Spatial correlation length

Hence, close to the critical point thermodynamic quantities at comparatively distant spatial locations become correlated. Especially in the case of liquid micro flows close to a phase transition, these considerations suggest that the correlation length and not the molecular diameter is the length scale determining the onset of deviations from macroscopic behavior. 

The correlation length corresponds to the spatial extent of the restoring force originating from an ordered region. When the temperature approaches the critical temperature Ttr, the restoring force vanishes. This can be formalized by letting diverge as... 

The bottleneck of this approach is obvious the expression for transition probabilities through collective variables of a whole system (total number of particles) means that rather rare fluctuations are taken into account only, whereas their spatial correlations are neglected (i.e., different parts of a system interact being separated by a distinctive distance - the correlation length). 

Summing this Section up, we would like to note that in the approach discussed here the introduction of stochasticity on a mesoscopic level restricts the applicability of a method by such statements of a problem where subtle details of particle interaction become unimportant. First of all, we mean that kinetic processes with non-equilibrium critical points, when at long reaction time the correlation length exceeds all other spatial dimensions. This limitation makes us consider in the next Section 2.3 the microscopic level of the kinetic description. 

These well-known results of the physics of phase transitions permit us to stress useful analogy of the critical phenomena and the kinetics of bimolec-ular reactions under study. Indeed, even the simplest linear approximation (Chapter 4) reveals the correlation length 0 - see (4.1.45) and (4.1.47), or 0 = /d for the diffusion-controlled processes. At t = 0 reactants are randomly distributed and thus there is no spatial correlation between them. These correlations arise in a course of the reaction, the correlation length 0 increases monotonously in time but 0 — 00 at t —> 00 only. Consequently, a formal difference from statistical physics is that an approach to the critical point is one-side, t0 —> 00, and the ordered phase is absent here. There is also evident correspondence between the parameter t in the theory of equilibrium phase transitions and time t in the kinetics of the bimolecular... 

The reaction radius ro does not enter the asymptotic relation (5.2.1). Its absence at t —> oo in the spirit of Section 5.1 could be interpreted as emergence of a new spatial scale - the correlation length . It arises in (5.2.11) through the terms u>mgm which play the role of the correlation sources. 

This estimate should be made more precise. To do it, let us use some results of the numerical solution of a set of the kinetic equations derived in the superposition approximation. The definition of the correlation length o in the linear approximation was based on an analysis of the time development of the correlation function Y(r,t) as it is noted in Section 5.1. Its solution is obtained neglecting the indirect mechanism of spatial correlation formation in a system of interacting particles, i.e., omitting integral terms in equations (5.1.14) to (5.1.16). Taking now into account such indirect interaction mechanism, the dissimilar correlation function, obtained as a solution of the complete set of equations in the superposition approximation... 

New reaction asymptotic law (2.1.78) emerges due to formation during the reaction course of a new spatial scale - the correlation length = Id- Similar to the case of immobile particles, we can expect here that at long times the coordinate r enters into the correlation function in a scaling form rj = r/Io, so that Y(r,t) —> Y(t, t), X (r,t) -> where the second variable... 

It is important to note that in the inhomogeneous gel, the average crosslinking density is not a relevant parameter for determining the frictional pore size of the gel. It is the spatial correlation length of the density fluctuations that determines the bulk frictional behavior of water in the gel. 

Figure 8.14 The product of the tension of the liquid-vapor surface, cr, and the isothermal compressibility, Kj-, identified by Egelstaff and Widom (1970) as proportional to the spatial correlation length. This combination was suggested as appropriate for the low density of the coexisting vapor phase.

Some difficulties in comparing the experimental kinetic data with the outer-sphere reorganization energy calculated from the Marcus formula (28) result from several assumptions made in this theory. The reactant was assumed to have a spherical shape with a symmetric charge distribution. No field penetration into the metal was considered. Also, the spatial dispersion of the dielectric permittivity of the medium was not taken into account. In fact, the positions and orientations of dipoles around a given ion are correlated with each other therefore the reorientation of one dipole, under the influence of the external field, changes to some extent the reorientation of other dipoles within the distance defined by the correlation length. 

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