Photon diffusion equation is a second order partial differential equation describing the time behavior of photon fluence rate distribution in a low-absorption high-scattering medium.
where is photon fluence rate (W/cm2), is del operator, is absorption coefficient (cm−1), is diffusion constant, is the speed of light in the medium (m/s), and is an isotropic source term (W/cm3).
Its main difference with diffusion equation in physics is that photon diffusion equation has an absorption term in it.
Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. 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.
The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a given region.
The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles. In mathematics, it is related to Markov processes, such as random walks, and applied in many other fields, such as materials science, information theory, and biophysics. The diffusion equation is a special case of the convection–diffusion equation when bulk velocity is zero. It is equivalent to the heat equation under some circumstances.
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of earth sciences. It is analogous to Ohm's law in electrostatics, linearly relating the volume flow rate of the fluid to the hydraulic head difference via the hydraulic conductivity. In fact, the Darcy's law is a special case of the Stokes equation for the momentum flux, in turn deriving from the momentum Navier-Stokes equation.
In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the quantization of the electromagnetic field by Lagrangian formulation described below.
The Grad–Shafranov equation is the equilibrium equation in ideal magnetohydrodynamics (MHD) for a two dimensional plasma, for example the axisymmetric toroidal plasma in a tokamak. This equation takes the same form as the Hicks equation from fluid dynamics. This equation is a two-dimensional, nonlinear, elliptic partial differential equation obtained from the reduction of the ideal MHD equations to two dimensions, often for the case of toroidal axisymmetry. Taking as the cylindrical coordinates, the flux function is governed by the equation,
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.
In fluid dynamics, eddy diffusion, eddy dispersion, or turbulent diffusion is a process by which fluid substances mix together due to eddy motion. These eddies can vary widely in size, from subtropical ocean gyres down to the small Kolmogorov microscales, and occur as a result of turbulence. The theory of eddy diffusion was first developed by Sir Geoffrey Ingram Taylor.
Spinodal decomposition is a mechanism by which a single thermodynamic phase spontaneously separates into two phases. Decomposition occurs when there is no thermodynamic barrier to phase separation. As a result, phase separation via decomposition does not require the nucleation events resulting from thermodynamic fluctuations, which normally trigger phase separation.
Photon transport in biological tissue can be equivalently modeled numerically with Monte Carlo simulations or analytically by the radiative transfer equation (RTE). However, the RTE is difficult to solve without introducing approximations. A common approximation summarized here is the diffusion approximation. Overall, solutions to the diffusion equation for photon transport are more computationally efficient, but less accurate than Monte Carlo simulations.
In fluid dynamics, aerodynamic potential flow codes or panel codes are used to determine the fluid velocity, and subsequently the pressure distribution, on an object. This may be a simple two-dimensional object, such as a circle or wing, or it may be a three-dimensional vehicle.
Gyrokinetics is a theoretical framework to study plasma behavior on perpendicular spatial scales comparable to the gyroradius and frequencies much lower than the particle cyclotron frequencies. These particular scales have been experimentally shown to be appropriate for modeling plasma turbulence. The trajectory of charged particles in a magnetic field is a helix that winds around the field line. This trajectory can be decomposed into a relatively slow motion of the guiding center along the field line and a fast circular motion, called gyromotion. For most plasma behavior, this gyromotion is irrelevant. Averaging over this gyromotion reduces the equations to six dimensions rather than the seven. Because of this simplification, gyrokinetics governs the evolution of charged rings with a guiding center position, instead of gyrating charged particles.
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, as in spinodal decomposition. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as statistics, probability theory, information theory, neural networks, finance, and marketing.
Within physics, the Hybrid Theory for photon transport in tissue uses the advantages and eliminates the deficiencies of both the Monte Carlo method and the diffusion theory for photon transport to model photons traveling through tissue both accurately and efficiently.
Diffusion current is a current in a semiconductor caused by the diffusion of charge carriers. This is the current which is due to the transport of charges occurring because of non-uniform concentration of charged particles in a semiconductor. The drift current, by contrast, is due to the motion of charge carriers due to the force exerted on them by an electric field. Diffusion current can be in the same or opposite direction of a drift current. The diffusion current and drift current together are described by the drift–diffusion equation.
Magnetic diffusion refers to the motion of magnetic fields, typically in the presence of a conducting solid or fluid such as a plasma. The motion of magnetic fields is described by the magnetic diffusion equation and is due primarily to induction and diffusion of magnetic fields through the material. The magnetic diffusion equation is a partial differential equation commonly used in physics. Understanding the phenomenon is essential to magnetohydrodynamics and has important consequences in astrophysics, geophysics, and electrical engineering.
Diffuse correlation spectroscopy (DCS) is a type of medical imaging and optical technique that utilizes near-infrared light to directly and non-invasively measure tissue blood flow. The imaging modality was created by David Boas and Arjun Yodh in 1995.
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