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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 ...
The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan , who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.
The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity , thermal conductivity , and electrical conductivity (by treating the charge carriers in a material as a gas). [ 2 ]
Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k, and gave a more precise value for it (1.346 × 10 −23 J/K, about 2.5% lower than today's figure), in his derivation of the law of black-body radiation in 1900–1901. [11]
A Markov process is called a reversible Markov process or reversible Markov chain if there exists a positive stationary distribution π that satisfies the detailed balance equations [12] =, where P ij is the Markov transition probability from state i to state j, i.e. P ij = P(X t = j | X t − 1 = i), and π i and π j are the equilibrium probabilities of being in states i and j, respectively ...
Despite the foregoing, there is a difference between the two quantities. The information entropy Η can be calculated for any probability distribution (if the "message" is taken to be that the event i which had probability p i occurred, out of the space of the events possible), while the thermodynamic entropy S refers to thermodynamic probabilities p i specifically.
Boltzmann tried for many years to "prove" the second law of thermodynamics using his gas-dynamical equation – his famous H-theorem. However the key assumption he made in formulating the collision term was " molecular chaos ", an assumption which breaks time-reversal symmetry as is necessary for anything which could imply the second law.
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