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Boltzmann's equation—carved on his gravestone. [1]In statistical mechanics, Boltzmann's equation (also known as the Boltzmann–Planck equation) is a probability equation relating the entropy, also written as , of an ideal gas to the multiplicity (commonly denoted as or ), the number of real microstates corresponding to the gas's macrostate:
A macrostate is characterized by a probability distribution of possible states across a certain statistical ensemble of all microstates. This distribution describes the probability of finding the system in a certain microstate.
The ensemble of microstates comprises a statistical distribution of probability for each microstate, and the group of most probable configurations accounts for the macroscopic state. Therefore, the system can be described as a whole by only a few macroscopic parameters, called the thermodynamic variables : the total energy E , volume V ...
The distribution of the grand canonical ensemble is called generalized Boltzmann distribution by some authors. [ 2 ] Grand ensembles are apt for use when describing systems such as the electrons in a conductor , or the photons in a cavity, where the shape is fixed but the energy and number of particles can easily fluctuate due to contact with a ...
Hirschman showed, [17] cf. Hirschman uncertainty, that Heisenberg's uncertainty principle can be expressed as a particular lower bound on the sum of the classical distribution entropies of the quantum observable probability distributions of a quantum mechanical state, the square of the wave-function, in coordinate, and also momentum space, when ...
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applications include many problems in the fields of physics, biology, [1] chemistry, neuroscience, [2] computer science, [3] [4] information theory [5] and ...
Bayesian inference models often apply the principle of maximum entropy to obtain prior probability distributions. [35] The idea is that the distribution that best represents the current state of knowledge of a system is the one with the largest entropy, and is therefore suitable to be the prior.
In contrast to the macrostate, which characterizes plainly observable average quantities, a microstate specifies all molecular details about the system including the position and momentum of every molecule. The more such states are available to the system with appreciable probability, the greater the entropy.