Kinetic Equation

[Telly Piatt]

Theequation was derived by Lev Landau in 1936[1] as an alternative to the Boltzmann equation in the case of Coulomb interaction. When used with the Vlasov equation, the equation yields the time evolution for collisional plasma, hence it is considered a staple kinetic model in the theory of collisional plasma. [2][3]

The equation is used primarily in Statistical mechanics and Particle physics to model plasma. As such, it has been used to model and study Plasma in thermonuclear reactors.[4][5][6] It has also seen use in modeling of Active matter .[7]

An equation in non-equilibrium statistical physics that is used in gas theory, aerodynamics, plasma physics, the theory of the passage of particles through matter, and the theory of radiation transfer. The solution of the kinetic equation determines the distribution function of the dynamical states of a single particle, which usually depends on time, coordinates and velocity.

In 1872 L. Boltzmann formulated the fundamental kinetic equation of gas theory. It is a non-linear integro-differential equation (see Boltzmann equation), which describes the motion of the molecules as a certain random process determined by collisions between pairs of molecules. The coefficients (effective cross sections) entering into the equation are calculated from the equations of classical mechanics. By studying the properties of the solutions of the kinetic equation, Boltzmann gave a molecular-kinetic interpretation of the second law of thermodynamics and established a statistical meaning of the notion of entropy (see Boltzmann $H$-theorem). In quantum statistical physics the Boltzmann equation is described in the simplest case by analogy with the classical case, but using quantum effective cross sections and taking symmetry requirements into account. For a relativistic gas the Boltzmann equation is stated in covariant form. A method has been developed for obtaining the kinetic equation of gas theory taking into account the correlation between the dynamical states of the molecules (see Bogolyubov chain of equations). Starting from the Liouville equation one can use this method to obtain in the lowest approximation the Boltzmann equation if one uses a power series expansion in terms of the density of the gas.

The power series expansion in the strength of the interaction energy leads to the Vlasov kinetic equation with a self-adapting field, and in the subsequent approximation in the spatially-homogeneous case to the Landau kinetic equation, describing the so-called "diffusion in velocity space" .

Few exact solutions of the non-linear kinetic equation are known. It is difficult to solve it numerically, even when using a computer. More thoroughly investigated is the linearized kinetic equation, which describes small deviations from the equilibrium solution of the non-linear equation. It is the same in form as the linear transport equations occurring in the theory of radiative and neutron transfer (see Radiative transfer theory), and also in the theory of the passage of particles through matter.

The theory of radiative transfer is close, with regard to its problems and methods of their solution, to that of neutron transfer. Calculations regarding nuclear reactors and protection against nuclear radiation have required the creation of effective methods for solving the kinetic equation of the transfer of neutrons and gamma-quanta, and have also promoted the creation of a mathematical theory of the linear kinetic equation. Existence and uniqueness theorems and the asymptotic behaviour of the non-linear Boltzmann equation have been widely studied [1]. In the case of the linear equation these theorems are obtained in the most general formulation of the mathematical theory of nuclear reactors [2], [3], [4]. A number of problems have been solved by analytic methods, and in the general case many approximate methods, convenient for programming on a computer, have been developed for its solution (see Transport equations, numerical methods).

The problem of the passage of charged particles through matter often reduces to the solution of the linear transport equation in the presence of anisotropic scattering or to the solution of the linearized Landau equation (for example, in the determination of the angular distribution of the particles scattered from the surface of a body, or in the determination of the energy spectrum of charged particles in matter).

Their study involves a wide mathematical palette ranging from modeling to theoretical study through numerical analysis. One of the main challenges is the study of the dynamic properties of these equations, particularly in a long time. Obtaining precise quantitative information is also a current challenge in theory. Recently significant progress has been made in the study of these models using tools from many areas of mathematics: PDE, Mathematical Physics, and Probability.

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In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

in C++, creating a function that calculates a value using the kinetic energy equation. But I'm not entirely sure how to go about showing 1 / 2. Wouldn't I have to make it a double because if I represent 1 / 2 as an integer it'll just display 5? What the compiler is really seeing would be 0.5, from which the zero is cut off? Here is the piece of code I have so far to calculating this equation: double KE = 1/2*mvpow(KE,2); Here's my code and what i'm trying to do.

I'm having trouble implementing the kinetic equation by Dyson & Simon for ammonia synthesis. Fugacity coefficients are to be calculated by correlations (= f(p,T)), so I added a Python script for the kinetic equation. However, I'm getting "Attempted to divide by zero" errors. If the variables for the fugacity coefficients are set to 1 manually, the solver finds a solution. Playing with the integration step size does not make a differene, unfortunately.

The goal of this first unit of The Physics Classroom has been to investigate the variety of means by which the motion of objects can be described. The variety of representations that we have investigated includes verbal representations, pictorial representations, numerical representations, and graphical representations (position-time graphs and velocity-time graphs). In Lesson 6, we will investigate the use of equations to describe and represent the motion of objects. These equations are known as kinematic equations.

There are a variety of quantities associated with the motion of objects - displacement (and distance), velocity (and speed), acceleration, and time. Knowledge of each of these quantities provides descriptive information about an object's motion. For example, if a car is known to move with a constant velocity of 22.0 m/s, North for 12.0 seconds for a northward displacement of 264 meters, then the motion of the car is fully described. And if a second car is known to accelerate from a rest position with an eastward acceleration of 3.0 m/s2 for a time of 8.0 seconds, providing a final velocity of 24 m/s, East and an eastward displacement of 96 meters, then the motion of this car is fully described. These two statements provide a complete description of the motion of an object. However, such completeness is not always known. It is often the case that only a few parameters of an object's motion are known, while the rest are unknown. For example as you approach the stoplight, you might know that your car has a velocity of 22 m/s, East and is capable of a skidding acceleration of 8.0 m/s2, West. However you do not know the displacement that your car would experience if you were to slam on your brakes and skid to a stop; and you do not know the time required to skid to a stop. In such an instance as this, the unknown parameters can be determined using physics principles and mathematical equations (the kinematic equations).

The kinematic equations are a set of four equations that can be utilized to predict unknown information about an object's motion if other information is known. The equations can be utilized for any motion that can be described as being either a constant velocity motion (an acceleration of 0 m/s/s) or a constant acceleration motion. They can never be used over any time period during which the acceleration is changing. Each of the kinematic equations include four variables. If the values of three of the four variables are known, then the value of the fourth variable can be calculated. In this manner, the kinematic equations provide a useful means of predicting information about an object's motion if other information is known. For example, if the acceleration value and the initial and final velocity values of a skidding car is known, then the displacement of the car and the time can be predicted using the kinematic equations. Lesson 6 of this unit will focus upon the use of the kinematic equations to predict the numerical values of unknown quantities for an object's motion.

There are a variety of symbols used in the above equations. Each symbol has its own specific meaning. The symbol d stands for the displacement of the object. The symbol t stands for the time for which the object moved. The symbol a stands for the acceleration of the object. And the symbol v stands for the velocity of the object; a subscript of i after the v (as in vi) indicates that the velocity value is the initial velocity value and a subscript of f (as in vf) indicates that the velocity value is the final velocity value.

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