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# Statistical Mechanics

Statistical mechanics is a branch of physics which applies the probability theory to the study of thermodynamic behaviors of a system with a large number of particles. It provides a framework for relating the microscopic properties of atoms to the macroscopic overall properties of the material. Therefore, statistical mechanics is very useful when explaining thermodynamic properties as a result of classical and quantum mechanical descriptions of statistical and mechanics at a microscopic level.

Statistical mechanics gives a molecular-level interpretation of macroscopic thermodynamic properties such as free energy, entropy, work and heat. The bulk material thermodynamic properties can be studied with regards to the individual molecules. The two central quantities in statistical thermodynamics are the Boltzmann factor and the partition function.

Boltzmann factor
P(E_i )= 1/Z exp^(-βE_i)

Where

β= 1/(k_B T)

Partition Function

Z= ∑_s e^(-βE_s )

Where

β= 1/(k_B T)

These equations are the basic equations of statistical mechanics. These equations were derived in 1870 when studying the Gas Theory by Ludwig Boltzmann. The fundamental postulate for statistical mechanics is

“Given an isolated system in equilibrium, it is found with equal probability in each of its accessible microstates. “ – [1]

This postulate is important due to its ability to allow one to conclude that for a system at equilibrium, the macrostate could result from the largest number of microstates which is also the most probable macrostate of the system.

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