In molecular kinetic theory in

physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

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

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

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

phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...

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

velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

\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

\begin
n(\mathbf,t) &= \int f(\mathbf, \mathbf, t) \,dv_x \,dv_y \,dv_z, \\
N(t) &= \int n(\mathbf, t) \,dx \,dy \,dz,
\end

where is the total number of particles and is the number density of particles – the number of particles per unit volume, or the

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

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 (: momenta or momentums; 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. ...

(p_x,p_y,p_z)

. Particle distribution functions are often used in

plasma physics Plasma () is a state of matter characterized by the presence of a significant portion of charged particles in any combination of ions or electrons. It is the most abundant form of ordinary matter in the universe, mostly in stars (including th ...

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 plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

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 th ...

. The basic distribution function uses the

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

k

and temperature

T

with the number density to modify the

normal distribution In probability theory and 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 ...

&= n\left(\frac\right)^ \exp\left(-\frac\right). \end

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 In physics and engineering, magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is a model of electrically conducting fluids that treats all interpenetrating particle species together as a single Continuum ...

may assume the particles to be in

thermodynamic equilibrium Thermodynamic equilibrium is a notion of thermodynamics with axiomatic status referring to an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...

. 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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

References

Statistical mechanics Dynamical systems {{statisticalmechanics-stub