Monoatomic Ideal Gas — Energy Fluctuations & Velocity Probabilities




Monoatomic Perfect Gas (N particles)

Below are (A) the mean-square fluctuation in energy in the canonical ensemble and (B) the probability that a chosen particle’s velocity component v_x lies in the interval [v_x,\,v_x+\Delta v_x]. Equations are provided in LaTeX and will be rendered by MathJax.

A. Mean-square fluctuation of the energy (canonical ensemble)

Start with the canonical partition function Z for the full system at temperature T. Define \beta \equiv 1/(k_B T). The canonical relations are

    \[ \langle E \rangle \;=\; -\frac{\partial \ln Z}{\partial \beta} \]

    \[ \langle (\Delta E)^2 \rangle \;=\; \langle E^2\rangle - \langle E\rangle^2 \;=\; \frac{\partial^2 \ln Z}{\partial \beta^2}. \]

Using the thermodynamic identity \partial/\partial \beta = -k_B T^2 \,\partial/\partial T (or equivalently using standard manipulations) one obtains the well-known relation

    \[ \boxed{\qquad \langle (\Delta E)^2 \rangle \;=\; k_B T^2\, C_V \qquad} \]

where C_V = \big(\partial \langle E\rangle/\partial T\big)_V is the heat capacity at constant volume.

For a classical monoatomic ideal gas

    \[ \langle E\rangle \;=\; \tfrac{3}{2} N k_B T \qquad\Rightarrow\qquad C_V \;=\; \tfrac{3}{2} N k_B . \]

    \[ \boxed{\qquad \langle (\Delta E)^2 \rangle \;=\; \tfrac{3}{2} N k_B^2 T^2 \qquad} \]

Relative fluctuation (useful scaling):

    \[ \frac{\sqrt{\langle(\Delta E)^2\rangle}}{\langle E\rangle} \;=\; \sqrt{\frac{2}{3N}} \;, \]

This shows energy fluctuations scale as N^{-1/2} and are negligible for macroscopic N.

B. Probability that a particle’s v_x lies in [v_x,\,v_x+\Delta v_x]

In the canonical ensemble for a classical ideal gas the single-particle momentum/velocity components are independent and Gaussian. The one-component Maxwell–Boltzmann probability density for v_x is

    \[ f_{v_x}(v_x) \;=\; \sqrt{\frac{m}{2\pi k_B T}}\; \exp\!\Big(-\frac{m v_x^2}{2 k_B T}\Big). \]

For a small interval \Delta v_x (infinitesimal approximation), the probability that a chosen particle has v_x in [v_x,\,v_x+\Delta v_x] is

    \[ \boxed{\qquad P\big(v_x\le v_x' < v_x+\Delta v_x\big) \;\approx\; f_{v_x}(v_x)\,\Delta v_x \;=\; \sqrt{\frac{m}{2\pi k_B T}}\,e^{-\frac{m v_x^2}{2k_B T}}\;\Delta v_x \qquad} \]

For the 3D speed (magnitude) distribution the Maxwell speed density is

    \[ f_v(v) \;=\; 4\pi\!\left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 \exp\!\Big(-\frac{m v^2}{2k_B T}\Big), \]

so the probability a chosen particle’s speed lies in [v,\,v+\Delta v] is f_v(v)\,\Delta v.

C. Probability that any of the N particles has v_x in the interval

Let

    \[ p \;=\; f_{v_x}(v_x)\,\Delta v_x \]

be the single-particle probability for that small interval. Assuming independent particles, the probability that none of the N particles lies in the interval is (1-p)^N. Thus the probability that at least one particle lies in the interval is

    \[ \boxed{\qquad P_{\text{(at least one)}} \;=\; 1-(1-p)^N \;\approx\; 1-e^{-N p} \quad(\text{for small }p)\qquad} \]

If p is very small and Np \ll 1 then P_{\text{(at least one)}}\approx N p (expected number of particles in the interval).

D. Optional: brief canonical derivation of \langle(\Delta E)^2\rangle = k_B T^2 C_V

From the partition function Z(\beta):

    \[ \langle E\rangle \;=\; -\frac{\partial \ln Z}{\partial \beta}, \qquad \langle E^2\rangle \;=\; \frac{1}{Z}\frac{\partial^2 Z}{\partial \beta^2} \;=\; \frac{\partial^2 \ln Z}{\partial \beta^2} + \left(\frac{\partial \ln Z}{\partial \beta}\right)^2. \]

Hence

    \[ \langle(\Delta E)^2\rangle \;=\; \frac{\partial^2 \ln Z}{\partial \beta^2}. \]

Noting \partial/\partial \beta = -k_B T^2 \partial/\partial T and recognizing C_V = \partial\langle E\rangle/\partial T yields the relation \langle(\Delta E)^2\rangle = k_B T^2 C_V.

E. Quick summary

  • \langle(\Delta E)^2\rangle = k_B T^2 C_V = \tfrac{3}{2}N k_B^2 T^2 for a monoatomic ideal gas.
  • Single-component velocity density: f_{v_x}(v_x)=\sqrt{\dfrac{m}{2\pi k_B T}} e^{-mv_x^2/(2k_B T)}.
  • Probability (small interval): P\approx f_{v_x}(v_x)\,\Delta v_x. For any of the N particles: 1-(1-p)^N\approx 1-e^{-Np}.

If you’d like, I can also:

  1. Provide the same page but with explicit numeric examples (choose m,T,N,v_x,\Delta v_x).
  2. Show the derivation of the Maxwell distribution from the canonical single-particle Hamiltonian step-by-step.
  3. Format this for printing (PDF-friendly) or convert to LaTeX source.