TransWikia.com

Random walk through Ising model

Physics Asked on January 2, 2021

Ising model consists of up spin and down spin or empty/filled space. Can we model random walk for different densities of packing through the Ising model?

2 Answers

There are ways to map the magnetization distribution problem of the Ising model to persistent random walks. The relevant references are:

R. Garcia-Pelayo, "Distribution of magnetization in the finite Ising chain," Journal of Mathematical Physics 50, 013301 (2009)

and

T. Antal, M. Droz, and Z. Racz, "Probability distribution of magnetization in the one-dimensional Ising model: effects of boundary conditions," Journal of Physics A: Mathematics and General 37, 1465 (2004)

Answered by Jose Perico Esguerra on January 2, 2021

Yes, we can imagine going through 1D Ising model as random walk, analogous to Maximal Entropy Random Walk, e.g. to model 2D Ising $wtimes infty$ as 1D for width $w$ slices ( https://arxiv.org/pdf/1912.13300 ).

For $M_{ij}=exp(-beta E_{ij})$ transfer matrix, where $E_{ij}$ is energy of $i-j$ edge (e.g. negative for the same spins in ferromagnet), you need to find dominant eigenvector $Mpsi = lambda psi$, then

stationary probability distribution is $Pr(i)propto (psi_i)^2$,

Stochastic matrix: $S_{ij}=Pr(x_t=j|x_{t-1}=i)= frac{M_{ij}}{lambda} frac{psi_j}{psi_i}$.

Sketch of derivation:

https://i.imgur.com/CW3Lvrk.png

Answered by Jarek Duda on January 2, 2021

Add your own answers!

Ask a Question

Get help from others!

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP