Thermodynamics & Statistical Mechanics
Thermodynamics studies energy and its transformations.
It describes macroscopic systems with a few variables.
Statistical mechanics links those variables to microscopic motion.
First, energy cannot be created or destroyed.
Therefore, the total energy of an isolated system is constant.
Second, entropy tends to increase in a closed system.
Consequently, processes move toward greater disorder.
Moreover, temperature measures the average kinetic energy of particles.
Thus, heat flows from hot to cold bodies.
In addition, pressure arises from particle collisions with walls.
Statistical mechanics begins with a large number of particles.
It assumes each microstate is equally likely.
Therefore, macroscopic properties are ensemble averages.
The microcanonical ensemble fixes energy, volume, and particle number.
It yields the fundamental relation S = k ln Ω.
The canonical ensemble allows energy to fluctuate with a heat bath.
It introduces the partition function Z.
From Z we derive free energy, entropy, and pressure.
Hence, thermodynamic quantities follow from microscopic models.
Phase transitions occur when free energy changes non‑analytically.
They are classified by order and symmetry breaking.
Critical points show universal behavior independent of material details.
Applications range from ideal gases to solids and magnets.
They also include black holes and quantum fluids.
Thus, the framework unifies diverse physical phenomena.
In summary, thermodynamics gives the rules of energy exchange.
Statistical mechanics explains those rules with particle statistics.
Together they provide a complete picture of matter at all scales.