Boltzmann

The Boltzmann distribution is named after Ludwig Boltzmann who first formulated it in 1868 during his studies of the statistical mechanics of gases in thermal equilibrium.

The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known as Maxwell–Boltzmann statistics (from statistical thermodynamics).

The iconic terse form of the equation on Boltzmann's tombstone is in fact due to Planck, not Boltzmann.

The ratio of a Boltzmann distribution computed for two states is known as the Boltzmann factor and characteristically only depends on the states' energy difference.

The Stefan–Boltzmann law gives the power emitted per unit area of the emitting body, :: main To derive the Stefan–Boltzmann law, we must integrate Ω over the half-sphere and integrate ν from 0 to ∞.

Kinetic theory of gases, average kinetic energy of translation, Boltzmann constant and absolute scale of temperature.

As a result, the Boltzmann factor for states of systems at negative temperature increases rather than decreases with increasing state energy.

Because of this Boltzmann distribution can be used to solve a very wide variety of problems.

Boltzmann argued that by averaging the behaviors of all the different molecules in a gas, one can understand macroscopic properties such as volume, temperature, and pressure.

Boltzmann considered a certain ideal heat engine with light as a working matter instead of gas.

Boltzmann later carried out significant investigations into the physical origins of this distribution.

Boltzmann treated a heat engine with light as a working matter.

Consistent with the Boltzmann definition, the second law of thermodynamics needs to be re-worded as such that entropy increases over time, though the underlying principle remains the same.

From Planck's constant h and the Boltzmann constant k, Wien's constant b can be obtained.

In 1877 Austrian physicist Ludwig Boltzmann described it more precisely in terms of the number of distinct microscopic states that the particles composing a macroscopic "chunk" of matter could be in, while still looking like the same macroscopic "chunk".

In 1877 Boltzmann visualized a probabilistic way to measure the entropy of an ensemble of ideal gas particles, in which he defined entropy to be proportional to the logarithm of the number of microstates such a gas could occupy.

In accordance with this philosophy, Mach opposed Ludwig Boltzmann and others who proposed an atomic theory of physics.

In Boltzmann's 1896 Lectures on Gas Theory, he showed that this expression gives a measure of entropy for systems of atoms and molecules in the gas phase, thus providing a measure for the entropy of classical thermodynamics.

In economics The Boltzmann distribution can be introduced to allocate permits in emissions trading.

In November 1900, Planck revised this first approach, relying on Boltzmann 's statistical interpretation of the second law of thermodynamics as a way of gaining a more fundamental understanding of the principles behind his radiation law.