Well, we did something called the Gibbs–Nozhkin equation when we started working with energy, and the answer is that the free energy is zero at equilibrium. The energy of a system is an expression of the energy of its constituent parts divided by their potential energy [in the energy-mass system]. Now, the free energy is one of the two equations for the free energy of an aggregate, and the other one is in terms of motion.
If Gibbs is a “nonzero” free energy, that is, if it is in equilibrium, then in a system with no kinetic energy and no potential energy, the system has zero free energy. Now, a little math shows that if the free energy is in equilibrium, as it is here, then the total kinetic potential energy of the system is equal to the density of particles in the system [mass density, a measure of the total kinetic energy of a system]. The kinetic energy also has to be zero [no kinetic energy, a measure of the total kinetic energy of the system without physical motion].
If we can make a simplified “zero free energy” system have zero free energy, in terms of free energy, the density of particles in the system is the same as the density of particles in the system, and the total free energy is the same as the kinetic energy.
Thus, since no energy is dissipated in the total kinetic energy of the system, the total kinetic energy is zero.
Gibbs free energy and Gibbs-Newton-law in the context of a thermodynamic system
What happens when you make the system have zero free energy and you try to look at some other temperature range? The most typical temperature range considered is that at which the mean free path of the system is equal to its total potential energy. Here, the mean free path is of the form:
The temperature in terms of mean free path is expressed in terms of the energy balance of the system.
There is a new equation, now:
Gibbs-Newton law in the context of quantum mechanics
The law in the above two cases for systems with zero free energy and zero mass is also true for systems with zero temperature. In physics, we have called these conditions the “equilibrium conditions” in terms of temperature and volume. This equation for the mean free path can also be written as where:
This equation is very similar to the previous one, except that we are dealing with a quantum mechanical system.
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