Computational Physics Lectures: Statistical physics and the Ising Model
Contents
Ensembles
Famous Ensembles
Canonical Ensemble
The partition function is a normalization constant
Helmoltz free energy, what does it mean?
Thermodynamical quantities
Thermodynamical quantities, the energy in the canonical ensemble
Energy and specific heat in the canonical ensemble
Magnetic moments and susceptibility in the canonical ensemble
Our model, the Ising model in one and two dimensions
Boltzmann distribution
Energy for a specific configuration
Configurations
Boundary conditions, free ends
Free ends and the energy
Free ends and energy
Periodic boundary conditions
Energy with PBC
Simple code for PBC
Summing up
Reorganizing
Our model, the Ising model in one and two dimensions
Ising model in one and two dimensions
List of configurations with energies and magnetic moment
Phase Transitions and Critical Phenomena
Phase Transitions and Critical Phenomena, correlation length
Classification of phase transitions
Second-order phase Transitions
Phase Transitions and Critical Phenomena
Eherenfest definition of phase Transitions
Phase Transitions and Critical Phenomena
The Ising Model and Phase Transitions
The Ising Model and Phase Transitions, mean magnetization
The Ising Model and Phase Transitions, correlation length
The Ising Model and Phase Transitions, critical behavior
The Ising Model and Phase Transitions, critical temperature
The Ising Model and Phase Transitions, correlation length
The Metropolis Algorithm and the Two-dimensional Ising Model
The Metropolis Algorithm and the Two-dimensional Ising Model
The Metropolis Algorithm and the Two-dimensional Ising Model, practical issues
Five possible energy differences
"The Metropolis Algorithm and the Two-dimensional Ising Model, elements of program":"https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Programs/ParallelizationMPI/MPIising.cpp"
Coding energy differences
Efficient expression for energy differences
Final energy difference
Change in magnetization
A small note
Initialization
"The Metropolis Algorithm and the Two-dimensional Ising Model, elements of program":"https://github.com/CompPhysics/ComputationalPhysics/blob/master/doc/Programs/ParallelizationMPI/IsingModel.cpp"
Two-dimensional Ising Model, energy per spin and specific heat
Two-dimensional Ising Model and analysis of spin values
Reorganizing
We can reorganize according to the number of spins pointing up, as shown in the table here
Number spins up
Degeneracy
Energy (FE)
Energy (PBC)
Magnetization
2
1
\( -J \)
\( -2J \)
2
1
2
\( J \)
\( 2J \)
0
0
1
\( -J \)
\( -2J \)
-2
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