The process of evolving to equilibrium is often characterized by a time scale called the e-folding time,τ. This time is used for processes which evolve exponentially toward a final state (equilibrium). In other words, if we examine an observable, X, associated with a system, (temperature or density, for example) then after a time, τ, the initial difference between the initial value of the observable and the equilibrium value, ΔXi, will have decreased to ΔXi/e where the number e ≈ 2.71828.
The concept of e-folding time may be used in the analysis of kinetics. Consider a chemical species A, which decays into another chemical species, B. We could depict this as an equation:
Let us assume that this reaction follows first order kinetics, meaning that the conversion of A into B depends only on the concentration of A, and the rate constant which dictates the velocity at which this happens,k. We could write the following reaction to describe this first order kinetic process:
This ordinary differential equation states that a change (in this case the disappearance) of the concentration of A, d[A]/dt, is equal to the rate constant k multiplied by the concentration of A. Consider what the units of k would be. On the left hand side, we have a concentration divided by a unit of time. The units for k would need to allow for these to be replicated on the right hand side. For this reason, the units of k, here, would be 1/time.
Because this is a linear, homogeneous and separable differential equation, we may separate the terms such that the equation becomes:
We may then integrate both sides of this equation, which results in the inclusion of the constant e:
where [A]f and [A]i are the final and initial concentrations of A. Upon comparing the ratio on the left hand side to the equation on the right hand side, we conclude that the ratio between the final and initial concentrations follows an exponential function, of which e is the base.
As mentioned above, the units for k are inverse time. If we were to take the reciprocal of this, we would be left with units of time. For this reason, we often state that the lifetime of a species that undergoes first order decay is equal to the reciprocal of k. Consider, now, what would happen if we were to set the time, t, to the reciprocal of the rate constant, k, i.e. t=1/k. This would yield
This states that after one lifetime (1/k), the ratio of final to initial concentrations is equal to about 0.37. Stated another way, after one lifetime, we have
which means that we have lost (1−0.37=0.63) 63% of A, with only 37% left. With this, we now know that if we have 1 lifetime passed, we have gone through 1 "e-folding". What would 2 "e-foldings" look like? After two lifetimes, we would have t = 1/k + 1/k = 2/k, which would result in
which says that only about 14% of A remains. It is in this manner that e-folding lends us an easy way to describe the number of lifetimes that have passed. After 1 lifetime, we have 1/e remaining. After 2 lifetimes, we have 1/e2 remaining. One lifetime, therefore, is one e-folding time, which is the most descriptive way of stating the decay.
Half-life is the time required for a quantity to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable atoms survive. The term is also used more generally to characterize any type of exponential decay. For example, the medical sciences refer to the biological half-life of drugs and other chemicals in the human body. The converse of half-life is doubling time.
A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.
The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit time. Reaction rates can vary dramatically. For example, the oxidative rusting of iron under Earth's atmosphere is a slow reaction that can take many years, but the combustion of cellulose in a fire is a reaction that takes place in fractions of a second. For most reactions, the rate decreases as the reaction proceeds. A reaction's rate can be determined by measuring the changes in concentration over time.
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, disintegration constant, rate constant, or transformation constant:
In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. The analogous solution for a charged, rotating body is given by the Kerr–Newman metric.
The classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity and can thereby move due to the conservation of momentum. It is credited to Konstantin Tsiolkovsky, who independently derived it and published it in 1903, although it had been independently derived and published by William Moore in 1810, and later published in a separate book in 1813. Robert Goddard also developed it independently in 1912, and Hermann Oberth derived it independently about 1920.
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent.
Compartmental models are a very general modelling technique. They are often applied to the mathematical modelling of infectious diseases. The population is assigned to compartments with labels – for example, S, I, or R,. People may progress between compartments. The order of the labels usually shows the flow patterns between the compartments; for example SEIS means susceptible, exposed, infectious, then susceptible again.
The Friedmann equations, also known as the Friedmann–Lemaître (FL) equations, are a set of equations in physical cosmology that govern cosmic expansion in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density ρ and pressure p. The equations for negative spatial curvature were given by Friedmann in 1924.
In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real number line.
In chemistry, the rate equation is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters only. For many reactions, the initial rate is given by a power law such as
In physical systems, damping is the loss of energy of an oscillating system by dissipation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include viscous damping in a fluid, surface friction, radiation, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes. Damping is not to be confused with friction, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping.
The principle of detailed balance can be used in kinetic systems which are decomposed into elementary processes. It states that at equilibrium, each elementary process is in equilibrium with its reverse process.
In physics, a mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique. The exact conservation law used in the analysis of the system depends on the context of the problem, but all revolve around mass conservation, i.e., that matter cannot disappear or be created spontaneously.
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form where and the integrands are functions dependent on the derivative of this integral is expressible as where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative.
Dilution is the process of decreasing the concentration of a solute in a solution, usually simply by mixing with more solvent like adding more water to the solution. To dilute a solution means to add more solvent without the addition of more solute. The resulting solution is thoroughly mixed so as to ensure that all parts of the solution are identical.
The elimination rate constantK or Ke is a value used in pharmacokinetics to describe the rate at which a drug is removed from the human system.
In physics and engineering, the time constant, usually denoted by the Greek letter τ (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system. The time constant is the main characteristic unit of a first-order LTI system. It gives speed of the response.
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.
The Langmuir adsorption model explains adsorption by assuming an adsorbate behaves as an ideal gas at isothermal conditions. According to the model, adsorption and desorption are reversible processes. This model even explains the effect of pressure; i.e., at these conditions the adsorbate's partial pressure is related to its volume V adsorbed onto a solid adsorbent. The adsorbent, as indicated in the figure, is assumed to be an ideal solid surface composed of a series of distinct sites capable of binding the adsorbate. The adsorbate binding is treated as a chemical reaction between the adsorbate gaseous molecule and an empty sorption site S. This reaction yields an adsorbed species with an associated equilibrium constant :
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