Effect of sample size when using periodic boundary conditions in 2D Ising model

Click For Summary
SUMMARY

The discussion centers on the impact of sample size (N) on the sharpness of phase transitions in the 2D Ising model using the Monte Carlo Metropolis algorithm. As N increases, the transition becomes sharper, approaching a step function, due to the exponential increase in correlation length as the phase transition is approached. This phenomenon occurs despite the use of periodic boundary conditions, which simulate an infinite lattice, because the finite size of the unit cell limits the accuracy of the simulation when the correlation length exceeds the lattice size.

PREREQUISITES
  • Understanding of the 2D Ising model
  • Familiarity with the Monte Carlo Metropolis algorithm
  • Knowledge of periodic boundary conditions
  • Concept of correlation length in statistical physics
NEXT STEPS
  • Research the definition and calculation of correlation length in the context of the Ising model
  • Examine the temperature dependence of phase transitions in finite versus infinite lattices
  • Explore the effects of different lattice sizes on simulation results in statistical mechanics
  • Investigate critical opalescence and its implications for phase transitions
USEFUL FOR

Physicists, computational scientists, and researchers interested in statistical mechanics, particularly those studying phase transitions and the Ising model.

DavidwN
Messages
1
Reaction score
0
Hi,

I'm currently using the Monte Carlo Metropolis algorithm to investigate the 2D Ising model.

I have an NxN lattice of points with periodic boundary conditions imposed. I was wondering if anyone could explain why the sharpness of the phase transition is affected by the size of N?
I.e. if N is small I get a slow transition and as N is increased, the transition approaches a step function.

I don't understand why this is as I am only considering nearest neighbour interactions and by using periodic boundary conditions surely I am effectively modelling an infinite lattice? So why does the size of the unit cell affect my results?

Thanks!
 
Physics news on Phys.org
You are experiencing critical opalescence. As you approach the phase transition, the correlation length increases exponentially. When this domain size reaches the size of your simulation, then the simulation breaks down, i.e. does not describe the physics correctly anymore.

http://en.wikipedia.org/wiki/Ising_critical_exponents

Try and find a definition of the correlation length and calculated that on your grid. Then compare the temperature dependence for different grid sizes to what one would expect for an infinite lattice.
 

Similar threads

Graduate Ising model open chain and periodic boundary conditions
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad How do irrational numbers give incommensurate potential periods?
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Convergence of lattice Ising model
  • · Replies 5 ·
Replies
5
Views
4K
Graduate The Role of Boundary Conditions in the 2D Ising Model: A Finite Lattice Study
  • · Replies 14 ·
Replies
14
Views
4K
Graduate Crystal model with periodic boundary conditions
  • · Replies 2 ·
Replies
2
Views
4K
Graduate Nucleation and growth in a 2D Ising Model with Monte Carlo Simulation
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Free Electron Model: Why periodic boundary conditions and what is L ?
  • · Replies 4 ·
Replies
4
Views
8K
Graduate Ising Model for Spins: External Magnetic Field Effect
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Learning 1D Ising Model With Nearest Site Interaction
  • · Replies 12 ·
Replies
12
Views
8K
Graduate Kirkendall effect: vacancy concentration and pair interaction energy
  • · Replies 1 ·
Replies
1
Views
2K