Non-equilibrium critical phenomena

Since a number of particles involved in any reaction event are small, a change in concentration is of the order of 1/K Therefore, we can use for the system with complete particle mixing the asymptotic expansion in this small parameter X/V. The corresponding van Kampen [73, 74] procedure (see also [27, 75]) permits us to formulate simple rales for deriving the Fokker-Planck or stochastic differential equations, asymptotically equivalent to the initial master equation (2.2.37). It allows us also to obtain coefficients Gij in the stochastic differential equation (2.2.2) thus liquidating their uncertainty and strengthening the relation between the deterministic description of motion and density fluctuations. 

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). 

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Self-organization manifests itself only in systems far from equilibrium and consisting of a large number of objects, whose cooperative behaviour is sometimes considered in terms of the non-equilibrium critical phenomena... 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

It is necessary to mention that an avalanche-like destruction is also observed in a NaDoS foam but it occurs at significantly higher pressure drops with respect to the equilibrium pressure pa (see Fig. 6.12,a). That is why it is important to distinguish between the destruction at equilibrium critical Apcr,e and at non-equilibrium critical Apcrne pressures since probably the causes are different. Foam destruction at Apcre is perhaps due to foam film rupture while at Apcr ne the destruction results from other phenomena occurring in the disperse... 

The new non-equilibrium thermodynamic theory of heterogeneous polymer systems [37] is aimed at giving a basis for an integrated description for the dynamics of dispersion and blending processes, structure formation, phase transition and critical phenomena. Our new concept is derived from these more general non-equilibrium thermodynamics and has been worked out on the basis of experiments mainly with conductive systems, plus some orienting and critical examples with non-con-ductive systems [72d]. The principal ideas of the new general non-equilibrium thermodynamical theory of multiphase polymer systems can be outlined as follows. 

III. Non-equilibrium in dynamic systems, critical phenomena Chairmen K.A. M0rch, M.E.H. yan Dongen... 

The invitations to participants suggested that the written papers concern Fast Adiabatic Phase Changes in Fluids and Related Phenomena. Particular topics suggested were Liquefaction shockwaves and Shock splitting Evaporation waves Condensation in Laval nozzles and turbines Stability in multiphase shocks Non-equilibrium and near-critical phenomena Nucleation in dynamic systems Structure of transition layers Acoustic phenomena in two-phase systems and Cavitation waves. All of these topics should have been treated with emphasis on physical results, new phenomena and theoretical models. Participants from fourteen nations took part in the Symposium and presented papers which were within the range of suggested topics. 

In 9.5 we turn again to the problem of the interfaces in three-phase equilibrium, as in Chapter 8, but now the three phases are near that limit of their coexistence at which they become one the tricritical point. In 9.6 we treat still another problem related to three-phase eqtdiibrium the interface between a non-critical phase a and another fluid phase that is itself near the critical point of separation into two phases and y. An example is the intnface between a liquid mixture and its equilibrium vapour when the former is near its oonsolute point. Finally, in f 9.7, we sketdi the applications to interfaces of the ideas and methods of the modem renormalization-group theory of critical phenomena. 

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