The Minimum Energy Principle

The principle of conservation of energy says that in the universe or more abstract in a closed system the energy is constant. This statement is different that the energy is tending to a minimum. We experience this principle in everyday mechanics. 

For example, if we raise a body and allow it to fall down, we would feel that due to the action of gravity the body will search a state where its energy is a minimum. However, here we are considering only the potential energy of the body. We will not take into account that the potential energy is meanwhile transformed into kinetic energy and that the floor where the body is coming at rest would warm up in the course of the crash. 

Further, we have withdrawn gravitational energy from the system that provides the gravitational energy. Therefore, in summary we have balanced the energy only for a part of the actors that are important to a body that is falling down. 

Instead, U(X + AX, Y + AY. + A...) = U(X, y.) = Ci. We divide now the system under consideration into two subsystems Ui and U2. Since the original system was chosen arbitrarily, we expect the same property of attaining a minimum energy also for the subsystems. However, the subsystems now can exchange energy 

On the other hand, if there should be a maximum of energy, than the tendency is reverse. Each subsystem would like to catch energy at the cost of the other subsystem. So, the particular subsystem that is stronger would accumulate energy, as long as it can suck energy from the other system. In the limiting case in one system a zero 

---

In the sequence of orbital energies shown above the 4s orbitals have a lower energy than the 3d orbitals and so they will be filled first in keeping with the minimum energy principle. For example, the electron configuration of the outer 10 electrons of calcium (atomic number Z = 20) is 3s 3p 3d 4s. In the filling of the electron orbitals for elements 21 to 29, there are two irregularities, one at 24 (chromium) and one at 29 (copper). Each of these elements contains one 4s electron instead of two. The reason... 

The minimum energy principle in the closed system is derived from the First and Second Laws of Thermodynamics. Since iQ = TdS, we have... 

Thus the minimum energy principle is also given for the continuum. 

In principle, we could find the minimum-energy crystal lattice from electronic structure calculations, determine the appropriate A-body interaction potential in the presence of lattice defects, and use molecular dynamics methods to calculate ab initio dynamic macroscale material properties. Some of the problems associated with this approach are considered by Wallace [1]. Because of these problems it is useful to establish a bridge between the micro-... 

The fact that each of the two variables, S and U may be expressed as a function of the other, indicates that the extremum principle could likewise be stated in terms of either entropy or energy. The alternative to the maximum entropy principle is a minimum energy principle, valid at constant entropy, as graphically illustrated for a one-component system in figure 1, below. 

The transformation from spheres to cyhnders is a peculiar example for the self-adjustment of the molecular conformation. The switching shape can be regarded as an example for the principle of quasi equivalency established by A. Klug for the self-assembly of biomolecules and viruses [145] for the sake of an improved intermolecular packing, the molecules adopt a conformation different from the minimum energy one. This also demonstrates that shape control does not mean a fully constrained structure. Similar to biomolecules, the combination of flexible macromolecules and self-assembly principles is a powerful strategy for preparation of molecules with well-defined but switchable shape [23]. 

Using eqn 1.12, and the principle of energy conservation, the photoelectric effect can be explained without difficulty. Let be the minimum energy required to liberate an electron from a particular solid, a quantity known as the work function. Then if each photon transmits its energy to just one electron, the maximum emitted energy must be... 

This potential is the same as a flat-bottomed container with infinitely high walls separating inside from outside. Here we will use the Uncertainty Principle to estimate the minimum energy later (in Chapter 6) we will find all of the possible energies and states for this system, using a differential equation known as... 

---