Including Coulomb interaction in a free energy calculation

[barrozoah]

Hi everyone!

I am trying to create a crude electron-hopping model to study conductivity in a biological wire composed of discrete sites. The model is pretty simple: imagine a line composed of sites. Electrons can hop from site to site with probabilities that depend on the free energy difference between those sites.

All these free energies were obtained numerically for the case where only one electron is in the system. Now I am considering the scenario where two electrons are in adjacent sites, and Coulomb repulsion would push them apart, changing their probability of hopping.

My question is how to add this extra Coulomb interaction to the free energy directly. I imagine that the internal energy and also the Hamiltonian change by a simple addition of a term, but I could not yet figure out how to account for an extra entropic difference that would be involved. Since I have the free energies obtained numerically for the case without this extra interaction, I wouldn't know how to decompose this back to a partition function so that it could be recalculated.

Thanks!

 

[eljose79]

Hello!

Your question is an interesting one and it raises some important points about modeling biological systems. The addition of Coulomb interaction to your model is definitely an important consideration, as it can significantly affect the behavior of the electrons in the system.

One way to incorporate this interaction is to consider the Coulomb energy as an additional term in the Hamiltonian. This would be a potential energy term that takes into account the repulsion between the two electrons. In terms of the free energy, this would correspond to adding an additional term to the internal energy and subtracting it from the entropy.

However, as you mentioned, the challenge here is to decompose the free energy back to a partition function. One way to approach this would be to use statistical mechanics principles and consider the system as a canonical ensemble, where the number of electrons and their energy are conserved. This would allow you to derive the partition function and calculate the free energy for the system with the additional Coulomb interaction.

Another approach could be to use computer simulations to model the behavior of the electrons in the system with and without the Coulomb interaction. This would allow you to directly compare the results and see the effect of the interaction on the system's behavior.

I hope this helps and good luck with your research!