Macrostates, microstates, and entropy

Gas at the same temperature and pressure can have many arrangements of particle positions and momenta. One detailed arrangement is a microstate. A macrostate is specified by macroscopic conditions such as temperature and pressure; many compatible micro-arrangements can correspond to it. A simple Boltzmann expression is S=kB ln Ω, where Ω counts compatible microstates. Boltzmann’s 1877 paper in English translation connects a distribution’s probability to the number of arrangements that realize it. MIT course notes distinguish macrostates and microstates and explain the relation.

A small counting example

Assume 10 distinguishable particles can each occupy a left or right box, independently and with equal chance. There are 2¹⁰=1,024 assignments. The 5–5 macrostate corresponds to choosing five of ten particles for the left box: 252 arrangements. “All left” has one arrangement, as does “all right.” This toy model shows why a balanced macrostate can correspond to more microstates; it is not a force pushing particles toward disorder. Ten particles are not a realistic gas, so do not transfer its exact probabilities to room air.

Local decrease and the second law

Cooling a refrigerator’s interior can lower the entropy there. But a refrigerator uses electricity and releases heat into the room, so its interior is not isolated. Choose a boundary that includes the contents, appliance, and room before considering the whole change. MIT’s second-law notes use expansion and thermal equilibration to discuss system boundaries.

Consider N ideal-gas particles initially confined to volume V in a rigid, insulated container. Removing the partition lets them expand freely into 2V. With no heat exchange or work, the initial and final equilibrium temperatures are the same. Each particle has twice the positional volume, so the microstate ratio is 2^N and ΔS=NkB ln 2. Free expansion is irreversible, but the state-function difference can be computed along a reversible path between the same equilibrium endpoints.

This simple form counts accessible microstates with equal weight under specified macroscopic constraints. If microstates have different probabilities, one count Ω is not enough; the probability distribution must be included in the statistical definition.

Conditions and limits

The key statement is that total entropy for an isolated system does not decrease in a spontaneous process. Statistical explanations depend on the selected system, constraints such as energy, and how states are counted. Equilibrium macrostates dominate in large systems, but this is a statistical tendency, not a claim that every microscopic motion points in one direction at every instant. Small systems can fluctuate.

Sources