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====== Maxwell–Boltzmann statistics ======

**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:

  - All directions in velocity space are equally probable (isotropy).
  - The total energy of the system is finite.
  - 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.

===== One-dimensional distribution =====

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.

===== Momentum space =====

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

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

{{:courses:ast301:mbd.webp?nolink&750|}}

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**.

===== Spherical representation in momentum space =====

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.

===== Phase space =====

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.

{{:courses:ast301:momentum-phase.webp?nolink&750|}}

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

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

These correspond to the number of particles per unit volume and the number streaming through a unit area per second, respectively.

===== Insights =====
  - 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 \(f_{MB}\) forms the basis for computing observable macroscopic quantities like density, flux, and pressure.

===== Inquiries =====
  - Derive the expression for the Maxwell–Boltzmann distribution in momentum space from the 1D velocity distribution.
  - Why does the most probable momentum occur at \(p>0\) instead of \(p=0\)?
  - How does increasing temperature affect the shape and peak of the distribution?
  - Explain the physical meaning of the phase-space distribution function \(f_{MB}\).
  - How can macroscopic observables like pressure or flux be obtained from \(f_{MB}\)?