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#### Robert S. Pierre

##### Guest

*Does diffusion of a spring depend on the spring’s equilibrium length?*

Say we perform a random walk on $\mathbb{Z}$ with two equal particles that are connected by a spring with spring constant $K$ and equilibrium length $a$, where at each time step each particle has probability $\frac{1}{2}$ of making a step $\pm1$. However, at each step a random number $q\in[0,1]$ is sampled in order to refuse the move if $$q\geq\exp\left(\Delta E\right),$$ where $\Delta E$ is the difference in elastic energy between the two time steps.

Does the diffusion coefficient depend on the equilibrium length $a$?

I know that the diffusion is independent of the spring constant $K$, so we find the same diffusion coefficient for all $K$, namely that of two independent particles. I assume the same holds for the equilibrium length, but is this true and how can I see this?

And furthermore, what role does the initial extension of the spring play? Say the initial length of the spring $x_0$ is not equal to $a$, then the whole system will need a while to equilibrize, as opposed to when we start with $x_0=a$.

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