Microcanonical Ensemble: Complete Guide to Probable Distribution of Energies

Prebable Distribution of Energies (Microcanonical Ensemble)

Equal a-priori probabilities, microcanonical probability density \(p(E)\), Dirac-delta formulation, and a worked two-particle example — Math in LaTeX for Blogger.

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Contents

• Prebable Distribution of Energies (microcanonical)
• Equal a priori probabilities
• Microcanonical probability formula
• Dirac delta viewpoint
• Nature of probability — two-particle example

Prebable Distribution of Energies (Microcanonical Ensemble)

Statement (short)

Let the energies of accessible states (microstates) lie between \(E\) and \(E+dE\). For an isolated system (fixed \(E,V,N\)) the probability of finding the system with energy in that small interval is constructed from the number (or density) of states \(\Omega(E)\).

Fundamental postulate

All accessible microstates of an isolated system are equally probable.

\[ = P_{\text{micro}} \;=\; \frac{1}{\Omega(E)} \]

Each accessible microstate has probability \(1/\Omega(E)\).

Equal a priori probabilities & normalization

If there are \(\Omega(E)\) microstates in the shell \(E\) to \(E+dE\), and each microstate has equal probability, the probability to find the system with energy in that shell is proportional to \(\Omega(E)\,dE\). Normalizing over all energies gives the microcanonical probability density.

\[ = p(E)\,dE \;=\; \frac{\Omega(E)\,dE}{\displaystyle\int \Omega(E')\,dE'} \]

This expression states that the probability to find the system with energy between \(E\) and \(E+dE\) equals the number of states in that interval divided by the total number of states (integrated over the allowed energy range).

Microcanonical probability — compact forms

Often one writes the probability in proportional form (useful for formal manipulations):

\[ = p(E) \;\propto\; \Omega(E) \]

Or, if one imposes an exact energy constraint using the Dirac delta, the microcanonical distribution can be written as a normalized delta-weighted measure on phase space (classical description). See the next section.

Dirac-delta formulation (brief)

When we want to restrict the ensemble strictly to energy \(E\) (an energy shell), we use the Dirac delta \(\delta(H-E)\) in the phase-space integral. This is the usual classical expression for the density of states on the energy surface.

\[ = \Omega(E) \;=\; \frac{1}{h^{3N}N!}\int \delta\big(H(q,p)-E\big)\; d^{3N}q\,d^{3N}p \]

Here the prefactor \(h^{3N}N!\) enforces quantum counting conventions (Gibbs correction) when comparing classical phase-space volumes to quantum state counts.

Nature of Probability — worked example (two particles)

Consider a simple classical model: two non-interacting particles of mass \(m\) constrained to move on a line segment of length \(L\). The phase space coordinates are \((x_1,p_1,x_2,p_2)\).

The microcanonical constraint (fixed total energy \(E\)) restricts the momentum coordinates by:

\[ = E \;=\; \frac{p_1^2}{2m} \;+\; \frac{p_2^2}{2m} \]

In the two-dimensional momentum plane \((p_1,p_2)\) this relation is the equation of a circle of radius \(\sqrt{2mE}\). Therefore all accessible momentum microstates (at that exact energy) lie on the circumference (energy shell). If we allow a small energy window \(E\) to \(E+dE\), we get a thin annulus (ring) in momentum space.

Geometric observation

• The radius of the energy circle in \(p\)-space is \( \sqrt{2mE} \).
• If we count states in a small window \(dE\), the accessible region becomes a ring whose area is related to \(\Omega(E)\,dE\).

Thus the microcanonical ensemble selects momentum points on (or near) this circle; the spatial coordinates \(x_1,x_2\) are free within \([0,L]\) and contribute a multiplicative factor \(L^2\) to the phase-space volume (or multiplicity) for this simple model.

\[ = \text{radius in }p\text{-space} \;=\; \sqrt{2mE} \]

\[ = \text{phase-space factor from positions} \;=\; L^2 \]

Combining momentum shell measure and the configuration space factor gives the multiplicity (up to constant prefactors) for this two-particle example.

Quick FAQs

Q: Why is \(\Omega(E)\) sometimes a density and sometimes a count?

A: In continuous classical phase space \(\Omega(E)\) is a density (states per energy) obtained from a delta function integral. For discrete quantum systems \(\Omega(E)\) is a count of energy eigenstates (or of states in a small energy window).

Q: What role does Dirac delta play?

A: It enforces the exact energy constraint by selecting the energy surface \(H=E\) inside the phase-space integral.

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