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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.
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:
Ludwig Boltzmann defined entropy as a measure of the number of possible microscopic states (microstates) of a system in thermodynamic equilibrium, consistent with its macroscopic thermodynamic properties, which constitute the macrostate of the system. A useful illustration is the example of a sample of gas contained in a container.
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.}
According to the fundamental postulate of statistical mechanics (which states that all attainable microstates of a system are equally probable), the probability p i will be inversely proportional to the number of microstates of the total closed system (S, B) in which S is in microstate i with energy E i.
If all the microstates are equiprobable (a microcanonical ensemble), the statistical thermodynamic entropy reduces to the form, as given by Boltzmann, = , where W is the number of microstates that corresponds to the macroscopic thermodynamic state. Therefore S depends on temperature.
where S is the entropy of the system, k B is the Boltzmann constant, and Ω the number of microstates. At absolute zero there is only 1 microstate possible ( Ω = 1 as all the atoms are identical for a pure substance, and as a result all orders are identical as there is only one combination) and ln ( 1 ) = 0 {\displaystyle \ln(1)=0} .
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.