Computational Physics Lectures: Statistical physics and the Ising Model

This means in turn that we could construct an array which contains all values of

$e^{\beta \Delta E}$ before doing the Metropolis sampling. Else, we would have to evaluate the exponential at each Monte Carlo sampling. For the two-dimensional Ising model there are only five possible values. It is rather easy to convice oneself that for the one-dimensional Ising model we have only three possible values. The main part of the Ising model program is shown here

/* 
   Program to solve the two-dimensional Ising model 
   The coupling constant J = 1
   Boltzmann's constant = 1, temperature has thus dimension energy
   Metropolis sampling is used. Periodic boundary conditions.
*/
#include <iostream>
#include <fstream>
#include <iomanip>
#include "lib.h"
using namespace  std;
ofstream ofile;
// inline function for periodic boundary conditions
inline int periodic(int i, int limit, int add) { 
  return (i+limit+add) % (limit);
}
// Function to read in data from screen  
void read_input(int&, int&, double&, double&, double&);
// Function to initialise energy and magnetization
void initialize(int, double, int **, double&, double&);
// The metropolis algorithm 
void Metropolis(int, long&, int **, double&, double&, double *);
// prints to file the results of the calculations  
void output(int, int, double, double *);

//  main program
int main(int argc, char* argv[])
{
  char *outfilename;
  long idum;
  int **spin_matrix, n_spins, mcs;
  double w[17], average[5], initial_temp, final_temp, E, M, temp_step;

  // Read in output file, abort if there are too few command-line arguments
  if( argc <= 1 ){
    cout << "Bad Usage: " << argv[0] << 
      " read also output file on same line" << endl;
    exit(1);
  }
  else{
    outfilename=argv[1];
  }
  ofile.open(outfilename);
  //    Read in initial values such as size of lattice, temp and cycles
  read_input(n_spins, mcs, initial_temp, final_temp, temp_step);
  spin_matrix = (int**) matrix(n_spins, n_spins, sizeof(int));
  idum = -1; // random starting point
  for ( double temp = initial_temp; temp <= final_temp; temp+=temp_step){
    //    initialise energy and magnetization 
    E = M = 0.;
    // setup array for possible energy changes
    for( int de =-8; de <= 8; de++) w[de+8] = 0;
    for( int de =-8; de <= 8; de+=4) w[de+8] = exp(-de/temp);
    // initialise array for expectation values
    for( int i = 0; i < 5; i++) average[i] = 0.;
    initialize(n_spins, double temp, spin_matrix, E, M);
    // start Monte Carlo computation
    for (int cycles = 1; cycles <= mcs; cycles++){
      Metropolis(n_spins, idum, spin_matrix, E, M, w);
      // update expectation values
      average[0] += E;    average[1] += E*E;
      average[2] += M;    average[3] += M*M; average[4] += fabs(M);
    }
    // print results
    output(n_spins, mcs, temp, average);
  }
  free_matrix((void **) spin_matrix); // free memory
  ofile.close();  // close output file
  return 0;
}

The Metropolis Algorithm and the Two-dimensional Ising Model, elements of program