Approach to zigzag and checkerboard patterns in spatially extended systems
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2025
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| author | Warambhe, Manoj C. Gade, Prashant M. |
| author_facet | Warambhe, Manoj C. Gade, Prashant M. |
| contents | Zigzag patterns in one dimension or checkerboard patterns in two dimensions occur in a variety of pattern-forming systems. We introduce an order parameter `phase defect' to identify this transition and help to recognize the associated universality class on a discrete lattice. In one dimension, if $x_{i}(t)$ is a variable value at site $i$ at time $t$. We assign spin $s_i(t)=1$ for $x_{i}(t)>x_{i-1}(t)$, $s_i(t)=-1$ if $x_{i}(t)<x_{i-1}(t)$, and $s_i(t)=0$ if $x_{i}(t)=x_{i-1}(t)$. The phase defect $D(t)$ is defined as $D(t)={\frac{\sum_{i=1}^N \vert s_i(t)+s_{i-1}(t)\vert} {2N}}$ for a lattice of $N$ sites with periodic boundary conditions. It is zero for a zigzag pattern. In two dimensions, $D(t)$ is the sum of row-wise as well as column-wise phase defects and is zero for the checkerboard pattern. The persistence $P(t)$ is the fraction of sites whose spin value did not change even once till time $t$. We find that $D(t)\sim t^{-δ}$ and $P(t)\sim t^{-θ}$ for the parameter range over which the zigzag or checkerboard pattern is realized. We observe that $δ=0.5$ and $θ=3/8$ for 1-d coupled logistic maps or Gauss maps, and $θ=0.22$ and $δ=0.45$ in 2-d logistic or Gauss maps. The exponent $θ$ matches with the persistence exponent at zero temperature for the Ising model, and $δ$ matches with the exponent for the Ising model at the critical temperature. This power-law decay is observed over a range of parameter values and not just critical point. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approach to zigzag and checkerboard patterns in spatially extended systems Warambhe, Manoj C. Gade, Prashant M. Statistical Mechanics Zigzag patterns in one dimension or checkerboard patterns in two dimensions occur in a variety of pattern-forming systems. We introduce an order parameter `phase defect' to identify this transition and help to recognize the associated universality class on a discrete lattice. In one dimension, if $x_{i}(t)$ is a variable value at site $i$ at time $t$. We assign spin $s_i(t)=1$ for $x_{i}(t)>x_{i-1}(t)$, $s_i(t)=-1$ if $x_{i}(t)<x_{i-1}(t)$, and $s_i(t)=0$ if $x_{i}(t)=x_{i-1}(t)$. The phase defect $D(t)$ is defined as $D(t)={\frac{\sum_{i=1}^N \vert s_i(t)+s_{i-1}(t)\vert} {2N}}$ for a lattice of $N$ sites with periodic boundary conditions. It is zero for a zigzag pattern. In two dimensions, $D(t)$ is the sum of row-wise as well as column-wise phase defects and is zero for the checkerboard pattern. The persistence $P(t)$ is the fraction of sites whose spin value did not change even once till time $t$. We find that $D(t)\sim t^{-δ}$ and $P(t)\sim t^{-θ}$ for the parameter range over which the zigzag or checkerboard pattern is realized. We observe that $δ=0.5$ and $θ=3/8$ for 1-d coupled logistic maps or Gauss maps, and $θ=0.22$ and $δ=0.45$ in 2-d logistic or Gauss maps. The exponent $θ$ matches with the persistence exponent at zero temperature for the Ising model, and $δ$ matches with the exponent for the Ising model at the critical temperature. This power-law decay is observed over a range of parameter values and not just critical point. |
| title | Approach to zigzag and checkerboard patterns in spatially extended systems |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2510.10723 |