Phase space. Phase space of a dynamic system with focal instability, showing one phase space trajectory.
Phase diagram. Overview[edit] Common components of a phase diagram are lines of equilibrium or phase boundaries, which refer to lines that mark conditions under which multiple phases can coexist at equilibrium.
Phase transitions occur along lines of equilibrium. Triple points are points on phase diagrams where lines of equilibrium intersect. Triple points mark conditions at which three different phases can coexist. For example, the water phase diagram has a triple point corresponding to the single temperature and pressure at which solid, liquid, and gaseous water can coexist in a stable equilibrium. Chaos theory.
A double rod pendulum animation showing chaotic behavior.
Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Chaotic behavior can be observed in many natural systems, such as weather and climate.[6][7] This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Introduction[edit] Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic dynamics[edit] The map defined by x → 4 x (1 – x) and y → x + y mod 1 displays sensitivity to initial conditions.
In common usage, "chaos" means "a state of disorder".[9] However, in chaos theory, the term is defined more precisely. The Structure of Scientific Revolutions. The Structure of Scientific Revolutions is a 1962 book about the history of science by Thomas S.
Kuhn. Its publication was a landmark event in the history, philosophy, and sociology of scientific knowledge and triggered an ongoing worldwide assessment and reaction in—and beyond—those scholarly communities. Kuhn challenged the then prevailing view of progress in "normal science. " Normal scientific progress was viewed as "development-by-accumulation" of accepted facts and theories. Kuhn argued for an episodic model in which periods of such conceptual continuity in normal science were interrupted by periods of revolutionary science. Wave function. However, complex numbers are not necessarily used in all treatments.
Louis de Broglie in his later years proposed a real-valued wave function connected to the complex wave function by a proportionality constant and developed the de Broglie–Bohm theory. The unit of measurement for ψ depends on the system. For one particle in three dimensions, its units are [length]−3/2. Wave function collapse. When the Copenhagen interpretation was first expressed, Niels Bohr postulated wave function collapse to cut the quantum world from the classical.[5] This tactical move allowed quantum theory to develop without distractions from interpretational worries.
Mathematical description[edit] Mathematical background[edit] Quantum state. Black hole information paradox. Artist's representation of a black hole Principles in action There are two main principles in play: Quantum determinism means that given a present wave function, its future changes are uniquely determined by the evolution operator.Reversibility refers to the fact that the evolution operator has an inverse, meaning that the past wave functions are similarly unique.
Causal dynamical triangulation. Causal dynamical triangulation (abbreviated as CDT) invented by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, and popularized by Fotini Markopoulou and Lee Smolin, is an approach to quantum gravity that like loop quantum gravity is background independent.
This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the spacetime fabric itself evolves. The Loops '05 conference, hosted by many loop quantum gravity theorists, included several presentations which discussed CDT in great depth, and revealed it to be a pivotal insight for theorists. It has sparked considerable interest as it appears to have a good semi-classical description. At large scales, it re-creates the familiar 4-dimensional spacetime, but it shows spacetime to be 2-d near the Planck scale, and reveals a fractal structure on slices of constant time. Introduction[edit] Black hole complementarity. Black hole complementarity is a conjectured solution to the black hole information paradox, proposed by Leonard Susskind and Larus Thorlacius,[1] and Gerard 't Hooft.[2] Leonard Susskind[3] proposed a radical resolution to this problem by claiming that the information is both reflected at the event horizon and passes through the event horizon and can't escape, with the catch being no observer can confirm both stories simultaneously.
According to an external observer, the infinite time dilation at the horizon itself makes it appear as if it takes an infinite amount of time to reach the horizon. Quantum gravity. Quantum gravity (QG) is a field of theoretical physics that seeks to describe the force of gravity according to the principles of quantum mechanics.
Induced gravity. Induced gravity (or emergent gravity) is an idea in quantum gravity that space-time background emerges as a mean field approximation of underlying microscopic degrees of freedom, similar to the fluid mechanics approximation of Bose–Einstein condensates. The concept was originally proposed by Andrei Sakharov in 1967. Sakharov observed that many condensed matter systems give rise to emergent phenomena which are identical to general relativity. Inhomogeneous electromagnetic wave equation. In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves, due to nonzero source charges and currents.
The source terms in the wave equations makes the partial differential equations inhomogeneous. Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. The equations follow from Maxwell's equations. Maxwell's equations[edit] For reference, Maxwell's equations are summarized below in SI units and Gaussian units. Higgs boson. The Higgs boson is named after Peter Higgs, one of six physicists who, in 1964, proposed the mechanism that suggested the existence of such a particle.
Although Higgs's name has come to be associated with this theory, several researchers between about 1960 and 1972 each independently developed different parts of it. In mainstream media the Higgs boson has often been called the "God particle", from a 1993 book on the topic; the nickname is strongly disliked by many physicists, including Higgs, who regard it as inappropriate sensationalism.[17][18] In 2013 two of the original researchers, Peter Higgs and François Englert, were awarded the Nobel Prize in Physics for their work and prediction[19] (Englert's co-researcher Robert Brout had died in 2011). A non-technical summary[edit]
Boson. In quantum mechanics, a boson (/ˈboʊsɒn/,[1] /ˈboʊzɒn/[2]) is a particle that follows Bose–Einstein statistics. Bosons make up one of the two classes of particles, the other being fermions.[3] The name boson was coined by Paul Dirac[4] to commemorate the contribution of the Indian physicist Satyendra Nath Bose[5][6] in developing, with Einstein, Bose–Einstein statistics—which theorizes the characteristics of elementary particles.[7] Examples of bosons include fundamental particles such as photons, gluons, and W and Z bosons (the four force-carrying gauge bosons of the Standard Model), the recently discovered Higgs boson, and the hypothetical graviton of quantum gravity; composite particles (e.g. mesons and stable nuclei of even mass number such as deuterium (with one proton and one neutron, mass number = 2), helium-4, or lead-208[Note 1]); and some quasiparticles (e.g.
Cooper pairs, plasmons, and phonons).[8]:130 Types[edit] Properties[edit] Supersymmetry. Electromagnetic radiation. The electromagnetic waves that compose electromagnetic radiation can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. Light. Photon. Wave–particle duality. Electromagnetic spectrum. Spacetime.
Complementarity (physics) In physics, complementarity is a fundamental principle of quantum mechanics, closely associated with the Copenhagen interpretation. It holds that objects governed by quantum mechanics, when measured, give results that depend inherently upon the type of measuring device used, and must necessarily be described in classical mechanical terms. Quantum nonlocality. Black body. As the temperature of a black body decreases, its intensity also decreases and its peak moves to longer wavelengths. Absolute zero. Thermal radiation. This diagram shows how the peak wavelength and total radiated amount vary with temperature according to Wien's displacement law.