Distribution function (physics)

:''This article describes the ''distribution function'' as used in physics. You may be looking for the related mathematical concepts of

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Eve ...

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...

.'' In molecular kinetic theory in

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

, a system's distribution function is a function of seven variables,

f(x,y,z,t;v_x,v_y,v_z)

, which gives the number of particles per unit volume in single-particle

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...

. It is the number of particles per unit volume having approximately the

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

\mathbf=(v_x,v_y,v_z)

near the position

\mathbf=(x,y,z)

and time

t

. The usual normalization of the distribution function is :

n(x,y,z,t) = \int f \,dv_x \,dv_y \,dv_z,

N(t) = \int n \,dx \,dy \,dz,

where, ''N'' is the total number of particles, and ''n'' is the

number density The number density (symbol: ''n'' or ''ρ''N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric num ...

of particles – the number of particles per unit volume, or the

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

divided by the mass of individual particles. A distribution function may be specialised with respect to a particular set of dimensions. E.g. take the quantum mechanical six-dimensional phase space,

f(x,y,z;p_x,p_y,p_z)

and multiply by the total space volume, to give the momentum distribution, i.e. the number of particles in the momentum phase space having approximately the

momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...

(p_x,p_y,p_z)

. Particle distribution functions are often used in

plasma physics Plasma ()πλάσμα
, Henry George Liddell, R ...

to describe wave–particle interactions and velocity-space instabilities. Distribution functions are also used in

fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...

and

nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies t ...

. The basic distribution function uses the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...

k

and temperature

T

with the number density to modify the

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

: :

f = n\left(\frac\right)^ \exp\left(\right).

Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is

m((v_x - u_x)^2 + (v_y - u_y)^2 + (v_z - u_z)^2)

, where

(u_x, u_y, u_z)

is the bulk velocity of the fluid. Distribution functions may also feature non-isotropic temperatures, in which each term in the exponent is divided by a different temperature. Plasma theories such as

magnetohydrodynamics Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...

may assume the particles to be in

thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...

. In this case, the distribution function is '' Maxwellian''. This distribution function allows fluid flow and different temperatures in the directions parallel to, and perpendicular to, the local magnetic field. More complex distribution functions may also be used, since plasmas are rarely in thermal equilibrium. The mathematical analogue of a distribution is a measure; the time evolution of a measure on a phase space is the topic of study in

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...

. Statistical mechanics Dynamical systems {{statisticalmechanics-stub