MC Simulation Lab 7: 2-Dimensional Lennard-Jones Simple Liquid (Off-Lattice) and Its Phase Transition

Lennard-Jones (vdW) interaction potential V(r)

Almost always involves a Markov process Move to a new configuration from an existing one according to a well-defined transition probability Simulation procedure 1.Generate a new “trial” configuration by making a perturbation to the present configuration 2.Accept the new configuration based on the ratio of the probabilities for the new and old configurations, according to the Metropolis algorithm 3.If the trial is rejected, the present configuration is taken as the next one in the Markov chain 4.Repeat this many times, accumulating sums for averages state k state k+1 David A. Kofke, SUNY Buffalo 2-Dimensional Lennard-Jones Simple Liquid (Off-Lattice)

Simulation box = square (2D) or cubic (3D) Number of particles N = 30 x 30 = 900 Density ρ = 0.8 particles/σ 2 Size of the box L = (N/ρ) 1/2 Cutoff radius r C = L/2 Temperature T = Make a suitable test to check for periodic boundary conditions. 2.Make sure that your acceptance rate is between 30 and 50%. 3.Equilibrate the system: Calculate the total energy U for each configuration, its average and the correlation function. 4.Obtain for different temperatures and the specific heat. 5.Check your result with the specific heat evaluated from the fluctuation analysis: mc ndim init nx ny density temp rcutoff nsteps nopt displmax seed graphswitch mc versus

/* Get command-line input */ if (argc != 13) { fprintf(stderr, "Usage: %s n_dim init n_x n_y density temp r_cutoff n_steps n_opt displ_max seed graph_switch\n", argv[0]); exit(1); } n_dim = atoi(argv[1]); init = atoi(argv[2]); n_x = atoi(argv[3]); n_y = atoi(argv[4]); density = atof(argv[5]); temp = atof(argv[6]); r_cutoff = atof(argv[7]); n_steps = atoi(argv[8]); n_opt = atoi(argv[9]); displ_max = atof(argv[10]); seed = atoi(argv[11]); graph_switch = atoi(argv[12]); sigma = 1.0; epsilon = 1.0; mc.c (part 1) mc ndim init nx ny density temp rcutoff nsteps nopt displmax seed graphswitch mc

/* Derived parameters. */ n_atoms = n_x * n_y; /* LJ parameters */ sigma2 = SQR(sigma); four_epsilon = 4.0 * epsilon; /* range of interactions. */ r_cutoff2 = SQR(r_cutoff); if (init == 1 && n_dim != 2 ) { printf("crystal builder only for n_dim = 2\n"); exit(1); } if (init == 1 && n_y % 2 !=0) { printf("n_y must be even for crystal builder\n"); exit(1); } if (init == 1 && density > 2.0 / sqrt(3.0) / sigma2) { printf("density must be lower or equal to %g\n", 2.0 / sqrt(3.0) / sigma2); exit(1); } mc.c (part 2)

Reduced units Unit of length = σ (i.e.  = 1) Unit of energy = ε (i.e.  = 1) Unit of mass = m (i.e. m = 1) All the other units descend from these. Unit of time = Unit of temperature =  / k B  LJ vdW interaction potential in these unit:

Lennard-Jones (vdW) interaction potential V(r)

/* Allocate memory. */ h = (double*) allocate_1d_array(n_dim, sizeof(double)); h_inv = (double*) allocate_1d_array(n_dim, sizeof(double)); r = (double**) allocate_2d_array(n_atoms, n_dim, sizeof(double)); s = (double**) allocate_2d_array(n_atoms, n_dim, sizeof(double)); if (init == 0) { /* build isotropic */ /* Create simple unit cell with specified number density. */ for (i = 0; i < n_dim; ++i) { h[i] = pow(n_atoms / density, 1.0 / n_dim); h_inv[i] = 1.0 / h[i]; } /* Build an isotropic initial configuration. */ build_isotropic(n_dim, n_atoms, sigma2, r, s, h, &seed); } else { /* build crystal 2d */ /* Create simple unit cell with specified number density. */ a = sqrt(2.0 / sqrt(3.0) / density); h[0] = n_x * a; h_inv[0] = 1.0 / h[0]; h[1] = n_y * sqrt(3.0) * 0.5 * a; h_inv[1] = 1.0 / h[1]; /* Build a hexagonal crystal initial configuration. */ build_crystal(n_dim, n_x, n_y, a, r, s, h); } mc.c (part 3)

#include "test.h" #include "build_crystal.h" void build_crystal(int n_dim, int n_x, int n_y, double a, double **r, double **s, double *h) { int i, j, k; k = 0; /* Loop over atoms. */ for (j = 0; j < n_y / 2; ++j) { for (i = 0; i < n_x; ++i) { /* Generate scaled and real atom position (2 atoms per unit cell). */ r[k][0] = -n_x * i * a; r[k][1] = -n_y * j * a * sqrt(3.0); s[k][0] = r[k][0] / h[0]; s[k][1] = r[k][1] / h[1]; ++k; r[k][0] = r[k-1][0] + a * 0.5; r[k][1] = r[k-1][1] + sqrt(3) * a * 0.5; s[k][0] = r[k][0] / h[0]; s[k][1] = r[k][1] / h[1]; ++k; } printf("\nConfiguration building completed\n"); fflush(NULL); return; /* Return zero if the initial condition is successfully constructed. */ } build_crystal.c

#include "test.h" #include "build_isotropic.h" #include "ran3.h" #define MAX_TRIALS void build_isotropic(int n_dim, int n_atoms, double sigma2, double **r, double **s, double *h, long *seed) { int i, i_atom, j_atom, n_trials, overlap_flag; double r2; static int first_call = 1; static double *s_pair, *r_pair; if (first_call) { s_pair = (double*) allocate_1d_array(n_dim, sizeof(double)); r_pair = (double*) allocate_1d_array(n_dim, sizeof(double)); first_call = 0; } /* Loop over atoms. */ for (i_atom = 0; i_atom < n_atoms; ++i_atom) { /* Generate trial position for atom i until an overlap-free set is found. */ n_trials = 0; do { build_isotropic.c (part 1)

/* Exit with nonzero return value if maximum number of trial insertions is exceeded. */ if (n_trials > MAX_TRIALS) { printf("Maximum number of trials exceeded in amorphous builder\n"); exit(1); } /* Generate scaled and real trial atom position. */ for (i = 0; i < n_dim; ++i) { s[i_atom][i] = ran3(seed) - 0.5; r[i_atom][i] = s[i_atom][i] * h[i]; } /* Check for overlaps between i_atom and the atoms already in the unit cell. */ overlap_flag = 0; for (j_atom = 0; j_atom < i_atom; ++j_atom) { r2 = 0.0; /* Compute distance between atoms i_atom and j_atom */ for (i = 0; i < n_dim; ++i) { s_pair[i] = s[j_atom][i] - s[i_atom][i]; s_pair[i] -= NINT(s_pair[i]); r_pair[i] = s_pair[i] * h[i]; r2 += SQR(r_pair[i]); } build_isotropic.c (part 2)

/* Exit if an overlap is found. */ if (r2 < sigma2) { overlap_flag = 1; break; } /* Increment trial counter. */ ++n_trials; } while (overlap_flag); /* Print out number of trials required for successful particle insertion. */ printf("i_atom = %d, n_trials = %d\n", i_atom, n_trials); fflush(NULL); } /* next atom. */ printf("\nConfiguration building completed\n"); fflush(NULL); return; /* Return zero if the initial condition is successfully constructed. */ } #undef MAX_TRIALS build_isotropic.c (part 3)

Initial configuration: Usually, start from a lattice. Historically, the FCC structure has been the starting configuration (for 3D). The lattice spacing is chosen so the appropriate liquid state density is obtained. During the course of the simulation, the lattice structure will disappear, to be replaced by a typical liquid structure. This process of “melting” can be enhanced by giving each molecule a small random displacement from its initial lattice point. A supercell is constructed repeating the conventional cubic unit cell of the FCC lattice N times along each direction. For 2D? We may choose to place the N particles randomly or on a cubic lattice. We are interested in equilibrium properties, hence our results MUST not depend on the initial configuration. If they do, our simulation is simply WRONG. Almost any lattice is suitable.

Units of the mass, density, and lattice constant For systems consisting of just one type of atom or molecule, it is sensible to use the mass of the molecule as a fundamental unit. m = 1 (dimensionless) With this convention: -Particle momenta and velocities become numerically identical (in MD). -Forces and accelerations become numerically identical (in MD). Lattice constant of a closely-packed FCC lattice (4 atoms in a cubic unit cell), Density of systems interacting via LJ potential  * =  3 (3D) or  2 (2D) Lattice constant of a closely-packed 2D lattice? Scaled unit: Lattice constant L = 1, atomic coordinates in [-0.5, 0.5]

/* Initialize graphics. */ antialias_switch = 1; radius = sigma / 2.0; resph = 8; if (graph_switch > 0 && n_dim == 2) { initialize_2d(&dpy, &win, doubleBuffer, &qobj, graph_switch, antialias_switch, h); // while (1) redraw_2d(dpy, win, doubleBuffer, qobj, n_atoms, r, h, radius, resph); } if (graph_switch > 0 && n_dim == 3) { initialize_3d(&dpy, &win, doubleBuffer, &qobj, graph_switch, antialias_switch, h); redraw_3d(dpy, win, doubleBuffer, qobj, n_atoms, r, h, radius, resph); } /* Compute initial potential energy */ pot_energy = compute_pot(n_dim, n_atoms, s, h, sigma2, four_epsilon, r_cutoff2); /* Print energy. */ fprintf(stderr, "\n\nInitial\n"); fprintf(stderr, "\n\nPotential energy = %12.8f\n", pot_energy); mc.c (part 4)

/* Initialize variables for MC optimization. */ n_trial = 0; n_accept = 0; sum_displ2 = 0; cpu_time = 0; displ_max_old = displ_max; displ_efficiency = 0; displ_cutoff = r_cutoff; /* Open thermodynamic file. */ fp = fopen("thermo.out", "a"); /* MC loop. */ for (i_step = 1; i_step <= n_steps; ++i_step) { /* Perform a MC cycle. */ mc_cycle(n_dim, n_atoms, r, s, displ_max, h, h_inv, temp, sigma2, four_epsilon, r_cutoff2, &seed, &pot_energy, &sum_displ2, &n_trial, &n_accept); /* Write thermodynamic quantities in file every cycle. */ fprintf(fp, "%d %12.8f\n", i_step, pot_energy); mc.c (part 5)

/* Optimize displ_max every n_opt_steps. */ if ((n_opt > 0) && (i_step % n_opt == 0)) { cpu_time_old = cpu_time; cpu_time = cpu(); elapsed_time = cpu_time - cpu_time_old; optimize_displacement(n_atoms, &displ_max, &displ_max_old, elapsed_time, &displ_efficiency, &sum_displ2, displ_cutoff); fprintf(stderr, "\n\ni_step = %d: Accepted moves = %12.8f, displ_max = %12.8f\n", i_step, (double)n_accept / (double)n_trial, displ_max); } /* Draw system. */ if (graph_switch && n_dim == 2) redraw_2d(dpy, win, doubleBuffer, qobj, n_atoms, r, h, radius, resph); else if (graph_switch && n_dim == 3) redraw_3d(dpy, win, doubleBuffer, qobj, n_atoms, r, h, radius, resph); } /* Print final energy. */ fprintf(stderr, "\n\nFinal\n"); fprintf(stderr, "\n\nPotential energy = %12.8f\n", pot_energy); fprintf(stderr, "\n\nAccepted moves = %12.8f\n", (double)n_accept / (double)n_trial); /* Close file */ fclose(fp); /* Exit */ return(0); } mc.c (part 2)

#include "test.h" #include "compute_pot.h" double compute_pot(int n_dim, int n_atoms, double **s, double *h, double sigma2, double four_epsilon, double r_cutoff2) { int i, i_atom, j_atom; double u, r_pair_mag2, rho2, rho6, rho12, factor; static int first_call = 1; static double *s_i, *s_pair, *r_pair; /* Allocate memory for local arrays the first time the routine is called. */ if (first_call) { s_i = (double*) allocate_1d_array(n_dim, sizeof(double)); s_pair = (double*) allocate_1d_array(n_dim, sizeof(double)); r_pair = (double*) allocate_1d_array(n_dim, sizeof(double)); first_call = 0; } /* Zero Lennard-Jones potential energy and forces. */ u = 0.0; /* Loop over first atom. */ for (i_atom = 0; i_atom < n_atoms - 1; ++i_atom) { /* Get i_atom scaled coordinates. */ for (i = 0; i < n_dim; ++i) s_i[i] = s[i_atom][i]; comput_pot.c (part 1)

/* Loop over second site. */ for (j_atom = i_atom + 1; j_atom < n_atoms; ++j_atom) { /* Compute pair separation vector and squared pair separation. */ r_pair_mag2 = 0.0; for (i = 0; i < n_dim; ++i) { s_pair[i] = s[j_atom][i] - s_i[i]; s_pair[i] -= NINT(s_pair[i]); r_pair[i] = h[i] * s_pair[i]; r_pair_mag2 += SQR(r_pair[i]); } /* Compute interaction if pair separation is less than the interaction cutoff. */ if (r_pair_mag2 < r_cutoff2) { /* Add contributions to potential energy and forces. */ rho2 = sigma2 / r_pair_mag2; rho6 = CUBE(rho2); rho12 = SQR(rho6); u += four_epsilon * (rho12 - rho6); } return u; /* Return potential energy. */ } comput_pot.c (part 2)

/* This routine carries out a single Monte Carlo cycle, comprising only of Monte Carlo atom move. Output: particle positions and potential energy are modified on output */ #include "test.h" #include "mc_cycle.h" #include "atom_move.h" void mc_cycle(int n_dim, int n_atoms, double **r, double **s, double displ_max, double *h, double *h_inv, double temp, double sigma2, double four_epsilon, double r_cutoff2, long *seed, double *u_pot, double *sum_displ2, int *n_trial, int *n_accept) { int i_move; /* Loop over trial Monte Carlo moves (we only move atoms here). */ for (i_move = 0; i_move < n_atoms; ++i_move) { /* Make trial move. */ ++(*n_trial); *n_accept += atom_move(n_dim, n_atoms, r, s, displ_max, h, h_inv, temp, sigma2, four_epsilon, r_cutoff2, seed, u_pot, sum_displ2); } return; } mc_cycle.c

/* This routine generates a trial displacement of an interaction atom and accepts it based on the Metropolis criterion. Output: the return value is 1 if the move was accepted and 0 otherwise. */ #include "test.h" #include "compute_pot_single.h" #include "periodic_boundary_conditions_single.h" #include "ran3.h" int atom_move(int n_dim, int n_atoms, double **r, double **s, double displ_max, double *h, double *h_inv, double temp, double sigma2, double four_epsilon, double r_cutoff2, long *seed, double *u_pot, double *sum_displ2) { int i, i_atom, accept; double u_old, u_new, delta_u; static int first_call = 1; static double *r_old, *s_old, *displ; /* Allocate memory for local arrays the first time the routine is called. */ if (first_call) { r_old = (double*) allocate_1d_array(n_dim, sizeof(double)); s_old = (double*) allocate_1d_array(n_dim, sizeof(double)); displ = (double*) allocate_1d_array(n_dim, sizeof(double)); first_call = 0; } atom_move.c (pt. 1)

/* Select an atom at random. */ i_atom = (int) (ran3(seed) * n_atoms); if (i_atom == n_atoms) i_atom -= 1; /* Store current position and potential energy of selected site. */ for (i = 0; i < n_dim; ++i) { r_old[i] = r[i_atom][i]; s_old[i] = s[i_atom][i]; } u_old = compute_pot_single(n_dim, i_atom, n_atoms, s, h, sigma2, four_epsilon, r_cutoff2); /* Generate random displacement within hypercube of size displ_max. */ for (i = 0; i < n_dim; ++i) { displ[i] = displ_max * (ran3(seed) - 0.5); r[i_atom][i] += displ[i]; } /* Apply periodic boundary conditions. */ periodic_boundary_conditions_single(n_dim, h, h_inv, r[i_atom], s[i_atom]); /* Compute trial potential energy. */ u_new = compute_pot_single(n_dim, i_atom, n_atoms, s, h, sigma2, four_epsilon, r_cutoff2); atom_move.c (pt. 2)

For a new configuration of the same volume V and number of molecules N, displace a randomly selected atom to a point chosen with uniform probability inside a cubic volume of edge 2  centered on the current position of the atom. Examine underlying transition probability to formulate the acceptance criterion ? Select an atom at random. Consider a region centered at it. Move atom to a point chosen uniformly in region. Consider acceptance of new configuration. 22 Step 1Step 2Step 3Step 4 general hitherto David A. Kofke, SUNY Buffalo 2-Dimensional Lennard-Jones Simple Liquid (Off-Lattice)

/* Compute change in potential energy and accept move using Metropolis criterion. */ delta_u = u_new - u_old; if (delta_u <= 0.0) accept = 1; else accept = ran3(seed) < exp(- delta_u / temp); /* Update system potential energy if trial move is accepted. Otherwise, reset variables. */ if (accept) { *u_pot += delta_u; for (i = 0; i < n_dim; ++i) *sum_displ2 += SQR(displ[i]); } else { for (i = 0; i < n_dim; ++i) { r[i_atom][i] = r_old[i]; s[i_atom][i] = s_old[i]; } /* Return 1 if trial move is accepted, 0 otherwise. */ return accept; } atom_move.c (pt. 3)

/* Compute potential energy of i_atom using an all pair search. */ #include "test.h" #include "compute_pot_single.h" double compute_pot_single(int n_dim, int i_atom, int n_atoms, double **s, double *h, double sigma2, double four_epsilon, double r_cutoff2) { int i, j_atom; double u, r_pair_mag2, rho2, rho6, rho12, factor; static int first_call = 1; static double *s_i, *s_pair, *r_pair; /* Allocate memory for local arrays the first time the routine is called. */ if (first_call) { s_i = (double*) allocate_1d_array(n_dim, sizeof(double)); s_pair = (double*) allocate_1d_array(n_dim, sizeof(double)); r_pair = (double*) allocate_1d_array(n_dim, sizeof(double)); first_call = 0; } /* Zero Lennard-Jones potential energy and forces. */ u = 0.0; /* Get i_atom scaled coordinates. */ for (i = 0; i < n_dim; ++i) s_i[i] = s[i_atom][i]; comput_pot_single.c (part 1)

/* Loop over atoms. */ for (j_atom = 0; j_atom < n_atoms; ++j_atom) { if (j_atom != i_atom) { /* Compute pair separation vector and squared pair separation. */ r_pair_mag2 = 0.0; for (i = 0; i < n_dim; ++i) { s_pair[i] = s[j_atom][i] - s_i[i]; s_pair[i] -= NINT(s_pair[i]); r_pair[i] = h[i] * s_pair[i]; r_pair_mag2 += SQR(r_pair[i]); } /* Compute interaction if pair separation is less than the interaction cutoff. */ if (r_pair_mag2 <= r_cutoff2) { /* Add contributions to potential energy and forces. */ rho2 = sigma2 / r_pair_mag2; rho6 = CUBE(rho2); rho12 = SQR(rho6); u += four_epsilon * (rho12 - rho6); } return u; /* Return potential energy of atom i_atom. */ } comput_pot_single.c (part 2)

/* This routine optimizes displacement moves for a Monte Carlo simulation by maximizing the displacement efficiency, defined as the cumulative squared displacement per cpu second. output: displ_max, displ_max_old, displ_efficiency, and sum_displ2 are modified on return. */ #include "test.h" #include "optimize_displacement.h" #include "displacement_efficiency.h" #define SCALE_FACTOR 1.1 void optimize_displacement(int n_atoms, double *displ_max, double *displ_max_old, double elapsed_time, double *displ_efficiency, double *sum_displ2, double displ_cutoff) { double displ_efficiency_old, damping_factor, deriv; /* Save old displacement efficiency. */ displ_efficiency_old = *displ_efficiency; /* Calculate new displacement efficiency. */ *displ_efficiency = displacement_efficiency(n_atoms, elapsed_time, sum_displ2); /* Include damping factor to prevent runaway at low densities. */ damping_factor = exp(- *displ_max / displ_cutoff); *displ_efficiency *= damping_factor; optimize_displacement.c (pt. 2)

fprintf(stdout, " displ_max_old = %g, displ_efficiency_old = %g / cpu s\n", *displ_max_old, displ_efficiency_old); fprintf(stdout, " displ_max = %g, displ_efficiency = %g / cpu s\n", *displ_max, *displ_efficiency); /* Estimate derivative of displacement efficiency with respect to maximum displacement. */ if (*displ_max == *displ_max_old) deriv = 0.0; else deriv = (*displ_efficiency - displ_efficiency_old) / (*displ_max - *displ_max_old); /* Adjust maximum displacement. */ *displ_max_old = *displ_max; if (deriv < 0.0) *displ_max /= SCALE_FACTOR; else *displ_max *= SCALE_FACTOR; fprintf(stdout, " new displ_max = %g\n\n", *displ_max); return; } #undef SCALE_FACTOR optimize_displacement.c (pt. 2)

/* This routine calculates the atom displacement efficiency for a Monte Carlo simulation, defined as the cumulative squared object displacement per cpu second. output: efficiency (return value) sum_displ2 is modified on return. */ #include "test.h" #include "displacement_efficiency.h" double displacement_efficiency(int n_atoms, double elapsed_time, double *sum_displ2) { double efficiency; /* Calculate efficiency, defined as the cumulative squared atom displacement per cpu second. */ efficiency = *sum_displ2 / (n_atoms * elapsed_time); /* Zero displacement accumulator. */ *sum_displ2 = 0.0; return efficiency; } optimize_efficiency.c

/* This routine calculates the atom displacement efficiency for a Monte Carlo simulation, defined as the cumulative squared object displacement per cpu second. output: efficiency (return value) sum_displ2 is modified on return. */ #include "test.h" #include "displacement_efficiency.h" double displacement_efficiency(int n_atoms, double elapsed_time, double *sum_displ2) { double efficiency; /* Calculate efficiency, defined as the cumulative squared atom displacement per cpu second. */ efficiency = *sum_displ2 / (n_atoms * elapsed_time); /* Zero displacement accumulator. */ *sum_displ2 = 0.0; return efficiency; } optimize_efficiency.c

#include "test.h" #include "periodic_boundary_conditions.h" void periodic_boundary_conditions(int n_dim, int n_atoms, double *h, double *h_inv, double **r, double **s) { int i, i_object; for (i_object = 0; i_object < n_atoms; ++i_object) { /* Compute scaled coordinate and apply periodic boundary conditions. */ for (i = 0; i < n_dim; ++i) { s[i_object][i] = h_inv[i] * r[i_object][i]; s[i_object][i] -= NINT(s[i_object][i]); } /* Recompute real coordinates accounting for periodic boundary conditions. */ for (i = 0; i < n_dim; ++i) r[i_object][i] = h[i] * s[i_object][i]; } return; } periodic_boundary_conditions.c

#include "test.h" #include "periodic_boundary_conditions_single.h" void periodic_boundary_conditions_single(int n_dim, double *h, double *h_inv, double *r, double *s) { int i; /* Compute scaled coordinate and apply periodic boundary conditions. */ for (i = 0; i < n_dim; ++i) { s[i] = h_inv[i] * r[i]; s[i] -= NINT(s[i]); } /* Recompute real coordinates accounting for periodic boundary conditions. */ for (i = 0; i < n_dim; ++i) r[i] = h[i] * s[i]; return; } periodic_boundary_conditions_single.c