Skip to main content

computational geometry - 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.




Comments

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...