Nonclassical exponents

Exponents derived from the analytic theories are frequently called classical as distinct from modem or nonclassical although this has nothing to do with classical versus quantum mechanics or classical versus statistical thermodynamics. The important thermodynamic exponents are defined here, and their classical values noted the values of the more general nonclassical exponents, determined from experiment and theory, will appear in later sections. The equations are expressed in reduced units in order to compare the amplitude coefficients in subsequent sections. 

The values of the exponents for ordinary critical poiats or bicritical poiats (where two phases become identical) are called nonclassical, because (unlike the exponents iu van der Waals and other classical equations) they are not multiples of 1/2. 

Actually, MC simulations should reflect the nonclassical critical fluctuations as well, thus allowing us to identify the critical exponents and the... 

Note that all these formulas also contain the result for the limiting case of short chains dynamics described by the Rouse model [139,140] if we formally put Ne N in these equations. As will be discussed later (Sect. 2.5), there occurs a crossover in the static critical behavior from mean-field-like behavior where ocR e-1/2 with e= 1 — x/X rit> Scon(0)ccN e to the nonclassical critical behavior with Ising model [73, 74] critical exponents cce-v, S, ii(0) oceT, vw0.63, 1.24. This crossover occurs, as predicted by the Ginzburg... 

Griffiths has given a phenomenological (Landau) treatment of tricritical points which expresses the free energy as a sixth-order polynomial in an order parameter (which is some suitable linear combination of the physical densities , e.g. the mole fractions). The scaling properties of the singular part of the polynomial lead to four numbers = 5/6, 2 = 4/6 = 2/3, 3 = 3/6 = 1/2, = 2/6 = 1/3, in terms of which various critical exponents are expressed. Because this is an analytic (mean field) formulation, these exponents are classical , but it is believed that for experimental tricritical points in three dimensions they should be. ( Nonclassical logarithmic factors may exist, but these do not alter the exponents.)... 

To distinguish between the classical and nonclassical Coulomb integrals, it is useful to introduce the concept of the extent of a Gaussian distribution. For a target accuracy of 10 in the evaluation, we introduce the extent of a Gaussian distribution of exponent p as [21]... 

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