Maxwell–Boltzmann statistics (MBS) describes how particles in a gas distribute themselves in momentum or velocity when the gas is in thermal equilibrium. The distribution arises from three fundamental constraints:

1. All directions in velocity space are equally probable (isotropy).
2. The total energy of the system is finite.
3. The total number of particles is fixed.

Condition (1) ensures a wide range of possible particle speeds, while (2) and (3) constrain the total spread of those speeds.

Consider a gas of identical particles of mass \(m\) in equilibrium at temperature \(T\). The probability \(\mathsf{P}(v_x)\,dv_x\) of finding a particle with velocity between \(v_x\) and \(v_x+dv_x\) is given by the Maxwell–Boltzmann distribution:

$$ \mathsf{P}(v_x)\,dv_x = \left(\frac{m}{2\pi kT}\right)^{1/2} e^{-mv_x^2/(2kT)}\,dv_x, $$

where \(k\) is Boltzmann’s constant. This Gaussian form shows that most particles have moderate velocities, while both very slow and very fast particles are less common.

—

Because \(p_x = m v_x\), the same distribution can be expressed in terms of momentum:

$$ \mathsf{P}(p_x)\,dp_x = \left(\frac{1}{2\pi m kT}\right)^{1/2} e^{-p_x^2/(2mkT)}\,dp_x. $$

Extending to three dimensions:

$$ \mathsf{P}(p)\,d^3p = \left(\frac{1}{2\pi m kT}\right)^{3/2} \exp\!\left(-\frac{p^2}{2mkT}\right)\,d^3p, $$

where \(p^2 = p_x^2 + p_y^2 + p_z^2\). The quantity \(\mathsf{P}(p)\) is dimensionless — the units of \(d^3p\) cancel those of \(\mathsf{P}(p)\). Thus, \(\mathsf{P}(p)\,d^3p\) gives the probability of finding a particle in a small volume of momentum space.

The distribution may also be written in terms of energy using \(E = p^2 / (2m)\):

$$ \mathsf{P}(p)\,d^3p = \left(\frac{1}{2\pi m kT}\right)^{3/2} e^{-E/(kT)}\,d^3p. $$

Hence, \(\mathsf{P}(p) \propto e^{-E/(kT)}\), showing that the probability of finding particles with energy greater than \(kT\) falls exponentially — this is the hallmark of Maxwell–Boltzmann statistics.

—

For an isotropic distribution, the probability of finding a particle in a spherical shell of radius \(p\) and thickness \(dp\) is

$$ \mathsf{P}(p)\,dp = 4\pi p^2 \left(\frac{1}{2\pi m kT}\right)^{3/2} e^{-p^2/(2mkT)}\,dp. $$

This 1D form represents a 3D gas compactly. The function peaks at a finite momentum \(p_{\text{max}} > 0\) because:

• at \(p = 0\), the \(p^2\) term drives the probability to zero;
• at large \(p\), the exponential suppression dominates.

The temperature dependence (\(T^{-3/2}\)) ensures that the curve flattens and broadens with increasing temperature, reflecting faster-moving particles.

—

The phase space of a system combines position space \((x, y, z)\) and momentum space \((p_x, p_y, p_z)\) into a 6D continuum. Each point in this space corresponds to one possible microstate of a particle.

If \(N\) particles occupy a total volume \(V\), their number density is \(n = N/V\). The phase-space distribution function for Maxwell–Boltzmann statistics is then

$$ f_{MB} = n\,\mathsf{P}(p) = \frac{N}{V} \left(\frac{1}{2\pi m kT}\right)^{3/2} \exp\!\left(-\frac{p^2}{2mkT}\right), $$

where \(f_{MB}\) has units of m\(^{-3}\)(Ns)\(^{-3}\) or equivalently (Js)\(^{-3}\). This quantity represents the number of particles per unit phase-space volume — often called the phase-space density or simply the distribution function.

—

Observable quantities can be derived by integrating \(f\) over momentum space:

• Number density:

$$ n(x,y,z) = \int f(x,y,z,p)\,d^3p $$

  (particles per cubic meter)

• Particle flux density:

$$ \phi_p(x,y,z,t) = \int v\,f\,d^3p $$

  (particles per square meter per second)

—

A closely related quantity in radiative processes is the specific intensity, \(J\), which measures the directional flux of particles or photons:

$$ J = \frac{dN}{dA\,d\Omega\,dE\,dt}, $$

the number of particles passing through area \(dA\) with energy between \(E\) and \(E+dE\) within solid angle \(d\Omega\) per unit time.

From the definition of \(f\):

• volume element in physical space: \(dV = v\,dt\,dA\),
• volume element in momentum space: \(dV_{mom} = p^2\,d\Omega_p\,dp\).

Hence the number of particles in phase space:

$$ dN = f\,p^2\,d\Omega_p\,dp\,v\,dt\,dA. $$

Comparing this with the definition of \(J\):

$$ J\,dA\,d\Omega\,dE\,dt = f\,p^2\,d\Omega_p\,dp\,v\,dt\,dA, $$

and assuming \(d\Omega = d\Omega_p\), we find

$$ J\,dE = f\,v\,p^2\,dp. $$

From special relativity, \(E^2 = (pc)^2 + (mc^2)^2\), so

$$ E\,dE = c^2 p\,dp, $$ and since \(v = p c^2 / E\), substituting gives

$$ J = p^2 f. $$

Thus, specific intensity is simply the distribution function multiplied by \(p^2\):

\begin{equation}\label{J} J = p^2 f. \end{equation}

—

The units of particle-specific intensity are m\(^{-2}\) s\(^{-1}\) J\(^{-1}\) sr\(^{-1}\), representing particle flux per unit energy per steradian.

For energy-specific intensity \(I(\nu)\):

$$ I(\nu) = \frac{E J(E)\,dE}{d\nu}. $$

Using \(E = h\nu\) and \(dE = h\,d\nu\):

$$ I = J\,h^2\nu. $$

Since \(J = p^2 f\) and \(p = h\nu / c\):

\begin{equation}\label{I} I(\nu) = \frac{h^4 \nu^3}{c^2} f. \end{equation}

This equation links the energy-specific intensity directly to the distribution function.

—

Liouville’s theorem states that for an observer moving with a stream of particles, the distribution function \(f\) — and hence \(J\) and \(I\) — remains constant along the trajectory:

$$ \frac{df}{dt} = 0. $$

This implies that the specific intensity is conserved along a ray in free space. Therefore, the surface brightness of an astronomical object seen by an observer is equal to its intrinsic brightness at the source:

\begin{equation}\label{B} B(\nu,\theta_s,\phi_s) = I(\nu,\theta_o,\phi_o), \end{equation}

where \((\theta_s,\phi_s)\) and \((\theta_o,\phi_o)\) denote angular coordinates at the source and at the observer, respectively.

—

• The Maxwell–Boltzmann distribution arises from the most probable energy partition among many identical, distinguishable particles.
• It predicts that few particles have very low or very high momenta; most lie near the peak at \(E \approx kT\).
• In three dimensions, \(P(p)\,dp \propto p^2 e^{-p^2/(2mkT)}\).
• The distribution function \(f_{MB}\) serves as the foundation for computing observable macroscopic quantities.
• Specific intensity and surface brightness are conserved along a ray path, illustrating the connection between thermodynamics and radiative transfer.

1. Derive the expression for the Maxwell–Boltzmann distribution in momentum space from the 1D velocity distribution.
2. Why does the most probable momentum occur at \(p>0\) instead of \(p=0\)?
3. How does increasing temperature affect the shape and peak of the distribution?
4. Explain the physical meaning of the phase-space distribution function \(f_{MB}\).
5. Show how Liouville’s theorem leads to conservation of surface brightness in astronomy.