Boltzmann's entropy formula

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: where is the Boltzmann constant (also written as simply ) and equal to 1.380649 × 10−23 J/K, and is the natural logarithm function (also written as , as in the image above). In short, the Boltzmann formula shows the relationship between entropy and the number of ways the atoms or molecules of a certain kind of thermodynamic system can be arranged. The equation was originally formulated by Ludwig Boltzmann between 1872 and 1875, but later put into its current form by Max Planck in about 1900. To quote Planck, "the logarithmic connection between entropy and probability was first stated by L. Boltzmann in his kinetic theory of gases". A 'microstate' is a state specified in terms of the constituent particles of a body of matter or radiation that has been specified as a macrostate in terms of such variables as internal energy and pressure. A macrostate is experimentally observable, with at least a finite extent in spacetime. A microstate can be instantaneous, or can be a trajectory composed of a temporal progression of instantaneous microstates. In experimental practice, such are scarcely observable. The present account concerns instantaneous microstates. The value of W was originally intended to be proportional to the Wahrscheinlichkeit (the German word for probability) of a macroscopic state for some probability distribution of possible microstates—the collection of (unobservable microscopic single particle) "ways" in which the (observable macroscopic) thermodynamic state of a system can be realized by assigning different positions and momenta to the respective molecules. There are many instantaneous microstates that apply to a given macrostate. Boltzmann considered collections of such microstates.

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