# Portal:Statistical mechanics

Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. Statistical mechanics is most commonly used in quantum mechanics and gas laws to describe the average behavior of particles and is often jokingly referred to as "Sadistical mechanics."

It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, therefore explaining physical phenomena like those of thermodynamics as a natural result of statistics and mechanics (classical and quantum) at the microscopic level. In particular, it can be used to calculate the thermodynamic properties of bulk materials from the spectroscopic data of individual molecules.

This ability to make macroscopic predictions based on microscopic properties is the main asset of statistical mechanics. In thermodynamics theories are governed by the second law of thermodynamics through the medium of entropy. However, Entropy in thermodynamics can only be known empirically, whereas in statistical mechanics, it is a function of the distribution of the system on its micro-states. In chemistry, predictions are made about a chemical's behavior based on its relation to other elements within the periodic table, but statistical mechanics not only allows for a derivation of many chemical properties, it also can give reliable data for tolerance levels and expected behavior in real-world scenarios.

In learning statistical mechanics, there are three statistical distributions (though in strict mathematical terms they are merely functions) that are essential to learn:

• Maxwell-Boltzmann
• Fermi-Dirac
• Bose-Einstein

This topic page is for organizing the development of Statistical mechanics content on Wikiversity.

If you are knowledgeable in any area Statistical mechanics, feel free to improve upon what you see, we would greatly appreciate your contributions.

## Maxwell-Boltzman Statistics

The Maxwell-Boltzman distribution is given by:

${\displaystyle f_{MB}(\epsilon )=Ae^{-{\frac {\epsilon }{kT}}}}$

where:

 ${\displaystyle f_{MB}}$ means function of Maxwell-Boltzman ${\displaystyle \epsilon }$ is the state of energy in a system of particles ${\displaystyle e}$ is the natural base ${\displaystyle k}$ is Boltzmann's constant 1.381*10-23J/K or 8.617*10-5eV/K ${\displaystyle T}$ is the absolute temperature of the system

Maxwell Boltzmann Statistics apply to systems where individual particles are distinguishable from each other, and identical to each other. The classical mechanics and all classical particles behave the rules of this distribution. This distribution can be used to solve for the ratio of atoms (or average number of atoms if a total number of atoms is given) in a given energy state.

## Fermi-Dirac Statistics

The distribution for Fermi-Dirac systems is given by:

${\displaystyle f_{FD}(\epsilon )={\frac {1}{e^{\alpha }e^{\frac {\epsilon }{kT}}+1}}}$

where:

 ${\displaystyle f_{FD}}$ means function of Fermi-Dirac ${\displaystyle \epsilon }$ is the state of energy in a system of particles ${\displaystyle e}$ is the natural base ${\displaystyle k}$ is Boltzmann's constant1.381*10-23J/K or 8.617*10-5eV/K  ${\displaystyle T}$ is the absolute temperature of the system ${\displaystyle \alpha }$ is an arbitrary value based on the number of particles in a system and normalizing their energy states

Fermi-Dirac statistics describes that of electrons, and other sub-atomic particles that exhibit a quantum "spin" of ${\displaystyle {\frac {2N+1}{2}}}$

## Bose-Einstein Statistics

The distribution for Bose-Einstein systems is given by:

${\displaystyle f_{BE}(\epsilon )={\frac {1}{e^{\alpha }e^{\frac {\epsilon }{kT}}-1}}}$

where:

 ${\displaystyle f_{BE}}$ means function of Bose-Einstein ${\displaystyle \epsilon }$ is the state of energy in a system of particles ${\displaystyle e}$ is the natural base ${\displaystyle k}$ is Boltzmann's constant1.381*10-23J/K or 8.617*10-5eV/K  ${\displaystyle T}$ is the absolute temperature of the system ${\displaystyle \alpha }$ is an arbitrary value based on the number of particles in a system and normalizing their energy states

Bose-Einstein statistics describes that of matter which exhibit an integer value of quantum "spin" (0,1,2,...)

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