Göran Wahnström, Department of Physics, Chalmers Lectures Ordinary differential equations Stochastic methods Partial differential equations Exercises/Home work problems Linear dynamics E1 Brownian dynamics E4 Metropolis algorithm Monte Carlo integration E3 H2a/H2b Non-linear dynamics Molecular dynamics E2 H1a/H1b Quantum structure H3a Quantum dynamics E5 H3b
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Göran Wahnström, Department of Physics, Chalmers
Lectures
Ordinary
differential
equations
Stochastic
methods
Partial
differential
equations
Exercises/Home work problems
Linear dynamics E1
Brownian dynamics E4
Metropolis algorithm
Monte Carlo integration E3
H2a/H2b
Non-linear dynamics
Molecular dynamics
E2
H1a/H1b
Quantum structure H3a
Quantum dynamics
E5
H3b
Ising model
Content:
Ising model
Ising model: Mean field solution
Ising model: Metropolis algorithm
Ising model: Numerical results
Binary alloy: CuZn
Binary alloy: Mean field solution
Göran Wahnström, Department of Physics, Chalmers
Ising model
a lattice model
one of the simpliest and non-trivial model
systems of interacting degrees of freedom
introduced by Lenz and Ising to model
phase transitions in magnetic materials in
the 1920s
solved exactly in 2D by Onsager 1944
has not been solved exactly in 3D (yet)
useful in condensed matter physics and field
theory
a (rather crude) model for magnetism
can be used to model binary alloys in
materials science
can be used to model adsorbed particles in
surface science
can be extended in many directions
Göran Wahnström, Department of Physics, Chalmers
Binary alloyFerromagnetic substance
low T high T
disordered stateordered state
Heat capacity
temperature
high Tlow T
paramagneticferromagnetic
Magnetization
temperature
Göran Wahnström, Department of Physics, Chalmers
Ising model – statistical thermodynamics
Göran Wahnström, Department of Physics, Chalmers
Ising model – the exact solution (2D)
Due to Onsager, 1944
Göran Wahnström, Department of Physics, Chalmers
Ising model – mean field solution (h=0)
Göran Wahnström, Department of Physics, Chalmers
Ising model – mean field solution (h=0)
Göran Wahnström, Department of Physics, Chalmers
Ising model – Metropolis algorithm
Göran Wahnström, Department of Physics, Chalmers
The Metropolis algorithm
1
acceptrej.
Göran Wahnström, Department of Physics, Chalmers
Equilibration
It is important to wait until
the system has equilibrated.
Göran Wahnström, Department of Physics, Chalmers
Equilibration
It is important to wait until
the system has equilibrated.
Göran Wahnström, Department of Physics, Chalmers
Equilibration Boundary conditions
To mimic a large system periodic boundary
conditions are commonly used.
It is important to wait until
the system has equilibrated.
Göran Wahnström, Department of Physics, Chalmers
Ising model – Numerical results
Göran Wahnström, Department of Physics, Chalmers
Ising model – Numerical results
Ising model – Exact results
Göran Wahnström, Department of Physics, Chalmers
Ising model – spatial correlations
T = 2.0 T = 2.4 T = 3.0
Göran Wahnström, Department of Physics, Chalmers
Binary alloyFerromagnetic substance
low T high T
disordered stateordered state
Heat capacity
temperature
high Tlow T
paramagneticferromagnetic
Magnetization
temperature
Göran Wahnström, Department of Physics, Chalmers
The Cu-Zn system – a binary alloy
Equilibrium phase diagram CuZn (-brass)
temperatureh
eat
cap
acit
y
Göran Wahnström, Department of Physics, Chalmers
Binary alloy – simple model
Göran Wahnström, Department of Physics, Chalmers
Binary alloy – mean field solution
Göran Wahnström, Department of Physics, Chalmers
Binary alloy – mean field solution
Göran Wahnström, Department of Physics, Chalmers
References:
Computational treatment of the Ising model:
- Giordano and Nakanishi, Computational Physics
- Koonin and Meredith, Computational Physics
Binary alloy:
- Kittel, Introduction to Solid State Physics
Advanced treatment of the Ising model:
- Huang, Statistical Mechanics
Göran Wahnström, Department of Physics, Chalmers
Göran Wahnström, Department of Physics, Chalmers
Lectures
Ordinary
differential
equations
Stochastic
methods
Partial
differential
equations
Exercises/Home work problems
Linear dynamics E1
Brownian dynamics E4
Metropolis algorithm
Monte Carlo integration E3
H2a/H2b
Non-linear dynamics
Molecular dynamics
E2
H1a/H1b
Quantum structure H3a
Quantum dynamics
E5
H3b
Variational Monte Carlo
Content:
Variational Quantum Monte Carlo
Multi-dimensional case
Example: Helium
Göran Wahnström, Department of Physics, Chalmers
Variational Quantum Monte Carlo
Göran Wahnström, Department of Physics, Chalmers
Multi-dimensional case
Göran Wahnström, Department of Physics, Chalmers
Example: Helium
Göran Wahnström, Department of Physics, Chalmers
Example: Helium
Göran Wahnström, Department of Physics, Chalmers
Variational Quantum Monte Carlo:
- Koonin and Meredith, Computational Physics
- Thijssen, Computational Physics
- P.J. Reynolds, J. Tobochnik, and H. Gould,
Computers in Physics, nov/dec 1990.
References:
Green’s function and Diffusion Monte Carlo:
- Koonin and Meredith, Computational Physics
- Thijssen, Computational Physics
Göran Wahnström, Department of Physics, Chalmers
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