Microstates and macrostates

The statistical viewpoint

Thermodynamics describes systems using macroscopic variables such as pressure, volume, temperature, and entropy. Statistical mechanics explains those macroscopic properties in terms of microscopic particles and probability.

A macrostate is a large-scale description of a system. For a gas, a macrostate might specify $N$, $V$, and $T$. A microstate gives the detailed positions and momenta, or quantum states, of all particles consistent with that macrostate.

Many different microstates can correspond to the same macrostate.

Multiplicity

The number of microstates corresponding to a macrostate is called multiplicity, often written $Omega$. Boltzmann connected entropy to multiplicity with

$S=k_BlnOmega$

This equation says that entropy measures how many microscopic ways a macroscopic state can occur.

The logarithm is important because entropies add for independent systems, while multiplicities multiply. If two independent systems have multiplicities $Omega_1$ and $Omega_2$, the combined multiplicity is

$Omega_{total}=Omega_1Omega_2$

Then

$S_{total}=k_Bln(Omega_1Omega_2)=S_1+S_2$

Coin model analogy

A simple analogy uses coin flips. Suppose 100 coins are tossed. A macrostate might be the number of heads. A microstate is the exact sequence of heads and tails.

There is only one microstate with 100 heads, but there are enormous numbers of microstates with about 50 heads. Therefore the macrostate near 50 heads is overwhelmingly more probable.

Similarly, a gas spread throughout a room is more probable than all molecules gathering in one corner because there are vastly more microstates corresponding to the spread-out macrostate.

Equilibrium as most probable macrostate

In statistical mechanics, equilibrium is the macrostate with overwhelmingly large multiplicity. Systems evolve toward equilibrium not because every microstate is forced in one direction, but because high-multiplicity macrostates dominate the possibilities.

For macroscopic systems with around $10^{23}$ particles, the probability difference is so enormous that equilibrium behavior appears deterministic.

Constraints and accessible microstates

Not all microstates are accessible. Conservation of energy, particle number, volume, and other constraints restrict what states are possible. Statistical mechanics counts or weights the microstates consistent with constraints.

In an isolated system with fixed energy, the microcanonical ensemble treats accessible microstates as equally likely.

Fluctuations

Small systems can fluctuate noticeably away from equilibrium. Large systems can also fluctuate, but significant fluctuations are extraordinarily unlikely. This explains why thermodynamics works so reliably for macroscopic systems.

Entropy can briefly decrease in small systems, but the second law emerges statistically for large systems.

Information and coarse-graining

A macrostate is a coarse-grained description. It ignores microscopic details. Entropy is related to how much microscopic information is missing when only the macrostate is known.

This does not mean entropy is subjective in a simple sense. For specified macroscopic constraints, multiplicity is a physical count of compatible microscopic possibilities.

The big idea

Statistical mechanics connects thermodynamics to counting and probability. A macrostate is a coarse description; a microstate is a detailed microscopic arrangement. Entropy measures multiplicity through $S=k_BlnOmega$, and equilibrium corresponds to the overwhelmingly most probable macrostate.