Critical exponent definition

The origin of this unusual behaviour is partly clarified from Fig. 6.34(a) where the relevant curves 2 demonstrate the same kind of the non-monotonous behaviour as the critical exponents above. Since, according to its definition, equation (4.1.19), the reaction rate is a functional of the joint correlation function, this non-monotonicity of curve 2 arises due to the spatial re-arrangements in defect structure. It is confirmed by the correlation functions shown in Fig. 6.34(a). The distribution of BB pairs is quasi-stationary, XB(r,t) X°(r) = exp[(re/r)3], which describes their dynamic aggregation. (The only curve is plotted for XB in Fig. 6.35(a) for t = 102 (the dotted line) since for other time values XB changes not more than by 10 per cent.) This quasi-steady spatial particle distribution is formed quite rapidly already at t 10° it reaches the maximum value of XB(r, t) 103. The effect of the statistical aggregation practically is not observed here, probably, due to the diffusion separation of mobile B particles. 

We will say that a function with asymptotic behavior like Eq. (10) has a critical exponent plog and that for a function that obeys Eq. (12), the exponent is p = =poo. It is clear that these definitions of critical exponents are just a useful notation. It means that the function does not obey a power law near a given point, but has logarithmic corrections in the first case and it goes faster than any power of (x — xo) in the second case. 

With these definitions we can go a step beyond Eq. (7) and define the critical exponent a for the near-threshold behavior of the bound-state energy... 

The self-avoiding walks for the GM2 fractal have the same exponents as for the 3-simplex. For small values of 6, it is straight forward to generate the explicit recursion equations on a computer, and determine the critical exponents. These were worked out by Elezovic et al [26] for 6=2 to 8, and these studies were extended to 6 = 9 by Bubanja et al [27]. The general form of recursion equations for the function A B C and ( definition of these is same as in Fig. 6) are of the form... 

Table 8.1 Definitions and values of the major critical exponents. The quantities Kt and Cy are the isothermal compressibility and constant volume specific heat capacity respectively, piiq and Pvap are the densities of the coexisting liquid (liq) and vapour (vap) phases, T the temperature, and and the critical temperature and pressure respectively.

In addition to the critical exponents v and yt, defined above, other exponents describe the dependence of the order parameter and its correlations on the distance from the critical point and on the field conjugate to the order parameter. The definitions of the most commonly used critical exponents are summarized in Table 1. These exponents are not all independent from each other. The four thermodynamic exponents a, p, y, 8 all derive from the... 

Alternatively, if the experimental uncertainty is interpreted as a coarse graining in a log( m) vs log(e) plot, then the determination of the exponents would be closer to the experimental one, if a limiting quotient definition of the critical exponent is used instead of the logarithmic derivative of equation (9). In this case f can take values in the range (0,1) for a wide set of ep values, thus lowering the Ising exponents. A more thorough examination of this situation is called for. 

From the knowledge of g and the tangent of p g ) and the definition of the critical exponents, values of the critical exponents can be obtained. 

The exponents in the above power laws are called critical exponents. Table4.1 compiles a selected number of them together with their definition, thermodynamic conditions, and van der Waals values. Here pi - pg is the density difference across the coexistence curve. This quantity is called order parameter. Notice that by construction the order parameter vanishes above Tc. In addition, 8T = T — Tc and 8P = P — Pel- Notice also that we have not yet talked about the heat capacity exponent a. The prime indicates the same critical exponent below Tc. The van der Waals theory yields the same values for the two exponents listed here, i.e. a = a ... 

Of course, the universality statement in the form of Eq. (11) is more a definition than a law until we have clarified how to determine in practice the universality class. Which parameters are relevant and change the universality class and which details are irrelevant for this classification Now the universality principle becomes unreliable. Special cases have often been found where, unexpectedly at that time, the exponents suddenly changed. In the Baxter modeP the critical exponents even varied continuously if a suitable parameter changed continuously. But these exceptions are relatively infi equent and occur mostly in rather complicated systems for a problem as simple as the ordinary liquid-gas critical point all three-dimensional materials and models have the same critical exponents, within some error bars. We hope that ordinary gelation also belongs to the simpler cases where universality holds and will give more complicated examples in Chap. C.V. and D. 

Fast extrusion furnace black with a particle size of 360 A, was used to verify different theoretical concepts of percolation which by definition predicts a rapid change in conductance when volume fraction of conductive particles attains a critical value. Figure 15.38 shows the effect of a carbon black addition to polychloroprene. Up to 30 phr carbon black, the conductivity of poly chloroprene is almost constant and then it increases linearly as concentration of carbon black increases. The following equation applies o = o (P - P Jwhere c is constant, P is concentration of conducting particles, Pc is percolation threshold, and P is exponent which accounts for cluster size."" When data from the Figure 15.38 are replotted as in Figure 15.39 it is evident that the percolation law is valid. 

These equations may be regarded as the definitions of the critical point exponents a, jS, y and S. 

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