**Authors: **Gerasymov О.I.

**Year: **2017

**Issue: **21

**Pages: **86-92

**Abstract**

It is well known that one-dimensional (1D) models can be an effective tool for solving many problems in statistical mechanics. For instance a particular attention in such areas as chemical reactions, random walks and aggregation problems has been paid to the role of dimensionality. We study the effects of low dimensional constrains of model reactive systems. We present an exactly solvable model of fluctuational dynamics in bimoleculary reactive, partially filled, 1D perfect lattice. A rigorous expressions have been obtained for the probability distribution function, average numbers of particles, mean square fluctuations, configurational entropy and statistical sum. The previous data for Ising model of 1D nonreactive lattice gas adsorption have been completed by getting a rigorous expression for configurational statistical sum. We found that in the case of vacancied chemically reactive lattice, like in the case of exclusion statistics , distribution function has a chiral form, expressed in terms of Jacobi polynomials or Gauss confluent functions. It is shown that the nonlinearity of the reaction radically change the expected mean-field behavior. We show considered system is nonergodic with respect to chemical dynamics, and has a steady state, with a not a mean-field ratio of the average numbers of particles, which approached asymptotically. Obtained results also contrastly display coupling between microscopic processes and collective behavior as described by the macrovariables

**Tags: **ergodicity; fluctuation dynamics; probability distribution functions; reactive 1D lattices; statistical mechanics on frustrated lattices; вероятностные функции распределения; реагирующие одномерные решетки; статистическая механика реагирующих решеток с вакансиями; флуктуационная динамика; эргодичность

**Bibliography**

- Nicolis G., Prigogine I. Self-organization in Nonequilibrium systems. New York: Wiley Interscience, 1977.
- Privman V.(Ed.) Non-equilibrium Statistical Mechanics in 1D. Cambridge: Cambridge University Press, 1997.
- Hill T. An introduction to statistical thermodynamics. New York: Dover Publ., 1986.
- Provata A., Turner J., Nicolis G. Journ. Stat. Phys, 1993, vol. 70, p. 1195.
- Prudnicov A., Brychkov Yu., Marichev O. Integrals and series. Vol.1.: Elementary functions. Vol.2.: Special functions. New York: Gordon and Breach S.P., 1986.
- Marro J., Dickman R. Non-equilibrium phase transitions in Lattice models. Cambridge: Cambridge University Press, 1999.
- Hikami K. Excusion statistics and chiral partition function. Physics and combinatorics. Singapore: World Scientific, 2002. (Ed.:A.Kirilov)
- Gerasymov O.I,Khudyntsev N.N.Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 75.
- Bowker M. The basis and applications of Heterogeneous Catalysis. Oxford: Oxford University Press, 1998.