Critical behaviors

Evidence of helium-like critical behavior has been reported by means of light scattering [49] and optical interferometry [50] experiments. However, other experiments have shown that the transition is mean-field. 

For example, it was argued that the bare correlation length of tilt an- 

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Jacob J, Kumar A, Anisimov M A, Povodyrev A A. and Sengers J V 1998 Crossover from Ising to mean-field critical behavior in an aqueous electrolyte solution Phys. Rev. E 58 2188... 

Garland C W and Nounesis G 1994 Critical behavior at nematic-smectic-A phase transitions Phys. Rev. E 49 2964-71... 

The proposed mechanism for the DD process is not intended to represent that of any actual catalytic reaction, but to simulate a generic bimolecular reaction. Monte Carlo simulations of the reaction mechanism described by Eqs. (21)-(25) have shown the existence of IPTs exhibiting a rich variety of critical behavior. 

This small figure may also be compatible with a logarithmic decay [34,57]. Also, the survival probability of a blob of A species embedded in a B sea (separated by a wall of empty sites) decreases in time with exponent 6 = 0.80 0.20 (see Eq. (6)), reminiscent of critical behavior of the ZGB model at the first-order IPT [34]. 

T. Aukrust, D. A. Browne, I. Webman. Critical behavior of an autocatalytic reaction model. Phys Rev A 47 5294-5301, 1990. 

I. Jensen. Critical behavior of the three dimensional contact process. Phys Rev A 45 R563-R566, 1992. 

J. Satulovsky, E. V. Albano. The influence of lateral interactions on the critical behavior of a dimer-monomer surface reaction model. J Chem Phys 97 9440-9446, 1992. 

I. Jensen. Non-equilibrium critical behavior on fractal lattices. J Phys A (Math Gen) 24 L1111-L1117, 1991. 

E. V. Albano. The critical behavior of dimer-dimer surface reaction models. Monte Carlo and finite-size scaling investigation. J Stat Phys 69 643-666,1992. 

E. V. Albano. Critical behavior of an irreversible multiple reaction process. Surf Sci 306 1A0-1A6, 1994. 

M. H. Kim, H. Park. Critical behavior of an interacting monomer-dimer model. Phys Rev Lett 73 2579-2582, 1994. 

Generally, the models used for simulation of living polymers can be divided roughly into two classes, focused on static or dynamic properties of the LP or GM. The static models are mainly designed to study equilibrium conformational properties of the polymer chains, critical behavior at the polymerization transition, and molecular weight distribution... 

H. Meirovitch, S. Livne. II. Critical behavior of single self-avoiding walks. J Chem Phys 4507 515, 1988. 

K. Binder. Critical behavior at surfaces. In C. Domb, ed. Phase Transitions and Critical Phenomena. London, Academic Press, 1989, Vol. 8, pp. 2-144. 

One prominent example of rods with a soft interaction is Gay-Berne particles. Recently, elastic properties were calculated [89,90]. Using the classical Car-Parrinello scheme, the interactions between charged rods have been considered [91]. Concerning phase transitions, the sohd-fluid equihbria for hard dumbbells that interact additionally with a quadrupolar force was considered [92], as was the nematic-isotropic transition in a fluid of dipolar hard spherocylinders [93]. The influence of an additional attraction on the phase behavior of hard spherocylinders was considered by Bolhuis et al. [94]. The gelation transition typical for clays was found in a system of infinitely thin disks carrying point quadrupoles [95,96]. In confined hquid-crystalline films tilted molecular layers form near each wall [97]. Chakrabarti has found simulation evidence of critical behavior of the isotropic-nematic phase transition in a porous medium [98]. 

M. L. Japas, J. M. H. Levelt Sengers. Critical behavior of a conducting ionic solution near its consolute point. J Chem Phys 94 5361-5368, 1994. 

Only in the limit Ixj/Gi 1 does one expect Ising-like critical behavior in Eq (7) due to the universality principle of critical phenomena, "... 

Recently efficient techniques were developed to simulate and analyze polymer mixtures with Nb/Na = k, k > I being an integer. Going beyond meanfield theory, an essential point of asymmetric systems is the coupling between fluctuations of the volume fraction (j) and the energy density u. This coupling may obscure the analysis of critical behavior in terms of the power laws, Eq. (7). However, it turns out that one can construct suitable linear combinations of ( ) and u that play the role of the order parameter i and energy density in the symmetrical mixture, ... 

Chapter 7 discusses a variety of topics all of which are related to the class of probabilistic CA (PCA) i.e. CA that involve some elements of probability in their state and/or time-evolution. The chapter begins with a physicist s overview of critical phenomena. Later sections include discussions of the equivalence between PCA and spin models, the critical behavior of PCA, mean-field theory, CA simulation of conventional spin models and a stochastic version of Conway s Life rule. 

Many more such relationships can be derived in a similar manner (see [ma85] or [stan71]). For our purposes here, it will suffice to merely take note of the fact that certain relationships among the critical exponents do exist and are in fact commonly exploited. Indeed, we shall soon sec that certain estimates of critical behavior in probabilistic CA system are predicated on the assumptions that (1) certain rules fall into in the same universality class as directed percolation, and (2) the same relationships known to exist among critical exponents in directed percolation must also hold true for PC A (see section 7.2). 

It is important to understand that critical behavior can only exist in the thermodynamic limit that is, only in the limit as the size of the system N —> = oo. Were we to examine the analytical behavior of any observables (internal energy, specific heat, etc) for a finite system, we would generally find no evidence of any phase transitions. Since, on physical grounds, we expect the free energy to be proportional to the size of the system, we can compute the free energy per site f H, T) (compare to equation 7.3)... 

Note that the critical behavior just described holds true strictly only in the thermodynamic limit i.c. only when the number of sites N oo. The above results are in fact obtained by extrapolating from finite system calculations. Kinzel... 

These properties include (l)a continuous decrease in the average density of turbulent sites as the threshold is approached from above (with exponent /3) (2) divergence of the average transient time (3) algebraic distributions of the sizes and durations of laminar clusters at the threshold (with exponents and < ). Houlrik, Webrnan arid Jensen [houl90] discuss the critical behavior of this map from a mean-field theoretic perspective. 

Bruce, A. D., Finite-size critical behavior in the Gibbs ensemble, Phys. Rev. E1997, 55, 2315-2320... 

Linear chain termination is not, however, a necessary condition for the critical behavior. Indeed, with mechanisms V and XII, chain termination is quadratic (v v,172), but critical transition does take place because hydroperoxide decomposes into radicals that contribute to chain propagation. As a result, v (v [ROOH])1/2 v, [ROOH]172, and v [ROOH] (see Equation (14.11)) which explains the critical behavior. 

Quasistationary (with respect to hydroperoxide) oxidation is possible for the mechanisms with a critical behavior (at [InH] > [InH]cr) as well as for ordinary mechanisms (at kdr > 1). 

When the inhibitor is able to interact with hydroperoxide (mechanism VI), the necessary conditions for critical behavior are fkq kd > 2k2en A ,, [RH] or kA > 2e k n [InH]i that is, the reaction of inhibitor with hydroperoxide must not be too fast. 

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