Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them. The analysis of autocorrelation is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal obscured by noise, or identifying the missing fundamental frequency in a signal implied by its harmonic frequencies. It is often used in signal processing for analyzing functions or series of values, such as time domain signals.
Brownian motion, or pedesis, is the random motion of particles suspended in a medium.
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.
The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles, all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.
In physics, a Langevin equation is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. One application is to Brownian motion, which models the fluctuating motion of a small particle in a fluid.
In physics, mean free path is the average distance over which a moving particle travels before substantially changing its direction or energy, typically as a result of one or more successive collisions with other particles.
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ (lambda) is a positive rate called the exponential decay constant:
In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval between two events on a world line is the change in proper time. This interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line.
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials. Basically, Ohm's law was well established and stated that the current J and voltage V driving the current are related to the resistance R of the material. The inverse of the resistance is known as the conductance. When we consider a metal of unit length and unit cross sectional area, the conductance is known as the conductivity, which is the inverse of resistivity. The Drude model attempts to explain the resistivity of a conductor in terms of the scattering of electrons by the relatively immobile ions in the metal that act like obstructions to the flow of electrons.
For detection systems that record discrete events, such as particle and nuclear detectors, the dead time is the time after each event during which the system is not able to record another event. An everyday life example of this is what happens when someone takes a photo using a flash - another picture cannot be taken immediately afterward because the flash needs a few seconds to recharge. In addition to lowering the detection efficiency, dead times can have other effects, such as creating possible exploits in quantum cryptography.
A Knudsen gas is a gas in a state of such low density that the average distance travelled by the gas molecules between collisions is greater than the diameter of the receptacle that contains it. If the mean free path is much greater than the diameter, the flow regime is dominated by collisions between the gas molecules and the walls of the receptacle, rather than intermolecular collisions with each other. It is named after Martin Knudsen.
The diffusion of plasma across a magnetic field was conjectured to follow the Bohm diffusion scaling as indicated from the early plasma experiments of very lossy machines. This predicted that the rate of diffusion was linear with temperature and inversely linear with the strength of the confining magnetic field.
The continuous stirred-tank reactor (CSTR), also known as vat- or backmix reactor, mixed flow reactor (MFR), or a continuous-flow stirred-tank reactor (CFSTR), is a common model for a chemical reactor in chemical engineering and environmental engineering. A CSTR often refers to a model used to estimate the key unit operation variables when using a continuous agitated-tank reactor to reach a specified output. The mathematical model works for all fluids: liquids, gases, and slurries.
Fluid mechanics is the branch of physics concerned with the mechanics of fluids and the forces on them. It has applications in a wide range of disciplines, including mechanical, civil, chemical and biomedical engineering, geophysics, oceanography, meteorology, astrophysics, and biology.
The plug flow reactor model is a model used to describe chemical reactions in continuous, flowing systems of cylindrical geometry. The PFR model is used to predict the behavior of chemical reactors of such design, so that key reactor variables, such as the dimensions of the reactor, can be estimated.
Lattice Boltzmann methods (LBM), originated from the lattice gas automata (LGA) method, is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations directly, a fluid density on a lattice is simulated with streaming and collision (relaxation) processes. The method is versatile as the model fluid can straightforwardly be made to mimic common fluid behaviour like vapour/liquid coexistence, and so fluid systems such as liquid droplets can be simulated. Also, fluids in complex environments such as porous media can be straightforwardly simulated, whereas with complex boundaries other CFD methods can be hard to work with.
Rotational diffusion is the rotational movement which acts upon any object such as particles, molecules, atoms when present in a fluid, by random changes in their orientations. Whilst the directions and intensities of these changes are statistically random, they do not arise randomly and are instead the result of interactions between particles. One example occurs in colloids, where relatively large insoluble particles are suspended in a greater amount of fluid. The changes in orientation occur from collisions between the particle and the many molecules forming the fluid surrounding the particle, which each transfer kinetic energy to the particle, and as such can be considered random due to the varied speeds and amounts of fluid molecules incident on each individual particle at any given time.
In physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals the time average. In this regime, any collection of random samples from a process must represent the average statistical properties of the entire regime. Conversely, a process that is not in ergodic regime said to be in non-ergodic regime.
Chapman–Enskog theory provides a framework in which equations of hydrodynamics for a gas can be derived from the Boltzmann equation. The technique justifies the otherwise phenomenological constitutive relations appearing in hydrodynamical descriptions such as the Navier–Stokes equations. In doing so, expressions for various transport coefficients such as thermal conductivity and viscosity are obtained in terms of molecular parameters. Thus, Chapman–Enskog theory constitutes an important step in the passage from a microscopic, particle-based description to a continuum hydrodynamical one.
The residence time of a fluid parcel is the total time that the parcel has spent inside a control volume. The residence time of a set of parcels is quantified in terms of the frequency distribution of the residence time in the set, which is known as residence time distribution (RTD), or in terms of its average, known as mean residence time.
