The PushASEP is interacting random walk on a 1D lattice. The interaction to the left and right is different. Moving to the right, particles are blocked from moving to an occupied space. Moving to the left, particle push particles aside when moving to an occupied site.
We, Jhih-Huang Li and I, computed the fluctuation statistics for the PushASEP on the ring. You can find the result here: Contour Integral Formulas for PushASEP on the Ring.
The result for the step initial conditions has an additional parameter \(\gamma\), compared to the periodic initial conditions. The additional term \(\gamma\) is due to the fact that the dynamics (in large scale) are translation invariant for the periodic initial conditions and not for step initial conditions. There is a key event for the case with step initial conditions that remove translation invariance, the shock from high vs low density of particles. The average speed of a particle in the system determine by the local density of the particles. The shock becomes a reference point in the large scale regime. Thus, in the result for the step initial conditions, we have a parameter \(\gamma \in \mathbb{T}\) that transforms the frontier to the reference frame of the shock.
The hydrodynamic limit is the function \(\rho(x,t)\) of the local density of the particles at positions \(n = [x L]\) at time \(T = [ t]\). The evolution is given by the Burgers Equation:
\[\frac{ d \rho}{dt} + \frac{d}{dx} \left[ p \rho (1 -\rho) - q \frac{\rho}{1-\rho}\right]= 0\]Let’s consider the numerical solution of the Burger’s equation with step initial conditions. We’ll use a Fourier transform (approximation) since the step intial data is not smooth.
Let the ring have period one so that solutions are periodic on \([0,1]\). For the step initial conditions, take
\[\rho(x, 0) = \begin{cases} 1, & \quad x \in [1/4, 3/4] \\ 0, & \quad x \notin [1/4, 3/4]\end{cases}\]We have a Fourier expansion
\[\rho(x, 0) = \sum_{k \in \mathbb{Z}} a_k e^{2 \pi i k x} = a_0 + \sum_{k=0} ( a_k + a_{-k}) \cos(2\pi k x) + i( a_k - a_{-k}) \sin( 2\pi kx)\]where the coefficients are given by
\[a_k = \int_{1/4}^{3/4} e^{2 \pi i k x} dx = \frac{1}{2 \pi i k} \left(e^{ 3 k \pi i /2} - e^{ k \pi i /2} \right)\]so that
\[a_k + a_{-k} = 2 \cos( 3 k \pi /2) - 2 \cos( k \pi/2)\] ]]>Today, I’m trying to use disqus to add comments to the blog to see if it is better. Click here for documentation.
The issue now is that disqus doesn’t support LaTeX. :(
]]>How about this?
\(\prod_{i=1}^{N}(x-x_i) = \sum_{k=0}^{N} (-1)^{N-k} e_{N-k}(x_1, x_2, \dots, x_N) x^{k}\).
The coefficients \(e_k(x_1, x_2, \dots, x_N)\) are called the elementary symmetric functions. For example, we have
\[\begin{equation} \begin{split} e_0(x_1, \dots, x_N) &=1\\ e_1(x_1, \dots, x_N) &= x_1 +x_2 + \cdots + x_N \end{split} \end{equation}\]This was failry confusing process. I tried several things. In the end, I created a file _layouts/bease.html, where I copied the file from the minima GitHub repository click here and then I added the following scritp that I found here:
<script type="text/javascript" async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML" async></script>
Also, I changed the _layouts/bease.html to the file given in this repository for the math-minima Jekyll theme. For some reason, I wasn’t able to get the math-minima theme to work. Instead, I just copied the file from their repo.
]]>Next: figure out how to add math equations and tags.
]]>Using “utteramnces” plug-in for GitHub.
Installtion: Official website. Note that it produces a html script at the bottom so you can create an html in your repo to add to your _includes folder.
More deatails: