Metropolis Monte Carlo

There are many different types of Monte Carlo. The one used in molecular simulation is called “Metropolis Monte Carlo”. This is named after Metropolis, who was one of the five authors of the famous 1953 paper that first introduced the method.

Metropolis Monte Carlo is based on random numbers and has a very simple algorithm.

1. Calculate the energy of the molecular system. Save this as the old energy.
2. Randomly choose a molecule in the system to move, and save its coordinates (the old coordinates).
3. Randomly move this molecule (e.g. randomly translate or rotate it). This results in the new coordinates.
4. Calculate the energy of the molecular system. Save this as the new energy.
5. Use the difference between the new and old energies in a Monte Carlo test. If this test passes, then keep the new coordinates, else, restore the old coordinates.

For Metropolis Monte Carlo, in a NVT (canonical) ensemble, the Monte Carlo test is;

exp( -(E_{new} - E_{old}) / kT ) >= random(0,1)

where E_{new} is the new energy, E_{old} is the old energy, kT is Boltzmann’s constant (k) multiplied by temperature in Kelvin (T), and random(0,1) is a uniform random number between 0 and 1. If you want to know where this test comes from, take a look at these slides.

Take a look at metropolis.py (e.g. by opening it in nano). Scroll down until you can find the code that implements the above test (it is copied below here);

# calculate the old energy
old_energy = calculate_energy();

# Pick a random atom (random.randint(x,y) picks a random
# integer between x and y, including x and y)
atom = random.randint(0, n_atoms-1);

# save the old coordinates
old_coords = ( coords[atom], coords[atom],
coords[atom] )

# Make the move - translate by a delta in each dimension
delta_x = random.uniform( -max_translate, max_translate )
delta_y = random.uniform( -max_translate, max_translate )
delta_z = random.uniform( -max_translate, max_translate )

coords[atom] += delta_x
coords[atom] += delta_y
coords[atom] += delta_z

# wrap the coordinates back into the box
coords[atom] = wrap_into_box(coords[atom], box_size)
coords[atom] = wrap_into_box(coords[atom], box_size)
coords[atom] = wrap_into_box(coords[atom], box_size)

# calculate the new energy
new_energy = calculate_energy();

accept = False;

# Automatically accept if the energy goes down
if (new_energy <= old_energy):
accept = True

else:
# Now apply the Monte Carlo test - compare
# exp( -(E_new - E_old) / kT ) >= rand(0,1)
x = math.exp( -(new_energy - old_energy) / kT )

if (x >= random.uniform(0.0,1.0)):
accept = True
else:
accept = False

if accept:
# accept the move
naccept += 1
total_energy = new_energy
else:
# reject the move - restore the old coordinates
nreject += 1

coords[atom] = old_coords
coords[atom] = old_coords
coords[atom] = old_coords
total_energy = old_energy

Find this code in metropolis.py (it is near the bottom of the script). You should be able to see that old energy is calculated using the line;

# calculate the old energy
old_energy = calculate_energy()

Note that calculate_energy() is a function that calculates the total energy of the molecular system.

An atom to move is chosen at random using the line;

# Pick a random atom (random.randint(x,y) picks a random
# integer between x and y, including x and y)
atom = random.randint(0, n_atoms-1)

Note that random.randint(0, n_atoms-1) is a function from the python random module that generates a random number between 0 and n_atoms - 1. This chooses the index of the random atom in the coords array, and saves that index in the variable atom.

The old coordinates of the chosen atom are saved into the array old_coords using the line;

# save the old coordinates
old_coords = ( coords[atom], coords[atom],
coords[atom] )

In this program, a random move is just a random translation along the x, y and z axis by a random value between -max_translate and max_translate Angstroms. This is performed using these lines;

# Make the move - translate by a delta in each dimension
delta_x = random.uniform( -max_translate, max_translate )
delta_y = random.uniform( -max_translate, max_translate )
delta_z = random.uniform( -max_translate, max_translate )

coords[atom] += delta_x
coords[atom] += delta_y
coords[atom] += delta_z

The krypton atoms are in a periodic box. This random translation may have moved them outside of the box, so the atoms have to be wrapped back into the box. This is achieved using these lines;

# wrap the coordinates back into the box
coords[atom] = wrap_into_box(coords[atom], box_size)
coords[atom] = wrap_into_box(coords[atom], box_size)
coords[atom] = wrap_into_box(coords[atom], box_size)

The new energy is calculated using this line, and stored in the variable new_energy;

# calculate the new energy
new_energy = calculate_energy()

The Metropolis Monte Carlo test is performed using these lines, which perform the test and store whether or not the test passed in the variable accept;

accept = False

# Automatically accept if the energy goes down
if (new_energy <= old_energy):
accept = True

else:
# Now apply the Monte Carlo test - compare
# exp( -(E_new - E_old) / kT ) >= rand(0,1)
x = math.exp( -(new_energy - old_energy) / kT )

if (x >= random.uniform(0.0,1.0)):
accept = True
else:
accept = False

If the energy goes down (new_energy is less than or equal to old_energy) then the test is automatically passed (accept is set to True, which means pass). Otherwise, the exponential of the difference of new_energy and old_energy, divided by kT is calculated, and stored in the variable x. This is compared to a random number between 0 and 1 (random.uniform(0.0,1.0)). If it is greater than or equal to the random number, then the test is passed. Otherwise the test fails (and accept is set equal to False, which means fail).

If the Metropolis Monte Carlo test is passed, then the following code is run;

if accept:
# accept the move
naccept += 1
total_energy = new_energy

This code increases the number of accepted moves (stored in naccept) by one. It also saves the new total energy of the system into the variable total_energy.

If the Metropolis Monte Carlo test is failed, then the following code is run;

else:
# reject the move - restore the old coordinates
nreject += 1

coords[atom] = old_coords
coords[atom] = old_coords
coords[atom] = old_coords
total_energy = old_energy

The number of rejected moves (stored in nreject) is increased by one. The old coordinates of the atom are restored from the old_coords array. The old total energy of the system is then saved into the variable total_energy.

That is Metropolis Monte Carlo :-)

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