Distribution function (physics)

This article is about the distribution function as used in physics. For the related mathematical concepts, see cumulative distribution function and probability density function.

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 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.

Definition

In the molecular kinetic theory, the distribution function is denoted as $f(t,x,y,z,v_{x},v_{y},v_{z})$. 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 $t$.

Mathematical Definition: The distribution function $f(t,x,y,z,v_{x},v_{y},v_{z})$ is a function of time t, position coordinates $(x,y,z)$, and velocity components $(v_{x},v_{y},v_{z})$. 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 $t$. More specifically, if we consider a very small region in phase space centered around the point $(x,y,z,v_{x},v_{y},v_{z})$, the quantity $f(t,x,y,z,v_{x},v_{y},v_{z})dxdydzdv_{x}dv_{y}dv_{z}$ represents the number of particles within that region.

Physical Interpretation:

• Position and Velocity: The function $f(t,x,y,z,v_{x},v_{y},v_{z})$ describes not just the location of the particles in space (given by $x,y,z$), but also their velocities (given by $v_{x},v_{y},v_{z}$). 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, $f(t,x,y,z,v_{x},v_{y},v_{z})$, 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.

Mathematical formulation

For the distribution function to be meaningful and consistent with the 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.

Normalization conditions

Particle Number Density $n(r,t)$

• The particle number density $n(r,t)$ at a given position $r=(x,y,z)$ and time $t$ 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: $n(r,t)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(r,v,t)\,dv_{x}\,dv_{y}\,dv_{z}$

• 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 $N(t)$

• The total number of particles N(t) in the system is obtained by integrating the particle number density $n(r,t)$ over the entire physical space: $N(t)=\int _{V}n(r,t)\,dx\,dy\,dz$
• Here, $V$ 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

$n(r,t)=\int f(r,v,t)\,dv_{x}\,dv_{y}\,dv_{z}$

• 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

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

• Substituting the expression for n(r,t), the total number of particles can also be expressed directly in terms of the distribution function: $N(t)=\int \int f(r,v,t)\,dv_{x}\,dv_{y}\,dv_{z}\,dx\,dy\,dz$
• 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 $f(t,x,y,z,v_{x},v_{y},v_{z})$ 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, 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 $T$ of the system and the mass $m$ of the particles.

Formula: $f=n\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left({-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}}\right)$

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 $u=(u_{x},u_{y},u_{z})$. When such a bulk flow exists, the distribution function must be adjusted to account for this overall motion of the system.

Modified Distribution Function: $f(v)=n\left({\frac {m}{2\pi kT}}\right)^{3/2}\exp \left(-{\frac {m}{2kT}}[(v_{x}-u_{x})^{2}+(v_{y}-u_{y})^{2}+(v_{z}-u_{z})^{2}]\right)$

Where $u_{x},u_{y},u_{z}$​ 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 $Tx,Ty,Tz$ along the $x,y,$ and $z$ axes, the distribution function takes the form: $f(v)=n\left({\frac {m}{2\pi kT_{x}}}\right)^{1/2}\left({\frac {m}{2\pi kT_{y}}}\right)^{1/2}\left({\frac {m}{2\pi kT_{z}}}\right)^{1/2}\exp \left(-{\frac {mv_{x}^{2}}{2kT_{x}}}-{\frac {mv_{y}^{2}}{2kT_{y}}}-{\frac {mv_{z}^{2}}{2kT_{z}}}\right)$

Where $T_{x},T_{y},T_{z}$ are the temperatures corresponding to the $x,y,$ and $z$ directions, respectively.