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A turning point of a differentiable function is a point at which the derivative has an isolated zero and changes sign at the point. [2] A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). A turning point is thus a stationary point, but not all stationary points are turning points. If ...
In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. One 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.
This sort of definition extends to differentiable maps between and , 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.
At the critical point, the order parameter susceptibility will usually diverge. An example of an order parameter is the net magnetization in a ferromagnetic system undergoing a phase transition. For liquid/gas transitions, the order parameter is the difference of the densities.
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; Quantum critical point ...
Solid carbon dioxide sublimes rapidly along the solid-gas boundary (sublimation point) below the triple point (e.g., at the temperature of −78.5 °C, at atmospheric pressure), whereas its melting into liquid CO 2 can occur along the solid-liquid boundary (melting point) at pressures and temperatures above the triple point (i.e., 5.1 atm, − ...
The critical point is described by a conformal field theory. According to the renormalization group theory, the defining property of criticality is that the characteristic length scale of the structure of the physical system, also known as the correlation length ξ, becomes infinite. This can happen along critical lines in phase space.
Infinite-period bifurcation in which a stable node and saddle point simultaneously occur on a limit cycle. [5] As the limit of a parameter approaches a certain critical value, the speed of the oscillation slows down and the period approaches infinity. The infinite-period bifurcation occurs at this critical value.