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For an open thermodynamic system in which heat and work are transferred by paths separate from the paths for transfer of matter, using this generic balance equation, with respect to the rate of change with time of the extensive quantity entropy , the entropy balance equation is: [54] [55] [note 1] = = ˙ ^ + ˙ + ˙ where = ˙ ^ is the net rate ...
When these probabilities are substituted into the above expression for the Gibbs entropy (or equivalently k B times the Shannon entropy), Boltzmann's equation results. In information theoretic terms, the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate.
The von Neumann entropy formula is an extension of the Gibbs entropy formula to the quantum mechanical case. It has been shown [ 1 ] that the Gibbs Entropy is equal to the classical "heat engine" entropy characterized by d S = δ Q T {\displaystyle dS={\frac {\delta Q}{T}}\!} , and the generalized Boltzmann distribution is a sufficient and ...
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 connection between thermodynamic entropy and information entropy is given by Boltzmann's equation, which says that S = k B ln W. If we take the base-2 logarithm of W, it will yield the average number of questions we must ask about the microstate of the physical system in order to determine its macrostate. [13]
As the entropy is a function of state the result is independent of the path. The above relation shows that the determination of the entropy requires knowledge of the heat capacity and the equation of state (which is the relation between P,V, and T of the substance involved). Normally these are complicated functions and numerical integration is ...
The Sackur–Tetrode equation is an expression for the entropy of a monatomic ideal gas. [1]It is named for Hugo Martin Tetrode [2] (1895–1931) and Otto Sackur [3] (1880–1914), who developed it independently as a solution of Boltzmann's gas statistics and entropy equations, at about the same time in 1912.
Thus the definitions of entropy in statistical mechanics (The Gibbs entropy formula = ) and in classical thermodynamics (=, and the fundamental thermodynamic relation) are equivalent for microcanonical ensemble, and statistical ensembles describing a thermodynamic system in equilibrium with a reservoir, such as the canonical ensemble, grand ...