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The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for
In either case, the partition function may be solved exactly using eigenanalysis. If the matrices are all the same matrix W , the partition function may be approximated as the N th power of the largest eigenvalue of W , since the trace is the sum of the eigenvalues and the eigenvalues of the product of two diagonal matrices equals the product ...
The partition function is commonly used as a probability-generating function for expectation values of various functions of the random variables. So, for example, taking β {\displaystyle \beta } as an adjustable parameter, then the derivative of log ( Z ( β ) ) {\displaystyle \log(Z(\beta ))} with respect to β {\displaystyle \beta }
In other words, each single-particle level is a separate, tiny grand canonical ensemble. By the Pauli exclusion principle, there are only two possible microstates for the single-particle level: no particle (energy E = 0), or one particle (energy E = ε). The resulting partition function for that single-particle level therefore has just two terms:
The denominator in equation 1 is a normalizing factor so that the ratios : add up to unity — in other words it is a kind of partition function (for the single-particle system, not the usual partition function of the entire system).
The terms in the bracket give the total partition function of the adsorbed molecules by taking a product of the individual partition functions (refer to Partition function of subsystems). The 1 / N A ! {\displaystyle 1/N_{A}!} factor accounts for the overcounting arising due to the indistinguishable nature of the adsorbates.
A polymer field theory is a statistical field theory describing the statistical behavior of a neutral or charged polymer system. It can be derived by transforming the partition function from its standard many-dimensional integral representation over the particle degrees of freedom in a functional integral representation over an auxiliary field function, using either the Hubbard–Stratonovich ...
Example of microcanonical ensemble for a classical system consisting of one particle in a potential well. Plot of all possible states of this system. The available physical states are evenly distributed in phase space, but with an uneven distribution in energy; the side-plot displays dv / dE .