Phase of matter in the context of "Chemical stability"

An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin and abide by Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas and extended to massive particles by Albert Einstein, who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate.

In physics, topological order describes a state or phase of matter that arises in a system with non-local interactions, such as entanglement in quantum mechanics, and floppy modes in elastic systems. Whereas classical phases of matter such as gases and solids correspond to microscopic patterns in the spatial arrangement of particles arising from short range interactions, topological orders correspond to patterns of long-range quantum entanglement. States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.

Technically, topological order occurs at zero temperature. Various topologically ordered states have interesting properties, such as (1) ground state degeneracy and fractional statistics or non-abelian group statistics that can be used to realize a topological quantum computer; (2) perfect conducting edge states that may have important device applications; (3) emergent gauge field and Fermi statistics that suggest a quantum information origin of elementary particles; (4) topological entanglement entropy that reveals the entanglement origin of topological order, etc. Topological order is important in the study of several physical systems such as spin liquids, and the quantum Hall effect, along with potential applications to fault-tolerant quantum computation.

In chemistry and chemical physics, a mesophase or mesomorphic phase is a phase of matter intermediate between solid and liquid. Gelatin is a common example of a partially ordered structure in a mesophase. Further, biological structures such as the lipid bilayers of cell membranes are examples of mesophases. Mobile ions in mesophases are either orientationally or rotationally disordered while their centers are located at the ordered sites in the crystal structure. Mesophases with long-range positional order but no orientational order are plastic crystals, whereas those with long-range orientational order but only partial or no positional order are liquid crystals.

Georges Friedel (1922) called attention to the "mesomorphic states of matter" in his scientific assessment of observations of the so-called liquid crystals. Conventionally a crystal is solid, and crystallization converts liquid to solid. The oxymoron of the liquid crystal is resolved through the notion of mesophases. The observations noted an optic axis persisting in materials that had been melted and had begun to flow. The term liquid crystal persists as a colloquialism, but use of the term was criticized in 1993: In The Physics of Liquid Crystals the mesophases are introduced from the beginning: