Računanje vjerojatnosti u kritičnim sustavima

master's thesis

• Title (english): Calculation of probabilities in critical systems
• Author: Lovro Šaravanja
• Mentor: Ivan Balog (mentor)
• Mentor: Zoran Rukelj (komentor)
• Committee member: Zoran Rukelj (predsjednik povjerenstva)
• Committee member: Ivan Balog (član povjerenstva)
• Committee member: Mihael Makek (član povjerenstva)
• Committee member: Željko Skoko (član povjerenstva)
• Committee member: Mario Novak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Physics); Zagreb
• Defense date and country: 2024-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

U ovom radu analizira se vjerojatnosna raspodjela magnetizacije O(Ns) modela analitičkim metodama neperturbativne renormalizacijske grupe (NPRG) kao i numerički, Monte-Carlo (MC) simulacijama. U uvodnom dijelu dan je pregled analitičkih metoda koje se temelje na pristupu renormalizacijske grupe. Predstavljen je NPRG pristup ili formalizam efektivnog djelovanja, koji je ishodište za postavljanje jednadžbe toka za vjerojatnosni potencijal, koji određuje vjerojatnosnu raspodjelu magnetizacije. Naglašavamo razlike između efektivnog i vjerojatnosnog potencijala. Fokusiramo se nadalje na N → ∞ limes, u kojemu je moguće naći egzaktan izraz kako za efektivni potencijal, tako i za vjerojatnosni potencijal. Ti izrazi se uspoređuju i analiziraju u različitim režimima. Također diskutiramo kako se postiže limes centralnog graničnog teorema, u režimu daleko od prijelaza u uređenoj fazi. Tamo su efektivni stupnjevi slobode sustava grozdovi konačne veličine koji su statistički neovisni jedni od drugih. S druge strane u uređenoj fazi efektivni stupnjevi slobode su spinski valovi, čiji se potpis može vidjeti u vjerojatnosnom potencijalu. U konačnici dajemo kratak pregled Wolffovog algoritma kao MC koji se može koristiti za nalaženje vjerojatnosnog potencijala. Prezentirani su rezultatu za Ns = 3, 4 kao i usporedba s analitičkim Ns → ∞.

Abstract (english)

In this work, the probability distribution of the magnetization in the O(Ns) model is analyzed using both analytical methods of the non-perturbative renormalization group (NPRG) and numerically, through Monte Carlo (MC) simulations. The introductory section provides an overview of analytical methods based on the renormalization group approach. The NPRG approach, or the effective action formalism, is presented as the foundation for deriving the flow equation for the probability potential, which determines the probability distribution of the magnetization. We emphasize the differences between the effective and probability potentials. We further focus on the N → ∞ limit, where it is possible to find exact expressions for both the effective potential and the probability potential. These expressions are compared and analyzed in different regimes. Additionally, we discuss how the central limit theorem is approached, in the regime far from the transition to the ordered phase. In this regime, the systems effective degrees of freedom are clusters of finite size that are statistically independent of one another. On the other hand, in the ordered phase, the effective degrees of freedom are spin waves, whose signature can be observed in the probability potential. Finally, we provide a brief overview of the Wolff algorithm as an MC method that can be used to determine the probability potential. Results for Ns = 3, 4 are presented, along with a comparison to the analytical Ns → ∞.

• Keywords: kritični sustavi; statistička teorija polja; neperturbativna renormalizacijska grupa; fazni prijelazi; raspodjela magnetizacije
• Keywords (english): critical systems; statistical field theory; non-perturbative renormalization group; phase transitions; magnetization distribution
• Language: croatian
• URN:NBN: urn:nbn:hr:217:709192
• Study programme: Title: The University Integrated Undergraduate and Graduate Programme in Physics; Research Physics; specializations in: Research Physics; Course: Research Physics; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: sveučilišni magistar fizike (sveučilišni magistar fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-01-31 12:18:39