PHY 341641 Thermodynamics and Statistical Physics Lecture 20

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PHY 341/641 Thermodynamics and Statistical Physics Lecture 20 Methodologies of statistical mechanics. (Chapter 5

PHY 341/641 Thermodynamics and Statistical Physics Lecture 20 Methodologies of statistical mechanics. (Chapter 5 in STP) • Phase transitions in spin systems • Mean field approximation 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 1

10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 2

10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 2

Ising model in various lattices – mean field solutions q=4/2 10/3/2020 q=6/2 Note: textbooks

Ising model in various lattices – mean field solutions q=4/2 10/3/2020 q=6/2 Note: textbooks differ with counting neighbors q without double counting. Here q implies “unique” neighbors. PHY 341/641 Spring 2012 -- Lecture 20 3

Self-consistency condition for mean field treatment 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20

Self-consistency condition for mean field treatment 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 4

Mean field self-consistency condition: H=0 Mean field solutions exhibit “critical behavior” (phase transition) at

Mean field self-consistency condition: H=0 Mean field solutions exhibit “critical behavior” (phase transition) at bc. Jq=1. 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 5

Mean field self-consistency condition for H=0: m 10/3/2020 T/Tc PHY 341/641 Spring 2012 --

Mean field self-consistency condition for H=0: m 10/3/2020 T/Tc PHY 341/641 Spring 2012 -- Lecture 20 6

Summary of results for mean field treatment of Ising model 10/3/2020 PHY 341/641 Spring

Summary of results for mean field treatment of Ising model 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 7

Summary of results for mean field treatment of Ising model (continued) 10/3/2020 PHY 341/641

Summary of results for mean field treatment of Ising model (continued) 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 8

Summary of results for mean field treatment of Ising model (continued) 10/3/2020 PHY 341/641

Summary of results for mean field treatment of Ising model (continued) 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 9

Summary of results for mean field treatment of Ising model (continued) Behavior for T

Summary of results for mean field treatment of Ising model (continued) Behavior for T near Tc where m is small 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 10

Summary of results for mean field treatment of Ising model (continued) -- behavior for

Summary of results for mean field treatment of Ising model (continued) -- behavior for T near Tc where m is small 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 11

Summary of results for mean field treatment of Ising model (continued) -- behavior for

Summary of results for mean field treatment of Ising model (continued) -- behavior for T near Tc where m is small 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 12

Behavior of magnetic susceptibility (in scaled units): 10/3/2020 PHY 341/641 Spring 2012 -- Lecture

Behavior of magnetic susceptibility (in scaled units): 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 13

Behavior of magnetic susceptibility (in scaled units) -- continued: 10/3/2020 PHY 341/641 Spring 2012

Behavior of magnetic susceptibility (in scaled units) -- continued: 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 14

Behavior of magnetization at Tc: 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 15

Behavior of magnetization at Tc: 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 15

Critical exponents: 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 16

Critical exponents: 10/3/2020 PHY 341/641 Spring 2012 -- Lecture 20 16

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