# Ideas for visualizing the shape of a random walk

Context:

In the context of (3D) random walks or polymer chains, a useful quantity for capturing and characterizing the shape of the walk or the conformation of the polymer in space is the gyration tensor $$T$$, which in words is the arithmetic mean of the second moment of particle positions along the chain, and visited positions in case of random walks.

Given that the gyration tensor is a symmetric 3-by-3 matrix, it can be diagonalized and written in the following form:

$$T = \begin{pmatrix} \lambda_x^2 & 0 & 0 \\ 0 & \lambda_y^2 & 0 \\ 0 & 0 & \lambda_z^2 \end{pmatrix} \tag{1}$$ and the axes chosen such that the diagonal elements follow the $$\lambda_z^2 \ge \lambda_y^2 \ge \lambda_x^2$$ relation.

The found eigen-spectrum and various function of $$T$$ can be used to define geometric descriptors such as:

• Radius of gyration (average spatial extent of the structure) $$R_g^2 = \lambda_z^2+\lambda_y^2+\lambda_x^2 \tag{2}$$

• The asphericity ($$0$$ when the structure of the walk or the distribution of the particles is spherically symmetric, and positive otherwise) $$b = \frac{1}{2}(3\lambda_z^2-R_g^2) \tag{3}$$

• Similarly, acylindrity ($$0$$ when there's cylindrical symmetry): $$c = \lambda_y^2 - \lambda_x^2 \tag{4}$$

• Relative shape anisotropy (takes values between $$0$$ and $$1$$, it reaches near $$1$$ for linear structures, line-like, and $$0$$ for highly symmetric ones), with a slight modification of $$T,$$ namely $$\hat{T}_{ij}=T_{ij}-\delta_{ij}\text{Tr}(T/3)$$ it can be written as ($$\text{Tr}$$ denoting the trace operation) $$\kappa^2 = \frac{3}{2}\frac{\text{Tr}(\hat{T}^2)}{(\text{Tr}\hat{T})^2}\tag{5}$$

• Nature of asphericity, describing the prolateness or oblateness of the structure, can be expressed as (varying between $$-1$$ to $$1$$, with $$-1$$ corresponding to the fully oblate case, and $$1$$ to the fully prolate one.) $$S=\frac{4 \text{det}\hat{T}}{\left(\frac{2}{3}\text{Tr}\hat{T}^2\right)^{3/2}} \tag{6}$$

Therefore we have all these quantities which together describe the overall geometric properties of the structure.

• Based on the so-defined geometric descriptors of a random walk (or polymer chain), would it be possible to visualize/mimic the overall structure in Mathematica? Intuitively, I imagine a parametric approach where one would start from a sphere for example, and by changing either of the geometric descriptors (e.g. as a manipulate parameter) adjusting the drawn structure, but I don't know how to compose a visualisation from the collection of these properties alone. Any ideas for how to create such a visualisation would be most welcome.

The cool aspect of it all is that the gyration tensor allows one to capture and characterize the essence of the geometric structure by abstracting away from the actual details of the system (whether it's a random walk, a polymer, etc.), therefore, any visualisation would be likely useful for a variety of systems.

• Example of a related paper in the context of RW's: "The Shapes of Random Walks" by J. Rudnick and G. Gaspari.
– user52181
May 1, 2019 at 15:23