2: Microscopic States and Macrostates

Section 1.2: Microscopic States and Macrostates

Statistical mechanics bridges the gap between the chaotic, detailed motion of individual particles and the predictable, averaged behavior of bulk matter. This connection hinges on two fundamental concepts: microscopic states (microstates) and macroscopic states (macrostates).

A microstate describes a system with complete microscopic detail. For a classical system of $N$ particles, this corresponds to a single, unique point in the $6N$-dimensional $\Gamma$-space (phase space), specifying every particle's position $(q_x, q_y, q_z)$ and momentum $(p_x, p_y, p_z)$ at a given instant. For a quantum system, a microstate is a unique, well-defined quantum state of the entire system, specified by a complete set of quantum numbers. Microstates are dynamically evolving: particles move and collide, causing the system to traverse countless microstates over time. Observing or calculating individual microstates for a macroscopic system (e.g., $10^{23}$ particles) is utterly impossible.

A macrostate, in contrast, describes the system using only a few macroscopic, experimentally accessible variables. These variables – like:

• Total energy ($E$)
• Volume ($V$)
• Number of particles ($N$)
• Magnetization ($M$)
• Pressure ($P$)

represent coarse-grained averages or sums over the microscopic constituents. Crucially, a single macrostate corresponds to a vast number (often an astronomically large number) of microstates that are consistent with its defining macroscopic constraints. For example, the macrostate defined by $E$, $V$, and $N$ encompasses every possible microscopic configuration where the particles collectively have the exact total energy $E$, occupy the volume $V$, and number $N$, regardless of which specific particles have which positions and momenta.

The core idea is that the macroscopic properties we measure (temperature, entropy, etc.) are not properties of individual microstates, but emerge from the statistical behavior of the enormous ensemble of microstates compatible with a given macrostate. The multiplicity ($\Omega$) of a macrostate is the number of microstates associated with it. Statistical mechanics postulates that an isolated system in equilibrium spends equal time in all accessible microstates. Consequently, the observed macrostate is overwhelmingly likely to be the one with the greatest multiplicity $\Omega$. This explains why systems evolve towards equilibrium: it's the macrostate encompassing the maximum number of microstates, making it statistically inevitable. Understanding this link between the microscopic (individual states) and the macroscopic (collective properties via multiplicity) is the foundation for deriving thermodynamics from mechanics.