where is the Boltzmann constant, T is temperature ( assumed to be a well-defined quantity ), is the degeneracy ( meaning, the number of levels having energy ; sometimes, the more general ' states ' are used instead of levels, to avoid using degeneracy in the equation ), N is the total number of particles and Z ( T ) is the partition function.

The plasma pressure can be calculated by the state equation of a perfect gas, where is the particle number density, the Boltzmann constant and the plasma temperature.

where is a constant resulting from the numerical solution of the maximization equation, k is the Boltzmann constant, h is the Planck constant, and T is the temperature ( in kelvins ).

* 1872 – Ludwig Boltzmann states the Boltzmann equation for the temporal development of distribution functions in phase space, and publishes his H-theorem

* 1978 – Peter Goldreich and Scott Tremaine present a Boltzmann equation model of planetary-ring dynamics for indestructible spherical ring particles that do not self-gravitate and find a stability requirement relation between ring optical depth and particle normal restitution coefficient

A special form of the Boltzmann equation

Using the relaxation time approximation for the Boltzmann equation and assuming steady-state conditions, the phonon thermal conductivity λ < sub > L </ sub > can be determined.

More modern developments relax these assumptions and are based on the Boltzmann equation.

The results of the quantum Boltzmann gas are used in a number of cases including the Sackur-Tetrode equation for the entropy of an ideal gas and the Saha ionization equation for a weakly ionized plasma.

* Boltzmann transport equation

If the charge density follows a Boltzmann distribution, then the Poisson-Boltzmann equation results.

The Poisson – Boltzmann equation plays a role in the development of the Debye – Hückel theory of dilute electrolyte solutions.

* Poisson – Boltzmann equation

* 1872-Ludwig Boltzmann states the Boltzmann equation for the temporal development of distribution functions in phase space, and publishes his H-theorem.

This is similar to the Vlasov equation, or the collisionless Boltzmann equation, in astrophysics.

The stationary states equation is satisfied by equal to any function of the Hamiltonian: in particular, it is satisfied by the Maxwell-Boltzmann distribution, where is the temperature and the Boltzmann constant.

Its application to dilute gases is called the Boltzmann equation.

The general theoretical description of a Langmuir probe measurement requires the simultaneous solution of the Poisson equation, the collision-free Boltzmann equation, and the continuity equation with regard to the boundary condition at the probe surface and requiring that, at large distances from the probe, the solution approaches that expected in an undisturbed plasma.

The model for the plasma bulk is based on 2d-fluid model ( zero and first order moments of Boltzmann equation ) and the full set of the Maxwellian equations leading to the Helmholtz equation for the magnetic field.

The Boltzmann distribution applies only to particles at a high enough temperature and low enough density that quantum effects can be ignored, and the particles are obeying Maxwell – Boltzmann statistics.

This effect can be illustrated with a Boltzmann distribution and energy profile diagram.

By treating the star as an idealized energy radiator known as a black body, the luminosity L and radius R can be related to the effective temperature by the Stefan – Boltzmann law:

If the group of atoms is in thermal equilibrium, it can be shown from thermodynamics that the ratio of the number of atoms in each state is given by a Boltzmann distribution:

This was followed by the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system can be discrete, and the 1900 quantum hypothesis of Max Planck.

The history of quantum chemistry essentially began with the 1838 discovery of cathode rays by Michael Faraday, the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by Max Planck that any energy radiating atomic system can theoretically be divided into a number of discrete energy elements ε such that each of these energy elements is proportional to the frequency ν with which they each individually radiate energy and a numerical value called Planck ’ s Constant.

It can be defined in terms of the Boltzmann constant as:

We can model the electrons at the sheath edge with a Boltzmann distribution, i. e.,

The classical ideal gas can be separated into two types: The classical thermodynamic ideal gas and the ideal quantum Boltzmann gas.

The ratio of oxidized to reduced molecules, /, is equivalent to the probability of being oxidized ( giving electrons ) over the probability of being reduced ( taking electrons ), which we can write in terms of the Boltzmann factors for these processes:

Speed distribution can be derived from Maxwell – Boltzmann distribution.

The thermionic emission current density, J, can be related to the work function of the emitting material and is a Boltzmann distribution given below, where A is a constant, Φ is the work function and T is the temperature of the material.

Since at temperature T the molecules have energies given by a Boltzmann distribution, one can expect the number of collisions with energy greater than E < sub > a </ sub > to be proportional to.

Because it can see the radiating temperature of an object as well as what that object is radiating at, the product of the radiation can be calculated using the Stefan – Boltzmann constant.

A collection of molecules forming a gas can be considered in an excited state if one or more molecules are elevated to kinetic energy levels such that the resulting velocity distribution departs from the equilibrium Boltzmann distribution.

Relative populations of rotational energy levels can be determined from Boltzmann factors and hence depend on the temperature of a sample.

In 1922, for example, Alfred J. Lotka referred to Boltzmann as one of the first proponents of the proposition that available energy can be understood as the fundamental object under contention in the biological, or life-struggle and therefore also in the evolution of the organic world.

It is also possible to write down relativistic Boltzmann equations for systems in which a number of particle species can collide and produce different species.

The Boltzmann distribution describes a system that can exchange energy with a heat bath ( or alternatively with a large number of similar systems ) so that its temperature remains constant.

If the system is large the Boltzmann distribution could be used ( the Boltzmann distribution is an approximate result )

Yet, one must consider Boltzmann to be the " father " of statistical thermodynamics with his 1875 derivation of the relationship between entropy S and multiplicity Ω, the number of microscopic arrangements ( microstates ) producing the same macroscopic state ( macrostate ) for a particular system.

Using the Boltzmann expression for the mean electron velocity given above with and setting the ion current to zero, the electron saturation current density would be

Classically, the ratio of probabilities that two states with an energy difference ΔE will be occupied by an electron is given by the Boltzmann factor:

* the right hand side represents the outgoing energy from the Earth, calculated from the Stefan – Boltzmann law assuming a constant radiative temperature, T, that is to be found,

He for the first time took the position that the Maxwell – Boltzmann distribution would not be true for microscopic particles where fluctuations due to Heisenberg's uncertainty principle will be significant.

Following the same procedure used in deriving the Maxwell – Boltzmann statistics, we wish to find the set of for which W is maximised, subject to the constraint that there be a fixed total number of particles, and a fixed total energy.