Computer Geometry – Ideas to Visualize the Shape of a Random Walk

The context:

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

Since the turning 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 chosen axes so that the diagonal elements follow the $ lambda_z ^ 2 ge lambda_y ^ 2 ge lambda_x ^ 2 $ relationship.

The clean spectrum found and various functions 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} $

  • Asphericity ($ 0 when the structure of the step or the distribution of the particles is symmetrical spherical, and positive otherwise) $ b = frac {1} {2} (3 lambda_z ^ 2-R_g ^ 2) tag {3} $

  • Similarly, acylindrity ($ 0 when there is cylindrical symmetry):
    $ c = lambda_y ^ 2 – lambda_x ^ 2 tag {4} $

  • Anisotropy of relative form (takes values ​​between $ 0 and $ 1 $he reaches near $ 1 $ for linear structures, in line form and $ 0 very symmetrical), with a slight modification of $ T, $ to know $ hat {T} _ {ij} = T_ {ij} – delta_ {ij} text {Tr} (T / 3) $ it can be written as ($ text {Tr} $ denoting the trace operation) $$ kappa ^ 2 = frac {3} {2} frac { text {Tr} ( hat {T} ^ 2)} {( text {Tr} hat {T}) ^ 2} tag {5} $$

  • The nature of asphericity, describing the prolateness or flattening of the structure, can be expressed in the form (varying between -1 $ at $ 1 $with -1 $ corresponding to the completely Oblate case, and $ 1 $ to the complete prolate.)
    $$
    S = frac {4 text {t} hat {T}} { left ( frac {2} {3} text {Tr} hat {T} ^ 2 right) ^ {3/2} } tag {6}
    $$

So we have all these quantities that together describe the global geometric properties of the structure.


  • On the basis of the geometric descriptors thus defined of a random walk (or of a polymer chain), would it be possible to visualize / imitate the global structure in Mathematica? Intuitively, I imagine a parametric approach in which one started from a sphere, for example, and by modifying one or other of the geometric descriptors (for example, a manipulated parameter) by adjusting the drawn structure, but I do not do not know how to compose a visualization from the collection of these properties only. Any idea of ​​how to create such a visualization would be welcome.

What's cool is that the Gyration Tensor captures and characterizes the essence of the geometric structure, ignoring the details of the system (whether it's a random walk, a polymer , etc.). visualization would probably be useful for a variety of systems.