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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:
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.
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} .
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.
(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.
N i is the expected number of particles in the single-particle microstate i, N is the total number of particles in the system, E i is the energy of microstate i, the sum over index j takes into account all microstates, T is the equilibrium temperature of the system, k B is the Boltzmann constant.