 | This article provides insufficient context for those unfamiliar with the subject.(February 2014) |
The radiation law is way of modeling human mobility (geographic mobility, human migration) and it gives better empirical predictions than the gravity model of migration which is widely used in this subject. [1]
Waves of migration due to displacement by war, or exploitation in the hope of geographical discoveries could be observed in the past, however with new technological advancements in transportation keep making it easier and cheaper to get to one place from another. With intercontinental flights we even can travel to another continent, on a business trip for instance, and come back within a few hours. Not only time but road networks and flight networks are being used more and more intensively also, and there is an increasing need to describe the patterns of human peoples' mobility and their effect on network usage, whether the network is a transportation, communication or some other type of network.
Radiation models appeared first in physics to study the process of energetic particles or waves travel through vacuum. The model in the social science describes the flows of people between different locations. Daily commuting is the major part of the flows, so modeling job seeking has to be an important part of the model and so it is in the radiation model. People look for jobs in every county starting with their own home county. The number of open jobs depends on the size of the resident population . The potential employment opportunity (e.g. conditions, income, working hour, etc.) is with the distribution of . Then, for each county job opportunities are assigned, which are random draws from the distribution. Individuals then chooses the job which is closest to their home county and provides the highest . Thus, they take into account the proximity to their home county and the benefits it can provide. This optimization gives the migration flows (called commuting fluxes) between counties across the country. This is analogous to the model in physics that describes the radiation and absorption process, that's why it's called the radiation model. An important feature of the model is that the average flux between two counties does not depend on the benefit distribution, the number of job opportunities and the total number of commuters. The fundamental equation of the radiation model gives the average flux between two counties,
where is the total number of commuters from county , and are the population in county and respectively, and is the total population in the circle centered at and touching excluding the source and the destination population. The model is not static as the Gravity model, and has clear implications which can be empirically verified.
The population density around Utah is much lower than around Alabama and so are the job opportunities, given that the population of the two states is the same. Thus, the fundamental equation implies that people from Utah have to travel further to find suitable jobs on average than people from Alabama, and indeed, this is what the data shows. The Gravity model gives bad predictions both on short and long distance commuting, while the prediction of the Radiation model is close to the census data. Further empirical testing shows [2] that the Radiation model underestimates the flow in case of big cities, but generalizing the fundamental equation the model can give at least as good predictions as the Gravity model.
In 1971 famed economist William Alonso produced a working paper that describes a mathematical model of human mobility. [3] In that manuscript Alonso remarks: "It is almost as if an urban area were a radioactive body, emitting particles at a steady rate[.]" In addition to many of the same mathematical terms used by Simini et al., Alonso's radiation model includes measures of climate (degree days) and wealth (per capita income) for both the emitting and receiving locales, but only includes the distance between these urban areas as opposed to a radial measure of intervening population density.
The most influential model to describe trade patterns, and in a similar way, describe human mobility is the gravity model of trade. The model predicts, that the migration flow is proportional to the population of the cities/countries, and it is reciprocal in a quadratic order in the distance between them. Although, it is an intuitive description of the flows, and it is used to describe gravitational forces in physics, in terms of migration it does not perform well empirically. Moreover, the model just simply assumes the given functional form without any theoretical background.
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, D. Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation.
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Non-equilibrium thermodynamics is concerned with transport processes and with the rates of chemical reactions.
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872. The classic example of such a system is a fluid with temperature gradients in space causing heat to flow from hotter regions to colder ones, by the random but biased transport of the particles making up that fluid. In the modern literature the term Boltzmann equation is often used in a more general sense, referring to any kinetic equation that describes the change of a macroscopic quantity in a thermodynamic system, such as energy, charge or particle number.
Trip distribution is the second component in the traditional four-step transportation forecasting model. This step matches tripmakers’ origins and destinations to develop a “trip table”, a matrix that displays the number of trips going from each origin to each destination. Historically, this component has been the least developed component of the transportation planning model.
Hydraulic head or piezometric head is a specific measurement of liquid pressure above a vertical datum.
Used in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar to that used in heat transfer to describe the flow of heat in a solid. The steady-state flow of groundwater is described by a form of the Laplace equation, which is a form of potential flow and has analogs in numerous fields.
The gravity model of international trade in international economics is a model that, in its traditional form, predicts bilateral trade flows based on the economic sizes and distance between two units. Research shows that there is "overwhelming evidence that trade tends to fall with distance."
The ADM formalism is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first published in 1959.
A vertex model is a type of statistical mechanics model in which the Boltzmann weights are associated with a vertex in the model. This contrasts with a nearest-neighbour model, such as the Ising model, in which the energy, and thus the Boltzmann weight of a statistical microstate is attributed to the bonds connecting two neighbouring particles. The energy associated with a vertex in the lattice of particles is thus dependent on the state of the bonds which connect it to adjacent vertices. It turns out that every solution of the Yang–Baxter equation with spectral parameters in a tensor product of vector spaces yields an exactly-solvable vertex model.
Sediment transport is the movement of solid particles (sediment), typically due to a combination of gravity acting on the sediment, and/or the movement of the fluid in which the sediment is entrained. Sediment transport occurs in natural systems where the particles are clastic rocks, mud, or clay; the fluid is air, water, or ice; and the force of gravity acts to move the particles along the sloping surface on which they are resting. Sediment transport due to fluid motion occurs in rivers, oceans, lakes, seas, and other bodies of water due to currents and tides. Transport is also caused by glaciers as they flow, and on terrestrial surfaces under the influence of wind. Sediment transport due only to gravity can occur on sloping surfaces in general, including hillslopes, scarps, cliffs, and the continental shelf—continental slope boundary.
In statistical thermodynamics, the UNIFAC method is a semi-empirical system for the prediction of non-electrolyte activity in non-ideal mixtures. UNIFAC uses the functional groups present on the molecules that make up the liquid mixture to calculate activity coefficients. By using interactions for each of the functional groups present on the molecules, as well as some binary interaction coefficients, the activity of each of the solutions can be calculated. This information can be used to obtain information on liquid equilibria, which is useful in many thermodynamic calculations, such as chemical reactor design, and distillation calculations.
In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces.
The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation, or (generic) scalar transport equation.
Diffusion is the net movement of anything generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical potential. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in spinodal decomposition.
The generalized Lotka–Volterra equations are a set of equations which are more general than either the competitive or predator–prey examples of Lotka–Volterra types. They can be used to model direct competition and trophic relationships between an arbitrary number of species. Their dynamics can be analysed analytically to some extent. This makes them useful as a theoretical tool for modeling food webs. However, they lack features of other ecological models such as predator preference and nonlinear functional responses, and they cannot be used to model mutualism without allowing indefinite population growth.
Pitzer equations are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water. They were first described by physical chemist Kenneth Pitzer. The parameters of the Pitzer equations are linear combinations of parameters, of a virial expansion of the excess Gibbs free energy, which characterise interactions amongst ions and solvent. The derivation is thermodynamically rigorous at a given level of expansion. The parameters may be derived from various experimental data such as the osmotic coefficient, mixed ion activity coefficients, and salt solubility. They can be used to calculate mixed ion activity coefficients and water activities in solutions of high ionic strength for which the Debye–Hückel theory is no longer adequate. They are more rigorous than the equations of specific ion interaction theory, but Pitzer parameters are more difficult to determine experimentally than SIT parameters.
In engineering, physics, and chemistry, the study of transport phenomena concerns the exchange of mass, energy, charge, momentum and angular momentum between observed and studied systems. While it draws from fields as diverse as continuum mechanics and thermodynamics, it places a heavy emphasis on the commonalities between the topics covered. Mass, momentum, and heat transport all share a very similar mathematical framework, and the parallels between them are exploited in the study of transport phenomena to draw deep mathematical connections that often provide very useful tools in the analysis of one field that are directly derived from the others.
In fluid dynamics, the radiation stress is the depth-integrated – and thereafter phase-averaged – excess momentum flux caused by the presence of the surface gravity waves, which is exerted on the mean flow. The radiation stresses behave as a second-order tensor.
Reynolds stress equation model (RSM), also referred to as second moment closures are the most complete classical turbulence model. In these models, the eddy-viscosity hypothesis is avoided and the individual components of the Reynolds stress tensor are directly computed. These models use the exact Reynolds stress transport equation for their formulation. They account for the directional effects of the Reynolds stresses and the complex interactions in turbulent flows. Reynolds stress models offer significantly better accuracy than eddy-viscosity based turbulence models, while being computationally cheaper than Direct Numerical Simulations (DNS) and Large Eddy Simulations.
