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{{Short description|Function of seven variables}}

{{about|the distribution function as used in physics|the related mathematical concepts|cumulative distribution function|and|probability density function}}

In molecular [[kinetic theory of gases|kinetic theory]] in [[physics]], a system's '''distribution function''' is a function of seven variables, <math>f(t, x,y,z, v_x,v_y,v_z)</math>, which gives the number of particles per unit volume in single-particle [[phase space]].<ref name="m713">{{cite journal | last=Hillery | first=M. | last2=O'Connell | first2=R.F. | last3=Scully | first3=M.O. | last4=Wigner | first4=E.P. | title=Distribution functions in physics: Fundamentals | journal=Physics Reports | volume=106 | issue=3 | date=1984 | doi=10.1016/0370-1573(84)90160-1 | pages=121–167 | url=https://linkinghub.elsevier.com/retrieve/pii/0370157384901601 | access-date=2025-07-25| url-access=subscription }}</ref> It is the number of particles per unit volume having approximately the [[velocity]] <math>\mathbf{v} = (v_x,v_y,v_z)</math> near the position <math>\mathbf{r} = (x,y,z)</math> and time <math>t</math>. The usual normalization of the distribution function is

{{Unreferenced|date=December 2009}}

<math display="block">\begin{align}

The distribution function is at the heart of [[molecular kinetic theory]]. This mathematical tool describes the statistical behavior of particles in a system. It provides a connection between the microscopic world of individual particles and the macroscopic properties observed in experiments. The distribution function gives a detailed account. It shows how particles are distributed across different positions and velocities in phase space at any given time. These quantities include [[Number density|particle density]], [[temperature]], and [[pressure]]. It contains all the information needed to describe the state of a system. This applies to both equilibrium and non-equilibrium conditions.

<math>n(\mathbf{r},t) &= \int f(\mathbf{r}, \mathbf{v}, t) \, dv_x \, dv_y \, dv_z, \\

<math>N(t) &= \int n(\mathbf{r}, t) \, dx \, dy \, dz,

\end{align} </math>

where {{math|''N''}} is the total number of particles and {{math|''n''}} is the [[number density]] of particles – the number of particles per unit volume, or the [[density]] 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, <math>f(x,y,z;p_x,p_y,p_z)</math> 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]] <math>(p_x,p_y,p_z)</math>.

== Definition ==

In the molecular kinetic theory, the distribution function is denoted as <math>f(t,x,y,z,v_x,v_y,v_z)</math>. It is defined in a [[seven-dimensional space]], which includes both the physical coordinates (position) and the velocity components, known as the [[phase space]]. The distribution function provides a detailed description of how the particles are distributed within this space at any given time <math>t</math>.

Particle distribution functions are often used in [[plasma physics]] to describe wave–particle interactions and velocity-space instabilities. Distribution functions are also used in [[fluid mechanics]], [[statistical mechanics]] and [[nuclear physics]].

'''Mathematical Definition''': The distribution function <math>f(t,x,y,z,v_x,v_y,v_z)</math> is a function of time t, position coordinates <math>(x,y,z)</math>, and velocity components <math>(v_x,v_y,v_z)</math>. This function tells us the number of particles per unit volume in phase space, which means it tells us how many particles are located in a small region around a specific position and velocity at time <math>t</math>. More specifically, if we consider a very small region in phase space centered around the point <math>(x,y,z,v_x,v_y,v_z)</math>, the quantity <math>f(t,x,y,z,v_x,v_y,v_z)dx dy dz dv_x dv_y dv_z</math> represents the number of particles within that region.

The [[Maxwell–Boltzmann distribution|basic distribution function]] uses the [[Boltzmann constant]] <math>k</math> and temperature <math>T</math> with the number density to modify the [[normal distribution]]:

'''Physical Interpretation''':

<math display="block">\begin{align}

'''Formula''': <math> f &= n\left(\frac{m}{2 \pi kT}\right)^{3/2} \exp\left({-\frac{m(v_x^2 + v_yv^2}{2 +k v_z^2)}{2kT}T}\right) </math>\\[2pt]

'''Modified Distribution Function''': <math> f(v) &= n \left(\frac{m}{2 \pi kT}\right)^{3/2} \exp\left(-\frac{m}{2kT}[(v_x-u_x)^2 + (v_y-u_y)^2 + (v_z-u_z)^2])}{2kT}\right) </math>.

\end{align} </math>

Related distribution functions may allow bulk fluid flow, in which case the velocity origin is shifted, so that the exponent's numerator is <math>m((v_x - u_x)^2 + (v_y - u_y)^2 + (v_z - u_z)^2)</math>, where <math>(u_x, u_y, u_z)</math> 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.

* '''Position and Velocity''': The function <math>f(t,x,y,z,v_x,v_y,v_z)</math> describes not just the ___location of the particles in space (given by <math>x, y, z</math>), but also their velocities (given by <math>v_x, v_y, v_z</math>). This dual dependence on both position and [[velocity]] is what makes the distribution function a powerful tool for analyzing the dynamics of a system.

* '''Phase Space''': The [[seven-dimensional space]] is called phase space. This space is formed by the position and velocity components. Each point in this space represents a unique state of a particle. This state is defined by the particle's position and velocity at a specific time. The distribution function gives a complete statistical description of the system. It specifies how many particles are in each possible state.

* '''Time Dependence''': The distribution function, <math>f(t,x,y,z,v_x,v_y,v_z)</math>, changes over time. This allows it to capture the evolution of the system. As the system evolves, the particles move through the phase space. They change their positions and velocities. The distribution function changes accordingly. This reflects the dynamic nature of the system.

[[Plasma (physics)|Plasma]] theories such as [[magnetohydrodynamics]] may assume the particles to be in [[thermodynamic equilibrium]]. In this case, the distribution function is ''[[Maxwell–Boltzmann distribution|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.

== Mathematical formulation ==

For the distribution function to be meaningful and consistent with the [[Physical property|physical properties]] of the system, it must satisfy certain normalization conditions. These conditions ensure that the distribution function correctly reflects the number density of particles and the total number of particles in the system.

The mathematical analogue of a distribution is a [[measure (mathematics)|measure]]; the time evolution of a measure on a phase space is the topic of study in [[dynamical systems]].

=== Normalization conditions ===

==== '''Particle Number Density <math>n(r,t)</math>''' ====

* The particle number density <math>n(r,t)</math> at a given position <math>r=(x,y,z)</math> and time <math>t</math> is the number of particles per unit volume in physical space. It is obtained by integrating the distribution function over all possible velocities at that position: <math>n(r,t) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(r,v,t) \, dv_x \, dv_y \, dv_z

</math>

* This integral sums up the contributions of particles with different velocities to the total number density at a specific point in space and time.

==== '''Total Number of Particles <math>N(t)</math>''' ====

* The total number of particles N(t) in the system is obtained by integrating the particle number density <math>n(r,t)</math> over the entire physical space: <math>N(t) = \int_V n(r,t) \, dx \, dy \, dz

</math>

* Here, <math>V</math> represents the volume of the system. If the system is unbounded, the integral extends over all space. This equation ensures that the distribution function accounts for all particles in the system.

=== Integral Equations for Normalization ===

The above definitions lead to the following integral equations that represent the normalization conditions for the distribution function:

==== Normalization to Particle Number Density ====

<math>n(r,t) = \int f(r,v,t) \, dv_x \, dv_y \, dv_z

</math>

* This equation shows that by integrating the distribution function over the velocity components, we obtain the number density of particles at a specific position r and time t.

==== Normalization to Total Number of Particles ====

<math>N(t) = \int n(r,t) \, dx \, dy \, dz

</math>

* Substituting the expression for n(r,t), the total number of particles can also be expressed directly in terms of the distribution function: <math>N(t) = \int \int f(r,v,t) \, dv_x \, dv_y \, dv_z \, dx \, dy \, dz

</math>

* This double integral first sums the contributions from all velocities to find the number density at each position, and then integrates over all positions to give the total number of particles.

== Special cases ==

The distribution function <math>f(t,x,y,z,v_x,v_y,v_z)</math> is a useful tool in molecular kinetic theory. It can describe many different physical systems. These include systems in thermal equilibrium and systems with complex flows and temperature variations. Some special cases of the distribution function are very important. These include the [[Maxwell–Boltzmann distribution|Maxwell-Boltzmann distribution]], shifted distributions for bulk fluid flow, and distributions that account for non-isotropic temperatures.

=== Maxwell-Boltzmann Distribution ===

The Maxwell-Boltzmann distribution is a fundamental result in statistical mechanics. It describes the distribution of particle velocities in a system that is in thermal equilibrium. When a system of particles reaches thermal equilibrium, the distribution function takes on a specific form. This form depends only on the temperature <math>T</math> of the system and the mass <math>m</math> of the particles.

'''Formula''': <math> f = n\left(\frac{m}{2 \pi kT}\right)^{3/2} \exp\left({-\frac{m(v_x^2 + v_y^2 + v_z^2)}{2kT}}\right) </math>

=== Shifted Distribution for Bulk Flow ===

In many physical situations, a system of particles may show bulk fluid flow. In this case, the particles are moving together with an average velocity. This average velocity is known as the bulk velocity. The bulk velocity is represented as <math> u = (u_x, u_y, u_z) </math>. When such a bulk flow exists, the distribution function must be adjusted to account for this overall motion of the system.

'''Modified Distribution Function''': <math> f(v) = n \left(\frac{m}{2\pi kT}\right)^{3/2} \exp\left(-\frac{m}{2kT}[(v_x-u_x)^2 + (v_y-u_y)^2 + (v_z-u_z)^2]\right) </math>

Where <math>u_x,u_y,u_z</math>​ are the components of the bulk velocity.

=== Non-Isotropic Temperatures ===

Some systems, like plasmas or anisotropic materials, may not have the same temperature in all directions. These systems have non-isotropic temperatures. The distribution function must take into account the different temperatures along the different spatial axes.

'''Modified Distribution Function''': For a system with non-isotropic temperatures <math> Tx,Ty,Tz </math> along the <math> x, y, </math> and <math> z </math> axes, the distribution function takes the form: <math> f(v) = n \left(\frac{m}{2\pi kT_x}\right)^{1/2} \left(\frac{m}{2\pi kT_y}\right)^{1/2} \left(\frac{m}{2\pi kT_z}\right)^{1/2} \exp\left(-\frac{mv_x^2}{2kT_x} - \frac{mv_y^2}{2kT_y} - \frac{mv_z^2}{2kT_z}\right)

</math>

Where <math>T_x, T_y,T_z</math> are the temperatures corresponding to the <math>x, y,</math> and <math>z</math> directions, respectively.

==References==