Wu-Yang monopole: numerical calculation

I have been reading the paper by Wu and Yang [1] in which they find the famous Wu-Yang monopole. In the paper there are solutions for three types of monopoles: one has an analytical form, which is the one most often quoted, but there are also two other monopoles with numerical solution only. In this post I use Python/numpy to perform numerical analysis on the latter solution. I use the same notation as in [1].
Wu and Yang obtain the following system of ordinary differential equations \begin{align} \frac{d\Phi}{d \xi} &= \psi\label{eq:20170625a}\\ \frac{d\psi}{d \xi} &= \psi + \Phi(\Phi^2-1)\label{eq:20170626a} \end{align} Here $\xi$ is given by $r = e^{\xi}$, with $r$ the distance to the origin. The right-hand side of \eqref{eq:20170625a}-\eqref{eq:20170626a} defines the vector field ($d\Phi/d\xi, d\psi/d\xi)$ in the $(\Phi, \psi)$ plane. Its integral curves can be seen in the next figure

The integral curves of the vector field defined by \eqref{eq:20170625a}-\eqref{eq:20170626a}.
The stationary points are marked in red.

I calculate the integral curve from the point $(\Phi,\psi) = (0,0)$ to $(1,0)$ using the numpy function solve_bvp [2].

The integral curves of the vector field defined by \eqref{eq:20170625a}-\eqref{eq:20170626a}.
The integral curve from the stationary point $(0,0)$ to $(1,0)$ is added in red.

$\Phi(\xi)$ can be seen in the next graph. One sees that $\Phi(\xi) \to 0$ for $\xi \to -\infty$ and $\Phi(\xi) \to 1$ for $\xi \to +\infty$

In the rest of this post I reproduce part of Table 1 in [1].

Variance of Markov Chain Monte Carlo

In a previous post, I discussed the bias of Markov Chain Monte Carlo (MCMC) simulation. In this post I will discuss the variance. Please see the previous post for information about the notation that I use.
If \begin{equation*} S =\frac{1}{N} \sum_{t=1}^N f(X_t) \end{equation*} then for large $N$, the variance of $S$ is

Bias in Markov Chain Monte Carlo

Markov Chain Monte Carlo (MCMC) simulation can be used to calculate sums \begin{equation}\label{eq:20170427a} I = \sum_a \pi_a f(a) \end{equation} One finds a Markov process $X_t$ with stationary distribution $\pi_a$, then the sum \eqref{eq:20170427a} is approximated by \begin{equation*} S =\frac{1}{N} \sum_{t=1}^N f(X_t) \end{equation*} One can prove that under certain assumptions, \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \sum_{t=1}^N f(X_t) = \sum_a \pi_a f(a) \end{equation*} This is Birkhoff's ergodic theorem. In this post I illustrate the behaviour of $ES$ for large $N$.

A calculation on moduli stabilization

In section 21.6 "Moduli stabilization and the landscape" in Zwiebach's string theory book, I read the sentence "Deriving the potential $V(R)$ associated with $R$ is a straightforward but technical calculation in general relativity". At this point I did not understand what the calculation was. I vaguely remembered a paper by Witten about instabilities in Kaluza-Klein spacetimes related to instantons. A calculation with instantons is indeed technical, but perhaps straightforward for experts. I found more information in a paper by Denef [1]. The calculation has nothing to do with instantons, but is indeed a straightforward calculation in differential geometry. In the rest of this blog post I set out the calculation in the form of a new exercise for Zwiebach's book.

Comment about particle on a circle

The wave function of a particle on a circle is a solution of the Schrödinger equation \begin{equation}\label{eq:20170129a} i \frac{\partial \psi}{\partial t} = - \frac{1}{2 m} \frac{\partial^2 \psi}{\partial x^2} \end{equation} with $x \in [0 , 2 \pi]$ and $\hbar = 1$. When \eqref{eq:20170129a} is solved in physics books, it is usually imposed that the wave function should be periodic [1]. I used to be puzzled why one has to impose the periodicity. After all, I thought, only the probability density function $|\psi|^2$ has physical meaning, so one could as well impose that \begin{equation}\label{eq:20170129b} \psi(2 \pi) = e^{ i \alpha} \psi(0) \quad\text{with}\quad \alpha\in\mathbb{R} \end{equation}

A magnetostatic exercise in 10 dimensions

I calculate the electromagnetic field generated by electrical currents in 10 spacetime dimensions (9 space and 1 time). The set up is as follows: the current flows down the positive $x_1$-axis, hits the origin and then spreads out isotropically in the $x_2 x_3 x_4$ subspace, see figure 1 and 2. I wanted to calculate this because in string theory a similar calculation is needed to obtain the Kalb-Ramond field generated by a string ending on a $D3$-brane [1]

A calculation in magnetostatics

I wanted to calculate the magnetic field generated by a current which flows down the positive $z$-axis, hits the origin and then spreads out radially over the $xy$ plane, see figure 1.