Equation for entropy11/26/2023 "Über die Mechanische Bedeutung des Zweiten Hauptsatzes der Wärmetheorie". ^ Boltzmann equation – Eric Weisstein’s World of Physics (states the year was 1872).^ See: photo of Boltzmann's grave in the Zentralfriedhof, Vienna, with bust and entropy formula.This is exact for an ideal gas of identical particles, and may or may not be a good approximation for other systems. The term Boltzmann entropy is also sometimes used to indicate entropies calculated based on the approximation that the overall probability can be factored into an identical separate term for each particle - i.e., assuming each particle has an identical independent probability distribution, and ignoring interactions and correlations between the particles. In every situation where equation (1) is valid,Įquation (3) is valid also - and not vice versa.īoltzmann entropy excludes statistical dependencies That is, equation (1) is a corollary ofĮquation (3) - and not vice versa. Ds De T - Dv - + P - Dt Dt and the energy conservation equation. In this endeavor we will use the Gibbs equation on a fluid element. Gibbs gave an explicitly probabilistic interpretation in 1878.īoltzmann himself used an expression equivalent to (3) in his later work and recognized it as more general than equation (1). In this chapter the precise form of the flow properties that rep- resent the sources of entropy will be derived from first principles. He interpreted ρ as a density in phase space - without mentioning probability - but since this satisfies the axiomatic definition of a probability measure we can retrospectively interpret it as a probability anyway. This reduces to equation (1) if the probabilities p i are all equal.īoltzmann used a ρlogρ formula as early as 1866. The microstates of such a thermodynamic system are not equally probable - for example, high energy microstates are less probable than low energy microstates for a thermodynamic system kept at a fixed temperature by allowing contact with a heat bath.įor thermodynamic systems where microstates of the system may not have equal probabilities, the appropriate generalization, called the Gibbs entropy, is: W is sometimes called the "thermodynamic probability" since it is an integer greater than one, while mathematical probabilities are always numbers between zero and one.īoltzmann's formula applies to microstates of the universe as a whole, each possible microstate of which is presumed to be equally probable.īut in thermodynamics it is important to be able to make the approximation of dividing the universe into a system of interest, plus its surroundings and then to be able to identify the entropy of the system with the system entropy in Classical thermodynamics. The "correction" in the denominator is due to the fact that identical particles in the same condition are indistinguishable. Where i ranges over all possible molecular conditions and ! denotes factorial. W can be counted using the formula for permutations (2) Boltzmann’s paradigm was an ideal gas of N identical particles, of which N i are in the i-th microscopic condition (range) of position and momentum. The value of W, specifically, is the Wahrscheinlichkeit, or number of possible microstates corresponding to the macroscopic state of a system - number of (unobservable) "ways" the (observable) thermodynamic state of a system can be realized by assigning different positions and momenta to the various molecules. Boltzmann in his kinetic theory of gases." To quote Planck, "the logarithmic connection between entropy and probability was first stated by L. The equation was originally formulated by Ludwig Boltzmann between 1872 to 1875, but later put into its current form by Max Planck in about 1900.
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