# ising_2d_simulation

ising_2d_simulation, a FORTRAN90 code which carries out a Monte Carlo simulation of a 2D Ising model, using gnuplot() to create graphics images of the initial and final configurations.

A 2D Ising model is defined on an MxN array of cells. Each cell can have a "charge" or "spin" of +1 or -1. A neighborhood of a cell is defined to be itself, and the four immediate neighbors to the north, south, east, and west. A cell may change its charge; the likelihood of doing so depends on whether the charge of the neighbors agrees with the charge of the cell.

This program begins with a random assignment of charges to the cells. It then computes the neighbor agreement at each cell. It then updates the entire configuration in one step, using the neighbor agreement to determine the likelihood of flipping.

Currently, the chance that a cell will "flip" is determined from a table of probabilities based on the number of agreeing neighbors:

 Agree Prob 1 2 3 4 4 0.98 0.85 0.5 0.15 0.02

### Usage:

ising_2d_simulation m n iterations thresh seed
where
• m and n are the number of rows and columns of cells. The default values are 10.
• iterations is the number of iterations to take. The default value is 15.
• thresh is a number between 0 and 1 that chooses the percentage of negative (red) charges. The default value is 0.5.
• seed is a seed for the random number generator. The default value is 123456789.

Note that the user might also want to vary the transition probabilities. This cannot currently be done on the command line, and requires editing the program and recompiling it.

### Languages:

ising_2d_simulation is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

duel_simulation, a FORTRAN90 code which simulates N repetitions of a duel between two players, each of whom has a known firing accuracy.

gamblers_ruin_simulation, a MATLAB program which simulates the game of gambler's ruin.

gnuplot_test, FORTRAN90 codes which write data and command files so that gnuplot() can create plots.

high_card_simulation, a FORTRAN90 code which simulates a situation in which you see the cards in a deck one by one, and must select the one you think is the highest and stop; the program uses gnuplot() for graphics.

ising_3d_simulation, a FORTRAN90 code which carries out a Monte Carlo simulation of a 3D Ising model.

mandelbrot, a FORTRAN90 code which generates an ASCII PPM image of the Mandelbrot set;

reactor_simulation, a FORTRAN90 code which is a simple Monte Carlo simulation of the shielding effect of a slab of a certain thickness in front of a neutron source. This program was provided as an example with the book "Numerical Methods and Software."

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### Reference:

1. American National Standard for Programming Language: Fortran - Extended,
American National Standards Institute, 1992,
pages 296-299.
2. Bryan Hayes,
The World in a Spin,
On the Ising model of ferromagnets and other physical systems,
American Scientist,
Volume 88, Number 5, September-October 2000, pages 384-388,

### Source Code:

Last revised on 19 July 2020.