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In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows: =, where g j is the degeneracy factor, or number of quantum states s that have the same energy level defined by E j = E s.
is the average number of particles in the set of states with energy , g i {\displaystyle g_{i}} is the degeneracy of energy level i , that is, the number of states with energy ε i {\displaystyle \varepsilon _{i}} which may nevertheless be distinguished from each other by some other means, [ nb 1 ]
The free energy F serves two roles: first, it provides a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); second, many important ensemble averages can be directly calculated from the function F(N, V, T).
The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.
In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by: [1] ¯ = ().
Similarly, the spectral energy density of signal x(t) is = | | where X(f) is the Fourier transform of x(t).. For example, if x(t) represents the magnitude of the electric field component (in volts per meter) of an optical signal propagating through free space, then the dimensions of X(f) would become volt·seconds per meter and () would represent the signal's spectral energy density (in volts ...
The generalized version of the partition function provides the complete framework for working with ensemble averages in thermodynamics, information theory, statistical mechanics and quantum mechanics. The microcanonical ensemble represents an isolated system in which energy (E), volume (V) and the number of particles (N) are all
An average order of σ(n), the sum of divisors of n, is nπ 2 / 6; An average order of φ(n), Euler's totient function of n, is 6n / π 2; An average order of r(n), the number of ways of expressing n as a sum of two squares, is π; The average order of representations of a natural number as a sum of three squares is 4πn / 3;