In thermodynamics, specifically in statistical mechanics, the Gibbs entropy formula is the standard formula for calculating the statistical mechanical entropy of a thermodynamic system,

$S = -k_B \sum_i p_i \ln p_i \,$ (1)

where the summation is taken over the possible states of the system as a whole (typically a 6N-dimensional space, if the system contains N separate particles). In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Thermodynamics, a thermodynamic system, originally called a working substance, is defined as that part of the universe that is under consideration An overestimation of entropy will occur if all correlations, and more generally if statistical dependence between the state probabilities are ignored. In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other These correlations occur in systems of interacting particles, that is, in all systems more complex than an ideal gas. These four properties that constitute an ideal gas can be easily remembered by the acronym RIPE which stands for - R andom Motion (molecules are in constant random motion

The Shannon entropy formula is mathematically and conceptually equivalent to equation (1); the factor of $k B$ out front reflects two facts: our choice of base for the logarithm, ^[1] and our use of an arbitrary temperature scale with water as a reference substance.

The importance of this formula is discussed at much greater length in the main article Entropy (thermodynamics). In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy

This S is almost universally called simply the entropy. It can also be called the statistical entropy or the thermodynamic entropy without changing the meaning. The Von Neumann entropy formula is a slightly more general way of calculating the same thing. In Quantum statistical mechanics, von Neumann entropy refers to the extension of classical Entropy concepts to the field of Quantum mechanics. The Boltzmann entropy formula can be seen as a corollary of equation (1), valid under certain restrictive conditions of no statistical dependence between the states. In Statistical thermodynamics, Boltzmann's equation is a probability equation relating the Entropy S of an ideal gas to the quantity W, which ^[2]

References

^ The Laws of Thermodynamics including careful definitions of energy, entropy, et cetera. Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy In Quantum statistical mechanics, von Neumann entropy refers to the extension of classical Entropy concepts to the field of Quantum mechanics. In Statistical thermodynamics, Boltzmann's equation is a probability equation relating the Entropy S of an ideal gas to the quantity W, which
^ Jaynes, E. T. (1965). Edwin Thompson Jaynes ( July 5, 1922 &ndash April 30, 1998) was Wayman Crow Distinguished Professor of Physics at Washington University Gibbs vs Boltzmann entropies. American Journal of Physics, 33, 391-8.

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