Phase rule

Definitions

  • Webster's Revised Unabridged Dictionary
    • Phase rule (Phys. Chem) A generalization with regard to systems of chemical equilibrium, discovered by Prof. J. Willard Gibbs. It may be stated thus: The degree of variableness (number of degrees of freedom) of a system is equal to the number of components minus the number of phases, plus two. Thus, if the components be salt and water, and the phases salt, ice, saturated solution, and vapor, the system is invariant, that is, there is only one set of conditions under which these four phases can exist in equilibrium. If only three phases be considered, the system is univariant, that is, the fixing of one condition, as temperature, determines the others.
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Usage

In literature:

The ruling phase of passion now rumbles fiercely in lowest depths.
"Symphonies and Their Meaning; Third Series, Modern Symphonies" by Philip H. Goepp
The fortunes of the society had assumed a desperate phase when John Sheares became its ruling spirit.
"Speeches from the Dock, Part I" by Various
It takes nothing for granted, and gives sensible notes and rules for every phase of culinary work.
"Mr. Edward Arnold's New and Popular Books, December, 1901"
I have given phases of this one trade in detail, because the same general rules govern all.
"Prisoners of Poverty Abroad" by Helen Campbell
Until the period of the Civil War, competition was the generally accepted rule in all phases of economic life.
"The Next Step" by Scott Nearing
The earlier phases of Chivalry with its elaborate rules, its laws written and unwritten, were past long before Cressy.
"The Children of Westminster Abbey" by Rose G. Kingsley
According to the Phase Rule (p. 16), a system consisting of one component in two phases has one degree of freedom, or is univariant.
"The Phase Rule and Its Applications" by Alexander Findlay
This is so general a rule that I have not included that part of the statutes covering this phase of the law.
"How to Collect a Doctor Bill" by Frank P. Davis
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In news:

RULES OF ENGAGEMENT, a comedy about the different phases of male/female relationships, as seen the through the eyes of an engaged couple, a long-time married pair and a single guy on the prowl.
Los Angeles Angels organization in the major league phase of the Rule 5 draft.
Possibility Cannot Be Ruled Out Japan Already in Recessionary Phase.
Moon in dark phase, Gemini sign is ruling.
), an attorney who represented dozens of farmers in the lawsuit against the CTSP's fourth-phase expansion, adding that if executive powers are allowed to bully judicial powers, the nation will be ruled by the policies of the executive aut...
The Environmental Protection Agency, meanwhile, is slowly getting in on regulation and has proposed rules that will require all producers to phase out venting by 2015 and capture their waste methane instead.
Tibetan protests against Chinese rule in new phase.
The European Commission (EC) has adopted rules prohibiting transport of heavy fuel in single hull tankers and setting an earlier date for phase-out of single hull tankers.
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In science:

If we extend it to products of expectation values by using the Leibniz rule, we obtain a well-defined Poisson bracket on the quantum phase space, whose elements are states parameterized by their expectation values of ˆq and ˆp together with all moments. (The quantum phase space is symplectic.
Higher time derivatives in effective equations of canonical quantum systems
While the phase transition has received the most attention (, , ), rules for shifting the threshold of other properties have also been found (e.g.
Random k-SAT and the Power of Two Choices
These two phases are clearly inverses of each other, since Rules 1 and 5 of Tn imply Rule 2 of Sn and vice versa, and in corresponding sections of the tree and staircase.
Recursive bijections for Catalan objects
They introduce a probability assumption, that the a priori distribution of phases is uniform, and then show that, by their construction, this implies that the probabilities of outcomes obey the usual QM rule.
Probability in Orthodox Quantum Mechanics: Probability as a Postulate Versus Probability as an Emergent Phenomenon
As a result, the set of kinematical singularities has, as a rule, a very complicated positioning in the multidimensional phase space.
The QCD and Standard Model Working Group: Summary Report from Les Houches
But for the algebra of Toeplitz operators, and other related classes of operators, we establish a kind of a Gibbs phase rule .
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
Here, at least if the functions are sufficiently smooth, the boundaries between phases have a simple structure and the phase diagrams satisfy simple rules that have the flavor of Gibbs’ phase rule .
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
However, we consider the extent to which they follow the generic phase rules of (smooth, complex) Toeplitz operators anyway.
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
The phase diagram for the Euclidean plane, Fig. 2 satisfies the generic phase rules away from the line B = 0.
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
The phase diagram in the hyperbolic plane, Fig. 3, satisfies the generic phase rules outside the shaded parabolic region.
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
The phase diagram of the Harper model, Fig. 4, is in serious conflict with the phase rule for (smooth, complex) Toeplitz operators: It is known , that for a full measure of values of the magnetic field (irrational, of course), the spectrum is a Cantor set.
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
Figures 2 and 3 satisfy the phase rules in the region of large magnetic fields, but fail to do so for small magnetic fields.
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
We now examine how the two figures might be modified to satisfy the phase rules everywhere.
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
How do the phase diagrams, Figs. 6 and 7, compare with what one finds in experiments on the quantum Hall effect? For large magnetic fields one finds phase diagrams that resemble both Figs. 2 and 6 and satisfy the phase rules.
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
The emergence of an insulating phase (with index 0) for small magnetic fields is in agreement with the phase rule and Fig. 6.
Fredholm Indices and the Phase Diagram of Quantum Hall Systems
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