440 13 Monte Carlo Methods in Statistical Physics
for(inty =0; y < n_spins; y++){
for(intx= 0; x < n_spins; x++){
E -= (double) spin_matrix[y][x]*
(spin_matrix[periodic(y,n_spins,-1)][x] +
spin_matrix[y][periodic(x,n_spins,-1)]);
}
}
}// end function initialise
In the functionoutputwe print the final results, spanning from the mean energy to the sus-
ceptibility. Note that we divide by all spins. All the thermodynamical variables we compute
are so-called extensive ones meaning that they depend linearly on the number of spins. Since
our results will depend on the size of the lattice, we need to divide by the total number of
spins in order to see whether quantities like the energy or the heat capacity stabilise or not
as functions of increasing lattice size. This is
voidoutput(intn_spins,intmcs,doubletemperature,doubleaverage)
{
doublenorm = 1/((double) (mcs));// divided by total number of cycles
doubleEaverage = average[0]norm;
doubleE2average = average[1]norm;
doubleMaverage = average[2]norm;
doubleM2average = average[3]norm;
doubleMabsaverage = average[4]norm;
// all expectation values are per spin, divide by 1/n_spins/n_spins
doubleEvariance = (E2average- EaverageEaverage)/n_spins/n_spins;
doubleMvariance = (M2average - MaverageMaverage)/n_spins/n_spins;
doubleM2variance = (M2average - MabsaverageMabsaverage)/n_spins/n_spins;
doubleMvariance = (M2average - MabsaverageMabsaverage)/n_spins/n_spins;
ofile << setiosflags(ios::showpoint | ios::uppercase);
ofile << setw(15) << setprecision(8) << temperature;
ofile << setw(15) << setprecision(8) << Eaverage/n_spins/n_spins;
ofile << setw(15) << setprecision(8) << Evariance/temperature/temperature;
// ofile << setw(15) << setprecision(8) << Maverage/n_spins/n_spins;
ofile << setw(15) << setprecision(8) << M2variance/temperature;
ofile << setw(15) << setprecision(8) << Mabsaverage/n_spins/n_spins << endl;
}// end output function
13.5.1Parallelization of the Ising Model
To parallelize the Ising model, or many Monte Carlo procedures is in general rather simple.
Here we show an example of a modified main program where we let different nodes perform
a given set of Monte Carlo samples. We have fixed the size of thegrid to a 40 × 40 lattice, but
the reading of these variables can easily be done by the master node, either by reading the
variables from the command line or via a user-defined file.
Note that every node has its own seed for the random number generators.
/
Program to solve the two-dimensional Ising model
with zero external field using MPI
The coupling constant J = 1
Boltzmann's constant = 1, temperature has thus dimension energy
Metropolis sampling is used. Periodic boundary conditions.
The code needs an output file on the command line.
/
#include"mpi.h"