Question about possible issues in GPATH documentation (expressions for s and z)

[孟令屾]

Dear PLUMED developers and users,

I am writing to ask about a possible issue in the GPATH documentation, specifically in the page:
https://www.plumed.org/doc-master/user-doc/html/GPATH/

It seems that the expressions for s and z in this documentation may not be correct.

1. For s:

   • The formula in the documentation appears to differ from the one given in the reference:
     G. Díaz Leines and B. Ensing, "Path Finding on High-Dimensional Free Energy Landscapes", Physical Review Letters 109 (2012).

   • Path Finding on High-Dimensional Free Energy Landscapes | Phys. Rev. Lett. 

   • and the derivation of the path collective variable can be found in:

   • SuppInfo.pdf

   • From the theoretical point of view, it seems unlikely that the correct result can be obtained using only two points (as in the documentation) on the circle.

2. For z:

   • The formula seems to have a similar issue, a v₃ was write as v₂ .
   • Additionally, there may be a dimensional inconsistency: it looks like there could be an extra square under the square root.

Could you please clarify:

• Are these discrepancies only an issue in the documentation, or are they also present in the actual source code of PLUMED?

Thank you very much for your time and for developing PLUMED.

Best regards,
Lingshen Meng