# How to introduce dimensionality in a dimensionless framework?

This question is an extension of this one. I have been told that to introduce dimensionality in a dimensionless quantity I need to multiply with suitable parameters. For instance, for velocity I have to: $$v'=v*(l/\tau)$$ where $$v$$ is the dimensionless velocity and $$l$$ is the step length and $$\tau$$ is the time step. But the reference I am using Random walks of molecular motors arising from diffusional encounters with immobilized filaments defines $$v=1-\gamma-\delta-0.5\epsilon$$ and the units of $$\epsilon'$$ is $$\tau^{-1}$$. where $$\epsilon'=\epsilon*\tau^{-1}$$.

My question is how all of this makes sense in dimensionality. In the exact dimensional analysis, we are adding quantities with dimensions $$\tau^{-1}$$ and getting a dimensional quantity of $$l/\tau$$. Furthermore, the diffusion coefficient in the same reference has been defined as: $$D=v^2/\epsilon^2$$ Now if I want dimensionality of $$D'$$ I will have to do: $$D'=D*l^2/\tau$$ However, If I use the dimensional quantities $$v'$$ and $$\epsilon'$$, the dimensionality for $$D'_\text{wrong}=v'^2/\epsilon'^2$$ will be $$\frac{l^2}{\tau^2}*{\tau^2}=l^2$$, which is wrong. Also to get proper dimensionality the last analysis suggests that $$\epsilon'=\epsilon \tau^{-0.5}$$ which is different than the aforementioned analysis. I am confused about why I am getting these inconsistencies.

• I think there must be an hidden $l$ in the $\epsilon$ or $\epsilon'$ definition. The probabilities are for moving of one lattice point, that is long $l$. Maybe your reference uses "crystal lattice units" all along? – patta Apr 23 at 9:57
• @patta My real question is if we give a dimension to suppose parameter $a$ and a dimension to parameter $b$, then a parameter $c=a.b$ must have a dimension that agrees to $dim\{a\}*dim\{b\}$ and it can not have any arbitrary dimensions? – Userhanu Apr 23 at 10:09
• You are right, dimensions should be well defined! I looked at the paper you cite, they are quite sloppy, the speed is always in "lattice sites units" while they don't write it. My way of work is to make the calculation first, fix the dimensions afterwards. – patta Apr 24 at 9:30
• @patta Thanks a lot for your help! I got totally confused in terms of dimensions there. – Userhanu Apr 24 at 10:33
• thanks! another thing from the paper you cite: in the plots they always express the position of the walker as $n$, that is, the $n$th site on the filament. Not meters or microns! Thus the position is dimensionless in their treatment – patta Apr 25 at 11:55