3.1 Microcanonical ensemble

Definition and concept

The microcanonical ensemble describes an isolated system where energy, volume, and particle number are all fixed. It's the most fundamental ensemble in statistical mechanics because every other ensemble (canonical, grand canonical) can be derived from it.

The core idea: if you know the total energy of an isolated system, you can figure out its thermodynamic properties by counting how many microscopic arrangements (microstates) are compatible with that energy.

Isolated systems

An isolated system exchanges neither energy nor matter with its surroundings. The three conserved quantities that define a microcanonical ensemble are:

• Energy $E$ (no heat or work exchange)
• Volume $V$ (rigid walls)
• Particle number $N$ (impermeable walls)

A perfectly insulated, sealed, rigid container is the textbook example. In practice, perfect isolation is impossible, but the framework is still essential for building up the theory. On a grand scale, the universe itself is sometimes treated as a microcanonical system since there's nothing "outside" it to exchange energy with.

Energy conservation principle

Because the system is isolated, its total energy doesn't change over time:

$\frac{dE}{dt} = 0$

This doesn't mean every particle keeps the same energy. Energy constantly redistributes among the system's internal degrees of freedom through collisions and interactions. The constraint is only on the total.

Equal a priori probability

This is the foundational postulate of statistical mechanics: all accessible microstates at a given energy are equally probable. There's no theoretical proof of this from first principles; it's an assumption justified by its success.

This postulate gives you a concrete rule for assigning probabilities. If there are $\Omega$ accessible microstates, each one has probability $1/\Omega$. The closely related ergodic hypothesis states that over long enough times, the system visits all accessible microstates, so time averages equal ensemble averages.

Mathematical formulation

Phase space

Phase space is the abstract space where each point represents a complete microscopic state of the system. For $N$ particles in three dimensions, you need 3 position coordinates and 3 momentum coordinates per particle, giving a $6N$-dimensional phase space.

A single point $(q_1, \ldots, q_{3N}, p_1, \ldots, p_{3N})$ specifies every particle's position and momentum. The microcanonical ensemble occupies a thin energy shell in this space: the set of all phase-space points where the Hamiltonian $H(p,q)$ equals the fixed energy $E$ (or falls within a narrow window $E$ to $E + \delta E$).

Density of states

The density of states $\Omega(E)$ counts how many microstates the system can access at energy $E$. For a classical system, it's defined as:

$\Omega(E) = \frac{1}{h^{3N} N!} \int \delta(E - H(p,q)) \, dp \, dq$

The factor $h^{3N}$ (where $h$ is Planck's constant) makes the count dimensionless and connects the classical expression to quantum mechanics. The $N!$ in the denominator corrects for the indistinguishability of identical particles (Gibbs' correction factor), which is necessary to make entropy an extensive quantity.

For most systems, $\Omega(E)$ increases extremely rapidly with energy. This rapid growth is what makes the microcanonical ensemble tractable: the overwhelming majority of microstates cluster near a single macroscopic condition.

Boltzmann's entropy formula

The bridge between counting microstates and thermodynamics is Boltzmann's entropy:

$S = k_B \ln \Omega(E)$

Here $k_B \approx 1.381 \times 10^{-23} \, \text{J/K}$ is Boltzmann's constant. This equation gives a statistical meaning to entropy: it measures how many microscopic arrangements are compatible with the macroscopic state. More microstates means higher entropy.

This formula also provides a statistical foundation for the second law of thermodynamics. An isolated system evolves toward the macrostate with the largest $\Omega$, which corresponds to maximum entropy.

Thermodynamic properties

Once you have $S(E, V, N)$, all equilibrium thermodynamic quantities follow from partial derivatives. This is how statistical mechanics generates thermodynamics from scratch.

Temperature derivation

Temperature is defined through:

$\frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_{V,N}$

Since $\Omega(E)$ typically grows with $E$, the derivative is positive, giving positive temperatures. However, in systems with a bounded energy spectrum (like a collection of spins in a magnetic field, where there's a maximum possible energy), $\Omega(E)$ can decrease at high energies. This leads to negative absolute temperatures, which are actually hotter than any positive temperature because energy flows from the negative-temperature system to any positive-temperature system.

Pressure calculation

Pressure comes from the volume derivative:

$P = T \left(\frac{\partial S}{\partial V}\right)_{E,N}$

This connects the microscopic picture (how the number of accessible states changes when you expand the container) to the macroscopic force per unit area on the walls.

Other thermodynamic variables

• Chemical potential: $\mu = -T \left(\frac{\partial S}{\partial N}\right)_{E,V}$, which tells you the free energy cost of adding one particle.
• Heat capacity: $C_V = \left(\frac{\partial E}{\partial T}\right)_V$, derived by inverting the temperature relation.
• Magnetic susceptibility: obtained from fluctuations in magnetization for spin systems.

Together, these derivatives provide a complete thermodynamic description from the single function $S(E, V, N)$.

Microcanonical partition function

Definition and significance

The microcanonical partition function is the density of states $\Omega(E, V, N)$. For a system with discrete energy levels $E_i$:

$\Omega(E, V, N) = \sum_i \delta(E - E_i)$

For a continuous classical system, the sum becomes the phase-space integral described above. This single quantity encodes all the statistical information about the isolated system.

Relation to entropy

The connection is direct:

$S = k_B \ln \Omega(E, V, N)$

This is the same Boltzmann formula, restated to emphasize that $\Omega$ plays the same role in the microcanonical ensemble that the partition function $Z$ plays in the canonical ensemble. Every thermodynamic property flows from taking derivatives of $S$.

Calculation methods

How you actually compute $\Omega$ depends on the system:

1. Direct counting for simple discrete systems (e.g., $N$ two-level spins: just count the number of ways to distribute energy among the spins).
2. Phase-space integration for continuous classical systems (e.g., the ideal gas, where the integral over momenta gives a known geometric result).
3. Saddle-point / Stirling approximations for large $N$, where exact counting is impractical but logarithmic approximations become extremely accurate.
4. Computational methods (Monte Carlo sampling, Wang-Landau algorithm) for complex interacting systems where analytical solutions don't exist.

Applications

Ideal gas

The ideal gas is the standard test case. For $N$ non-interacting particles in a box of volume $V$ with total energy $E$:

The accessible phase-space volume is the surface area of a $3N$-dimensional sphere of radius $\sqrt{2mE}$, multiplied by $V^N$. Working through the calculation (using the formula for the surface area of a high-dimensional sphere and applying Stirling's approximation) yields the Sackur-Tetrode equation for the entropy. From there, taking derivatives reproduces:

• The ideal gas law: $PV = Nk_BT$
• The equipartition theorem: each quadratic degree of freedom carries average energy $\frac{1}{2}k_BT$

This is a powerful consistency check: purely statistical reasoning recovers the empirical gas law.

Paramagnetic systems

Consider $N$ non-interacting spin-$\frac{1}{2}$ particles in an external magnetic field $B$. Each spin has energy $+\mu B$ or $-\mu B$. If $n$ spins point up, the total energy is $E = (N - 2n)\mu B$.

The density of states is just the binomial coefficient $\Omega = \binom{N}{n}$. Using Stirling's approximation on $\ln \Omega$ and differentiating gives the temperature, which can become negative when more than half the spins are in the higher energy state. This system is one of the cleanest examples of negative temperature physics.

Quantum systems

For systems with discrete energy levels (atoms, molecules, quantum harmonic oscillators), the microcanonical approach involves counting the number of ways to distribute energy quanta among the available states. This naturally leads into:

• Bose-Einstein statistics for indistinguishable bosons (integer spin)
• Fermi-Dirac statistics for indistinguishable fermions (half-integer spin)

The microcanonical framework shows why quantum statistics differ from classical counting: the indistinguishability of particles fundamentally changes how many distinct microstates exist.

Limitations and assumptions

Finite vs. infinite systems

The microcanonical ensemble is defined for finite systems, but most calculations use the thermodynamic limit ($N \to \infty$, $V \to \infty$, with $N/V$ held constant). In this limit, energy fluctuations become negligible relative to the total energy, and the ensemble becomes equivalent to the canonical ensemble.

For small systems (nanoparticles, small atomic clusters), finite-size effects matter. Surface contributions to entropy can't be ignored, and the equivalence between ensembles may break down.

Quantum vs. classical considerations

Classical statistical mechanics fails when the thermal de Broglie wavelength becomes comparable to the interparticle spacing. This happens at low temperatures, high densities, or for light particles (electrons, helium atoms). Signatures of this breakdown include:

• Zero-point energy: particles retain kinetic energy even at $T = 0$
• Quantum degeneracy: occupation of energy levels is restricted by particle statistics
• Heat capacities that deviate from classical predictions (e.g., the freezing out of vibrational modes in diatomic gases)

Ergodic hypothesis

The assumption that time averages equal ensemble averages is not universally valid. Systems can fail to be ergodic due to:

• Integrable dynamics (too many conserved quantities restrict the trajectory)
• Glassy behavior (the system gets trapped in metastable states for astronomically long times)
• Many-body localization in quantum systems

When ergodicity breaks down, the microcanonical ensemble may not correctly predict the system's long-time behavior.

Relation to other ensembles

Microcanonical vs. canonical

Equivalence of ensembles

In the thermodynamic limit, all ensembles give the same thermodynamic predictions. The relative energy fluctuations in the canonical ensemble scale as $1/\sqrt{N}$, which vanishes for large $N$, so fixing $T$ versus fixing $E$ makes no practical difference.

This equivalence can break down near first-order phase transitions, where the energy distribution becomes bimodal, and in systems with long-range interactions (like gravitational systems), where the thermodynamic limit itself can behave unusually.

Historical context

Boltzmann's contributions

Ludwig Boltzmann developed the statistical interpretation of entropy in the 1870s. He introduced the idea that macroscopic irreversibility arises from the overwhelming number of disordered microstates compared to ordered ones. His formula $S = k_B \ln \Omega$ is engraved on his tombstone in Vienna.

Boltzmann faced strong opposition, particularly from Ernst Mach and Wilhelm Ostwald, who questioned the existence of atoms. The eventual acceptance of atomic theory vindicated his approach.

Development of statistical mechanics

• Maxwell (1860s) developed the kinetic theory of gases and the Maxwell speed distribution.
• Boltzmann (1870s) generalized this to the full statistical framework and introduced the H-theorem.
• Gibbs (1902) formalized ensemble theory in his monograph, introducing the canonical and grand canonical ensembles.
• Einstein and Planck (early 1900s) applied statistical mechanics to quantum phenomena (specific heats of solids, blackbody radiation), revealing the limits of classical statistics.

Experimental relevance

Measuring microcanonical quantities

Directly measuring $\Omega(E)$ is rarely possible. Instead, experimentalists access microcanonical information indirectly:

• Heat capacity measurements reveal how the density of states grows with energy.
• Spectroscopy probes individual energy levels in atoms, molecules, and solids.
• Single-molecule and single-particle experiments (optical traps, ion traps) can track individual microstates in small systems.

Realizing isolated systems

True isolation is an idealization, but several experimental platforms come close:

• Ultracold atomic gases in optical or magnetic traps, where coupling to the environment is extremely weak
• Trapped ions in ultra-high vacuum, which can maintain coherence for seconds or longer
• Nuclear spin systems, where spin-lattice relaxation times can be very long, allowing the spin subsystem to behave as effectively isolated
• Space-based experiments, which exploit the natural vacuum and thermal isolation of space

Computational methods

Monte Carlo simulations

Monte Carlo methods estimate $\Omega(E)$ by randomly sampling phase space. The basic procedure:

1. Generate a random configuration of the system.
2. Compute its energy.
3. Accept or reject the configuration based on a sampling criterion.
4. Accumulate statistics over many samples.

The Metropolis algorithm is the most common variant for canonical simulations, while the Wang-Landau algorithm is specifically designed to compute the microcanonical density of states by iteratively flattening the energy histogram.

Molecular dynamics approaches

Molecular dynamics (MD) simulates the actual time evolution of particles by numerically integrating Newton's equations of motion. Because total energy is conserved in a properly integrated trajectory, MD naturally samples the microcanonical ensemble.

Key considerations include choosing a stable integrator (the velocity Verlet algorithm is standard), selecting an appropriate time step, and running long enough to adequately sample phase space. MD also gives access to dynamical properties (diffusion coefficients, correlation functions) that pure Monte Carlo cannot.