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/τ) where v is the dimensionless velocity and l is the step length and τ 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−γ−δ−0.5ϵ and the units of ϵ′ is τ−1. where ϵ′=ϵ∗τ−1.
My question is how all of this makes sense in dimensionality. In the exact dimensional analysis, we are adding quantities with dimensions τ−1 and getting a dimensional quantity of l/τ. Furthermore, the diffusion coefficient in the same reference has been defined as: D=v2/ϵ2 Now if I want dimensionality of D′ I will have to do: D′=D∗l2/τ However, If I use the dimensional quantities v′ and ϵ′, the dimensionality for D′wrong=v′2/ϵ′2 will be l2τ2∗τ2=l2, which is wrong. Also to get proper dimensionality the last analysis suggests that ϵ′=ϵτ−0.5 which is different than the aforementioned analysis. I am confused about why I am getting these inconsistencies.
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