Phase transitionIn chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. A phase of a thermodynamic system and the states of matter have uniform physical properties. During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure.
LondonLondon ('lʌndən ) is the capital and largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary down to the North Sea and has been a major settlement for two millennia. The City of London, its ancient core and financial centre, was founded by the Romans as Londinium and retains its medieval boundaries. The City of Westminster, to the west of the City of London, has for centuries hosted the national government and parliament.
Stochastic processIn probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.
City of LondonThe City of London, widely referred to simply as the City, is a city, ceremonial county and local government district that contains the ancient centre and constitutes, alongside Canary Wharf, the primary central business district (CBD) of London and one of the leading financial centres of the world. It constituted most of London from its settlement by the Romans in the 1st century AD to the Middle Ages, but the modern area named London has since grown far beyond the City of London boundary.
London Bridge stationLondon Bridge is a central London railway terminus and connected London Underground station in Southwark, south-east London. It occupies a large area on three levels immediately south-east of London Bridge, from which it takes its name. The main line station is the oldest railway station in London fare zone 1 and one of the oldest in the world having opened in 1836. It is one of two main line termini in London to the south of the River Thames (the other being Waterloo) and is the fourth-busiest station in London, handling over 50 million passengers a year.
London OvergroundLondon Overground (also known simply as the Overground) is a suburban rail network serving London and its environs. Established in 2007 to take over Silverlink Metro routes, it now serves a large part of Greater London as well as the home county of Hertfordshire, with 113 stations on nine different routes. The Overground forms part of the United Kingdom's National Rail network but it is under the concession control and branding of Transport for London (TfL). Operation has been contracted to Arriva Rail London since 2016.
Transport in LondonLondon has an extensive and developed transport network which includes both private and public services. Journeys made by public transport systems account for 37% of London's journeys while private services accounted for 36% of journeys, walking 24% and cycling 2%. London's public transport network serves as the central hub for the United Kingdom in rail, air and road transport. Public transport services are dominated by the executive agency for transport in London: Transport for London (TfL).
Museum of LondonThe Museum of London is a museum in London, covering the history of the city from prehistoric to modern times, with a particular focus on social history. It was formed in 1976 by amalgamating collections previously held by the City Corporation at the Guildhall Museum (founded in 1826) and of the London Museum (founded in 1911). From 1976 to 2022, its main site was located in the City of London on London Wall, close to the Barbican Centre, as part of the Barbican complex of buildings created in the 1960s and 1970s to redevelop a bomb-damaged area of the city.
Chaos theoryChaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization.
Linear systemIn systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.
Linear time-invariant systemIn system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined below. These properties apply (exactly or approximately) to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = (x ∗ h)(t) where h(t) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication).
Impulse responseIn signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (δ(t)). More generally, an impulse response is the reaction of any dynamic system in response to some external change. In both cases, the impulse response describes the reaction of the system as a function of time (or possibly as a function of some other independent variable that parameterizes the dynamic behavior of the system).
Phase (matter)In the physical sciences, a phase is a region of material that is chemically uniform, physically distinct, and (often) mechanically separable. In a system consisting of ice and water in a glass jar, the ice cubes are one phase, the water is a second phase, and the humid air is a third phase over the ice and water. The glass of the jar is another separate phase. (See .) More precisely, a phase is a region of space (a thermodynamic system), throughout which all physical properties of a material are essentially uniform.
Phase diagramA phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous states) occur and coexist at equilibrium. 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.
Stochastic calculusStochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces.
Supersymmetric theory of stochastic dynamicsSupersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches.
Control theoryControl theory is a field of control engineering and applied mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any delay, overshoot, or steady-state error and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required.
Stochastic differential equationA stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale.
Complex systemA complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication systems, complex software and electronic systems, social and economic organizations (like cities), an ecosystem, a living cell, and ultimately the entire universe.
Period-doubling bifurcationIn dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves. A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system.
