Connection between Free Energy and Action

[physengineer]

Could anyone please, explain the connection between the Free Energy and the Action?
I know the definitions of each but not sure about the connection.

I read somewhere that the free energy is the same as the action when calculated "on-shell". What does this "on-shell" thing mean in this context?

Thanks a lot in advance!

 

[Relay]

What do you think "Free Energy" and "Action" mean?
You need to provide a lot more information if you want people to know of what you are speaking of.
As it stands; for me "Free Energy" is internet fun and "Action" is movement of an solid.

 

[SpectraCat]

I don't think the free energy can ever be the same as the action .. they have different units. Action is (joules)*(sec), while free energy is just joules.

Action is a bit tricky to define .. it is generally easier to understand in the general sense. The principle of least action tells us that for classical mechanics, the path followed by a system that is moving between two configurations is the path of least action. You can look at the http://en.wikipedia.org/wiki/Action_%28physics%29" to get a mathematical definition of the action (which may or may not make much sense to you, depending on your background in mathematics and classical physics).

Another point is that in physics, free-energy is a thermodynamic concept relating to statistical averages over large numbers of particles. You can calculate the Gibbs free energy or the Helmholtz free energy for an ensemble of particles, but I don't know what either of them have to do with the action. Usually you would use the action when you will be tracking the trajectories of individual particles, and the free energy to express thermodynamic averages for systems containing too many particles to track individually.

It is possible that there may be a deeper connection .. certainly there is a preference in natural systems to evolve so as to minimize the free energy of the system. That sounds sort of analogous to the principle of least action ... it may be that the tendency to minimize the free energy is some sort of statistical manifestation of the principle of least action. That would make sense to me, but at the moment I do not know how to derive the connection. I will check my statistical thermodynamics books tomorrow to find out.