Corresponding-states function integrals

This relation defines a time-dependent column vector a. Because = 1, Eq. (7-50) implies afa = 1 a is a unit vector. This is true of all state vectors that correspond to normalized state functions. Substitution of (7-50) into (7-49), subsequent multiplication by u, and integration yield the Schrodinger equation (sometimes called the equation of motion ) for the component ar... 

As we have seen before, exact differentials correspond to the total differential of a state function, while inexact differentials are associated with quantities that are not state functions, but are path-dependent. Caratheodory proved a purely mathematical theorem, with no reference to physical systems, that establishes the condition for the existence of an integrating denominator for differential expressions of the form of equation (2.44). Called the Caratheodory theorem, it asserts that an integrating denominator exists for Pfaffian differentials, Sq, when there exist final states specified by ( V, ... x )j that are inaccessible from some initial state (.vj,.... v )in by a path for which Sq = 0. Such paths are called solution curves of the differential expression The connection from the purely mathematical realm to thermodynamic systems is established by recognizing that we can express the differential expressions for heat transfer during a reversible thermodynamic process, 6qrey as Pfaffian differentials of the form given by equation (2.44). Then, solution curves (for which Sqrev = 0) correspond to reversible adiabatic processes in which no heat is absorbed or released. 

Thus, the assumption that an integration of a differential expression over a closed path is zero leads to a conclusion that an integration between two different, but fixed, states is independent of path. But, this property coincides with those we have ascribed to state functions. Thus, we have shown that a differential for which equation (A 1.17) is true must correspond to the differential of some state function. 

The nine-fold integration for the gain term is over both pre-collision velocities and over the second (all but the first) post-collision velocity. Both pre-collision states are folded with their corresponding distribution function. 

As a prelude to evaluating the Fourier transform (9.8) let us consider first the function B (J ) and assume that the manifold / constitutes a continuum of states. In this case the summation over / corresponds to the integral... 

Thus, to the degree that the corresponding-states theory for the direct correlation function integrals is accurate, the results are applicable to any system. In practice, we have found that when the species are at... 

At the core of the analysis and methods that are discussed in this Chapter is the consistent consideration of the fact that the form of each resonance wavefunction is = fl I o+Xas (Eq. (4.1) of text), if necessary, the extension to multi-dimensional forms is obvious. Depending on the formalism, the coefficient a and the asymptotic part, Xas, are functions of either the energy (real or complex) or the time. The many-body square-integrable, %, represents the localized part of the decaying (unstable) state, i.e., the unstable wavepacket which is assumed to be prepared at f = 0. its energy, Eo, is real and embedded inside the continuous spectrum, it is a minimum of the average value of the corresponding state-specific effective Hamiltonian that keeps all particles bound. 

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