LR_RW

This Monte Carlo simulation program provides a numerical implementation of Lévy Flights, also known as long-range random walks (LR-RW).

Introduction

A Lévy flight is a type of random walk where the step lengths follow a heavy-tailed probability distribution. Unlike ordinary (short-range) random walks, in which steps are typically of similar magnitude, Lévy flights occasionally include extremely long jumps, allowing the walker to explore space in a nonlocal manner.

Mathematically, the step-length distribution of a Lévy flight in $d$ dimensions is given by

$$ P(r) \sim \frac{1}{r^{d + \sigma}}, $$

where $r$ is the step length and $\sigma > 0$ is the Lévy exponent that controls the heaviness of the tail. Smaller $\sigma$ corresponds to a higher probability of long jumps, and when $\sigma \to 2$, the process approaches the standard Gaussian random walk.

Lévy flights play an important role in various physical and biological systems, such as anomalous diffusion, turbulent transport, and search processes.

• src_demo: Fix the distance $R$, and measure how many steps are taken.

• src_n_site: Fix the number of steps $N$, and measure the end-to-end distance as well as the range of the random walk (i.e., how many sites are visited after $N$ steps).

• src_chi_k: Use a lattice with periodic boundary conditions (PBC). Fix the number of steps to $L^d$, and measure the effective susceptibility $\chi_k$.

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How to use

1. Clone the repository:

   git clone https://github.com/Tensofermi/LR_RW
   cd LR_RW

2. Configure model selection in the Makefile:

   SRC_DIR := src_chi_k
   # SRC_DIR := src_demo
   # SRC_DIR := src_n_site

   Uncomment the desired model and comment out the others.

3. Set parameters in input.txt and run the simulation:

   ./run.sh

4. For advanced simulations, including job submissions on HPC systems, refer to the /lsub, /qsub, and /data directories in our related project: Zoo_of_Classical_ON_Spin_Model