Dielectric Materials with Memory II

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 Volume 1 | Issue 1 Journal of Atomic and Nuclear Physics SCHOLARLY PAGES Research Article Open Access Dielectric Materials with Memory II: Free Energies in Non- Magnetic Materials S Glasgow1* and JM Golden2 1Department of Mathematics, Brigham Young University, Provo, Utah, USA 2School of Mathematical Sciences, Dublin Institute of Technology, Dublin, Ireland

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 denoted by lowercase and uppercase boldface characters respectively and scalars by ordinary script. The real line A general theory of free energies for dielectric mate- + is denoted by  , the non-negative reals by  and the rials under isothermal conditions was given recently [1]. ++ − strictly positive reals by  . Similarly,  is the set The present work presents more detailed formulae and −− of non-positive reals and  the strictly negative reals. results for free energies in non-magnetic isotropic mate- Complex quantities arise in the frequency domain so we rials. We shall derive expressions for the minimum, max- have complex vector spaces for which the dot product in- imum and other intermediate free energy functionals volves using the complex conjugate of objects in the dual which also have an extremum property. Two approaches 2 space. The magnitude squared, denoted by . refers to the are possible: (1) To present the developments as in me- dot product of objects with their complex conjugates. chanics [2], based explicitly on thermodynamics; and (2) To use the method given independently [3-5] for di- General Relations electric materials with memory, but with quite diﬀerent Consider a rigid non-magnetic isotropic dielectric methods, notation and terminology. In this work, as in material subject to a varying electric field. Let the body [1], the fist approach will be adopted but with reference 3 under consideration occupy a volume ⊂ . A typi- to the connection between the two methods. The inter- cal point in B is x while t is a given time. The electric field mediate free energies are analogous to quantities known on this region is E(x,t), with electric displacement denot- in mechanics [2]. They are new and particularly physical- ed by D(x,t). The magnetic field and induction contri- ly relevant in the context of dielectrics, in that they apply butions are neglected. The space variables are generally to memory models that are standard for such materials, omitted. but not usually applied to viscoelastic behaviour.

Applications of the results of this work, from a phys- *Corresponding author: S Glasgow, Department of Math- ical point of view, include the establishment of bounds ematics, Brigham Young University, Provo, Utah 84602, on the level of dissipation in the material, using the fact USA, E-mail: [email protected] that the total dissipation associated with the minimum(- Received: January 20, 2017; Accepted: December 04, 2017; maximum) free energy is an upper(lower) bound on the Published online: December 06, 2017 actual physical dissipation. Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At On the matter of notation, vectors and tensors are Nucl Phys 1(1):16-29

Copyright: © 2017 Glasgow S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

• Page 16 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES

ered in [3-5], namely a passive, homogeneous isotropic Let ψ (t) be any free energy of the material, for the non-magnetic dielectric. All kernels are scalar quantities. isothermal case, and D(t) the rate of dissipation. Then We have the first and second laws of thermodynamics can be writ- 1 ∞∞ − tt ten as [1] (t) = er( t) ∫∫E.,E( s) G12 ( s u) r( u) dsdu (4.1a) . 2 00 ψ +D =D.E, D ≥ 0 . (3.1) ∞∞ 1 tt t + 3 a = −  E :  =  e (t) E( s) . G ( s , u )E ( u) dsdu , (4.1b) Let  be defined by 2 ∫∫ + 00 Et (s) = E( t−∈ s) , s  . (3.2) 1  (t) = - Gt E( ) .E ( t) , (4.1c) Equation (3.2) gives the history and also the current e 2 ∞ value E(t) of the electric field. A given future continua- where (4.1b) follows from (4.1a) by partial integration is denoted by tions. Also t −− ∂∂∂ E(s) = E( ts−∈) , s  . (3.3)  tttt E(uu) = E( ) = - E( u) = - Er ( u) . (4.2) Let us define the free enthalpy [1] as ∂∂∂ttt   = ψ − D.E . (3.4) G( su,) = G( su ,) - G∞ , and In terms of this quantity, (3.1) becomes ∂2 .. G( su,) ≡ G( su ,) , (4.3a)  +≥DD = -D.E, 0 . (3.5) ∂∂su 12 t + The relative history Er is defined by limG( su ,) = G∞ , u∈  , (4.3b) s→∞ Ett(s) = E( st) -E ( ) . (3.6) r ∂ + limG( su ,) = 0, u∈  , (4.3c) A relative future continuation is also defined by (3.6) s→∞ ∂ −− u for s ∈  . ∂ + limG( su ,) = 0, u∈  , (4.3d) We define the equilibrium free enthalpy e (t) to be s→∞ ∂ + s that given for the static history Et (s) = E( t) , s ∈  . G su, = G us , , (4.3e) Therefore, ( ) ( )  with similar limits at large u holding for fixed s. The ee(tt) = ( E( )) . (3.7) parameter G∞ in (4.1a) is defined by (4.3b). Relation emphasizing that the function e (t) depends only (3.8) gives the linear constitutive relation on E(t). ∞ Dt = G E t + G′ u Et u du A. Required properties of a free enthalpy ( ) ∞ ( ) ∫ ( ) r ( ) (4.4a) 0 Let us state the characteristic properties of a free en- ∞ thalpy, provable within a general framework [1,6-8]: = ′ t G0E+( t) ∫ G( u) E( u) du (4.4b) ∂ ()t 0 P1: = − D(t) . (3.8) ∞ ∂Λ()t t t = GE+( t) G ( u) E ( u) du (4.4c) P2: For any history E and current value E t ∞ ∫ a a ( ) 0 t ∞ (Eaa ,E(tt)) ≥ e( E a( )) . (3.9) t = ∫G( u)E ( u) du , (4.4d) P3: Condition (3.5) holds. 0 These will be referred to as the Graffi conditions by where analogy with those for a free energy in mechanics ([2] Gu( ) = G( 0, u) , (4.5a) for example). GG0 = ( 0,0) = G( 0) , (4.5b) A Linear Memory Model  Gu( ) = Gu( ) - G∞ , (4.5c) A special case of the linear model introduced in [1] is described in this section. This is the material consid- d G′( u) = Gu( ) . (4.5d) du a t Thequantity G(s) is the relaxation function of the ma- This quantity would be denoted as E− (s) in [3-5], while the t terial. In writing the final form of (4.4), we are assuming future continuation, defined by (3.3), would be written as E+ (s) . t The future continuation is referred to in [3] as the recovery field. that E(−∞) = E ( ∞) vanishes; we furthermore assume

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 • Page 17 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES that it goes to zero sufficiently strongly so that various where, using (4.4b), one sees that the polarization P(t) integrals, introduced below, exist. The quantity G∞ is re- is given by the causal relationship ∞ t lated to the relaxation function through t P(t) = ∫∫ G′′( u) E( u) du = G( t − s) E( s) ds . (4.15) G∞ = Gs( ,∞) = G( 0, ∞∞) = G( ) . (4.6) 0 −∞ We deduce from (3.5), (4.1a) and the time derivative Then, (4.11a) becomes 1 2 of (4.1b) that Wt( )φ0( t) += Wint ( t), φ∈ 00( t) E(t) , (4.16) ∞∞ 2 1 tt (4.7a) D( t) = E(s) . G12( s , u) + G( s , u) E ( u) dsdu φ t 2 ∫∫ where 0 ( ) is the quantity introduced in (4.10) and 00 Wt( ) is given by ∞∞ int = tt+ , (4.7b) ∫∫Err(s) . G112 ( s , u) G212 ( s ,E u) ( u) dsdu t 00 =  Wint ( t) ∫ P( s) .E ( s) ds , (4.17) where subscripts refer to partial diﬀerentiation with −∞ respect to the first or second argument. Equation (4.7b) is the accumulation of energy (density) transferred to follows from (4.7a) by partial integrations, using (4.1a), the medium at the point under consideration, from the t (4.3), and the fact that Er (0) vanishes. Relation (4.1a), beginning of the pulse-medium interaction until time t. expressed in terms of histories, becomes We can write (4.12) as 1  (t) = Gtttt E( ) .E( ) -D( ) .E( ) (4.8) ψ int (t) +=D ( t) Wtint ( ) , (4.18) 2 0 ∞∞ where ψ (t) is given by the integral term in (4.9a). 1 int − tt ∫∫E(s) . G12 ( s ,E u) ( u) dsdu . A. Minimal states 2 00 From (3.4), (4.2) and (4.8), we deduce that The fundamental definition of the state of a material ∞∞ with memory at time t is the history of the independent ψφ= − tt field variable and its current value (Et, E(t)). Also, diﬀerent (t) 0 ( t) ∫∫ E( s) . G12 ( s ,E u) ( u) dsdu (4.9a) 00 histories may be members of the same minimal state [1]. ∞∞ t t 1 Two states (E11 ,E (t)) , (E22 ,E (t)) are equivalent, or in the = − tt U( t) ∫∫Err( s) . G12 ( s ,E u) ( u) dsdu , (4.9b) same minimal state if from a time t onwards, we have 2 00 D1(ts+=) D 21( ts +) if E( ts +=) E 2( ts +) , so s ≥ 0 (4.19) 1 ∞∞ = U t − Ett s . G s ,E u u dsdu , (4.9c) ( ) ∫∫ ( ) ( ) ( ) Where D1, D2 are defined by (4.4) for these states. 2 00 Then it follows that Where ∞ tt 11′ +− =≥ φ = = − G( s u)(E12( u) E( u)) du 0, s 0 (4.20) 00(t) GttUt E( ) .E( ) , ( ) D( tt) .E( ) Gtt∞ E( ) .E ( ) (4.10) ∫ 220 The integral terms, with the negative signs included, A fundamental distinction between materials is that are non-negative [1]. The total work done by the electric for certain relaxation functions, namely those with only field up to timet t is isolated singularities (in the frequency domain), the set W( t) = ∫ D ( u) .E( u) du (4.11a) of minimal states is non-singleton, while if some branch −∞ cuts are present in the relaxation function, the material t has only singleton minimal states [2,9]. For relaxation = D(t) .E( t) − ∫ D( u) .E ( u) du . (4.11b) functions with only isolated singularities, there is a maxi- −∞ mum free energy that is less than the work function W(t) and also a range of related intermediate free energies, Under the assumption that ψ (−∞) = 0 , an integra- which are discussed later. On the other hand, if branch tion of (3.1) over (−∞,t) yields cuts are present, the maximum free energy is W(t). ψ (t) +=D ( t) Wt( ) , (4.12) In this work, we will deal exclusively with the case

Where t where the relaxation function has only isolated singular- D (t) = ∫ D( u) du ≥ 0 (4.13) ities, so that the minimal states will be non-singleton. −∞ For such materials, the free energy functional is posi- is the total dissipation up to time t. We take G0 to be tive semi-definite ([2], page 152). ∈ equal to the vacuum permittivity 0 [3-5], giving t t Note that the statement that (E11 ,E (t)) and (E22 ,E (t)) =∈ + t D(t) 0 E( tt) P( ) , (4.14) are equivalent is the same as the assertion that (Ed ,0) is

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 • Page 18 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES equivalent to the zero state (0,0), where 0 is the zero in We can write the Fourier transforms of ′ and 3 Gu( )  (and also the zero history), while Gu ( ) in (4.4) as t tt Ed (s) = E12( ss) − E( ) . (4.21) G++′(ω) =−==− Gcs ′′( ωω) iG ( ) χ( ω) χχcs( ωω) i ( ) , (5.6) A functional of (Et, E(t)) which yields the same val- Gω = G  ωω − iG ue for all members of the same minimal state will be re- + ( ) cs( ) ( ) . ferred to as a functional of the minimal state (abbreviat- The quantity χ+ (ω) is the susceptibility, denoted by ed to FMS) or as a minimal state variable. χ (ω) in [3-5]. By partial integration, one can show that Et ,E t Et ,E t  Let ( 11( )) , ( 22( )) be any equivalent states. G+∞′ (ω) =−−+ ( GG0 ) iωωG+ ( ) , (5.7) Then, a free energy is a functional of the minimal state if giving, in particular, that ψψEtt ,E(tt) = E ,E ( ) . ( 11 ) ( 2 2 )  GGsc(ω) = χs ( ω) = − ωω( ) . (5.8) It is not necessary that a free energy have this property, though it holds for the minimum and all related free Observe that ∞ energies introduced later. ′′= = − G+∞(0) ∫ G( s) ds G G0 . (5.9) Kernels and Field Variables in the Frequency 0 Domain Using (5.2), we see that 2 ′ For any fL∈ () , we denote its Fourier transform G+ (ωω) = χ++( ) = G′ (− ω) = χ +(−ω) , (5.10) ∈ 2 b fLF () by which will be useful later. By considering periodic be- ∞ havior in Est ( ) , we can show that a consequence of the ω =ξξ−ξiω fF ( ) ∫ fe( ) d (5.1a) second law is that ([2], page 140) −∞ GG′ (ω) = χ ( ωω) = − (ωω) > 0, 0 < <∞, (5.11) = ff(ωω) + ( ) , (5.1b) ss c +− > ∞ It follows from (5.11) that GG∞ 0 [1]. We have − ωξ = ∈> > ω= ξξi G00 0 , so that G∞ 0 . f+ ( ) ∫ f( ) ed, (5.1c) 0 B. The complex frequency plane and the function ∞ H(ω) f(ω) = f( ξξ) ed−iωξ , (5.1d) − ∫ We will be considering frequency domain quantities, −∞ defined by analytic continuation from integral defini- Only real valued functions will be considered so that tions, as functions on the complex ω plane, denoted by ffFF(ω) =−∈ ( ωω) ,  (5.2) Ω, where Where the bar denotes the complex conjugate. For Ω++ = {ωω ∈Ω | Im[ ] ∈ } complex values of ω , (5.2) becomes Ω(+) = {ωω ∈Ω | Im[ ] ∈++}. ff(ωω) = ( − ) . (5.3) FF − (−) + Similarly, Ω and Ω are the lower half-planes in- Functions defined on  are identified with func- −− cluding and excluding the real axis, respectively. tions on R which vanish identically on  . For such functions, The quantities f± (ω) , defined by (5.1), are analytic in Ω( ) respectively ([2], page 547). Thus, the quantity fF(ω) = f+ ( ω) = fcs( ωω) − if ( ), (5.4) ∓ (−) G+′ (ω) is analytic on Ω . It will be assumed that G+′ (ω) is analytic on  and thus on Ω−, or more precisely, on Where fc (ω) , fs (ω) are respectively the Fourier co- sine and sine transforms. A property of Fourier trans- an open set containing Ω−. Also, it will be taken to be forms which will be used later is analytic at infinity. This function is defined by analytic ff(00) ( ) continuation in regions of Ω+ where the Fourier integral ffωω  → , so that   → (5.5) + ( ) ωω→∞ s ( ) →∞ ′ iωωdoes not converge. The quantity Gs (ω) has singularities in both Ω(+) and Ω(−) that are mirror images of each oth- If ( ) is non-zero. er. It goes to zero at the origin and must also be analytic A. The kernel G(u) there. A quantity central to our considerations is defined by 2 b HG(ωωωω) = ′ ( ) = χ (ω) = − ωωG ( ) The quantity fF (ω) , defined by (5.1), would be denoted in [3-5] ss c. (5.13) ˆ ω by 2πωf (− ) or 2πωfˆ ( ) for real . It is a non-negative, even function of the frequency

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 • Page 19 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES and goes to zero quadratically at the origin. The relation where U(t) and φ0 (t) are defined by (4.10). We see (see (5.5)) from (6.1) that W(t) can be cast in the form (4.9) by put- ting iGlimωω+′( ) = lim ωω G ′′( ) = G( 0) (5.14) ωω→∞ →∞ s = − yields G12 ( su,) G12 ( u s) . (6.2) In terms of frequency domain quantities, we find that GHH′(0) = ( ∞=) ∞ . (5.15) [1] For the model considered later, H(ω) goes to zero at 1 ∞ = − ttω ω ωω large ω so that H∞ = 0. The Fourier transforms of the Wt( ) Ut( ) ∫ E.rr++( ) H( ) E( ) d , t 2π −∞ history and continuation are denoted by E+ (ω) and t c ∞ E− (ω) respectively . These are particular examples of t 1 tt (5.1c) and (5.1d). The quantity E+ (ω) is analytic on = φ0 (t) + E.++(ω) Hd( ω) E( ωω) . (6.3) t ∫ (−) (+) 2π −∞ Ω and E− (ω) is analytic on Ω . Both are assumed to be analytic on an open set including  . It is further Thequantity W(t)-D(t)·E(t) can be shown to obey the assumed that they are analytic at infinity. properties specified in subsection III A of a free enthalpy, The Fourier transforms of the relative history and with zero dissipation. Because of the vanishing dissipa- continuation, defined by (3.6), have the form tion, it must be the maximum free energy associated with the material or greater than this quantity, an observation ttE(t) which follows from (4.12). For relaxation functions with Er±±(ωω) = E( )  , (5.16) iω  only isolated singu-larities, as introduced later, there is a where the notation ω± is defined asω ± iα, α > 0. The maximum free energy which is a functional of the mini- parameter α is assumed to tend to zero after any integra- mum state and is less that W(t). tions have been carried out (see for example [2], page Consider the scalar product on the function space of 551). We have electric fields, defined by [2,5,9-12] ∞∞ td tt t  ω= ω ωω+ ω ω tt t t t t E+( ) E ++( ) = -i E( ) E( ti) = - Er+( ) (5.17) E ,E = E ,E = Es . G s− u E u dsdu dt ( 12) ( 1FF 2) ∫∫ 1( ) ( ) 2( ) (6.4a) The second relation follows from (3.2) and an inte- −∞ −∞ t 1 ∞ gration by parts in the Fourier integral defining E+ (ω) . t t tt t t (6.4b) = E1F(ω) .Hd( ω) E2F( ωω) = (E21 ,E) = (E 2FF ,E 1) π ∫ Also, based on arguments from [1] and [2], page 146, for 2 −∞ t example, we can express the constitutive equation (4.4) Recalling (5.1), we see that the quantity EF (ω) is the in the form Fourier transform of Et (s) , s ∈  in the time domain. i ∞ H (ω) Also, the subscript F in the bracket notation indicates D(tGt) = E( ) + Et (ωω)d ∞+π ∫ ω r that the frequency domain version is being used. The re- −∞ lated norm is ∞ 22 1 H (ω) t tt t t = GtEE( ) −  (ωω)d (5.18) (E ,E) = E = EF . (6.5) ∞+π ∫ ω 2 r −∞ In the notation of (6.5), the work function is given by ∞ ω t 2 i H ( ) t = + = GtEE( ) + (ωω)d Wt( ) Ut( ) Er (6.6a) 0 π ∫ ω + −∞ 2 =φ + t The Work Function 0 (t) E , (6.6b) t −− Various forms for the work function, defined by (4.11), where Er (s) = 0, s ∈  in (6.6a) and −− are given for the general tensor case in [1]. Those required in Et (s) = 0, s ∈  in (6.6b). this work for scalar kernels are summarized here. The analogous mechanics version of these relations may be found in Factorization of H (ω) [2], page 153, for example. It can be shown that ∞∞ ′ ω  ω 1 tt We consider materials such that G+ ( ) (or G + ( ) ) W( t) =−− U( t) G( u s)E( u) .E ( s) dsdu (+) 2 ∫∫ 12 rr has only a finite number of isolated singularities in Ω . 00 Thus, H(ω) has only isolated singularities in Ω(±), which 1 ∞∞ are mirror images of each other in the real axis, as as- =−−φ tt 0 (t) ∫∫ G12 ( u s)E( u) .E ( s) dsdu , (6.1) cribed to ′ ω before (5.13)). This means that it can be 2 Gs ( ) 00 put in the form of a ratio of polynomials. We will take the c t ω singularities to be a finite number of simple poles. The Referring to footnote a, we see that Er± ( ) would be denoted ± ˆ t quantity H(ω) has a finite number of zeros in Ω (ω), also in [3-5] by 2Eπωr (− ) .

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 • Page 20 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES mirror images of each other. It is real and non-negative pathways leading from the minimum to the maximum on  , an even function of ω and therefore a function of free energy. ω2, in view of its analyticity about the origin. It vanishes The most general free energy and rate of dissipation quadratically at the origin. This non-negative quantity arising from these factorizations is given by can be factorized in general as outlined in [2,13]. Thus, N NN ψt = λψ t , Dt= λ D t , λ= 1, λ ≥ 0 we have that ( ) ∑ff( ) ( ) ∑∑ff( ) f f . (7.4) f=1 ff=11=

HH+−(ωω) ( ) = H( ω) , (7.1) A particular case of this linear form is the free energy proposed in [11]. where Remark VII.3. The set of all free energies at time t HH(ω) = ( −= ωω) H( ) (7.2a) ±  associated with a given material is denoted by Φ(t). The 2 boundary of this set is where one or more of the fun- HH(ω) = ± ( ωω) , ∈  . (7.2b) damental properties listed in subsection III A is break- The factorization (7.1) is unique up to multiplication ing down; an example, which will arise later, would be iα by a phase factor e , where α is a constant. Relation (7.2a) where, for a given non-zero history, the rate of dissipa- reduces this arbitrariness to multiplication by a factor ±1. tion is zero, so that a small shift in its kernel parameters The factorization (7.1) is equivalent to a homogeneous results in it becoming negative. Riemann-Hilbert problem on the half-plane [1,3-5]. All the ψ f (t) emerge from extremum arguments. This is apparent in the case of the minimum free energy The quantity H+(ω) (H−(ω)) has all its singularities in Ω(+) (Ω(−)) and all its zeros in Ω+ (Ω−) ([2], page 239). for a given state, which is obtained by finding an optimal t − There are many other factorizations, obtained by inter- continuation Em (s) , s ∈  yielding the maximum recoverable work from this state. Also, the maximum changing some or all of the zeros of H+(ω) and H−(ω), while retaining the same singularity structure. The differ- free energy is determined by finding the optimal histo- t + ent factorizations are labeled by the subscript or super- ry EM (s) , s ∈  which minimizes the work done to script f. We have achieve the given state. It must be an equivalent state to the given history. In other words, the two histories must Hω = HHff ωω, (7.3a) ( ) +−( ) ( ) be in the same minimal state. The intermediate free en- ff f ergies are also obtained from an extremum principle, but HH±±(ω) = ( −= ωω) H( ) . (7.3b)  involving an optimization of the history/continuation Certain types of exchanges must be excluded to en- Et(s), s ∈  , as we shall see in section VIII. sure that (7.3b) are true [11], ([2], page 338). Each factorization generally yields a different free energy, though The relevant theoretical developments motivating the there may be exceptions. All these free energies are FMSs results of section VIII are presented in [2], chapter 15 for [2]. The factorization with no exchange of zeros, which mechanics; they also apply to dielectrics. We will pres- is that given by (7.1), yields the minimum free energy ent an abbreviated version of these arguments, and also demonstrate that the functionals which emerge are in ψm(t). fact free energies, with the required properties. It can be Remark VII.1. Each exchange of zeros, starting from shown that all these quantities, including the intermedi- these factors, can be shown to yield a free energy which is ate functionals, are on the boundary of Φ(t) for the mate- greater than or equal to the previous quantity ([2], page rial ([2], page 365), which is consistent with the fact that 363). they emerge from an extremum principle. This property f Note that the zeros of H± (ω) at the origin play no is discussed briefly below. part in these exchanges. Remark VII.2. A particularly interesting choice of dThe minimum and maximum free energies are given also in ψ f (t) , which we denote by ψ M (t) , is obtained by in- [5]. The intermediate free energies, introduced in this work, terchanging all the zeros. This can be identified as the have not been discussed previously in the context of dielec- maximum free energy among all those that are FMSsd. trics, but are analogous to those derived in mechanics in for It is less than the work function, which is not a FMS for example [2,10-12]. materials with only isolated singularities [2]. eThis behavior, while often used to describe memory behavior of dielectrics, is not usually applied to viscoelastic mate- If there are N different factorizations of H(ω), then f = rials. The singularity structure given by (9.4) corresponds to N is chosen to denote the maximum free energy. exponential decay with sinusoidal behavior in the time domain, while viscoelastic materials are generally modeled by simple Note that there are several (indeed many, for a large exponential decay, which in the frequency domain yields sim- number of isolated singularities) different zero exchange ple poles on the positive imaginary axis.

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 • Page 21 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES

The Free Energy Associated with a Particular The optimization problem which yields the free en- Factorization ergy functional associated with a given factorization of t t H(ω) is given as follows ([2], pages 346, 350). We mini- t 2 Let us define the quantities ER (s) and Ed (s) by mize ERF , subject to the constraint the relation t tt ∞ H ftωω'E ' ER (s) =+∈ E( s) Ed ( ss) ,  , (8.1) ft 1 − ( ) dF ( ) u− (ωω) = + d ' = 0. (8.9) t π−∫ ωω where E (s) is the given history, which is zero for 2'i −∞ −− t s ∈  . The quantity ER is typically non-zero on  The resultant free energy is t and E follows from (8.1). In the frequency domain, the 2 d ψφ= + t relation becomes ff(tt) 0 ( ) E , (8.10) t t t t t tt where E is the history/continuation E that is ERF (ω) ≡+ E RR+−( ω) E( ω) =+ E +( ω) EdF ( ωω) , ∈ , (8.2) f RF the solution of the constrained minimization problem. t ω where the quantities ER± ( ) are particular cases of The formulation of this problem is discussed at length 22= (5.1c) and (5.1d). Consider the norm ERR EF in [2], chapter 15 and earlier papers. In particular, it is , defined by (6.5). This can be written in a form similar to shown that each ψ f (t) has all the required properties the quantity W(∞) in [1], from which the minimum free of a free energy, which will in any case be shown later for energy was deduced. We have, from (6.4b) and (6.5), that ∞ the present context. 2 1 Et =++  EE.EEttωωωωωωHdtt RF ∫ ++( ) dF ( ) ( ) ( ) dF ( ) . (8.3) 2π −∞ Using (8.6), (8.7c) and (8.9) in (8.5) together with Let us choose a particular factorization of H(ω), as proposition 1 of [1], we find that given by (7.3). Also, we define [1] f t ft ft H−(ωω)EdF ( ) = q++( ω) − qR ( ω) , (8.11a) fωωωω t −−ft ft H−+( )Eq( ) −( ) q +( ) , (8.4a) ∞ 2 t 2 1 ft ft ∞ = ω− ωω ft ERF ∫ q−+( ) qR ( ) d (8.11b) ft 1 H−+(ωω'E) ( ') 2π qωω = d ' −∞ ± ( ) ∫  . (8.4b) 2'π−i −∞ ωω ∞ 1 ft 22ft ft = q(ω) +q ( ωω) d , (8.11c) q ω ∫ − R+ where the ± ( ) are analytic in Ω , respectively. The 2π −∞ ft ∓ f singularities of q (ω) are the same as those of H− (ω) , as − ft (−) ft may be perceived by closing the contour in (8.4b) on Ω . where qR+ (ω) is given by (8.8). Since qR- (ω) t Singularities on the real axis are excluded by assumption. depends only on the given history E+ (ω) , the solution 2 Using the factorization in (8.4), we can express ERF to the minimization problem is obtained by choosing tt in the form ERF (ωω) = Ef ( ) to give that

∞ 2 ft 2 1 Et = qft (ω) −+ qft ( ω) Hdf( ω) E t ( ωω) . (8.5) qR+ (ω) = 0, (8.12) RF 2π ∫ −+ −dF −∞ yielding Let us further define ∞ 2 2 1 f t ft ft Et = qft (ωω) d . (8.13) H−(ωω)EdF ( ) = u−+( ω) − u( ω), f ∫ − 2π −∞ ∞ ft From (8.11a) and (8.12), we obtain that ft 1 H− (ωω'E) dF ( ') uωω = d ' (8.6) ft ft ft ft ± ( ) ∫  ω ω= ω ω −= ωω ω H−( )EdF ( ) HH−( ) ERF ( ) −+( ) E( ) q +( ) 2'π−i −∞ ωω . (8.14) Note that, by virtue of Cauchy’s integral formula, Then, (8.4a) gives ft ft f t ftω ω= ft ωω= ft ω u−(ω) = qdd −( ω) + H −−( ωω) E( ) (8.7a) HH−( )ERF ( ) −−( ) Ef ( ) q( ) , (8.15) ft ft f t = qqR−(ω) −+ −( ω) H −−( ωω) Ed ( ) , (8.7b) and ft ∞ f t q−(ω) 1 H−+(ωω'E) ( ') ft ft ft ft Et ωω = = d ' u+(ω) = qd +( ω) = qR ++( ωω) − q( ) , (8.7c) f ( ) ff∫ + .(8.16) H−−(ω) 2'π−iH ( ω) −∞ ωω ft Where qd ± (ω) are the quantities defined by (8.4b) The observation after (8.4) about the singularities of ft f t q− (ω) and H− (ω) yield that the singularity structure but with E'+ (ω ) replaced by t f of E f (ω) is determined by the zeros of H− (ω) . Et(ω ') = E tt( ωω ') − E( ') . Also, d +R ++ Remark VIII.1. Note that for f ≠ 1, N, the quantity ∞ ft f − + ft 1 H−+(ωω'E) R ( ') H− (ω) has some of its zeros in Ω and some in Ω . For f ωω= f t qR± ( )  d '. (8.8) − ∫ = 1, all the zeros of H− (ω) are in Ω , so that E f (ω) has 2'π−i −∞ ωω

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− singularities only in Ω and its inverse Fourier transform ft − limωω q+ ( ) = itK ( ) , (8.19a) is thus non-zero only on  . This latter quantity is the ω →∞ f negative of the optimal continuation, yielding the mini- ft limωω q− ( ) = iK( tH) − E ( t) , (8.19b) mum free energy, which is discussed for the general tensor ω →∞ ( f sr ) case in [1]. It can be shown that2 both the constraint (8.9) t 11∞ and the minimization of ERF result in the same rela- ft ±=−ωω ∫ q+ ( )dt Kf ( ) , (8.19c) tion, namely (8.16) for f = 1. 22π −∞ f + For f = N, all the zeros of H− (ω) are in Ω , which gives ∞ t + 11ft that E f (ω) has singularities only in Ω . Therefore, its in- q(±=−−ωω)d (KEf ( tH) sr ( t)) . (8.19d) + ∫ verse Fourier transform is non-zero only on  and is 22−∞ the optimal history yielding the maximum free energy. In t 2 this case, the quantity ERF to be minimized is the work Remark VIII.2. For the material considered in section function and (8.9) is the condition that the optimum his- IX, the quantity H(ω) vanishes for large values of ω. Thus, tory and the given history are in the same minimal state. Hsr in (8.18) and (8.19) is zero. Also, ∞ f t ω H− (ω) For intermediate cases, E f ( ) is non-zero on  and = 0, (8.20) t 2 ∫ ω minimizes ERF , subject to the constraints (8.9), for var- −∞ ious factorizations. These constraints are transitional be- as can be seen by closing the contour on Ω(+), so that, tween the two extremes discussed above. recalling (5.16), it is clear that (8.18c) is replaced by We will see that all of these cases result in free energies ∞ 1 ft that are on the boundary of Φ(t), defined by remark VII.3. Kf (tH) = −+(ω) E( ωω) d. (8.21) 2π ∫ This property is apparent for the minimum and maximum −∞ free energies. For these and all intermediate cases, we will The work function, given by (6.3), takes the form [1] ∞ see that the corresponding rates of dissipation are posi- 1 ft 22ft Wt( ) =++ φ( t)  qq( ω) ( ωω) d . (8.22) 0 ∫ −+ tive semi-definite rather than positive definite quadratic 2π −∞ forms. They are zero for certain non-zero histories so that From (4.12), (8.17) and (8.22), we deduce that the to- small variations in their kernels may lead to negative rates tal dissipation corresponding to the minimum free ener- of dissipation, for these histories, in contradiction to the gy is given by 2 second law, as given by (3.1). t 1 ∞ = = ft ωω. (8.23) Dtff( ) ∫∫ Dudu( ) q+ ( ) d Thus, the fundamental physical content of the con- −∞ 2π −∞ strained optimizations is that they lead to free energies on Diferentiating this relation with respect to t and us- the boundary of Φ(t). ing (8.18a), (8.19c), gives The free energy (8.10) has the form 2 2 ∞ Dtff( ) = K ( t) (8.24) 1 ft ψf (tt) = φ0 ( ) + q− ( ωω) d (8.17a) ∫ Since Kf(t), given by (8.21) can vanish for non-zero his- 2π −∞ tories, Df(t) is a positive semi-definite rather than a positive 1 ∞ definite quadratic form, as noted in remark VIII.1. =φ + ttω ω ωω 0 (td) ∫ Eff( ) .H( ) E ( ) , (8.17b) 2π −∞ We can re-express these results in terms of relative by virtue of (8.13) and (8.15). We have the following histories [1]. Let us put relations ([1,10,13,14] for example) ft f ft ft ft P(ω) = H−+( ωω) Er ( ) = pp −( ω) − +( ω) , d ft ft q++(ω) =−− it ωω q( ) Kf ( ) , (8.18a) ∞∞ft ωω′′f dt ft 11P ( ) ft H− ( ) p±±(ω) = d ωω′′ = q( ) + dtωE( ) . (8.25) ∫∫′   22π−i −∞ ωω π−∞ ωω′′( − ω) d ft ft f + q−−(ω) =−−+ i ωω q( ) Kf ( tt) HE −( ω) ( ) , (8.18b) dt By closing the contour on Ω , it emerges that f ∞ H ω 1 ft ft − ( ) = ftω ωω p−−(ωω) = q( ) − E(t) , (8.26a) Kfr(td) ∫ H−+( ) E( ) (8.18c) iω 2π −∞ ft ft ∞ f p++(ωω) = q( ) . (8.26b) 1 H− (ω) = − E t (ωω)d . (8.18d) ∫ + Also, the work function, given by (8.22), becomes 2πi −∞ ω ∞ 1 22 and Wt( ) =++ Ut( )  pft (ω) pft ( ωω) d , (8.27) ∫ −+ 2π −∞

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while (8.17b) and (8.23) are given by ∞ 2 ∞ 1 ft 1 2 =−+ St( ) q− (ωω) d (8.35b) ψ = + ft ωω 2π ∫ f (t) Ut( ) ∫ p− ( ) d , (8.28a) −∞ 2π −∞ ∞ 2 1 ft ∞ = F e (t) + Pd− (ωω) , (8.35c) 1 ft 2 ∫ = ωω. (8.28b) 2π −∞ Dtf ( ) ∫ p+ ( ) d 2π −∞ where Equation (8.24) is unchanged. St( ) = D( t) .E ( t) −φ0 ( t) , (8.36) We now briefly present double integral forms of ψ (t), f and F (t) is given by (4.1c). It is easy to show that F (t) D (t) and D(t), which are analogous to (though simpler e f f obeys the Graffi conditions listed in subsection III A. than) the corresponding formulae in [1] for the mini- Property P2 is immediately apparent. The relation (4.12) mum free energy in the full tensor case. Using (8.17b) holds by virtue of (8.17), (8.22) and (8.23). The time de- and (8.28a), we can write ψf(t) in the forms ∞∞ tt rivative of (4.12) yields (3.1) with Df(t) given by (8.24), i EM++(ω).,( ωω) E( ω) ψ= φ + f 12 2ωω f (t) 0( t) 2 +− dd12 (8.29a) 4π∫∫ ωω− which is equivalent to P3. Property P1 can be proved −∞ −∞ 12 with the aid of (8.35b), by showing that ∞∞ ttω ωω ω i EMrf++( 1).,( 12) E r( 2) ∂St( ) = Ut( ) + +− dωω12 d (8.29b) = π2 ∫∫ ωω− D(t) . (8.37) 4 −∞ −∞ 12 ∂E(t) ff Mf (ωω12,) = HH+−( ω1) ( ω 2) , (8.29c) Remark VIII.3. It was shown in [14] (see also [2], page 340) that q ft ω , defined by (8.4), is a function of Also, D (t), given by (8.24), can be expressed as − ( ) f the minimal state in the sense defined after (4.21). This i ∞∞ Dt = Ettω., M ωω E ω dd ω ω result transfers to the present context without alteration. f( ) 2 ∫∫ rf++( 1) ( 12) r( 2) 1 2 (8.30) 4π −∞ −∞ From (8.17), we deduce that ψf(t) is a function of the min- From (8.23) and (8.28b), we deduce that imal state as defined by (4.22). ∞∞ ttω ωω ω i EM++( 1).,f ( 12) E( 2) For the minimum free energy ψ (t), we denote ψ (t), Dt( ) = ddωω m int f π2 ∫∫ ωω−+− 12 4 −∞ −∞ 12 introduced in (4.18), as ψrec(t), given by

∞∞ tt ∞ 2 i EMrf++(ω1).,( ωω 12) E r( ω 2) 1 t = − −+ ddωω12 ψ(t) = qd− (ωω) , (8.38) π2 ∫∫ ωω− . (8.31) rec ∫ 4 −∞ −∞ 12 2π −∞ One finds that t where q− (ω) is defined by (8.4) for f = 1. This is the ∞∞ tt EM++(ω).,( ωω) E( ω) recoverable energy defined in [3]. The quantity Dm (t) 1f 12 2ddωω = 0 (8.32) ∫∫ ωω+−− 12 , given by (8.23) for f = 1, is that referred to as the ir- −∞ −∞ 12 recoverable energy in [3]. Corresponding quantities by integrating over ω2 for example and closing the ψφ(tt) − ( ) and D (t) , for each f, may be similarly − f f f 0 f contour on Ω , since H+ (ω) and E+ (ω) have no labeled. singularity in the lower half plane. This allows us to write (8.29a) in the explicitly convergent form The optimal history/continuation, given by the in- ∞∞ ttω ωω− ωω ω verse transform of (8.1), can be shown to have a discon- i EM++( 1).,ff( 12) M( 21 ,) E( 2) ψ(t) = φ( t) + ddωω f 0 π2 ∫∫ ωω− 12. (8.33) 4 −∞ −∞ 12 tinuity at the origin which is infinite ifH sr = 0, while at t The fact that the integral term is non-negative implies = ±∞, it is non-zero ([1,2], page 354). that [1] A Detailed Dielectric Model  f f′′ f ff   (ω) = iH+( ω) H −( ω) −≥ H +−( ωω)  H( ) 0, We have ω ∈ ω  . (8.34) H (ω) = ωωχs ( ) =−−≥ ( χχ++( ω) ( ω)) 0 , ω ∈  , (9.1) 2i By invoking (5.17), we see that the quantity Et ω r+ ( ) by virtue of (5.11). As in [3-5], we take the suscepti- ω  t ( ) bility to be modeled by a sum of Lorentz oscillatorse may be replaced by iE+ ω in the above double integral expressions. N 2 fnω pn χ (ω) = , (9.2) The free enthalpy F (t) corresponding to ψ (t) may be + ∑ 22 f f n=1 ωωnn−−iγ ω deduced from (3.4) to be in terms of the oscillator strength fn, the plasma fre- F (t) = ψ ( t) − Dt( ) .E ( t) (8.35a) ff quency ωpn, the resonant frequency ωn and the damping

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rate γn of each oscillator. All of these are positive param- zeros may also coalesce. Indeed, we cannot exclude the eters. It is assumed that possibility that some of the zeros in (9.8) have a power γ 2 higher than unity, even if all the singularities are simple ω 2 > n . (9.3) n 4 poles. For simplicity, it is assumed that this does not hap- pen for our choice of parameters. The singularities of χ+(ω) are of course isolated. They are simple poles at Let us define for l = 1, 2, . . ., 2N −2, 1 η +  1   l 1 ζ=[i γσ + ] , ζ′ = [i γσ −=−] ζ, (9.4) ζ l +1  ,l odd  n2 nn n2 nn n  ,l odd  η′ 2 l  αβll =  , = 2  22 ζ ′l ,l even   σγn = 4ω nn − n = 1, 2, …, N. η′ ,l even 2  l , (9.10)  2  Thus, they occur in pairs, equidistant from the posi- and tive imaginary axis, as required by (5.3). Separating the ′ poles in (9.2), we can write αζ2 N −=1 N , αζ2 N = N . (9.11) N 2 fnω pn 11 We can write (9.5) as χ+ (ω) = − . (9.5) 22NN ∑ σζωω−−ζ ′ ggll(+) n = 1 n nn χ (ω) = = − , αl ∈Ω , (9.12) + ∑∑ω −α ω +α In the time domain, this translates into ll = 11 = 11 N 2 where (5.10) has been used and fnω pn ζ ζ ′ + Gs′( ) = i eisn − eis , s ∈  , (9.6) ∑ ( )  ω 2  n = 1 σ n fi pi l +1 which are decaying exponentials multiplying sine −=,il , odd, σ i 2 functions [15], so that there will be oscillatory behavior gl=  2 (9.13) superimposed on the exponential decay. From (9.2) and fiω pi l ,il = , even. (9.1), we have  σ i 2  N 22 In this notation, (9.6) becomes fnωωpnγ n ω = 2N H ( ) ∑ 2 , (9.7) 2 2 22 isαl n = 1 G′ s = i le (ωωnn− ) + γ ω ( ) ∑ g (9.14) l = 1 which can be written as a ratio of two (factored) poly- 2N −isαl nomials: − = − ∏N 1 ωωωω −−−−ηηηη′′ i∑ g le . (9.14b) 2 l = 1 ( llll)( )( )( ) HH(ωω) = l = 1 1 ∏−N (ωω−−ζ )( ζζ′′) ω ζ ω − (9.8a) l = 1  llll( )( ) Using (9.14b) in (4.20), we see that two electric field N histories are in the same minimal state if and only if 2 Hf1 = ∑ nω pnnγ , ω ∈  , (9.8b) tt n=1 E11++(αα) = E 21( ) , lN= 1,2,...,2 . (9.15) where the denominator of (9.8a) uses the notation of Thus, a minimal state is defined by the quantities (9.4) while the numerator is factored to yield the zeros of (e ,e ,...e ), where H(ω); these must occur also in pairs, as in (9.4), so that 1 1 2N = t α = ηηll′ = − for each l. We have explicitly included the fact el (t) E+ ( 1 ) , lN 1,2,...,2 . (9.16) that H(ω) vanishes quadratically at the origin. Note that Relation (5.3) gives the smaller number of zeros reflects the fact that H(ω) −2 tt behaves as ω for large ω. el(t) = E++(αα ll) =−= E( ) , lN 1,2,...,2 . (9.17)

The factorization can be carried out by inspection. Formula (5.17) for ωα = 1 yields We obtain el(t) =−+ itαll e( ) E( t) . (9.18) ∏−−N −1 (ωωηη)( ′) Hh(ω) = l = 1 ll, − 1 N ′ Equation (4.4b) can be written as ∏−−l = 1 (ωωζζ ll)( ) 22NN Dt =−=+ GtiEgg e t GtiE e t (−−) ( ) ( ) 00( ) ∑∑ll( ) ( ) l l( ) , (9.19) ll = 1 = 1 ζηll∈Ω, l = 1, 2, ..., ∈Ω , lN = 1, 2, ... -1. (9.9) with the aid of (9.14), so that HHH+−−(ωω) = ( ) = ( − ω) , 2N 2 Dt = GE ti − g e t. (9.20) = ( ) 0 ( ) ∑ ll( ) where hH11 . l = 1 The most general case of a rational function, which If a free energy or free enthalpy is to depend only on is considered in [2,11], can be obtained from (9.8) by al- the el(t), l = 1,2,...,2N, then the most general forms of  lowing singularities to coalesce. As a result of this, some G( su, ) and G12 ( su, ) in (4.2) and (4.9) are

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2N 2N αα− R  iij si u = ω i G( su,) = − ∑ Ceij , (9.21) h1 ∑ , (9.28d) ij, = 1 i = 1 ωα+ i where (7.3) has been used. The relationship 2N αα− G su, = −= Ce′′iij si u, C Cαα 2N 12 ( ) ∑ ij ij ij i j , = ij, = 1 ∑ Ri 0 , (9.29) i = 1 where C and C′ are non-negative hermitian matri- − must hold since H (ω) tends to hω 1 at large ω. ces. Then, (4.9) gives that − 1 Therefore, we can also write (9.26b), (9.28c) and (9.28d) 2N 1 as ψφ(tt) = ( ) + C′ e . e 2N 0 ∑ ij i j αiiR 2 ij, = 1 Hh− (ω) = 1 ∑ i = 1 ωα- i 1 2N = U( t) + Ce . e . (9.22) 22NN ∑ ij i j ααiiRRii 2 ij, = 1 Hh+ (ω) = = h (9.30) 11∑∑ωα-+ ωα Relation (4.5a) gives ii = 1 ii = 1 2N g j We identify also a much larger class of factorizations ∑ Cij = − , j= 1, 2,..., 2 N. (9.23) of H(ω), determined by interchanging particular βl in i = 1 α j (9.26a) with βl in (9.28a). These diﬀerent factorizations Taking the complex conjugate of this relation yields are labeled by the subscript or superscript f. Thus, 2N 22N − f gi ∏−ω ζ C = − , i = 1, 2,..., 2 N . (9.24) f l = 1 ( l ) ∑ ij Hh(ωω) = (9.31a) j = 1 αi − 1 2N ∏−ll = 1 (ωα) Equation (3.1) is satisfied if 1 2N Dt = Γ e .e , (9.25) (9.31b) ( ) ∑ ij i j ∑ ωα 2 ij, = 1 1 2N α R f Γ=−ij i(ααi − j) C ij, i , j = 1, 2, ..., 2N . ii = h1 ∑ (9.31c) i = 1 ωα− i The matrix with components Γij must also be hermitian and non-negative. (−−11) ( ) αβll∈Ω , lN = 1, 2, ..., 2 ∈Ω , lN-2 = 1, 2, ..., 2 , Let us now consider the free energies ψ (t). We can f f ff f ζ l = λβll+−(1λλ l) β l , l = 0 or 1, (9.31d) put (9.9) in the form 22N − ∏−22N − α ζ f ∏−ll = 1 (ωβ) ff li = 1 ( l ) ωω= RH = (ωα−=) −=( ω) |ωα h α Hh− ( ) 1 2N (9.26a) ii i 1 i2N . (9.31e) ∏−ωα ∏−l = 1 (α ilα ) ll = 1 ( ) l ≠ i f 2N R Observe that (9.29) also holds for the R , a property = ω i h1 ∑ , (9.26b) which has been used in writing (9.31c). There are 2 N−2 i = 1 ωα− i diﬀerent factorizations. Referring to the discussion in (−−) ( ) λ f αβll∈Ω , lN, = 1, 2, ..., 2 ∈Ω , lN = 1, 2, ..., 2 -2 , sections VII and VIII, we note that if all the l are equal to one, then ψ (t) is the minimum free energy ψ (t), while where f m 22N − ∏−(αβ) if all are zero, we obtain the maximum free energy ψM(t). R = ωα − Hh ω| = α l = 1 il ii( ) −=( ) ωα i 1 i2N . (9.27) All other possibilities yield functionals that are interme- ∏−l = 1 (αα il) li ≠ diate between these two extremes. These observations Also, follow from remark VII.1. Observe that

22N − 2N ff ∏−ll = 1 (ωβ) ff 2 RRij H(ω) = HH+−( ωω) ( ) = H1 ( ω) Hh+ (ωω) = 1 (9.28a) ∑ 2N ij, = 1(ωα−−ij)( ωα) ∏−ll = 1 (ωα) ff 22N − 2N RR  ∏+(ωγ) 2 ij 11 ll = 1 = H1ω ∑ − = − h1ω α−− α ωα ωα− . (9.32) 2N (9.28b) ij, = 1 ij( i) ( j ) ∏−ll = 1 (ωα) Now, we have the relation 2N R ∞ i 1 H (ω′) = h1ω ∑ (9.28c) χ+ (ωω) = d ′ . (9.33) i = 1 ωα− i ∫ − π −∞ ωω′′( − ω) which follows by applying an integration over  to

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 • Page 26 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES

(−) (9.1) and completing the contour over Ω . The integral 2N ff 1 RRij over χ ω vanishes. Thus, we have, on integrating over = U( t) + ∑ Cfije i( t) .e j( t) , C fij = 2 iH1 . (9.42) + ( ) 2 αα− Ω+, ij, = 1 ij H ∞ ω′ 2N RRff11 χ−ωω1 ij ′ Comparing (9.34b) with (9.12), we see that + ( ) = ∫ − ∑ d (9.34a) π ωω′− ij, = 1 αα− ωα ′′−− ωα −∞ ij i j 22NNff ff ααiiRR j jiRR j ff 2N α RR gi = -2iH11∑∑ = -2iH , (9.43) ii j αα−− αα = -2iiH ∑ jj = 1 ij = 1 ij ij, = 1(αij−− α)( ωαi ) . (9.34b) f ft where (9.29), for the Ri , has been used. The relation The quantity q− (ω) , defined by (8.4), may be evalu- (4.5a), which yields that Gf (0, s) = Gs′( ) , where the lat- ated by closing the contour on Ω(−), giving 2 ter quantity is given by (9.14), can be confirmed with the ∞ ft 2N f 1 H−+(ωω′′)Ee( ) αR ( ω) ft ωω′ ii i aid of (9.41) and (9.43). Relation (9.23), which is equiva- q+ ( ) = ∫ + dh = 1 ∑ . (9.35) 2πi ω′ −− ω = ωα −∞ i 1 i lent to (4.5a), can be confirmed for fC . From (8.4a), we have Recall from (9.16) that the quantities e1, i = 1,2,...,2N ft ft f t q+(ω) = q −( ω) − H −+( ωω)E ( ) represent a particular minimal state and all the ψf(t) are manifestly FMSs. 2N αEEtt( αωω) − ( ) f ii++ From (8.18d), (8.21), (9.31b), and (9.31c), we have = h1 ∑ Ri , (9.36) i = 1 ωα− i 22NN   α RRff, (9.44) t (+) Kf(t) = - ih11∑∑iii e(t)  = hii e ( t)  which has singularities at those of E+ (ω) in Ω but ii = 1  = 1  none in Ω(−). These explicit relations allow analytic con- so that t 22NN22 tinuation of q± (ω) to the whole complex plane, ex- Dt = Hα RRff e t = H e t (9.45a) cluding singular points. From (8.26a), (9.18), (9.31) and f( ) 11∑∑iii( ) ii( ) ii = 1 = 1 ft (9.35), we see that the quantity P− (ω) is given by 2N f ∞ ft 2N f = HR e tt .e 1 Ht−+(ωω′′)Ee( ) R  ( ) 1 ∑ ii( ) j( ) (9.45b) ft ωωr ′ ii . (9.37) P− ( ) = ∫ + d = ih1 ∑ ii, = 1 2πi −∞ ω′ −− ω i = 1 ωαi 2N The optimal history/continuation in the frequency ff = H1 ∑ ααijiRR jie(tt) .e j( ) . (9.45c) domain, which is a special case of (8.16), is given by for- ii, = 1 mulae which are generalizations of those in [2], page 366. This is in agreement with (9.25). Relations (9.38) and We deduce from (8.17b) and (9.35) that (9.45) can be derived also from (8.29) and (8.30). 2N ff ααijiRR j Note that Df(t) vanishes if ψφf(t) = 01( t) + iH eij(t) .e ( t) (9.38a) ∑ 2N ij, = 1 αα− ij α R f ∞∞ ∑ iie i(t) = 0 . (9.46) 1 i = 1 = φ (t) − Etf( s) . G( s ,E s) t( s) ds ds , (9.38b) 0 ∫∫ 1 12 1 2 2 1 2 Solutions to this equation will exist for non-zero val- 2 00 Where ues of ei(t). Therefore, (9.45b) is a positive semi-definite 2N ff αα RR isisαα− rather than a positive definite quadratic form, so that the Gf ( s, s) = -2 iH iji je ij12. (9.39) 12 1 2 1 ∑ αα− associated matrix will have some zero eigenvalues. If one ij, = 1 ij of these zero eigenvalues were to become slightly nega- Also, from (8.28a) and (9.37), tive, then the second law would no longer hold. Thus, the ff free energy ψ (t) is on the boundary of Φ(t), defined in 2N RR f ij remark VII.3. This observation is of course a special case ψ f(t) = U( t) + iH1 ∑ eij(t) .e ( t) , (9.40a) ij, = 1 ααij− of that after (8.24), and relates to remark VIII.1. 1 ∞∞ A. Sinusoidal histories for non-magnetic materials = − tf t U( t) ∫∫E( s1) . G( s 12 ,E s) ( s 2) ds 12 ds , (9.40b) 2 00 Let us consider the formulae (9.44) and (9.38) for his- Where tories ff 2N ωω− −− RRijisisαα− t i00( ts) i( ts) + G f ( s, s) = -2 iH e ij12. (9.41) E(se) = A+∈ A e , s  (9.47) 12 1∑ αα− ij, = 1 ij and We write (9.38a) and (9.40a) in the form (9.22), giv- itωω00−it t ee ing E+ (ω) = A+ A . (9.48) ii(ωω++00) ( ωω) 1 2N ψφt = t+ Ctte .e , f ( ) 0 ( ) ∑ fij i( ) j ( ) giving that el(t), defined by (9.16), has the form 2 ij, = 1

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 • Page 27 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES

ωω− eeit00it so that et( ) = A + A . (9.49) 2N RRff l d iH1 ij ii(ωα00++ll) ( ωα) ωω00X c ( ) = ∑ . (9.58) dω0 2 ij, = 1 (ααij− ) The relation (9.38a) becomes 22 A.A 2itω A.A −2itω0 22 −−ee0 ααααjj ff ii 2N αα ωαωα00++ij( ) ωαωα00 ++ ij( ) . ijiRR j ( ) ( ) + ++ ψφ(t) = ( t) + iH 2 22 2 f 01∑ αα− (9.50) ij, = 1 ijA.A A.A (ωα−+) ωα+−(ωα) ωα ++ 00ii( 00jj) ( ) ωαωα++ ωαωα ++ ( 00ij)( ) ( 00 ij)( ) Also, using (9.30), we see that the quantity ∆f(ω ), de- Also, from (9.44), K (t), which determines the rate of 0 f fined by (8.34), is given by dissipation, has the form 2N  ω −itω0 11 2N it0 ∆−fω αα ff f ee ( 01) = -iH ∑ ijiRR j22 α R + ij, = 1 ωα−− ωα Kf(th) = -1 ii A A . (9.51) ( 00ij) ( ) (ωαωα00−−ij)( ) ∑ αω+− αω  i = 1 ( ii00) ( ) 2N αα RRff11 1  1 1 1  = -iH iji j −−  −  Remark IX.1. General formulae for the minimum free 1 ∑   . (9.59) ij, = 1 ααij− ωαωαωα00 −− i i 0 − j(ωα0 − j ) ωαωα00−−j energy and the associated rate of dissipation were given    f for sinusoidal histories in [1]. These generalize immedi- Now, by virtue of (9.29), for the Ri , it follows that ately to apply to ψ (t) and D (t) by simply replacing H (ω) 22NNff f f ± ααjjRRij (or the tensor version of these quantities used in [1]) with ∑∑ = , f jj =1 αα−− =1 αα H± (ω) . ij ij 22NNff Thus, the formulae for ψ (t) and K (t) are as follows ααijRRjj f f = . (9.60) ωωf − 2 ∑∑ ff22it00it αα−− αα ψφf (t) = 0( tBe) ++ 10( ω) A.A++ Be1 ( ω0) A.A B20( ω) A , ij =1 ij =1 ij We deduce the relation 2 ff1 i ff 2 ff 22NNRR 2 α αα RR ωω′ +− ω f ij αi j ijij BH10( ) =  X+−( 0) ( 0) , (9.52) ∆(ω ) = -iH +−2  01∑∑αα− 22ααωαωα− −−, (9.61) ω ij, = 1 ij(ωα0 − i ) ωα− ij, = 1 ( ij)( 00 i)( j) 2 0 ( 0 j ) f ffd Using the symmetry of ∆ (ω0) with respect to ω0, we B20(ω) =  ωω0 Xc ( 0) +∆ ( ω 0) , dω find that 0 d 2 ωω− f ffit00it ∆ ω = -  ωωX Dtff( ) = K ( t) Kf (tHeHe) = +−(ωω00) A + ( )A ( 0)  00c ( ) , dω

f where ∆ (ω) is given by (8.34). It will now be shown 2N ff ααijiRR j11 ++iH  that relations (9.50) and (9.51) can also be obtained from 1 ∑ αα− , (9.62) ij, = 1 ij(ωαωα00−−ij)( ) ( ωαωα 00 ++ ij)( ) these, but after somewhat difficult manipulations. We  see from (9.26), (9.28) and remark IX.1 that and therefore

ff 2N ff 2N f ααijiRR j 11 2 RRij f ff 2 B20(ω ) = iH 1∑ + H−(ω0) = HH +−( ωω0) ( 0) = H 10 ω . (9.53) αα− ωαωα−− ωαωα ++, (9.63)  ∑ ij, = 1 ij( 00ij)( ) ( 00 ij)( ) ij, = 1 (ωαωα00−−ij)( ) which agrees with the constant term (proportional to Referring to (9.34), we have 2 2N ff |A| ) in (9.50). α jiRR j X++(−ωω0) = X( 01) = 2iH ∑ , (9.54) ij, = 1 (ααij−−)( ωα0 j) Summary so that The main results are listed below. 2N RRffα ijαi j X+ (ω01) = -iH ∑ − (9.55a) 1. An isothermal theory of free energies, based on ij, = 1 (ααij− ) ωαωα00−−ij continuum thermodynamics and corresponding to lin- ff 2N RR α−+ α ω αα ear constitutive relations with memory, is presented for i j( i j) 0 2 ij = -iH1 ∑ , (9.55b) isotropic non-magnetic materials. For a standard choice ij, = 1 (ααij−)( ωαωα00 −+ i)( i) of relaxation function, the minimum and maximum free We obtain energies are given explicitly. 2N ff 2 αα RR 1 i f iji j 2. The central new result of this work is the determina- X+−(ωω0) + H ( 01) = -iH ∑ , (9.56) 2 ω  αα− ωαωα −+ 0 ij, = 1 ( ij)( 00 i)( j) tion of a family of intermediate free energy functionals, ω which agrees with the coefficient of A.Ae2it0 in (9.50). for the same relaxation function. These are analogous, From (9.55a), we see that though not identical, to those known within a mechanics

ff2 2 framework. However, the assumed singularity structure d 2N RR α α ωω ij i + j 00X+ ( ) = iH 1∑ 22 (9.57) of the frequency domain version of the relaxation func- dω ij, = 1 αα− ωα− 0 ( ij) ( 0 i ) (ωα0 + j ) tion (the susceptibility, for dielectrics) is more relevant,

Glasgow and Golden. J At Nucl Phys 2017, 1(1):16-29 • Page 28 • Citation: Glasgow S, Golden JM (2017) Dielectric Materials with Memory II: Free Energies in Non-Magnetic Materials. J At Nucl Phys 1(1):16-29 SCHOLARLY PAGES in a physical sense, to dielectrics than to mechanics, as 6. BD Coleman, EH Dill (1971) On the thermodynamics of discussed in footnote 5, and this singularity structure is electromagnetic fields in materials with memory. Archive for Rational Mechanics and Analysis 41: 132-162. the basis of the new results. 7. BD Coleman, EH Dill (1971) Thermodynamic restrictions on 3. An important aspect of these developments, from the constitutive equations of electromagnetic theory. Zeitschrift a physical point of view, is that all of these quantities are für Angewandte Mathematik und Physik 22: 691-702. solutions of a constrained minimization problem, lead- 8. V Berti, G Gentili (1999) The minimum free energy for iso- ing to free energy functionals that are on the boundary of thermal dielectrics with memory. Journal of Non-Equilibri- the set of all free energies. Going outside of this bound- um Thermodynamics 24: 154. ary would yield functionals that do not obey properties 9. L Deseri, M Fabrizio, M Golden (2006) The concept of a imposed by the laws of thermodynamics. minimal state in viscoelasticity: New free energies and applications to PDEs. Archive for Rational Mechanics and References Analysis 181: 43-96. 1. S Glasgow, JM Golden, Dielectric materials with memory I: 10. M Fabrizio, J Golden (2002) Maximum and minimum free Minimum and general free energies. energies for a linear viscoelastic material. Quaterly of Ap- plied Mathematics 60: 341-381. 2. G Amendola, M Fabrizio, M Golden (2012) Thermodynam- ics of Materials with Memory: Theory and Applications. 11. J Golden (2005) A proposal concerning the physical dissi- Springer, New York. pation of materials with memory. Quaterly of Applied Math- ematics 63: 117-155. 3. S Glasgow, M Meilstrup, J Peatross, et al. (2007) Real-time recoverable and irrecoverable energy in dispersive-dissipa- 12. J Golden (2007) A proposal concerning the physical dissi- tive dielectrics. Phys Rev E Stat Nonlin Soft Matter Phys pation of materials with memory: The non-isothermal case. 75: 016616. Mathematics and Mechanics of Solids 12: 403-449. 4. S Glasgow, M Ware (2009) Real-time dissipation of op- 13. J Golden (2000) Free energies in the frequency domain: The tical pulses in passive dielectrics. Physical Review A 80: scalar case. Quaterly of Applied Mathematics 58: 127-150. 043817. 14. L Deseri, G Gentili, JM Golden (1999) An explicit formula 5. SA Glasgow, J Corson, C Verhaaren (2010) Dispersive di- for the minimum free energy in linear viscoelasticity. Jour- electrics and time reversal: Free energies, orthogonal spec- nal of Elasticity 54: 141-185. tra and parity in dissipative media. Phys Rev E Stat Nonlin 15. M Fabrizio, A Morro (2003) Electromagnetism of continu- Soft Matter Phys 82: 011115. ous media. Oxford University Press.

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