Computational Techniques and Simulations

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**Monte Carlo Methods**- This is a way to simulate any kind of system's physical properties at
a nonzero temperature. For a magnetic spin model, we use random numbers to
select how to make small changes to an initial configuration of the spin
variables. If the change causes the energy of the system to stay the same
or decrease, the change in the configuration is accepted. If the change
causes the energy to increase by DE, then it is accepted with a probability of
exp(-DE/kT), where k is Boltzmann's constant and T is the temperature.
Repeating this hundreds of thousands of times generates a sequence of configurations that are in thermodynamic equilibrium for that temperature. Then any kind of averages for that temperaure can be determined, such as the total magnetization.

Try out the xmc visualization program found in the zipped file at this link xmc. (You will need a c-compiler, unix system and X11 libraries.)

**Spin Dynamics---Integration of Equations of Motion**- This is a way to find the time evolution of the magnetism.
This method starts from either a configuration taken from the Monte Carlo
scheme, or from some other interesting selected configuration, such as a
vortex or pair of vortices. The known classical equations of motion are then
integrated by some scheme for differential equations such as Runge-Kutta.
Then we can find either the motion of the individual vortices, or the
overall time-dependent fluctuations in the magnetization.
Try out the xmd simulation program found in the zipped file at this link xmd. (You will need a c-compiler, unix system and X11 libraries.)

**Combined Monte Carlo -- Spin Dynamics Simulations**- By using the output from the Monte Carlo as input for the Spin Dynmaics,
it is possible to simulate how the system behaves at a nonzero temperature.
This is an important method because it gives us the time-dependent properties
as a function of the system temperature. We can also do things like measure
the vortices and their time-dependent motions, depending on how hot the system
is. The results for the time-dependent spins can be analyzed also by making
various Fourier transforms of the data, to generate so-called dynamic
correlation functions, which can be measured in real materials
by neutron scattering experiments.
**Numerical Diagonalization**- This is a method to find the
*linear*excitations (or spinwaves) in a magnetic system, which itself is always*nonlinear*. This means, it is a method to find how the magnetic degrees of freedom vibrate about some initially chosen static spin configuration. The initial configuration can either be a ferromagnetically aligned configuration of the spins, or, even better, it could be a nonuniformly magnetized state such as a vortex. The classical equations of motion are*linearized*about the initial configuration, leading to a huge matrix to be diagonalized. The output gives the normal frequencies and modes of oscillation. This information then tells us about the stability of the initial configuration, as well as how the magnetization could be affected by temperature, which would excite these excitations. **Pencil and Paper**- Sometimes no-tech is not only better, but necessary. The nonlinear equations associated with the description of the vortices and solitons have many fascinating properties that can be best elucidated by this technique. We also need this to be able to give the computer simulations appropriate inputs, and even more importantly, to have purely mathematical models with which to compare and interpret the computer simulations.

Last update: September 8, 1997.

email to --> wysin@phys.ksu.edu