What type of energy should I use when describing the amount of energy available to increase entropy of a system? There are many different specific definitions of energy within different physics fields: In thermodynamics we have at least U (system internal energy), F (Helmholtz Free Energy), G (Gibbs Free Energy), H (Enthalpy). Or we have the relativistic stress-energy-momentum tensor, etc.
Various forms or configurations of energy seem to be included/excluded in these different definitions.
I'm looking for the most appropriate "type" of energy to use in quantifying the potential to disrupt (disorder) an ordered system. That is, some amount of "available" energy that can cause increase of entropy of the system.  I do not want to get more specific than what I've said about what kind of ordered system it is. Just a configuration of matter and energy that has less than that amount of matter/energy's maximum possible entropy so has the potential to be disordered.
Disordering energy/matter-with-energy could come from outside the system I suppose, or be energy/mass within the system. So I guess we're talking about a thermodynamically open system, but thermodynamics tends to scope out energy that's locked up in/as the mass of matter.
For the most general case of describing energy available to disorder any ordered system, what type of energy definition should I use and why? 
 A: Rearranging the definition of the Helmholtz free energy, 
\begin{equation}
F = U-TS,
\end{equation}
we obtain
\begin{equation}
S = \frac{U-F}{T},
\end{equation}
so that the amount of entropy is determined by the difference in the internal and Helmholtz free energies. Qualitatively, then, if the Helmholtz free energy can significantly change, it can affect the entropy.
From 

a configuration of matter and energy that has less than that amount of matter/energy's maximum possible entropy

it sounds like $U$ is fixed, so if mechanical work is irrelevant, the main quantifier of how the entropy will change is the Helmholtz free energy:
\begin{equation}
TdS = -dF -SdT,
\end{equation}
assuming $dU=0$. Internal rearrangements of the system may release or absorb heat, contributing to $dT$.
If mechanical work is relevant, from the definition of the Gibbs free energy, enthalpy, and the fundamental thermodynamic relation,
\begin{equation}
TdS = dH-dG-SdT.
\end{equation}
These relations are equivalent (with $dU=0$), but they illustrate that in the most general case, no single energy quantity is sufficient to quantify how entropy may change.
A: The definition of "energy" is clear in the context of fundamental laws: mechanics and field theories (both quantum or classic). In such treatments, the fundamental mathematical model can be expressed by a Hamiltonian. This function is numerically equal to the energy of the system. Lagrangian formulations can always be related to the Hamiltonian form. Relativistic theories fall in this category.
Averaged, macroscopic or mesoscopic theories, or even coarse grained theories, are for example fluid dynamics. In such theories, there is a "macroscopic" energy and heat, and the macroscopic energy can be degraded to heat (e.g. by friction).
Thermodynamics is probably the most complex situation. Considering only the "true" thermodynamics, i.e. the "equilibrium" one, the internal energy U corresponds to the average of the Hamiltonian (in the canonical system). 
The other forms of energy have a clear mathematical meaning. The intuitive explanation is roughly the following. The enthalpy is an attempt to define an energy without its mechanical part, when this mechanical part is only the work performed by pressure on a piston. The "free energy" (Helmholtz and Gibbs) are an attempt to subtract the part of energy which is "not available" because it is in the form of heat. All of them are not "energy", indeed, they are not called "energy".
Now, the question asks what kind of energy generates "disorder". It is know that a system can show a regular motion at low energy, but it becomes chaotic at higher energy. In this case, we are considering the Hamiltonian (fundamental) energy, or the internal energy. Something similar applies to ordered structures, which become disordered at high temperature, e.g. a crystal which melts. However, we must always consider each problem separately. For example, if are discussing the motion of a system of planets, our Hamiltonian will not take into consideration the thermal energy of the material constituting the planets; in this case, what matters is the Hamiltonian energy, but not the internal energy.
Finally, as already emphasized, it is not generally possible to say how much the increase of energy of a system increases the entropy. It depends on the system and how the energy is provided. The only situation that is sure is when the energy is provided as heat, and slowly enough... but this simply gives us the definition of entropy and nothing else.
