Fundamentals of Statistical Mechanics Using the Boltzmann Distribution

The objective of this chapter is an examination of the energy distribution

of matter for the highest probability, considering that only phenomena with the

highest possibility are possible for a large number of masses (atoms or molecules in

a macroscopic object). The thermodynamic temperature T is the parameter for the

broadening of the energy distribution and the population in a state with an energy of

E is proportional to Ω exp(-E⁄(kBT)), where Ω is the number of states and kB is the

Boltzmann constant. The average energy and specific heat are discussed using the

Boltzmann distribution. The relationship between the gas pressure, volume, and

temperature (ideal gas law) is obtained from the average of the one-dimensional

kinetic energy. The work efficiency of the Carnot engine, using gas pressure, is also

discussed.

Keywords: Adiabatic expansion, Bose-Einstein condensation, Carnot cycle, Entropy, Fermi degeneracy, Free energy, Ideal gas law, Specific heat, Temperature, Thermal energy, Thermal equilibrium, Vapor pressure.

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