lab 4. calculate by metropolis monte carlo method
DESCRIPTION
Lab 4. Calculate by Metropolis Monte Carlo method when (x) = A x exp (-x) for 0 < x < 4 (a 1D problem). T he Metropolis Monte Carlo algorithm can be used for a general case when the limiting distribution is not a Boltzmann distribution. r (x). x. - PowerPoint PPT PresentationTRANSCRIPT
Lab 4. Calculate <x> by Metropolis Monte Carlo methodwhen (x) = A x exp(-x) for 0 < x < 4 (a 1D problem)
x
(x)
The Metropolis Monte Carlo algorithm can be used for a general case when the limiting distribution is not a Boltzmann distribution.
Almost always involves a Markov processMove 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.
new
old
U
U
e
e
state k
state k+1
Metropolis Monte Carlo Molecular Simulation (Review)
David A. Kofke, SUNY Buffalo
Choices of Metropolis for canonical ensembles (review)N. Metropolis et al. J. Chem. Phys. 21, 1087 (1953)
1. Detailed balance condition
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rracc
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2. Symmetric :
3. Limiting probability distribution for canonical ensemble = Boltzmann
Transition probability & acceptance probability acc should satisfy:
4. Define the acceptance probability as:
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rracc
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Accept all the downhill moves.
Accept uphill moves only whennot too uphill.
naturally satisfies this condition.
Metropolis Monte Carlo algorithm for general cases
1. Detailed balance condition
newnewoldold xxArrPxxArrP exp )(),'( ; exp )(),(
)(
)(
)(
)'(
)'(
)'(
rr
rr
r
r
rracc
rracc
Transition probability & acceptance probability acc should satisfy:
)()'( if 1
)()'( if )(
)'(
rr
rrr
r
oldold
newnew
xxA
xxA
r
r
rracc
rracc
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rr
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exp
)(
)'(
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)'(
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)'(
)'( rracc
Accept all the moves toward
more probable state.
Accept moves toward less probable state according to the probability ratio (< 1).
naturally satisfies this condition.
2. Symmetric
3. Limiting probability distribution
4. Define the acceptance probability as:
case 1D in this /1)'()'( rrrr
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 centered on the current position of the atom.
Examine underlying transition probability
to formulate the acceptance criterion
Displacement trial move. 1. Specification (Review)
?
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.
Step 1 Step 2 Step 3 Step 4
general hitherto
David A. Kofke, SUNY Buffalo
mc: Metropolis MC code
input- x0 : initial position of the particle (initial microstate)- n_steps: number of MC steps - n_thermo: frequency with which we write the observables (here: x, x2)- displ_max: maximum allowed displacement of the particle during a trial move- seed : random number generator seed
- block_size: size of data block to perform statistics on the observables (x, x2)
output files:- x.dat (position of the particle, text file)
- x.bin (position of the particle, binary file- x2.dat (squared position of the particle, text file)
Lab 4. Calculate <x> by Metropolis Monte Carlo methodwhen (x) = A x exp(-x) for 0 < x < 4 (a 1D problem)
(ignore for now)
mc.c (part 1) int main(int argc, char *argv[]){int i, j, i_step, n_steps, n_thermo;double x0, k, temp, displ_max, x, pot_energy;double *x_b, *x2_b, x_avg, x2_avg;long seed;int block_size, n_blocks, i_block;int n_trial, n_accept;FILE *fp, *fp2, *fp3, *f_order;
/* Get command-line input */ if (argc != 7) { fprintf(stderr, "Usage: %s x0 n_steps n_thermo displ_max seed block_size\n", argv[0]); exit(1); }
x0 = atof(argv[1]); n_steps = atoi(argv[2]); n_thermo = atoi(argv[3]); displ_max = atof(argv[4]); seed = atoi(argv[5]); block_size = atoi(argv[6]); k = 1.0; temp = 1.0;
(Ignore for now or use 1 for the block size.)
mc.c (part 2) /* Compute number of blocks and allocate memory */ if (n_steps % block_size != 0) { printf("n_steps must be a multiple of block_size\n"); exit(1); } n_blocks = n_steps / block_size; x_b = (double *) allocate_1d_array(n_blocks, sizeof(double)); x2_b = (double *) allocate_1d_array(n_blocks, sizeof(double));
/* Initialize block data */ for (i = 0; i < n_blocks; ++i) { x_b[i] = 0.0; x2_b[i] = 0.0; }
/* Initialize variables for MC optimization. */ n_trial = 0; n_accept = 0;
/* Open thermodynamic file. */ fp2 = fopen("x.dat", "w"); fp3 = fopen("x2.dat", "w"); f_order = fopen("x.bin", "w");
(Ignore for now or use 1 for the block size.)
mc.c (part 3 – main MC loop) /* MC loop. */ i_block = 0; x = x0; /* x0 corresponds to the initial microstate. */ for (i_step = 1; i_step <= n_steps; ++i_step) {
/* Perform a MC cycle. */ mc_cycle(&x, displ_max, temp, k, &seed, &pot_energy, &n_trial, &n_accept);
/* Write thermodynamic quantities in file every cycle. */ if (i_step % n_thermo == 0) { fprintf(fp2, "%d %12.8f\n", i_step, x); fprintf(fp3, "%d %12.8f\n", i_step, x*x); fwrite(&x, sizeof(double), 1, f_order); }
/* Block accumulation. */ x_b[i_block] += x; x2_b[i_block] += x*x;
/* Block average. */ if (i_step % block_size == 0) { x_b[i_block] /= (double) block_size; x2_b[i_block] /= (double) block_size; printf("bloc %d: x = %12.8f, x2 = %12.8f\n", i_block, x_b[i_block], x2_b[i_block]); printf("----------------------------------\n"); ++i_block; } }
(Ignore for now or use 1 for the block size.)
/* 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(double *x, double displ_max, double temp, double k, long *seed, double *u_pot, int *n_trial, int *n_accept){ int i_move;
/* Make trial move. */ ++(*n_trial); *n_accept += atom_move(x, displ_max, temp, k, seed, u_pot);
return;}
mc_cycle.c
atom_move.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 "atom_move.h"#include "ran3.h"
int atom_move(double *x, double displ_max, double temp, double k, long *seed, double *u_pot){ int accept; double p_old, p_new, delta_p; double displ, x_old;
x_old = *x; p_old = x_old * exp(-x_old);
/* Generate random displacement within hypercube of size displ_max. */ displ = displ_max * (ran3(seed) - 0.5); *x += displ;
/* Compute change in potential energy and accept move using Metropolis criterion. */ if (*x > 4.0 || *x < 0.0) p_new = 0.0; else p_new = (*x) * exp(-(*x));
delta_p = p_new / p_old;
if (delta_p >= 1.0) accept = 1; else accept = ran3(seed) < delta_p;
/* Update system potential energy if trial move is accepted. Otherwise, reset variables. */ if (!accept) { *x = x_old; }
/* Return 1 if trial move is accepted, 0 otherwise. */ return accept;}
Consider a region of (displ_max) centered at x_old to choose x_new.
(1-body 1-dimension case)
)()'( if 1
)()'( if )(
)'(
rr
rrr
r
)'( rracc
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 centered on the current position of the atom.
Examine underlying transition probability
to formulate the acceptance criterion
Displacement trial move. 1. Specification (Review)
?
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.
Step 1 Step 2 Step 3 Step 4
general hitherto
David A. Kofke, SUNY Buffalo
Metropolis Monte Carlo algorithm for general cases
1. Detailed balance condition
newnewoldold xxArrPxxArrP exp )(),'( ; exp )(),(
)(
)(
)(
)'(
)'(
)'(
rr
rr
r
r
rracc
rracc
Transition probability & acceptance probability acc should satisfy:
)()'( if 1
)()'( if )(
)'(
rr
rrr
r
oldold
newnew
xxA
xxA
r
r
rracc
rracc
rr
rr
exp
exp
)(
)'(
)(
)'(
)(
)'(
)'( rracc
Accept all the moves toward
more probable state.
Accept moves toward less probable state according to the probability ratio (< 1).
naturally satisfies this condition.
2. Symmetric
3. Limiting probability distribution
4. Define the acceptance probability as:
case 1D in this /1)'()'( rrrr
2) hist: calculates the histogram of x (i.e the probability to find x in [0,4]).input:
- x.dat or x.bin (output from mc)- ascii: read the .dat file (1) or read the .bin file (0)- n_bins: number of bins in the histogram, each bin having a size:
output:- hist.dat: the histogram file with elements
i.e. probability to find the particle at position x in [xi, xi + x]
Analysis 1 (Tools are provided in a subdirectory analysis.)
1) mean_sample: computes the sample mean of x and compare to the exact value. input:
- x.dat or x.bin (output from mc)- ascii: read the .dat file (1) or read the .bin file (0)
output:
- mean value
- sample mean and
with (integration by parts)
= 0.5 (Pacc = 95%, <x> = 1.6731)
= 8 (Pacc = 37%, <x> = 1.6760)) = 200 (Pacc = 1.5%, <x> = 1.6701)
1000 consecutive x values of Markov chain for different step sizes () among the total number of values or steps M = 2106
Histogram of x values for = 8 & a fit with (x) = A x exp(-x)
x
(x)
Correlation is significant for = 0.5 and 200 and negligible for = 8.
4) block: calculates correlation time using block method (successive blocks of size 2n, n = 0,..,16)
for n = 1: New serie with M/2 values (block size 21)
for n = k : new serie with M/2k values (each value is a block of size 2k)
input: - x.dat or x.bin (output from mc) - ascii: read the .dat file (1) or read the .bin file (0)output: - block.dat: uses columns 1 (n) and 8 (correlation time)
3) time_corr: calculates correlation time using autocorrelation function C(k) (single time origin)
input: - x.dat or x.bin (from mc) - ascii: read the .dat file (1), read the .bin file (0) - k_max: maximum separation between data considered to evaluate time correlation output: - time_corr.dat: time correlation function C(k) - time_integrated.dat: the integrated correlation time (k)
Analysis 2 (Tools are provided in a subdirectory analysis.)
Example of data correlation: 1D harmonic oscillator (review)
where (fluctuation around the average),
which measure whether the fluctuations are related for x values l measurement apart.
Autocorrelation functions for the complete data sets (400,000 steps) of x for three different step sizes = 1; 10; 200 (acceptance rate = 0.70; 0.35; 0.02)
e-1
correlation time (short)correlation time (long) correlation time (long)
In generating these results so far, x was measured every MC step.
k
Autocorrelation function & Integrated correlation time intA
uto
corr
ela
tion
fun
ctio
n C
(k)
Accu
mul
ated
in
t
k
Block Methodco
rre
latio
n tim
e
k (block size = 2k)