Description

We report our development and subsequent analysis of a Monte-Carlo Markov-Chain (MCMC) simulation for a 2-dimensional lattice following the Ising model, capable of studying the thermodynamic properties of the system. We reproduced plots for the magnetisation, susceptibility, magnetic susceptibility, and specific heat per spin for a system with external magnetic field, $𝐻 = 0$, and for a system with external magnetic field, $𝐻 = 1$. For the system absent of any external fields, we observe a phase transition between the magnetic and paramagnetic phase, with a critical temperature comparable to literature values. For the case of an external magnetic field, the system does not appear to undergo a similar phase transition, as indicated by our results, due to the impact of the introduced asymmetry. Overall, our simulation appears to be capable of accurately measure the thermodynamical properties of an Ising model, and reproduce the theoretically expected figures in both presence and absence of an external magnetic field.

Basics

• We define x_i as a state of the system (e.g. the position and velocity of all particles. Or the spin configuration). That’s what the integral runs over.
• We assume that the "next” state x_{i+1} in a chain only depends on the current state x_i, not on whatever happened before. (The system has no memory)
• We define T(x → x') the probability to go from state x to state x'. (e.g. the probability to flip 1 spin in our lattice)

Specific Goals

The following are themeasured quantities that this code gives as an output, and in addition plots them against the corresponding temperature of the system.

Main algorithm

main_file.py: Main program contains the simulation and relevant parameters. It allows the user to measure thermodynamic properties of a lattice following the Ising Model, for different numbers of particles and values of the coupling constant and external magnetic field. For more information regarding its proper use we refer to its docstring description.

Functions file

functions.py: File contains all the relevant functions used by the simulation. We note that the ising_mc_simu function performs the simulation on its own and can be used idependently.