State function, integrating

To proceed further, it is convenient to represent the grand canonical partition function for the redefined SB Hamiltonian (Eq. 12) in terms of a coherent-state functional integral [29] over Grassmann fermionic and complex hosonic fields as... 

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. 

According to the Caratheodory theorem, the existence of an integrating denominator that creates an exact differential (state function) out of any inexact differential is tied to the existence of points (specified by the values of their x, s) that cannot be reached from a given point by an adiabatic path (a solution curve), Caratheodory showed that, based upon the earlier statements of the Second Law, such states exist for the flow of heat in a reversible process, so that the theorem becomes applicable to this physical process. This conclusion, which is still another way of stating the Second Law, is known as the Caratheodory principle. It can be stated as... 

Application of the condition for exactness shows that both derivatives equal zero, so equation (2.45) must be exact. Thus, we have determined that when an ideal gas is involved, T is an integrating denominator for Sqrev, and the right hand side of equation (2.45) is the total differential for a state function that we will represent as dS.cc... 

The Caratheodory theorem establishes the existence of an integrating denominator for systems in which the Caratheodory principle identifies appropriate conditions — the existence of states inaccessible from one another by way of adiabatic paths. The uniqueness of such an integrating denominator is not established, however. In fact, one can show (but we will not) that an infinite number of such denominators exist, each leading to the existence of a different state function, and that these denominators differ by arbitrary factors of . Thus, we can make the assignment that A F (E ) = = KF(E) = 1. 

A1.4 State Functions and Exact Differentials Inexact Differentials and Line Integrals... 

Equation (A 1.16) shows that the integral evaluated over any two paths that connect states 1 and 2 must be equal. The value of the integral, AZ, cannot depend upon the path but must be associated with the choice of states so that AZ Zi — Z. This is consistent with our earlier definition of a state function. 

Suppose then, we encounter a general differential expression and want to know whether it is associated with a state function. The behavior of this differential expression integrated over a closed path provides a means to answer this question. Two possibilities need be considered. 

We have already established that an integral of a differential expression associated with a state function is zero over a closed path. Now, we must consider whether the converse of that statement is true. That is, if the integral of a general differential expression over a closed path is found to be zero, this expression is the differential expression of some state function. To answer the question, let us reconsider the example described in Figure (A 1.1) and equations (A 1.13) and (A 1.14) and assume that equation (A 1.17) is true for all closed (cyclic) paths. Then, for a path 1 and a path 3 that connect the same two states, 1 and 2,... 

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. 

By similar reasoning, one can show that differential expressions for which equation (Al.18) is true must yield integrals between two fixed states whose values depend upon the path. Such differential expressions cannot be associated with state functions because of the dependence upon path. Therefore, equations (Al.17) and (Al.18) distinguish between differentials that can ultimately be associated with state functions and that cannot. Expressions for which equation (Al.17) is true are called exact differentials while those for which equation (Al.18) is true are called inexact differentials. 

Clearly, the differential obtained, namely, d S = SqJT is exact and S, the entropy, is a thermodynamic state function, that is, it is independent of the path of integration. While Eq. (88) was obtained with the assumption of an ideal gas, the result is general if reversible conditions are applied. 

Fig. 26. Total and final rovibrational state specific integral cross-sections for the H + H2O(00)(0) —> H2(i>i, ji) + OH(j2) reaction as a function of translational energy. The total cross-sections in crosses were calculated in the atom—triatom coordinates.

Polarization fluctuations of a certain type were considered in the configuration model presented above. In principle, fluctuations of a more complicated form may be considered in the same way. A more general approach was suggested in Refs. 23 and 24, where Eq. (16) for the transition probability has been written in a mixed representation using the Feynman path integrals for the nuclear subsystem and the functional integrals over the electron wave functions of the initial and final states t) and t) for the electron ... 

Ax = / — x is the ionization potential from the lower state of the line and 0.75 eV is the electron detachment potential of H. [M+/H] = [M/H] + [v], where x is the degree of ionization which changes negligibly while it is close to one, and the electron pressure cancels out. A9 can be identified with A9f obtained by optimally fitting neutral lines with different excitation potentials to one curve of growth (see Fig. 3.13), or deduced from red-infrared colours. As a refinement, a small term [0] should be added to the rhs of Eq. (3.59) to allow for an increase of the weighting function integral towards lower effective temperatures. 

We see such parts as EXP, GAIN, HIPA5S, INTEG, MULT, SIN, and SQRT. The blocks perform the stated function on the input waveform. For example, with the SQRT function, the output voltage is the square root of the input voltage. With the INTEG function, the output waveform is the integral over time of the input waveform. 

The convolution operation is a way of describing the product of two overlapping functions, integrated over the whole of their overlap, for a given value of their relative displacement (Bracewell 1978 Hecht 2002). The symbol is often used to denote the operation of convolution. The convolution theorem states that the Fourier transform of the product of two functions is equal to the convolution of their separate Fourier transforms... 

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