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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:
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.
The collisionless Boltzmann equation, where individual collisions are replaced with long-range aggregated interactions, e.g. Coulomb interactions, is often called the Vlasov equation. This equation is more useful than the principal one above, yet still incomplete, since f cannot be solved unless the collision term in f is known.
Furthermore, the prescription to find the equilibrium distributions of statistical mechanics—such as the Boltzmann distribution—by maximising the Gibbs entropy subject to appropriate constraints (the Gibbs algorithm) can be seen as something not unique to thermodynamics, but as a principle of general relevance in statistical inference, if ...
The H-theorem is a natural consequence of the kinetic equation derived by Boltzmann that has come to be known as Boltzmann's equation. The H-theorem has led to considerable discussion about its actual implications, [6] with major themes being: What is entropy? In what sense does Boltzmann's quantity H correspond to the thermodynamic entropy?
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: [53] [54] [note 1] = = ˙ ^ + ˙ + ˙ where = ˙ ^ is the net rate ...
Boltzmann went beyond Maxwell by applying his distribution equation to not solely gases, but also liquids and solids. Boltzmann also extended his theory in his 1877 paper beyond Carnot, Rudolf Clausius , James Clerk Maxwell and Lord Kelvin by demonstrating that entropy is contributed to by heat, spatial separation, and radiation. [ 27 ]
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]