# 9. Exercise sheet#

Some general remarks about the exercises:

• For your convenience functions from the lecture are included below. Feel free to reuse them without copying to the exercise solution box.

• For each part of the exercise a solution box has been added, but you may insert additional boxes. Do not hesitate to add Markdown boxes for textual or LaTeX answers (via Cell > Cell Type > Markdown). But make sure to replace any part that says YOUR CODE HERE or YOUR ANSWER HERE and remove the raise NotImplementedError().

• Please make your code readable by humans (and not just by the Python interpreter): choose informative function and variable names and use consistent formatting. Feel free to check the PEP 8 Style Guide for Python for the widely adopted coding conventions or this guide for explanation.

• Make sure that the full notebook runs without errors before submitting your work. This you can do by selecting Kernel > Restart & Run All in the jupyter menu.

• For some exercises test cases have been provided in a separate cell in the form of assert statements. When run, a successful test will give no output, whereas a failed test will display an error message.

• Each sheet has 100 points worth of exercises. Note that only the grades of sheets number 2, 4, 6, 8 count towards the course examination. Submitting sheets 1, 3, 5, 7 & 9 is voluntary and their grades are just for feedback.

NAME = ""
NAMES_OF_COLLABORATORS = ""


Exercise sheet 9

## 9.1. Running code on the compute cluster: lattice scalar field#

Goal: Learn how to scale up simulations by transitioning from running them in the notebook to stand-alone scripts on the compute cluster.

In lecture 7 we implemented the Metropolis-Hastings simulation of a lattice scalar field using the following code:

import numpy as np
rng = np.random.default_rng()
import matplotlib.pylab as plt
%matplotlib inline

def potential_v(x,lamb):
'''Compute the potential function V(x).'''
return lamb*(x*x-1)*(x*x-1)+x*x

def neighbor_sum(phi,s):
'''Compute the sum of the state phi on all 8 neighbors of the site s.'''
w = len(phi)
return (phi[(s+1)%w,s,s,s] + phi[(s-1)%w,s,s,s] +
phi[s,(s+1)%w,s,s] + phi[s,(s-1)%w,s,s] +
phi[s,s,(s+1)%w,s] + phi[s,s,(s-1)%w,s] +
phi[s,s,s,(s+1)%w] + phi[s,s,s,(s-1)%w] )

def scalar_action_diff(phi,site,newphi,lamb,kappa):
'''Compute the change in the action when phi[site] is changed to newphi.'''
return (2 * kappa * neighbor_sum(phi,site) * (phi[site] - newphi) +
potential_v(newphi,lamb) - potential_v(phi[site],lamb) )

def scalar_MH_step(phi,lamb,kappa,delta):
'''Perform Metropolis-Hastings update on state phi with range delta.'''
site = tuple(rng.integers(0,len(phi),4))
newphi = phi[site] + rng.uniform(-delta,delta)
deltaS = scalar_action_diff(phi,site,newphi,lamb,kappa)
if deltaS < 0 or rng.uniform() < np.exp(-deltaS):
phi[site] = newphi
return True
return False

def run_scalar_MH(phi,lamb,kappa,delta,n):
'''Perform n Metropolis-Hastings updates on state phi and return number of accepted transtions.'''
total_accept = 0
for _ in range(n):
total_accept += scalar_MH_step(phi,lamb,kappa,delta)

def batch_estimate(data,observable,k):
'''Devide data into k batches and apply the function observable to each.
Returns the mean and standard error.'''
batches = np.reshape(data,(k,-1))
values = np.apply_along_axis(observable, 1, batches)
return np.mean(values), np.std(values)/np.sqrt(k-1)

lamb = 1.5
kappas = np.linspace(0.08,0.18,11)
width = 3
num_sites = width**4
delta = 1.5  # chosen to have ~ 50% acceptance
equil_sweeps = 1000
measure_sweeps = 2
measurements = 2000

mean_magn = []
for kappa in kappas:
phi_state = np.zeros((width,width,width,width))
run_scalar_MH(phi_state,lamb,kappa,delta,equil_sweeps * num_sites)
magnetizations = np.empty(measurements)
for i in range(measurements):
run_scalar_MH(phi_state,lamb,kappa,delta,measure_sweeps * num_sites)
magnetizations[i] = np.mean(phi_state)
mean, err = batch_estimate(np.abs(magnetizations),lambda x:np.mean(x),10)
mean_magn.append([mean,err])

plt.errorbar(kappas,[m for m in mean_magn],yerr=[m for m in mean_magn],fmt='-o')
plt.xlabel(r"$\kappa$")
plt.ylabel(r"$|m|$")
plt.title(r"Absolute field average on $3^4$ lattice, $\lambda = 1.5$")
plt.show()


The goal will be to reproduce and extend its output.

(a) Turn the simulation into a standalone script latticescalar.py (similar to ising.py) that takes the relevant parameters e.g.

\$ python3 latticescalar.py -l 1.5 -k 0.12 -w 3 -n 1000


for $$\lambda=1.5$$, $$\kappa=0.12$$, $$w=3$$, and $$1000$$ measurements, together with optional parameters $$\delta$$ and numbers of sweeps, as command line arguments and stores the relevant simulation outcomes in an hdf5-file.

(b) Write a bash script job_latticescalar.sh that submits an array job to the cluster for $$w=3$$ and $$2000$$ measurements and $$\lambda = 1.0, 1.5, 2.0$$ and $$\kappa = 0.08, 0.09, ..., 0.18$$ (so 33 simulations in total). Submit the job to the hefstud slurm partition (do not run all 33 in parallel).

(c) Load the stored data into this notebook and reproduce the plot above (with $$\lambda = 1$$ and $$\lambda=2$$ added).