Teorija perkolacije

master's thesis

• Title (english): Percolation theory
• Author: Valentina Krklec
• Mentor: Rudi Mrazović (mentor)
• Committee member: Rudi Mrazović (predsjednik povjerenstva)
• Committee member: Ivica Nakić (član povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Goranka Nogo (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-12-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Teorija perkolacije opisuje fazni prijelaz. Postoji prirodni parametar s obzirom na kojeg se ponašanje sustava drastično mijenja. Takav prirodan parametar u teoriji perkolacije nazivamo kritična vjerojatnost. Postoje točno dvije faze procesa koje ovise o odnosu vjerojatnosti p koja označava da je brid otvoren i kritične vjerojatnosti \(p_c\). Kada je vjerojatnost p manja od kritične vjerojatnosti \(p_c\) tada se proces nalazi u subkritičnoj fazi, a inače u superkritičnoj fazi. U subkritičnoj fazi ne postoji otvoreni beskonačni klaster, dok u superkritičnoj fazi postoji. Ako je vjerojatnost postojanja beskonačnog otvorenog klastera strogo pozitivna tada gotovo sigurno postoji točno jedan takav klaster. Iz osnovnog procesa teorije perkolacije možemo razviti znatno kompleksnije procese, te samim time znatno proširiti njihove primijene u biologiji, fizici, kemiji i astronomiji.

Abstract (english)

Percolation theory describes a phase transition process. There is a natural parameter which describes sudden change in the behavior of the system. Such natural parameter in percolation theory is called critical probability. There are two stages of the percolation process that are determined by the value of critical probability \(p_c\) and the probability p of a fixed edge being open. When the probability p is less than the critical probability \(p_c\) then the process is called the subcritical phase, otherwise it is known as supercritical phase. Open infinite cluster does not exist in the subcritical phase while in the supercritical phase it exists. If the probability for the existance of an infinite open cluster is strictly positive then there is almost certainly unique. From the basic process of the percolation theory we can develop much more complex processes, and thus significantly expand the range of their application in biology, physics, chemistry and astronomy.

• Keywords: kritična vjerojatnost; subkritična faza; superkritična faza; beskonačni otvoreni klaster
• Keywords (english): critical probability; subcritical phase; supercritical phase; infinite open cluster
• Language: croatian
• URN:NBN: urn:nbn:hr:217:039741
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-12-15 10:43:48