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

# Pieter Jan De Smet Calculations

Various topics in mathematics and physics

## Monday, May 1, 2017

## Friday, April 28, 2017

### 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 $N$.

## Wednesday, March 8, 2017

### 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.

## Friday, February 3, 2017

### Comment about particle on a circle

The wave function of a particle on a circle is a solution of the SchrÃ¶dinger 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}

## Thursday, January 19, 2017

### 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]

## Wednesday, January 11, 2017

### 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.

## Friday, January 6, 2017

### Lorentz invariance of string theory in the light-cone gauge

On page 261 in his book [1] Zwiebach writes ''There is much at stake in this calculation. It is in fact, one of the most important calculations in string theory. [...]
The calculation is long and uses many of our previously derived results''. Then Zwiebach states the result
\begin{align}
\left[ M^{- I}, M^{- J}\right] = &- \frac{1}{\alpha'\ { p^+ }^2} \sum_{m=1}^{\infty}
\left(\alpha^I_{-m} \alpha^J_m - \alpha^J_{-m} \alpha^I_m \right)\nonumber\\
&\times \left\{ m \left[1 - \dfrac{1}{24} (D-2) \right] + \dfrac{1}{m} \left[ \dfrac{1}{24} (D-2) + a \right] \right\}\label{eq:20170103}
\end{align}
This is the commutator of two Lorentz transformations in the light-cone gauge.
The commutator should be zero for string theory to be Lorentz invariant.
The calculation of the commutator is indeed very tedious and after scribling too many pages I gave up.
I googled for a quick way to obtain \eqref{eq:20170103}. Here are the more interesting
results that I found.

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