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Particle-particle collisions in materials give rise to a particle distribution in energy and momentum in such a way that a most probable distribution is realized. I will show that an evaporating liquid generates a molecular imbalance in the equilibrium energy distribution. The molecular collisions by their inherent nature are able to repair the imbalance and in so doing it is shown that the liquids cool down. Similarly an external electric field creates substantial imbalance in the momentum distribution for the electrons in ionic semiconductors. Electron-electron collisions are able to restore the imbalance and as a consequence, similar to the evaporating liquid, the electron gas loses thermal energy and cools down.

I discuss in this paper results which arise in some materials as a consequence of particle-particle collisions. When two particles in a material collide, the energy and linear momentum are conserved. The total of the energy and linear momentum of the particles before and after the collision therefore remains unchanged. Within the laws of statistical mechanics such collisions increase the entropy and are responsible for preserving the most probable distribution for the particles over the microstates of the system. For the purposes of this paper, I will consider two examples: example one consists of molecules in a fluid such as water which is kept in a container. All sides of the container are insulated against heat exchanges except one side which is exposed to the outside environment. The second example is composed of electrons in the conduction band of an ionic semiconductor. The wide gap Nitrides are particularly suitable for the purposes of this paper. I will show that particle-particle collision in example one is responsible for the cooling of the liquid by a process known as “evaporation”. In the second example I will demonstrate that electron in the ionic semiconductor could also cool down from its thermal equilibrium value when subjected to a strong electric field. Energy and momentum are two independent thermodynamic quantities but they are different in character. The main difference between them is that the momentum is a vector quantity and an energy scalar. In spite of this fundamental difference, I have shown in this paper that the process of randomization of these two quantities, under external forces, produces a common result of material “cooling”. There are not any papers discussing specifically this common property. The result in example one [

The distribution of water molecules in thermal equilibrium in example one and of electrons in example two is given by Maxwell-Boltzmann’s distribution [

where

The distribution given by (2) is called the “displaced Maxwell Boltzmann’s distribution” and was first proposed by H. Frohlich and Paranjape for conduction electrons when a semiconductor is subjected to a strong electric field [

There are different ways of obtaining the electron distribution function when it is under a strong electric field. One method is by expanding the distribution function in Legendre series and solving each term of the series by using the Boltzmann’s transport equation. This is a difficult and tedious method with the added limitation that one does not know how many terms in the series are needed for a realist approximation to the distribution function for a given electric field. The most popular method is the numerical evaluation of distribution function by Monte Carlo ’s procedure. The procedure is effective for a small number of electrons say of the order of 100/cc. Any number beyond this would need the computer capacity for beyond the capacities of present day computers. We may have to wait a long time for the arrival of the quantum computers for use of Monte Carlo calculations when a large number of electrons are considered. Someone may ask “Do we need a large number of electrons or can the small number give realistic result for the distribution unction?” Unfortunately we need a large number of electrons if the randomization is to be achieved within a short time. Both these methods are not specific for solving the effect of a strong electric field. Use of Druvystein distribution may appear useful as it is in specific suited when high electric field is imposed. But the method is suited when the collisions occur between electrons and heavy atomic ions, a situation not applicable to my calculation. The displaced Maxwell distribution function, which I use, is particularly suited for obtaining the distribution function under strong electric field. The electron temperature and electron drift velocity, which I try to calculate appear as parameters in the distribution function and therefore they can be easily evaluated.

Let us now consider example “one” which we introduced earlier in this paper. Water is stored in a container which is insulated on all sides except one of its side. The exposed side is in contact with the external environment which is kept dry and therefore practically free of water vapor. The energy distribution at temperature

We have at some stage in life have experienced the cooling discussed above when we rub alcohol on the skin. The cooling of the skin is exactly for the same reason as the cooling we have mentioned in example one. The cooling of water by the process of evaporation was used in ancient times by utilizing porous vats for water storage. Porosity of the pots helped to enhance the effective surface area exposed to the environment and as a consequence generated increased evaporation. The vats were kept in airy surroundings. This is the most practical and inexpensive way to cool water for those who did not have access to refrigerators and for those who like hiking in deserts.

The “second” example, as mentioned earlier in this paper, consists of an assembly of electrons in the conduction band of an ionic semiconductor. The energy distribution of electrons for a non-degenerate, n-type semiconductor is given by Maxwell-Boltzmann’s distribution (1). The temperature, a factor embedded in the distribution function, defines the equilibrium temperature of the electrons. If I disturb the system by an intense external D.C. electric field, the particle distribution will assume a new form. The changes depend on the degree of the external disturbance. According to Frohlich and Paranjape [

For a n-type ionic semiconductor I have selected AlN because as a wide gap semiconductor it can sustain high electric fields without the occurrence of electrical breakdown. Let the vibrational frequency of the AlN lattice be

tion

electrons with the ionic lattice is minimal as they cannot emit a phonon and have low probability of phonon absorption when the temperature is lower than the phonon frequency. The low energy electrons will propagate under the action of the D.C. field without much hindrance from the electron-phonons interaction and therefore gather large linear momentum. Lower the temperature, higher is the momentum acquisition. In fact by increasing the electric field, there is no theoretical limit to the linear momentum that low energy electrons can acquire. The situation with the high energy electrons is considerably different. These electrons can emit as well as absorb phonons. The probability of phonon emission is as stated earlier proportional to

I now turn my attention to thermal energy contents of the conduction electrons. The thermal energy of the electrons is determined by the difference in rates at which the conduction electrons acquire and lose thermal energy. The low energy electrons can only gain energy from the lattice by the absorption of the phonons while the high energy electrons can mainly lose energy. Both the rates also depend on the magnitude of linear momentum. The role of the electron-electron collisions is to balance the momentum between high and low energy electrons. Thus the energy loss by the high energy electrons is enhanced as they receive momentum from the electric field as well as momentum from the low energy electrons. The energy gain by the low energy electrons is reduced by electron-electron collisions. The enhancement of the energy loss and reduction of energy gain arise as a result of the electron-electron collisions. The difference between gain and loss of the thermal energy gives rise to the electron gas either “cooling” or “heating”. I have shown in this paper that the disparity in momentum arises when the electric filed is high. Although there are limits to the magnitudes of high electric fields that can be imposed, the significant cooling of electron temperature can occur in AlN in transient electric fields and in D.C. electric fields in the range of 100 - 150 kV. In this range, I have shown [

V. V.Paranjape, (2015) Randomization of Energy and Momentum in Statistical Mechanics. Journal of Modern Physics,06,2198-2201. doi: 10.4236/jmp.2015.615223