# Homework Help: Fluctuation Dissipation Theorem

1. Dec 21, 2015

### fayled

1. The problem statement, all variables and given/known data
We have a system of two coupled Langevin equations
dr/dt=kr-yrr+nr(t)
dp/dt=kpr-ypp+np(t)
where the ki,yi are constants and ni(t) are noise terms satisfying <ni(t)>=0 and <ni(t')ni(t'')>=qiδ(t'-t'') (this is zero if the two indices differ).

The physical background of these is that it is a model of protein synthesis - r represents the concentration of mRNA and p the concentration of a protein.

2. Relevant equations
The concentration of mRNA should obey Poisson statistics in the steady state, which means that in the long time limit we have <r>=<r2>-<r>2.

3. The attempt at a solution
I can solve the first equation for <r> and <r2> and apply the above condition to obtain qr=2kr. I can also then plug my expression for r(t) into the second to obtain expressions for <p> and <p2>, however p does not obey Poisson statistics at long times My issue is in finding the value of qp - the answer should be qp=2kpkr/yr but I cannot understand how to extract this from the condition that only r obeys Poisson statistics at long times.

2. Dec 26, 2015