Maxwell equations response theory

The spectra s (v) will be described here in terms of a linear-response theory. We shall employ the specific form [GT, VIG] of this theory, called the ACF method, which previously was termed the dynamic method. The latter is based on the Maxwell equations and classical dynamics. A more detailed description of this method is given in Section II. Taking into attention the central role of the model suggested here, we, for the sake of completeness, give below a brief list of the main assumptions employed in our variant of the ACF method. 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

These are the Maxy ell-Stefan diffusion equations for multicomponent systems. These equations are named after the Scottish physicist James Clerk Maxwell and the Austrian scientist Josef Stefan who were primarily responsible for their development (Maxwell, 1866, 1952 Stefan, 1871). These equations appeared, in more or less the complete form of Eq. 2.1.15, in an early edition of the Encyclopedia Britannica (incomplete forms had been published earlier) in a general article on diffusion by Maxwell (see Maxwell, 1952). In addition to his major contributions to electrodynamics and kinetic theory. Maxwell wrote several articles for the encyclopedia. Stefan s 1871 paper is a particularly perceptive one and anticipated several of the multicomponent interaction effects to be discussed later in this book. 

These examples, and others like them, allow us to discern three distinct levels of model building, though admittedly the boundary between them is blurred. In particular, the level of such modeling might be divided into (i) fundamental laws, (ii) effective theories and (iii) constitutive models. Our use of the term fundamental laws is meant to include foundational notions such as Maxwell s equations and the laws of thermodynamics, laws thought to have validity independent of which system they are applied to. As will be seen in coming paragraphs, the notion of an effective theory is more subtle, but is exemplified by ideas like elasticity theory and hydrodynamics. We have reserved constitutive model as a term to refer to material-dependent models which capture some important features of observed material response. 

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