2.1 Microstates, Macrostates, and Ensemble Averages

Microstates vs Macrostates

Microstates: Specific Configurations of Particles

A microstate is one exact configuration of every particle in a system at a single instant. To fully specify a microstate, you need the position and momentum of every particle. For a gas of $N$ particles, that means recording $(x, y, z)$ and $(p_x, p_y, p_z)$ for each one. In classical mechanics, this corresponds to a single point in $6N$-dimensional phase space.

Because even tiny changes in one particle's position or momentum produce a different microstate, the number of microstates available to a macroscopic system is astronomically large.

Macrostates: Macroscopic Descriptions of Systems

A macrostate is the system described only by its measurable thermodynamic variables: temperature $T$, pressure $P$, volume $V$, internal energy $U$, and so on. A macrostate tells you nothing about where any individual particle is or how fast it's moving. Two containers of gas at the same $(P, V, T)$ are in the same macrostate even though their particles are almost certainly in completely different arrangements.

Relationship between Microstates and Macrostates

A single macrostate can be realized by an enormous number of distinct microstates. The number of microstates $\Omega$ that correspond to a given macrostate is called its multiplicity, and it directly determines how probable that macrostate is.

• Many different particle arrangements produce the same values of $P$, $V$, and $T$.
• Macrostates with larger $\Omega$ are overwhelmingly more likely to be observed. At thermodynamic scales ($N \sim 10^{23}$), the most probable macrostate dominates so completely that fluctuations away from it are negligible.
• Macroscopic properties emerge from the collective, statistical behavior of huge numbers of particles. Statistical thermodynamics exists precisely to formalize this connection.

Ensemble Averages and Thermodynamic Properties

Ensembles: A Statistical Tool

An ensemble is a (conceptual) collection of a very large number of copies of the same system, all sharing the same macroscopic constraints but each sitting in a different microstate. You choose the ensemble type based on what's held fixed:

• Microcanonical ensemble (NVE): fixed particle number, volume, and total energy. Models an isolated system.
• Canonical ensemble (NVT): fixed particle number, volume, and temperature. The system can exchange energy with a heat bath.
• Grand canonical ensemble ($\mu$VT): fixed chemical potential, volume, and temperature. The system can exchange both energy and particles with a reservoir.

Computing Ensemble Averages

An ensemble average of some property $A$ is the weighted sum of that property across all microstates:

$\langle A \rangle = \sum_i P_i \, A_i$

where $P_i$ is the probability of microstate $i$ and $A_i$ is the value of $A$ in that microstate. In the canonical ensemble, the probability follows the Boltzmann distribution:

$P_i = \frac{e^{-E_i / k_B T}}{Q}$

where $Q = \sum_i e^{-E_i / k_B T}$ is the canonical partition function. The partition function acts as a normalizing constant and turns out to encode essentially all thermodynamic information about the system.

Thermodynamic Properties as Ensemble Averages

Key thermodynamic quantities map directly onto ensemble averages:

• Internal energy: $U = \langle E \rangle$, the ensemble average of the total energy.
• Pressure: $P = \langle -\partial E / \partial V \rangle$, related to how microstate energies change with volume (connected to the virial).
• Entropy: related to the distribution of probabilities across microstates. In the canonical ensemble, $S = -k_B \sum_i P_i \ln P_i$ (the Gibbs entropy).

The Ergodic Hypothesis

The ergodic hypothesis states that, for a system at equilibrium, the time average of a property equals its ensemble average:

$\langle A \rangle_{\text{time}} = \langle A \rangle_{\text{ensemble}}$

This is what makes ensemble averages physically meaningful. Instead of needing to observe millions of copies of your system, you can (in principle) track one system long enough and get the same result. The hypothesis holds for most systems at equilibrium, and it's the foundational assumption that lets statistical mechanics make contact with experiment.

Entropy and Number of Microstates

The Boltzmann Equation

The entropy of a system is directly tied to the number of accessible microstates through the Boltzmann equation:

$S = k_B \ln \Omega$

• $S$ is the entropy.
• $k_B = 1.381 \times 10^{-23}$ J/K is the Boltzmann constant, which sets the scale between microscopic energies and macroscopic temperature.
• $\Omega$ is the number of microstates accessible to the system at a given energy.

The logarithm is essential here: it makes entropy an extensive (additive) property. If you combine two independent subsystems with multiplicities $\Omega_1$ and $\Omega_2$, the total multiplicity is $\Omega_1 \cdot \Omega_2$, but the total entropy is $S = k_B \ln(\Omega_1 \Omega_2) = k_B \ln \Omega_1 + k_B \ln \Omega_2 = S_1 + S_2$. Without the logarithm, entropy wouldn't add up the way thermodynamics requires.

Numerical example: For a system with $\Omega = 10^{23}$ microstates:

$S = k_B \ln(10^{23}) = (1.381 \times 10^{-23}\;\text{J/K})(23 \ln 10) = (1.381 \times 10^{-23})(52.96) \approx 7.31 \times 10^{-22}\;\text{J/K}$

This is a tiny entropy, reflecting that $10^{23}$ microstates is actually quite small by thermodynamic standards. Real macroscopic systems have multiplicities more like $10^{10^{23}}$.

Entropy and the Second Law of Thermodynamics

The second law states that the entropy of an isolated system never decreases. Statistical thermodynamics reveals why: the system evolves toward the macrostate with the largest $\Omega$ simply because that macrostate is overwhelmingly more probable.

• Spontaneous processes move the system toward higher-multiplicity macrostates.
• At equilibrium, the system occupies the macrostate that maximizes $\Omega$ (subject to constraints like fixed total energy).
• The second law is fundamentally a statistical law. A decrease in entropy isn't impossible, just fantastically improbable for macroscopic systems.

A classic example: two gases initially separated by a partition will mix spontaneously when the partition is removed. The mixed state has far more accessible microstates than the separated state, so $\Omega$ increases and entropy rises. The reverse process (spontaneous unmixing) would require the system to find its way back to a vanishingly small subset of microstates, which is statistically negligible for $N \sim 10^{23}$ particles.