Discrete and continuous description of physical phenomena

Abstract

The values of many phenomena in the Nature $z$ are determined in some discrete set of times t_n, separated by a small interval $\Delta t$ (which may also represent a coordinate, etc.). Let the $z$ value in neighbour point $t_{n+1}=t_n+\Delta t$ be expressed by the evolution equation as $z(t_{n+1})= z(t_n+\Delta t)=f(z(t_n))$. This equation gives {\it\underline{a discrete description}} of phenomenon. Considering phenomena at $t\gg \Delta t$ this equation is transformed often into the differential equation allowing to determine $z(t)$ -- {\underline{\it continuous description}}. It is usually assumed that the continuous description describes correctly the main features of a phenomenon at values $t>> \Delta t$. In this paper I show that the real behavior of some physical systems can differ strongly from that given by the continuous description. The observation of such effects may lead to the desire to supplement the original evolutionary model by additional mechanisms, the origin of which require special explanation. We will show that such construction may not be necessary -- simple evolution model can describe different observable effects. This text contains no new calculations. Most of the discussed facts are well known. New is the treatment of the results.