What is "critical point"

critical point \crit"ic*al point\ (Physics), That combination of volume and pressure, at the critical temperature of the substance, at which the liquid and gaseous phases of a given quantity of a substance have identical values for their densities and other properties.

n. 1 (context thermodynamics English) The temperature and pressure at which the vapour density of the gas and liquid phases of a fluid are equal, at which point there is no difference between gas and liquid. 2 (context mathematics English) A maximum, minimum or point of inflection on a curve; a point at which the derivative of a function is zero or undefined. 3 A juncture at which time a critical decision must be made.

n. a crisis situation or point in time when a critical decision must be made; "at that juncture he had no idea what to do"; "he must be made to realize that the company stands at a critical point" [syn: juncture, crossroads]

Critical point may refer to:

• Critical phenomena in physics
• Critical point (mathematics), in calculus, the points of an equation where the derivative is zero
• Critical point (set theory), an elementary embedding of a transitive class into another transitive class which is the smallest ordinal which is not mapped to itself
• Critical point (thermodynamics), a temperature and pressure of a material beyond which there is no longer any difference between the liquid and gas phases
• Critical point (network science)
• Construction point, in skiing, a line the represents the steepest point on a hill

In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent example is the liquid-vapor critical point, the end point of the pressure-temperature curve that designates conditions under which a liquid and its vapor can coexist. At the critical point, defined by a critical temperature T and a critical pressure p, phase boundaries vanish. Other examples include the liquid–liquid critical points in mixtures.

In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0 or undefined. For a differentiable function of several real variables, a critical point is a value in its domain where all partial derivatives are zero. The value of the function at a critical point is a critical value.

The interest of this notion lies in the fact that the points where the function has local extrema are critical points.

This definition extends to differentiable maps between R and R, a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called bifurcation points.

In particular, if C is a plane curve, defined by an implicit equation f(x,y) = 0, the critical points of the projection onto the x-axis, parallel to the y-axis are the points where the tangent to C are parallel to the y-axis, that is the points where $\frac{\partial f}{\partial y}(x,y)=0.$ In other words, the critical points are those where the implicit function theorem does not apply.

The notion of a critical point allows the mathematical description of an astronomical phenomenon that was unexplained before the time of Copernicus. A stationary point in the orbit of a planet is a point of the trajectory of the planet on the celestial sphere, where the motion of the planet seems to stop before restarting in the other direction. This occurs because of a critical point of the projection of the orbit into the ecliptic circle.

In network science, a critical point is a value of average degree, which separates random networks that have a giant component from those that do not (i.e. it separates a network in a subcritical regime from one in a supercritical regime). Considering a random network with an average degree  < k >  the critical point is

 < k >  = 1

where the average degree is defined by the fraction of the number of edges (e) and nodes (N) in the network, that is $<k>=\frac{e}{N}$.

In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself.

Suppose that j : N → M is an elementary embedding where N and M are transitive classes and j is definable in N by a formula of set theory with parameters from N. Then j must take ordinals to ordinals and j must be strictly increasing. Also j(ω)=ω. If j(α)=α for all ακ, then κ is said to be the critical point of j.

If N is V, then κ (the critical point of j) is always a measurable cardinal, i.e. an uncountable cardinal number κ such that there exists a \{A \vert A \subseteq \kappa \land \kappa \in j (A) \} \,. Generally, there will be many other <κ-complete, non-principal ultrafilters over κ. However, j might be different from the ultrapower(s) arising from such filter(s).

If N and M are the same and j is the identity function on N, then j is called "trivial". If transitive class N is an inner model of ZFC and j has no critical point, i.e. every ordinal maps to itself, then j is trivial.