PHY 341641 Thermodynamics and Statistical Physics Lecture 19
- Slides: 17
PHY 341/641 Thermodynamics and Statistical Physics Lecture 19 Methodologies of statistical mechanics. (Chapter 5 in STP) Ising model systems a. Definition b. Exact solution for 1 -dimension c. Mean field solution for 1 -d d. Ising model for 2 dimensions 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 1
12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 2
Ising model 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 3
Exact solution for 1 -dimensional Ising system of N spins with periodic boundary conditions (s. N+1=s 1) 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 4
b. J=2 b. J=1 M/N b. J=0 b. H 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 5
Mean field approximation for 1 -dimensional Ising model 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 6
Mean field partition function and Free energy: 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 7
One dimensional Ising model with periodic boundary conditions: b=1 J=1 H 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 8
Two-dimensional Ising model – paramagnetic ferromagnetic phase transition 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 9
Behavior of Elliptic integral function K 1(x) Re[K 1(x)] 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 10
12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 11
Magnetic properties of 2 -dimensional Ising model 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 12
Simulation of 2 -dimensional Ising model with Monte Carlo sampling stp_Ising 2 D. jar 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 13
Ising model in various lattices – mean field solutions q=4 12/29/2021 q=6 PHY 341/641 Spring 2012 -- Lecture 19 14
Self-consistency condition for mean field treatment 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 15
Mean field self-consistency condition: H=0 Mean field solutions exhibit “critical behavior” (phase transition) at bc. Jq=1. 12/29/2021 PHY 341/641 Spring 2012 -- Lecture 19 16
Mean field self-consistency condition for H=0: m 12/29/2021 T/Tc PHY 341/641 Spring 2012 -- Lecture 19 17
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