2D Ising Model (analytical expressions)

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SUMMARY

The discussion focuses on simulating the 2D Ising Model using the Metropolis algorithm and comparing results with analytical expressions for mean energy, magnetization, specific heat, and magnetic susceptibility. The user successfully identified expressions for mean energy, magnetization, and specific heat from Huang's book but encountered issues with the mean energy plot. The user seeks the analytical expression for magnetic susceptibility, which is derived from the magnetization expression M=[1-\sinh^{-4}(2/k_B T)]^{1/8} and the susceptibility formula \chi = \frac{- ^2}{k_B T}.

PREREQUISITES
  • Understanding of the 2D Ising Model
  • Familiarity with the Metropolis algorithm
  • Knowledge of statistical mechanics concepts such as magnetization and susceptibility
  • Proficiency in mathematical expressions involving hyperbolic functions
NEXT STEPS
  • Research the analytical expression for magnetic susceptibility in the 2D Ising Model
  • Explore numerical methods for validating simulation results against analytical predictions
  • Study the derivation of specific heat in the context of the Ising Model
  • Investigate the impact of external magnetic fields on magnetization and susceptibility
USEFUL FOR

Physicists, computational scientists, and students interested in statistical mechanics, particularly those working on simulations of the Ising Model and related phenomena.

Orion_PKFD
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Hi all,

I am doing a program to simulate the 2D Ising Model under the metropolis algorithm. In order to check my results I would like to compare them with the analytical expressions for the mean energy, magnetization, specific heat and magnetic susceptibility.

I already found the expressions for the mean energy, magnetization and specific heat. However, when I plot the expression for the mean energy the result does not look right. I used the one in Huang's book. Concerning the magnetic susceptibility, I couldn't find any analytical expression...

Anyone could give me some help? Thanks.

Best regards!
 
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If you have the expression for the magnetisation for a given applied field you should be able to go from there to the susceptibility...
 
Hi,

Thank you for your reply. I am only interested in the case with B=0.

The expression for the magnetization is M=[1-\sinh^{-4}(2/k_B T)]^{1/8}.

The susceptibility is obtained from \chi = \frac{<M^2>- <M>^2}{k_B T}.

But I don't know the analytical expression for \chi...
 

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