Polymer Physics - Radius of Gyration

[ChrisJ]

I was not sure if this was the best place for this, it could fit here, in the Chemisty section or the Programming section. So feel free to move if needed.

Essentially I have been modeling polymers in python and using a Monte Carlo, Metropolis type algorithm, to minimise its energy into equilibrium from its intial state governed by a 3D random walk (not on a lattice).

I don't know why I am having so much trouble with this, probably because of different sources calling it different things, or using same notation but using it to refer to different bits. But anyway, all I am trying to do is to calculate the Radius of Gyration for my polymer once it has reached equilibrium.

I have tried to program these two equivalent expressions (below) for the radius of gyration, given in the same textbook, but a couple things confuse me which they don't explain.

$R_g^2 = \frac{1}{N+1} \sum_{i=0}^N \left < ( \textbf{r}_i - \textbf{r}_G )^2 \right >$
where $\textbf{r}_G$, the centre of mass, is given by $\textbf{r}_G = \frac{1}{N+1} \sum_{i=1}^N \textbf{r}_i$

And also
$R_g^2 = \frac{1}{2(N+1)^2} \sum_{i,j=0}^N \left < ( \textbf{r}_i - \textbf{r}_j )^2 \right >$

where the angle brackets represent averaging, but I am confused with what is being averaged? In the notation is is clear that $R_g^2$ is a scalar, but following the equation I end up with a vector still, so it must come out of the averaging,

but if one finds the difference between two vectors, you end up with a vector, if one squares a vector you end up with a vector, and then it asks to take the average... Is the radius of gyration only applicable to many instances of a polymer configuration. Like for example if one did 100 different configurations, then the radius of gyration is applicable to that set of 100 configurations, like with the rms distance in random walks?

I was told by my tutor to find the Radius of gyration of my system in equilibrium and plot vs N, yet it takes like 2-3 hours of the code running to get one polymer configuration from its starting point to equilibrium. So to get a decent set of configurations for N, and then do it x10 to get a plot vs N is going to take like a week of my computer @ 100% running code.

Any guidance on the Radius of Gyration is appreciated.

 

[bahamagreen]

Per Wikipedia-radius of gyration-molecular applications:

"Since the chain conformations of a polymer sample are quasi infinite in number and constantly change over time, the "radius of gyration" discussed in polymer physics must usually be understood as a mean over all polymer molecules of the sample and over time. That is, the radius of gyration which is measured as an average over time or ensemble:"

Wiki's equation shows the summation within the angular brackets...?

"where the angular brackets <...> denote the ensemble average."

 

[mjc123]

if one squares a vector you end up with a vector,

No. The dot product of a vector with itself is the square of the length of the vector. The cross product of a vector with itself is zero. (r_i - r_G)² must mean |r_i - r_G|²

 

[ChrisJ]

No. The dot product of a vector with itself is the square of the length of the vector. The cross product of a vector with itself is zero. (r_i - r_G)² must mean |r_i - r_G|²

Its funny. I know that relationship like the back of my hand, yet spent hours last night trying to figure this crap out, so caught up in a mix between formal mathematical notation and programming language notation, and the fact that if one just programs the equation using arrays of length three for vectors, squaring an array is a perfectly valid operation (where the components are simply squared). And my textbook doesn't help when it mixes notation, in the same chapter/section from the one I had in my post (standard ()^2 brackets) and ||^2 for different things. Was very tired last night!

Makes sense.

 

[ChrisJ]

Per Wikipedia-radius of gyration-molecular applications:

"Since the chain conformations of a polymer sample are quasi infinite in number and constantly change over time, the "radius of gyration" discussed in polymer physics must usually be understood as a mean over all polymer molecules of the sample and over time. That is, the radius of gyration which is measured as an average over time or ensemble:"

Thanks, the averaging over time bit makes more sense to me in my situation. I've gotten so used to relying on textbooks over the internet that sometimes I don't even think to check places like wikipedia.

Thanks, makes a bit more sense now.

Wiki's equation shows the summation within the angular brackets...?

My textbook showed them both, saying they were equivalent.