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Boltzmann's distribution is an exponential distribution. Boltzmann factor (vertical axis) as a function of temperature T for several energy differences ε i − ε j.. In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution [1]) is a probability distribution or probability measure that gives the probability that a system will be in a certain ...
Using the equipartition theorem, given that the energy is evenly distributed among all three degrees of freedom in equilibrium, we can also split () into a set of chi-squared distributions, where the energy per degree of freedom, ε is distributed as a chi-squared distribution with one degree of freedom, [13] = ()
As an example: the partition function for the isothermal-isobaric ensemble, the generalized Boltzmann distribution, divides up probabilities based on particle number, pressure, and temperature. The energy is replaced by the characteristic potential of that ensemble, the Gibbs Free Energy.
The Boltzmann distribution itself is one of the most important tools in applying statistical mechanics to real systems, as it massively simplifies the study of systems that can be separated into independent parts (e.g., particles in a gas, electromagnetic modes in a cavity, molecular bonds in a polymer).
Boltzmann also extended his theory in his 1877 paper beyond Carnot, Rudolf Clausius, James Clerk Maxwell and Lord Kelvin by demonstrating that entropy is contributed to by heat, spatial separation, and radiation. [27] Maxwell–Boltzmann statistics and the Boltzmann distribution remain central in the foundations of classical statistical mechanics.
Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics we use the Thomas–Fermi approximation (gas in a box) and go to the limit of a very large trap, and express the degeneracy of the energy states as a differential, and summations over states as integrals.
A four-level laser energy diagram. Here, there are four energy levels, energies E 1, E 2, E 3, E 4, and populations N 1, N 2, N 3, N 4, respectively. The energies of each level are such that E 1 < E 2 < E 3 < E 4. In this system, the pumping transition P excites the atoms in the ground state (level 1) into the pump band (level 4).
Total emitted energy, , of a black body as a function of its temperature, .The upper (black) curve depicts the Stefan–Boltzmann law, =.The lower (blue) curve is total energy according to the Wien approximation, = / ()