How to Prove sqrt(<Rg^2>) = sqrt(Lζ/3)?

[jaobyccdee]

With <Rg^2>=1/N [sum[(Ri-Rc)^{2}>] where Rc is the center of mass, =1/N sum Ri, and provided that <R^2>=2Lζ .Show that sqrt(R^{2})=sqrt(L ζ /3)

 

[Matterwave]

It would perhaps help if you defined all the variables, and what the problem is actually asking...

 

[jaobyccdee]

<Rg^2> is the radius of gyration. Ri-Rc is the distance between the monomers and the center of the polymer. The problem is that give <Rg^2>=1/N Sum<( Ri-Rc )^2>, and that Rc=1/N sum Ri. proof that sqrt(<Rg^2>) = sqrt(Lζ/3). Actually i was working on it, and there is a step that i m not sure, and it's that if 1/N Sum<(Ri-1/N Sum(Ri)>^2 ==1/(2N^2) <sum of [i,j] (Ri-Rj)>^2