What is more important energy or entropy
e. What are the free energy and chemical potential?
Most people have learned in school that every physical system aims at the lowest energy state. But that is not entirely true; it's only part of the truth. In reality, a many-particle system strives for the state of minimal energyand at the same time maximum disorder. To take this fact into account, "free energy" was introduced.
Here F symbolizes the "free energy", E the "normal energy", T the temperature of the system and S the entropy. Entropy is a measure of the disorder in the system. Entropy zero means absolute order, for example in a perfect crystal at absolute temperature zero; a higher entropy corresponds to a higher disorder.
Statistical mechanics, both quantum mechanical and classical, says that a system always strives for a minimum of "free energy". When a system reaches the minimum of free energy, it is in thermal equilibrium. This means that the system has reached a steady state and its macroscopic properties (such as temperature and pressure) no longer change over time as long as it remains isolated. In most cases, however, reducing energy also reduces clutter and vice versa. So there is a kind of tug-of-war between the two mechanisms.
One can initially imagine a process in which both the energy and the disorder decrease. As long as the decrease in energy compensates for the decrease in entropy, the "free energy" also decreases. Thus, it is a spontaneous process that can take place without external intervention, since the system strives for a minimum "free energy".
If, for example, a paramagnet makes a transition to the ferromagnetic state, it reduces its internal disorder quite considerably. Previously, all elementary magnetic moments were randomly oriented in space, while in the ferromagnetic phase they are all oriented parallel to one another. On the other hand, the internal energy of the paramagnet is also reduced, since the energy stored in the magnetic field of the elementary magnetic moments depends on the distances between neighboring north and south poles. These are minimized in the parallel configuration. In some materials (e.g. iron) the reduction in internal energy is large enough to justify the reduced disorder. These materials can reduce their free energy by aligning their elementary magnetic moments in parallel and thus obtain a large magnetic moment. They are commonly known to us as magnets. In the case of most other materials with elementary magnetic moments, the reduction of the internal energy in the case of parallel alignment is not sufficient to justify the high degree of order. These materials never become ferromagnetic, although they contain elementary magnetic moments. They remain paramagnetic forever.
The opposite case is also implemented in ferromagnets. A ferromagnetic material can significantly increase its internal disorder by generating magnetization waves (also called spin waves). The inner energy is only slightly increased. This point will be discussed in more detail in a later section.
Another important parameter of many particle systems, which particularly affects the Bose Einstein condensation, is the chemical potential. The chemical potential is defined for a system in thermal equilibrium. It indicates how much the "free energy" changes when a particle is added to or removed from the system.
In a system in which the number of particles is fixed, in which no particles can be created or destroyed out of nothing. The chemical potential assumes a certain value, which depends on the number of particles and other parameters (pressure, temperature). This situation occurs in all normal gases made up of atoms or molecules.
But there are also systems in which particles can be generated out of the vacuum. A good example of such a system is the photon gas of a "black body". A black body is a body whose walls absorb any photons that hit it. However, these walls also emit thermal radiation for their part. In this system, photons are continuously generated and destroyed.
Consequently, the chemical potential of such a system, in thermal equilibrium, is always zero. The reason for this is very simple. Since the number of particles is now variable, the system can choose the number of particles. It will then of course look for the number of particles at which its "free energy" is minimal. So the system is in thermal equilibrium, in a minimum of free energy in relation to the number of particles. And from school mathematics it should be known that the slope of a function is always zero in its minimum.
The point is that the creation and annihilation of particles always takes a certain amount of time and does not happen instantaneously. This fact opens up the possibility of changing the chemical potential of systems with a non-constant number of particles. We will see later that this will be an important step in the creation of a Magnon Bose Einstein condensate.
Now that the "free energy" and the chemical potential have been introduced, the next section deals with the question:
What are the distribution functions in many particle systems?
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