Homogeneous of degree one

The substitutions can be made because the extensive thermodynamic variables in the equations are homogeneous of degree one.d Thus, dividing the equation by n converts the extensive variable to the corresponding molar intensive variable. For example, to prove that equation (3.48) follows from equation... 

The exchange functional E fp] is well known to be homogeneous of degree one in coordinate scaling [1,2], that is, the Levy-Perdew relation [34],... 

These operational distinctions between extensive and intensive avoid ambiguities that can occur in other definitions. Some of those definitions merely say that extensive properties are proportional to the amount of material N in the system, while intensive properties are independent of N. Other definitions are more specific by identifying extensive properties to be those that are homogeneous of degree one in N, while intensive properties are of degree zero (see Appendix A). 

In this book we restrict ourselves to extensive properties that are homogeneous of degree one in the amount of material. Specifically, for a multicomponent system containing component mole numbers N, N2,. .., we will use only those extensive properties F that are related to their intensive analogs/ by... 

Careful examination of these two conditions indicates that the first rule of linearity, called scalar multiplication or homogeneity of degree one [21,27], is contained in the second rule of linearity, called additivity or Boltzmann superposition. This duplication can simply be shown for all scalars that are rational numbers [1]. Therefore only one mathematical requirement for linearity exists if a reasonable form of continuity requirement is enforced, and that is Boltzmann superposition. It can be shown also that scalar multiplication in no way implies superposition. In fact scalar multiplication is simply a homogeneity condition of degree one in the constitutive law, and many non-linear differential equations, functions, and functionals are homogeneous but not linear. 

The same result can be obtained from an application of Euler s theorem, explained in more detail in Appendix 1. The thermodynamic quantities, Z, are homogeneous functions of degree one with respect to mole numbers.c At constant T and p, one can use Euler s theorem to write an expression for Z in terms of the mole numbers and the derivatives of Z with respect to the mole numbers. The result isd... 

The extensive thermodynamic variables are homogeneous functions of degree one in the number of moles, and Euler s theorem can be used to relate the composition derivatives of these variables. 

For example, there is an isomorphism from to the vector space P3 of homogeneous polynomials of degree one in three variables (x, y, z), given by... 

Recall the vector space of homogeneous polynomials in two variables defined in Section 2.2. The vector space V is the tensor product ofP and P, denoted P P. In other words, the elements of V are precisely the linear combinations of terms of the form p(u, v)g (x, y), where is a homogeneous polynomial of degree one and q is a homogeneous polynomial of degree two. Note that, given an element r(M, u, x, y) of P 0p2 there are many different ways to write it as a linear combination of products. For example,... 

It is of the utmost importance to recognize that balances can only be made on extensive variables or quantities. If you double the system, an extensive measure is doubled, whereas an intensive measure remains the same. Thus the mass of two identical bricks is twice the mass of one, but the density, or mass per unit volume, remains the same in the duplicated system because we have doubled both the mass and the volume. The first (mass) behaves like a homogeneous function of degree one, the second (volume) of degree zero. Thus in the simple example used in Example 1, we did not make our balance on the concentration, moles per unit volume = c, but on the amount, moles - Vc. 

The substitution moment Isy differs from the equilibrium moment ley by first order terms of the expansion. Since g is a homogeneous function of degree one-half of the atomic masses [24], the second term on the right-hand side of Eq. 91a is, by Euler s theorem ... 

A function satisfying the requirement (1.4.19) is said to be homogeneous in a. of degree one. It is of interest to reexpress this requirement in another form. For this purpose set a — 1 + c and examine the special case for which c - 0. According to (1.4.19) we demand... 

Thus, a homogeneous function of degree one must satisfy the very stringent requirement specified by Eq. (1.4.22), which is known as Euler s Theorem. 

This relationship is known as Euler s Theorem for Homogeneous Functions of Degree One. However, in addition to the dependence on the x,- the function F may also display a dependence on parameters such as pressure P or temperature T that, of course, remains unaffected by the above manipulations. 

In order to understand the relationship G = p,N, let us investigate the theory of homogeneous functions. If a function, /(xi... xm) is homogeneous function of degree one in the variables xi... xm then scaling the Xj by a constant will result in the scaling of / by the same constant... 

Notice that the same problem would be encountered for the ratio <7n / N) in cases where the external potential prevents the thermodynamic potential to be a homogeneous function of degree one in either s or Sy and these two quantities would be increased by noninteger factors. These examples show that for confined fluids one has to be cautious to approach the thermodynamic limit properly. 

Note that SI yo h) is inherently a homogeneous functional of degree one. It means that... 

This relationship is known as Euler s theorem for homogeneous functions of degree one. This is a theorem of great importance in thermodynamics, as will become evident later. 

The potential V(r) is closely related to the average potential defined by Slater [59, 60] we call it the Ehrenfest potential. For Ehrenfest potentials that are homogeneous of degree minus one in r, (16) gives the usual Coulombic virial relation between the total potential energy and the total kinetic energy, —V=2T. 

This paper deals with non-linear homogeneous constitutive equations of degree one, a type of behavior that until recently [1] has not been mentioned in the field of mechanics. This type of... 

Before proceeding in the development of homogeneous constitutive equations of degree one, a special case of equation (4.1) is worth mentioning. Note that when the kernel functionals are independent of the variables i = l, n, then the integrations can be performed and the result is... 

The development of constitutive equations which are homogeneous to degree one is quire simple and can be done by simply imposing restrictions or constraints on equation (4.1). Recall... 

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