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I am doing a molecular dynamics simulation of Leonard Jones 6-12 potential. But instead of converging to a particular value. It always stays between -5.58 to -5.62. The standard value is -5.517. The problem is that however long it runs (I have tested up to 30000 steps) it does not converge to even 1 digit after the decimal point. It changes between -5.58 to -5.62 but never reaches -5.517. My timestep is 0.005 (as given in reference value). All my other parameters are the same as the value in reference. I am using a velocity-Verlet algorithm. The standard reference is available here.

I have used velocity rescaling and that's why kinetic energy remains almost constant, at least up to 2 digits after decimal always. Total energy change is quite small after a few steps, the change is mainly due to fluctuation in potential energy i.e. in the range -5.58 to -5.62.

The code to reproduce the error is below.

import numpy as np
import matplotlib.pyplot as plt
from numba import jit
import os
import sys

# All globals are compile time constants
# recompilation needed if you change this values
# Better way: hand a tuple of all needed vars to the functions
# params=(NSteps,deltat,temp,DumpFreq,epsilon,DIM,N,density,Rcutoff)

# Setting up the simulation
NSteps =100000 # Number of steps
deltat = 0.005 # Time step in reduced time units
temp   = 0.851# #Reduced temperature
DumpFreq = 100 # Save the position to file every DumpFreq steps
epsilon = 1.0 # LJ parameter for the energy between particles
DIM     =3
N       =500
density =0.776
Rcutoff =3



params=(NSteps,deltat,temp,DumpFreq,epsilon,DIM,N,density,Rcutoff)

#----------------------Function Definitions---------------------

#------------------Initialise Configuration--------

error_model=True
#If it is really required to search for division by zeros (additional cost)?

@jit(nopython=True,error_model="numpy")
def initialise_config(N,DIM,density):
    velocity = (np.random.randn(N,DIM)-0.5)


    # Set initial momentum to zero
    COM_V = np.sum(velocity)/N     #Center of mass velocity
    velocity = velocity - COM_V    # Fix any center-of-mass drift

    # Calculate initial kinetic energy
    k_energy=0
    for i in range (N):
        k_energy+=np.dot(velocity[i],velocity[i])
    vscale=np.sqrt(DIM*temp/k_energy)
    velocity*=vscale

    #wrong array initialization (use tuple)
    #Initialize with zeroes
    coords = np.zeros((N,DIM))

    # Get the cooresponding box size
    L = (N/density)**(1.0/DIM)

    """ Find the lowest perfect cube greater than or equal to the number of
     particles"""
    nCube = 2

    while (nCube**3 < N):
        nCube = nCube + 1



    # Assign particle positions
    ip=-1
    x=0
    y=0
    z=0

    for i in range(0,nCube):
        for j in range(0,nCube):
            for k in range(0,nCube):
                if(ip<N):
                    x=(i+0.5)*(L/nCube)
                    y=(j+0.5)*(L/nCube)
                    z=(k+0.5)*(L/nCube)
                    coords[ip]=np.array([x,y,z])
                    ip=ip+1
                else:
                    break
    return (coords/L),velocity,L


@jit(nopython=True)
def wrap(pos,L):
    '''Apply perodic boundary conditions.'''

    #correct array indexing
    for i in range (len(pos)):
        for k in range(DIM):
                if (pos[i,k]>0.5):
                    pos[i,k]=pos[i,k]-1
                if (pos[i,k]<-0.5):
                    pos[i,k]=pos[i,k]+1


    return (pos)    


@jit(nopython=True,error_model="numpy")
def LJ_Forces(pos,acc,epsilon,L,DIM,N):
    # Compute forces on positions using the Lennard-Jones potential
    # Uses double nested loop which is slow O(N^2) time unsuitable for large systems
    Sij = np.zeros(DIM) # Box scaled units
    Rij = np.zeros(DIM) # Real space units

    #Set all variables to zero
    ene_pot = np.zeros(N)
    acc = acc*0
    virial=0.0

    # Loop over all pairs of particles
    for i in range(N-1):
        for j in range(i+1,N): #i+1 to N ensures we do not double count
            Sij = pos[i]-pos[j] # Distance in box scaled units
            for l in range(DIM): # Periodic interactions
                if (np.abs(Sij[l])>0.5):
                    Sij[l] = Sij[l] - np.copysign(1.0,Sij[l]) # If distance is greater than 0.5  (scaled units) then subtract 0.5 to find periodic interaction distance.

            Rij   = L*Sij # Scale the box to the real units in this case reduced LJ units
            Rsqij = np.dot(Rij,Rij) # Calculate the square of the distance

            if(Rsqij < Rcutoff**2):
                # Calculate LJ potential inside cutoff
                # We calculate parts of the LJ potential at a time to improve the efficieny of the computation (most important for compiled code)
                rm2      = 1.0/Rsqij # 1/r^2
                rm6      = rm2**3
                forcefact=(rm2**4)*(rm6-0.5) # 1/r^6
                phi      =4*(rm6**2-rm6) 

                ene_pot[i] = ene_pot[i]+0.5*phi # Accumulate energy

                ene_pot[j] = ene_pot[j]+0.5*phi # Accumulate energy

                virial     = virial-forcefact*Rsqij # Virial is needed to calculate the pressure
                acc[i]     = acc[i]+forcefact*Sij # Accumulate forces
                acc[j]     = acc[j]-forcefact*Sij # (Fji=-Fij)
    #If you want to get get the best performance, sum directly in the loop intead of 
    #summing at the end np.sum(ene_pot)
    return 48*acc, np.sum(ene_pot)/N, -virial/DIM # return the acceleration vector, potential energy and virial coefficient


@jit(nopython=True,error_model="numpy")
def Calculate_Temperature(vel,L,DIM,N):

    ene_kin = 0.0

    for i in range(N):
        real_vel = L*vel[i]
        ene_kin = ene_kin + 0.5*np.dot(real_vel,real_vel)

    ene_kin_aver = 1.0*ene_kin/N
    temperature = 2.0*ene_kin_aver/DIM

    return ene_kin_aver,temperature


# Main MD loop
@jit(nopython=True,error_model="numpy")
def main(params):
    NSteps,deltat,temp,DumpFreq,epsilon,DIM,N,density,Rcutoff=params
    # Vectors to store parameter values at each step

    ene_kin_aver = np.ones(NSteps)
    ene_pot_aver = np.ones(NSteps)
    temperature = np.ones(NSteps)
    virial = np.ones(NSteps)
    pressure = np.ones(NSteps)


    pos,vel,L = initialise_config(N,DIM,density)
    acc = (np.random.randn(N,DIM)-0.5)
    volume=L**3

    # Open file which we will save the outputs to
    # Unsupported operations have to be in an objectmode block
    # or simply write the outputs at the end in a pure Python function
    """
    if os.path.exists('energy2'):
        os.remove('energy2')
    f = open('traj.xyz', 'w')
    """
    for k in range(NSteps):

        # Refold positions according to periodic boundary conditions
        pos=wrap(pos,L)

        # r(t+dt) modify positions according to velocity and acceleration
        pos = pos + deltat*vel + 0.5*(deltat**2.0)*acc # Step 1

        # Calculate temperature
        ene_kin_aver[k],temperature[k] = Calculate_Temperature(vel,L,DIM,N)

        # Rescale velocities and take half step
        chi = np.sqrt(temp/temperature[k])
        vel = chi*vel + 0.5*deltat*acc # v(t+dt/2) Step 2

        # Compute forces a(t+dt),ene_pot,virial
        acc, ene_pot_aver[k], virial[k] = LJ_Forces(pos,acc,epsilon,L,DIM,N) # Step 3

        # Complete the velocity step 
        vel = vel + 0.5*deltat*acc # v(t+dt/2) Step 4

        # Calculate temperature
        ene_kin_aver[k],temperature[k] = Calculate_Temperature(vel,L,DIM,N)

        # Calculate pressure
        pressure[k]= density*temperature[k] + virial[k]/volume

        # Print output to file every DumpFreq number of steps
        """
        if(k%DumpFreq==0): # The % symbol is the modulus so if the Step is a whole multiple of DumpFreq then print the values

            f.write("%s\n" %(N)) # Write the number of particles to file
            # Write all of the quantities at this step to the file
            f.write("Energy %s, Temperature %.5f\n" %(ene_kin_aver[k]+ene_pot_aver[k],temperature[k]))
            for n in range(N): # Write the positions to file
                f.write("X"+" ")
                for l in range(DIM):
                    f.write(str(pos[n][l]*L)+" ")
                f.write("\n")

        #Simple prints without formating are supported
        """
        if (k%150==0):
            print("step: ",k,"KE: ",ene_kin_aver[k],"PE :",ene_pot_aver[k],"\n Total Energy: ",ene_kin_aver[k]+ene_pot_aver[k])
            #print("\rStep: {0} KE: {1}   PE: {2} Energy:  {3}".format(k, ene_kin_aver[k], ene_pot_aver[k],ene_kin_aver[k]+ene_pot_aver[k]))
            #sys.stdout.write("\rStep: {0} KE: {1}   PE: {2} Energy:  {3}".format(k, ene_kin_aver[k], ene_pot_aver[k],ene_kin_aver[k]+ene_pot_aver[k]))
            #sys.stdout.flush()

    return ene_kin_aver, ene_pot_aver, temperature, pressure, pos

#------------------------------------------------------    
ene_kin_aver, ene_pot_aver, temperature, pressure, pos = main(params)




# Plot all of the quantities
def plot():
    plt.figure(figsize=[7,12])
    plt.rc('xtick', labelsize=15) 
    plt.rc('ytick', labelsize=15)
    plt.subplot(4, 1, 1)
    plt.plot(ene_kin_aver,'k-')
    plt.ylabel(r"$E_{K}", fontsize=20)
plt.subplot(4, 1, 2)
plt.plot(ene_pot_aver,'k-')
plt.ylabel(r"$E_{P}$", fontsize=20)
plt.subplot(4, 1, 3)
plt.plot(temperature,'k-')
plt.ylabel(r"$T$", fontsize=20)
plt.subplot(4, 1, 4)
plt.plot(pressure,'k-')
plt.ylabel(r"$P$", fontsize=20)
    plt.show()

plot()

I have tried with different timesteps but no result. I think I have used reduced units everywhere. Epsilon is taken as 1 and sigma is never used (always taken as 1). Box size is the length of the cube grid required (L) to make a system of required reduced density (density as a parameter in the code). It is calculated by the system and no. of particles and density (reduced) is taken as input.

I would also like to know how to calculate the radial distribution function using this code. I know its definition but can't understand how to implement it in code. As far as I can understand, my origin will be (0,0,0) as I have at first in initialise_config function made all the coordinates with respect to the center of mass. But how to implement a function to efficiently calculate the number of particles at a distance $r$? Making a histogram corresponding to the radial distance of each LJ particle from origin may work but I am not sure how it will produce the negative parts in the $\rho$ axis in the following plot.
.

enter image description here

Any help in implementing this will be much appreciated. I don't have any clue on how to sort these 2 problems.

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  • $\begingroup$ I would suggest that you first test your algoright with a simpler problem. For example, try with a spring-mass system that has an analytic solution that you can use to compare. $\endgroup$ – nicoguaro Mar 31 at 16:32
  • $\begingroup$ @nicoguaro Thank you. I shall try MD with the spring-mass system. But here the algo used is standard velocity verlet . For comparison, in this case, I am using reference data (all my parameters are same as the reference) mmlapps.nist.gov/srs/LJ_PURE/md.htm $\endgroup$ – Endeavour Apr 1 at 6:39
  • $\begingroup$ The point is to test only the integrator. If you are not able to solve for a simpler problem I doubt you are capable of a more complex one. $\endgroup$ – nicoguaro Apr 1 at 13:40

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