The problem here is the way Gibbs is describing the dynamics of the field and the way in which Gibbs applies the energy.
Here is a schematic of how the process is supposed to work from a standard model perspective:
In the schematic above, the particles are referred to as a dynamical system and the energy is a field. If you take the derivative of a system with a specific time derivative, you get a specific field, which is actually just the sum of the derivatives of the particles. (It’s the same thing as saying you get the change in amplitude of the field by multiplying the change in the field by the rate of change of the field).
Basically, then, if you have a system in which you don’t have the energy as being an energy, then (just as in classical mechanics) the system will oscillate back and forth or spin around.
What exactly is the Gibbs of a system?
This depends on what you mean by Gibbs of the system. If you don’t know what that means, look at how the second law of thermodynamics describes it:
At equilibrium, the amount of heat emitted by the system is equal to the amount of energy it had at equilibrium. This is the thermodynamic Gibbs.
If you have thermodynamic Gibbs of a system, then this is the heat going into the heat exchanger (this is a Gibbs of a heat exchanger, not the Gibbs of energy of the system). (This is called the total heat of the system).
This means that if the field is changing from heat coming from the reservoir to heat coming out of the reservoir, there is a certain amount of energy that ends up in the heat exchanger.
The net result of all of this is that any given heat exchanger will be doing some work, so to speak.
When does the heat flow into the heat exchanger stop?
This depends on the total heat of the system. The amount of heat that flows into the heat exchanger is a function of the total heat of the system. This means that energy in one direction cannot be dissipated in another. For example, if we have a thermodynamic Gibbs of 2x, it implies the heat flow is a balance function of the total heat of the reservoir, the total heat of the fluid that flows into the reservoir, and the total heat of the fluid that flows out. The amount of heat goes to zero if the heat flow is balanced.
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