Isingov model

master's thesis

• Title (english): Ising model
• Author: Anamarija Pavlina
• Mentor: Davor Horvatić (mentor)
• Committee member: Davor Horvatić (predsjednik povjerenstva)
• Committee member: Željka Milin Šipuš (član povjerenstva)
• Committee member: Zoran Rukelj (član povjerenstva)
• Committee member: Vanja Wagner (član povjerenstva)
• Committee member: Danko Radić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-12-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

Isingov model je matematički model uveden kao pojednostavljeni prikaz feromagnetizma u statističkoj fizici i kao takav bavi se fizikom faznih prijelaza. Model sadrži varijable diskretnih vrijednosti ±1, koje predstavljaju magnetske dipolne momente atoma – spinove. Osnovna pretpostavka modela je interakcija spina samo s njegovim najbližim susjedima. U jednodimenzionalnom Isingovom modelu pokazuje se nepostojanje faznoga prijelaza, osim pri temperaturi apsolutne nule. Landauova teorija faznih prijelaza dovodi do zaključka da se u dvodimenzionalnom Isingovom modelu javlja fazni prijelaz drugoga reda pri prelasku sistema iz feromagnetske u paramagnetsku fazu. Monte Carlo simulacijom generira se kvadratna rešetka dvodimenzionalnog Isingovog modela te se računaju termodinamičke veličine prosječne energije, magnetizacije i toplinskog kapaciteta, čija interpretacija dovodi do zaključka o pojavi i vrsti faznoga prijelaza.

Abstract (english)

The Ising model is a mathematical model introduced as a simplified representation of ferromagnetism in statistical physics and, as such, is concerned with the physics of phase transitions. The model consists of variables with discrete values ±1 that represent magnetic dipole moments of atoms - spins. The model assumes that each spin has only nearestneighbour interaction. Phase transition in the one-dimensional Ising model does not occur except at absolute zero temperature. Landau’s theory of phase transitions leads to the conclusion that in the two-dimensional Ising model, a phase transition of the second order occurs during the system’s transition from the ferromagnetic to the paramagnetic phase. Monte Carlo simulation generates a square lattice of the two-dimensional Ising model and calculates the thermodynamic quantities of average energy, magnetization and heat capacity, the interpretation of which leads to a conclusion about the occurrence and type of phase transition.

• Keywords: Isingov model; fazni prijelazi; prijelazna temperatura; Landauova teorija; magnetizacija; Monte Carlo simulacija
• Keywords (english): Ising model; phase transitions; transition temperature; Landau theory; magnetization; Monte Carlo simulation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:178440
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra edukacije matematike i fizike (magistar/magistra edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-12-19 11:15:05