Convex hull: Difference between revisions

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In [[quantum physics]], the [[state space]] of any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points are [[positive-semidefinite matrix|positive-semidefinite operators]] known as pure states and whose interior points are called mixed states.{{sfnp|Rieffel|Polak|2011}} The [[Schrödinger–HJW theorem]] proves that any mixed state can in fact be written as a convex combination of pure states in multiple ways.{{sfnp|Kirkpatrick|2006}}

=== Materials science Thermodynamics===

[[File:Mg–C convex hull.png|thumb|Convex hull of [[magnesium]]–[[carbon]] compounds.{{sfnp|Kim|Kim|Koo|Lee|2019}} Mg₂C₃ is expected to be unstable as it lies above the lower hull.]]

A convex hull in [[thermodynamics]] was identified by [[Josiah Willard Gibbs]] (1873),{{sfnp|Gibbs|1873}} although the paper was published before the convex hull was so named.

In a set of energies of several [[Stoichiometry|stoichiometries]] of a material, only those measurements on the lower convex hull will be stable. When removing a point from the hull and then calculating its distance to the hull, its distance to the new hull represents the degree of stability of the phase.<ref>{{harvtxt|Hautier|2014}}; {{harvtxt|Fultz|2020}}</ref>

===Thermodynamics===

A convex hull in [[thermodynamics]] was identified by [[Josiah Willard Gibbs]] (1873),{{sfnp|Gibbs|1873}} although the paper was published before the convex hull was so named. For a collection of a single species of molecules, the average energy, volume, and [[entropy]] per molecule form a three-dimensional [[real vector space]], in which a homogeneous state of the collection of molecules is a point. Because energy, volume, and entropy are [[extensive properties]], a mixture of homogeneous states is a [[convex combination]] of homogeneous states, with each coefficient being the number fraction of molecules in a homogeneous state. All mixtures constitute the convex hull of homogeneous states. Mixtures also constitute the convex hull of homogeneous states in similar spaces, including the plane of [[chemical composition]] and [[Helmholtz free energy]].{{sfnp|Kittel|Kroemer|1980}}

==History==