $H_{-}^f\left(\omega \right)$ are in Ω⁻, so that ${\text{E}}_f^t\left(\omega \right)$ has singularities only in Ω⁻ and its inverse Fourier transform is thus non-zero only on ${\mathcal{R}}^{-}$. 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 ${\Vert {\text{E}}_{RF}^t\Vert }^2$ result in the same relation, namely (8.16) for f = 1.

For f = N, all the zeros of $H_{-}^f\left(\omega \right)$ are in Ω⁺, which gives that ${\text{E}}_f^t\left(\omega \right)$ has singularities only in Ω⁺. Therefore, its inverse Fourier transform is non-zero only on ${\mathcal{R}}^{+}$ and is the optimal history yielding the maximum free energy. In this case, the quantity ${\Vert {\text{E}}_{RF}^t\Vert }^2$ 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, ${\text{E}}_f^t\left(\omega \right)$ is non-zero on $\mathcal{R}$ and minimizes ${\Vert {\text{E}}_{RF}^t\Vert }^2$, 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

${\psi }_f\left(t\right)={\varphi }_0\left(t\right)+\frac{1}{2\pi }{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left|{\text{q}}_{-}^{ft}\left(\omega \right)\right|}}^{2}d\omega (8.17a)$

$={\varphi }_0\left(t\right)+\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\overline{{\text{E}}_f^t}}\left(\omega \right).\mathrm{H}\left(\omega \right){\text{E}}_f^t\left(\omega \right)d\omega ,(8.17b)$

by virtue of (8.13) and (8.15). We have the following relations ([1,10,13,14] for example)

$\frac{d}{dt}{\text{q}}_{+}^{ft}\left(\omega \right)=-i\omega {\text{q}}_{+}^{ft}\left(\omega \right)-{\text{K}}_f\left(t\right),(8.18a)$

$\frac{d}{dt}{\text{q}}_{-}^{ft}\left(\omega \right)=-i\omega {\text{q}}_{-}^{ft}\left(\omega \right)-{\text{K}}_f\left(t\right)+{\text{H}}_{-}^f\left(\omega \right)\text{E}\left(t\right),(8.18b)$

${\text{K}}_f\left(t\right)=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\text{H}}_{-}^f\left(\omega \right)}{\text{E}}_{r+}^t\left(\omega \right)d\omega ,(8.18c)$

$=-\frac{1}{2\pi i}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{{\text{H}}_{-}^f\left(\omega \right)}{\omega }}{\dot{E}}_{+}^t\left(\omega \right)d\omega \left(8.18d\right)$

and

$\underset{\left|\omega \right|\to \infty }{\lim }\omega {\text{q}}_{+}^{ft}\left(\omega \right)=i{\text{K}}_f\left(t\right),(8.19a)$

$\underset{\left|\omega \right|\to \infty }{\lim }\omega {\text{q}}_{-}^{ft}\left(\omega \right)=i\left({\text{K}}_f\left(t\right)-H_{sr}\text{E}\left(t\right)\right),(8.19b)$

$\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\text{q}}_{+}^{ft}}\left(\pm \omega \right)d\omega =-\frac{1}{2}{\text{K}}_f\left(t\right),(8.19c)$

$\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{\text{q}}_{+}^{ft}}\left(\pm \omega \right)d\omega =-\frac{1}{2}\left({\text{K}}_f\left(t\right)-H_{sr}\text{E}\left(t\right)\right).(8.19d)$

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)

${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{{\text{H}}_{-}^f\left(\omega \right)}{\omega }}=0,(8.20)$

as can be seen by closing the contour on Ω⁽⁺⁾, so that, recalling (5.16), it is clear that (8.18c) is replaced by

${\text{K}}_f\left(t\right)=\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}{H}_{-}^f}\left(\omega \right){\text{E}}_{+}^t\left(\omega \right)d\omega \text{ (8}\text{.21)}$

The work function, given by (6.3), takes the form [1]

$W\left(t\right)={\varphi }_0\left(t\right)+\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left[{\left|{\text{q}}_{-}^{ft}\left(\omega \right)\right|}^2+{\left|{\text{q}}_{+}^{ft}\left(\omega \right)\right|}^2\right]}d\omega \text{ (8}\text{.22)}$

From (4.12), (8.17) and (8.22), we deduce that the total dissipation corresponding to the minimum free energy is given by

$D_f\left(t\right)={\displaystyle \underset{-\infty }{\overset{t}{\int }}{D}_f}\left(u\right)du=\frac{1}{2\pi }{{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left|{\text{q}}_{+}^{ft}\left(\omega \right)\right|}}^{2}d\omega \text{ (8}\text{.23)}$

Differentiating this relation with respect to t and using (8.18a), (8.19c), gives

$D_f\left(t\right)={\left|{\text{K}}_f\left(t\right)\right|}^2\text{ (8}\text{.24)}$

Since K_f(t), given by (8.21) can vanish for non-zero histories, D_f(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

${\text{P}}^{ft}\left(\omega \right)=H_{-}^f\left(\omega \right){\text{E}}_{r+}^{ft}\left(\omega \right)={\text{ p}}_{-}^{ft}\left(\omega \right)-{\text{p}}_{+}^{ft}\left(\omega \right)$

${\text{p}}_{\pm }^{ft}\left(\omega \right)=\frac{1}{2\pi i}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{{\text{P}}^{ft}\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-{\omega }^{\mp }}}d{\omega }^{\prime }={\text{ q}}_{\pm }^{ft}\left(\omega \right)+\frac{1}{2\pi }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{H_{-}^f\left({\omega }^{\prime }\right)}{{\omega }^{\prime }\left({\omega }^{\prime }-{\omega }^{\mp }\right)}d{\omega }^{\prime }\text{E}}\left(t\right)\text{ (8}\text{.25)}$