Stochastic storage models and noise-induced phase transitions

Abstract

The most frequently used in physical application diffusive (based on the Fokker-Planck equation) model leans upon the assumption of small jumps of a macroscopic variable for each given realization of the stochastic process. This imposes restrictions on the description of the phase transition problem where the system is to overcome some finite potential barrier, or systems with finite size where the fluctuations are comparable with the size of a system. We suggest a complementary stochastic description of physical systems based on the mathematical stochastic storage model with basic notions of random input and output into a system. It reproduces statistical distributions typical for noise-induced phase transitions (e.g. Verhulst model) for the simplest (up to linear) forms of the escape function. We consider a generalization of the stochastic model based on the series development of the kinetic potential. On the contrast to Gaussian processes in which the development in series over a small parameter characterizing the jump value is assumed [Stratonovich R.L., Nonlinear Nonequilibrium Thermodynamics, Springer Series in Synergetics, vol.59, Springer Verlag, 1994], we propose a series expansion directly suitable for storage models and introduce the kinetic potential generalizing them. PACS. 05.70.Ln Nonequilibrium and irreversible thermodynamics -05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion -05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin etc)

Modelling Coagulation Systems: A Stochastic Approach

“…The factor 1/2 arises because one accounts one coagulation act twice. Substituting (17) in (16) (17) is not quite correct from the point of view of traditional storage model. There is a dependence of λb(x) in (17) on ω(x) (through n(x)=ω(x)N), on t (through the time dependence in n(x,t), β(m(t),x(t))) and on m (through β(m,x)) which is in contradiction to the primary suppositions of (Prabhu, 1980).…”

“…There is a dependence of λb(x) in (17) on ω(x) (through n(x)=ω(x)N), on t (through the time dependence in n(x,t), β(m(t),x(t))) and on m (through β(m,x)) which is in contradiction to the primary suppositions of (Prabhu, 1980). Nevertheless we start from (16) supposing that its solution satisfies (17). It was this approximation that led to (18) which in its turn yields the Smolukhovsky equation of the free coagulation…”

“…The full series (31) is, of cource, equivalent to the full series of Gaussian scheme, but usually we intend to truncate the series at some <A k (x)> ; this form (contrary to the usual method implying the truncating at the term x k ) seems to be more convenient, for example, for investigating chaotic systems because it is quite natural to investtigate their characteristics (in particular, K n (q)) as arising from some averaging procedure over the areas of parameter space which yields either constant or smoothly varying in q coefficient functions. Applying (27) to (31) get (Shpyrko, Ryazanov, 2006) R…”

“…Combining (15)- (17) with (10) (40) For β(m,x) ≈ m a with 1<a<1 (40) corresponds to fractional terms in the series (29) and fractional derivatives in (40) which is the manifestation of the fractal character of cluster formation (Nigmatullin, 1992). Thus the relations (15)(16)(17)(18)(19)(20) used in (Ryazanov, 1991) correspond to the extension of the storage model with generalized Tailor series of fractional derivatives. The explicit form of Φ in (40) allows us to solve this equation specifying the kernel β(m,x).…”

“…The relation (17) corresponds to the assumption of instantaneous stirring of the system between successive coagulation acts resulting in quasihomogeneous state of the system and thus in the statistical identity of all clusters which (equation (20)) ensures the coincidence of the kinetic equation (16) with the Smolukhovsky equation (19) (if r=0 and (17) holds). One can detalize the model adding some function b 1 (x) to (17):…”