Extensibility parameter

As a first attempt to modify the code to be able to run simulations on SiH4-H2 discharges, a hybrid PlC/MC-fluid code was developed [264, 265]. It turned out in the simulations of the silane-hydrogen discharge that the PIC/MC method is computationally too expensive to allow for extensive parameter scans. The hybrid code combines the PIC/MC method and the fluid method. The electrons in the discharge were handled by the fluid method, and the ions by the PIC/MC method. In this way a large gain in computational effort is achieved, whereas kinetic information of the ions is still obtained. 

By analogy with Hamilton s principle of least action, the simplest proposition that could solve the thermodynamic problem is that equilibrium also depends on an extremum principle. In other words, the extensive parameters in the equilibrium state either maximize or minimize some function. 

The idea is developed by postulating a function of the extensive parameters that tends to a maximum for any composite system that approaches a state of equilibrium on removal of an internal constraint. This function, to be called the entropy S, is defined only for equilibrium states where it assumes definite values. The basic problem of thermodynamics may therefore be considered solved once the entropy is specified in terms of a fundamental relation as a function of the extensive parameters. ... 

This property is described by the statement that entropy is a homogeneous first-order function of the extensive parameters. The expression is readily interpreted to define molar entropy (s = S/N), internal energy (it) and... 

Once the total entropy of a composite system has been formulated as a function of the various extensive parameters of the subsystems, the extrema of this total entropy function may in principle be located by direct differentiation and classified as either maxima, minima or inflection points from the sign of the second derivative. Of these extrema, only the maxima represent stable equilibria. 

From the definition of temperature, pressure and chemical potential as partial derivatives it follows that these variables may be written as functions of the extensive parameters ... 

These relationships that express intensive parameters in terms of independent extensive parameters, are called equations of state. Because of the homogeneous first-order relationship between the extensive parameters it follows that multiplication of each of the independent extensive parameters by a scalar A, does not affect the equation of state, e.g... 

The surface BCDE represents a segment of the surface defined by the fundamental equation characteristic of a composite system with coordinate axes corresponding to the extensive parameters of all the subsystems. The plane Uo is a plane of constant internal energy that intersects the fundamental surface to produce a curve with extremum at A, corresponding to maximum entropy. Likewise So is a plane of constant entropy that produces a curve with extremum A that corresponds to minimum energy at equilibrium for the system of constant entropy. This relationship between maximum entropy... 

In both the entropy and energy representations the extensive parameters are the independent variables, whereas the intensive parameters are derived concepts. This is in contrast to reality in the laboratory where the intensive parameters, like the temperature, are the more easily measurable while a variable such as entropy cannot be measured or controlled directly. It becomes an attractive possibility to replace extensive by intensive variables as... 

By way of illustration consider a binary composite system characterized by extensive parameters Xk and Xf in the two subsystems and the closure condition Xk + X k — Xk. The equilibrium values of Xk and X k are determined by the vanishing of quantities defined in the sense of equation (3) as... 

Response to the driving force is defined as the rate of change of the extensive parameter Xk, i.e. the flux Jk = (dXk/dt). The flux therefore stops when the affinity vanishes and non-zero affinity produces flux. The product of affinity and associated flux corresponds to a rate of entropy change and the sum over all k represents the rate of entropy production,... 

In order to formulate entropy production, local entropy is assumed to depend on local extensive parameters Xk by the same functional relationship that exists at equilibrium. Thus... 

Now consider a pair of reservoirs in equilibrium with respect to the extensive parameters Xj and Xk, with instantaneous values of Xj and Xk. Let 6Xj denote a fluctuation from the instantaneous value. The average value of 6Xj is zero, but the average of its square (SXj)2 = ((6Xj)2) 0. Likewise, the average correlation moment (5Xj5Xk) / 0. 

To put the previous statement into perspective it is necessary to stipulate that any macrosystem with well-defined values of its extensive parameters is made up of myriads of individual particles, each of which may be endowed with an unspecified internal energy, within a wide range consistent with all external constraints. The instantaneous distribution of energy among the constituent particles, adding up to the observed macroscopic energy, defines a microstate. It is clear that any given macrostate could arise as the result of untold different microstates. 

To establish [5] the thermodynamic meaning of the index of probability r/ = (ip — H)/9, it is assumed that the distribution g = ev changes as the condition of the system changes, but always subject to the normalization condition f evdfl = 1. It follows that the derivative d f evdQ, = 0. It is assumed that both 9 and tp, as well as extensive parameters may be altered, such that... 

The microcanonical ensemble in quantum statistics describes a macroscopi-cally closed system in a state of thermodynamic equilibrium. It is assumed that the energy, number of particles and the extensive parameters are known. The Hamiltonian may be defined as... 

The intensive parameter Lj, conjugate to the extensive parameter Xi is defined by... 

During each phase transition of the type illustrated here, both of the intensive parameters P and T remain constant. Because of the difference in density however, when a certain mass of liquid is converted into vapour, the total volume (extensive parameter) expands. From the Gibbs-Duhem equation (8.8) for one mole of each pure phase,... 

In the most important stirring operation - the homogenization of liquid mixtures - the convective transport of liquid balls (macro-mixing) is of predominant importance. Thus, this process depends to a large degree on space geometry and type of stirrer. It is influenced by the extensive parameters such as stirrer speed, n, and stirrer diameter, d. Here, the similarity with respect to fluid dynamics is given by Re = n d2 p/p= idem. 

The statistical polymer method proposed recently by the author [6] considers polymeric systems as sets of assemblages possessing structures averaged over all these of polymers containing the same numbers of monomeric units, i.e., statistical polymers. For the case when one is interested in the evaluation of the weight distribution and/or additive (extensive) parameters like energy, entropy, etc., one can consider statistical polymers instead branched cross-linked ones. 

In [6] the author had shown that the values of an additive (extensive) parameter of a nonequilibrium system can be represented a combination of the values of that parameter of the subsystems with U — 0 and U2 — Umax ... 

This phenomenon is associated with the level of entropy production due to the irreversibility of the process. Entropy is not conserved it is the extensive parameter of heat. 

An extremum principle minimizes or maximizes a fundamental equation subject to certain constraints. For example, the principle of maximum entropy (dS)v = 0 and, (d2S)rj < 0, and the principle of minimum internal energy (dU)s = 0 and (d2U)s>0, are the fundamental principles of equilibrium, and can be associated with thermodynamic stability. The conditions of equilibrium can be established in terms of extensive parameters U and. S, or in terms of intensive parameters. Consider a composite system with two simple subsystems of A and B having a single species. Then the condition of equilibrium is... 

For the entropy and internal energy, the canonical variables consist of extensive parameters. For a simple system, the extensive properties are S, U, and V. and the fundamental equations define a fundamental surface of entropy S = S(U,V) in the Gibbs space of S, U, and V. 

Universal quantities T and p related to the whole system are not always appropriate for description of a nonequilibrium system due to its probable spatial inhomogeneity. In general, the nonequilibrium system needs to be characterized using local extensive parameters—for example, the Gibbs potential g(T(r, t), p(r, t)) per unit mass of the system matter. In this case. 

Consider an extensive quantity B = pb, where p is the density of the matter and b is the quantity of B per unit mass. According to the Ostro gradsky theorem, the local quantity of any extensive parameter B(r, t) = p(r, t)b(r, t) in a macroscopic system obeys the balance equation... 

Neither sources nor sinks exist for the extensive parameters character ized by the conservation laws. Therefore, for example, the spatial distribu tion of a nontransforming chemical component is described (due to necessity of the mass balance) by equation... 

It is easy to apply the Ostrogradsky theorem for deriving the entropy fluxes and the thermodynamic forces that initiate these fluxes. In fact, the balance of entropy (it also is an extensive parameter, S = ps) is... 

---