Understand the "Reweighting the results" in the "Belfast tutorial: Umbrella sampling"

[Zhang Cheng]

Dear All,

In the Section "Reweighting the results" of Tutorial "Belfast tutorial: Umbrella sampling"

https://plumed.github.io/doc-v2.4/user-doc/html/belfast-4.html

I generated two COLVAR plots (raw COLVAR files attached)

1) KAPPA=10.0 AT=-2 (left)

2) KAPPA=0.0 AT=-2 (right)

Can I confirm my understanding?

1) The "AT" argument defines the position of the restraint. That is reason that the phi values fluctuates along the -2 in the left plot. 

For the right plot where KAPPA=0, the AT should not affect the restraint because k/2*(x-x0)^2 always equals to 0. But why the phi values also are around -2 ?

2) Is the unit of phi (purple dots) degree (i.e. °) ? So it could be from -360° to 360° ?

Is the unit of "restraint-phi.bias" (green dots) kJ/mol ?

In this tutorial, it is said 

The keyword KAPPA determines the hardness of the spring constant and its units are [Energy units]/[Units of the CV].

However, I think its units should be [Energy units]/([Units of the CV]^2), so in this case, should it be kJ/(mol*degree^2)?

3)  In the equation k/2*(x-x0)^2, k (i.e. KAPPA) and x0 (i.e. AT) are defined. Is the remaining variable "x" a random value at different time points? Why in the left plot, most green dots are within the range of 0 to 5 ?

4) I also plot the psi only for the KAPPA=10 (left) and KAPPA=0 (right) as shown below. They are not exactly the same, but because they look similar, this indicates that the restraint on phi does not affect the psi values. 

Thank you very much!

Yours sincerely

Cheng