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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 number of microstates Ω that a closed system can occupy is proportional to its phase space volume: () = (()) = where (()) is an Indicator function. It is 1 if the Hamilton function H ( x ) at the point x = ( q , p ) in phase space is between U and U + δU and 0 if not.
The large number of particles of the gas provides an infinite number of possible microstates for the sample, but collectively they exhibit a well-defined average of configuration, which is exhibited as the macrostate of the system, to which each individual microstate contribution is negligibly small.
In what has been called the fundamental postulate in statistical mechanics, among system microstates of the same energy (i.e., degenerate microstates) each microstate is assumed to be populated with equal probability = /, where is the number of microstates whose energy equals to the one of the system.
If our system is in state , then there would be a corresponding number of microstates available to the reservoir. Call this number Ω R ( s 1 ) {\displaystyle \;\Omega _{R}(s_{1})} . By assumption, the combined system (of the system we are interested in and the reservoir) is isolated, so all microstates are equally probable.
The mathematical basis with respect to the association entropy has with order and disorder began, essentially, with the famous Boltzmann formula, = , which relates entropy S to the number of possible states W in which a system can be found. [13]
This approach shows that the number of available macrostates is N + 1. For example, in a very small system with N = 2 dipoles, there are three macrostates, corresponding to N ↑ = 0 , 1 , 2. {\displaystyle N_{\uparrow }=0,1,2.}
(a) Single possible configuration for a system at absolute zero, i.e., only one microstate is accessible. Thus S = k ln W = 0. (b) At temperatures greater than absolute zero, multiple microstates are accessible due to atomic vibration (exaggerated in the figure). Since the number of accessible microstates is greater than 1, S = k ln W > 0.