Search results
Results From The WOW.Com Content Network
Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions.
These three equations, along with the free energy in terms of the partition function, = , allow an efficient way of calculating thermodynamic variables of interest given the partition function and are often used in density of state calculations. One can also do Legendre transformations for different systems. For example, for a system with a ...
The Helmholtz free energy has a special theoretical importance since it is proportional to the logarithm of the partition function for the canonical ensemble in statistical mechanics. (Hence its utility to physicists ; and to gas-phase chemists and engineers, who do not want to ignore p dV work.)
The equations below (in terms of free energy) may be restated in terms of the canonical partition function by simple mathematical manipulations. Historically, the canonical ensemble was first described by Boltzmann (who called it a holode ) in 1884 in a relatively unknown paper. [ 2 ]
The partition function and the free energy are intimately linked to phase transitions, for which there is a sudden change in the properties of a physical system. Mathematically, a phase transition occurs when the partition function vanishes and the free energy is singular (non-analytic).
Given a set of random variables taking on values , and some sort of potential function or Hamiltonian (,, …), the partition function is defined as = ((,, …)) The function H is understood to be a real-valued function on the space of states {,,}, while is a real-valued free parameter (conventionally, the inverse temperature).
The partition function for the ... From the preceding analysis it is clear that the characteristic state function of this ensemble is the Gibbs free energy, ...
In particular, if the partition function P satisfies ... so its characteristic function is the Gibbs free energy. State functions are those which tell about the ...