First-order derivatives critical case

It is believed that the site-occupation probability, p5, plays a role of concentration in real systems. If that is the case, the hyperbolic relation between Dc and ps predicted by Eq. (114) is in marked contrast to the corresponding Dc vs. C diagram of real systems, where Dc is known to converge to the ideal value, Dco, with C—o [48,81,86,98,99]. Clearly, the abnormality in the critical behavior of the percolation model is derived from the abnormality in the chemical machinery of the intermolecular reaction (Sect. 5) which is in proportion to the first order of C, more exactly to M0ps2, the same order as the cyclization rate, so that the concentration terms cancel out each other, resulting in the expression of Eq. (114) without the y term. 

Finally, taking into account only the electrostatic quadrupole-quadmpole interactions, the ratio of the values of the critical temperature for the quantum case and that for the classical case were also derived. The ratio of the classical transition temperature to that of its (7=1) quantum analogue is found to be f without the crystal-field effects. This result could be derived for a quite general quadratic Hamiltonian of the type (2.1). However, it was assumed that the transition is continuous, and the reported ratio no longer holds for a first-order transition. It is suggested [141] that this result may be relevant to the N2 system. 

Commonly in thermodynamics, the situation is different. In thermodynamics, the isodynamic path is fixed by the first total differential dU = 0 or some other thermodynamic function. The stability is guaranteed by the total differential of second order, d t/. If d t/ > 0 then the state is stable and if d 7 < 0, the state is not stable. In these cases for the thermodynamic analysis, the total differentials of first order and second order are sufficient. Obviously, the limiting case of d C = 0 arises when the system passes from stable to unstable. This situation can be addressed as a critical state. Here, for the thermodynamic analysis, the third derivative enters the game. As pictorially explained in Example 1.14, fluctuations are likely in critical phenomena, e.g., the critical opalescence. 

Figure 1 shows that one must distinguish first-order phase transitions [where first derivatives of the appropriate thermodynamic potential F, such as the enthalpy U = [9(y3F)/9/3]p, where p = I/HbT, Hb being Boltzmann s constant, or volume V = dF/dp)T exhibit a jump] and second-order transitions, where U,V are continuons, bnt second derivatives are singular (1,2,7). The classical example for the latter case is the gas-liquid critical point, where the specific heat Cp = dU/dT)p or the isothermal compressibility kt = - VV) dVldp)T diverge. 

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