Statistical analogues

This formulation provides the microcanonical statistical analogue e-4> 7 of the empirical temperature. The width of the microcanonical energy distribution... 

Equation (37) is the quantum statistical analogue of Liouville s equation. To find the quantum analogue of the classical principle of conservation of phase density the solution to (37) is written in the form... 

As in the classical expression (25) the quantity ip can be inferred directly as representing the statistical analogue of the Helmholtz free energy. The average behaviour of the canonical ensemble thus obeys the laws of thermodynamics. 

The statistical analogue of entropy for the canonical ensemble follows as S = —kfj. Since fj = trace rje 1, the entropy expression becomes... 

In equilibrium and nonequilibrium thermodynamics, the spatial organization of a system is reflected in the configurational entropy. As has been discussed by Denbigh [113] and Jaynes [114], there are several statistical analogues of the entropy, each of which has the desired thermodynamic properties. For example, the function 5, defined by... 

Statistical analogues of the entropy and Helmholtz free energy... 

The same result may be obtained, for a variation between any other pair of quantum states. The canonical distribution (11 24), which we have derived on strictly statistical grounds with the help of our basic postulate, therefore results in the function A being at a minimumf when a body of fixed volume is at equilibrium in a thermostat. The thermodynamic function which has this property is the Helmholtz free energy A. It seems probable, therefore, that A as previously defined, is a statistical analogue of the free energy of thermodynamics. Further grounds for this will be discussed in the next section. 

Comparison of statistical analogues with thermodynamic functions... 

We thus conclude that 8 satisfies all the requirements of the thermo-d3Hiamic entropy. Of course the latter is only defined as a change of entropy, and therefore any statistical analogue which purports to give the absolute value of entropy, as in the case of 8 is satisfactory provided that its differential obeys (11 40). 

It was shown subsequently that 8 and 8" possess all of the properties of entropy and the primes were therefore deleted. On the other hand, since thermodynamics deals only with changes of entropy, it is clear that the statistical analogues would have been equally satisfactory if they had contained an additive constant. Let us write such a constant in the form —i( ln Then the entropy analogue... 

F-value is sometimes refereed to as F-ratio, used to test the significance of factor effects. It is statistically analogue to Taguchi s signal-to-noise ratio for control factor effect vs. the experimental error. The F-ratio uses information based on sample variances (mean squares) to define the relationship between the power of the control factor effects (a type of signal) and the power of the experimental error (a type of noise) (Fowlkes and Creveling (1995)). 

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