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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:
The probability P s that the system occupies microstate s is =. Thus, as shown above, the partition function plays the role of a normalizing constant (note that it does not depend on s ), ensuring that the probabilities sum up to one: ∑ s P s = 1 Z ∑ s e − β E s = 1 Z Z = 1. {\displaystyle \sum _{s}P_{s}={\frac {1}{Z}}\sum _{s}e^{-\beta ...
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.
Treatments on statistical mechanics [2] [3] define a macrostate as follows: a particular set of values of energy, the number of particles, and the volume of an isolated thermodynamic system is said to specify a particular macrostate of it. In this description, microstates appear as different possible ways the system can achieve a particular ...
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 ...
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 ...
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 ...
In quantum mechanics, this notion, due to von Neumann, is a way of assigning a probability distribution over the results of each complete set of commuting observables. In classical mechanics, the ensemble is instead written as a probability distribution in phase space ; the microstates are the result of partitioning phase space into equal-sized ...