Thermodynamics describes the macroscopic behavior of physical systems in terms of quantities such as energy, temperature, entropy, and pressure. Its laws are empirical and surprisingly universal, applying across a wide range of physical systems without reference to microscopic details. Statistical mechanics provides the theoretical framework that connects these macroscopic laws to the underlying dynamics of many-particle systems.
At the microscopic level, a system is described by a large number of degrees of freedom evolving according to classical or quantum mechanics. Rather than tracking individual trajectories, statistical mechanics characterizes the system through probability distributions on phase space, allowing macroscopic observables to be defined as ensemble averages.
Microstates, Macrostates, and Ensembles
A microstate specifies the complete microscopic configuration of a system, while a macrostate is defined by a small number of macroscopic variables such as total energy, volume, and particle number. Many distinct microstates may correspond to the same macrostate.
To relate microscopic dynamics to macroscopic behavior, statistical mechanics introduces ensembles, which are probability distributions over microstates consistent with specified macroscopic constraints. Common examples include the microcanonical, canonical, and grand canonical ensembles.
Entropy and the Second Law
In statistical mechanics, entropy is interpreted as a measure of the number of microscopic configurations compatible with a given macrostate. For a discrete set of microstates with probabilities \(p_i\), the entropy is given by
This statistical definition provides a microscopic explanation for the second law of thermodynamics: macroscopic systems evolve toward states with higher entropy because such states correspond to overwhelmingly many microscopic configurations.
Thermodynamics as a Macroscopic Theory
Thermodynamics emerges as a closed, macroscopic description in which detailed microscopic information is no longer required. Its laws impose constraints on energy exchange, work, heat, and entropy production, and they place fundamental limits on efficiency and achievable performance.
From an engineering perspective, thermodynamics functions as a constraint theory rather than a dynamical one. It determines what processes are possible, what losses are unavoidable, and what performance limits cannot be exceeded, regardless of the specific mechanisms involved.
Together, statistical mechanics and thermodynamics provide a bridge between microscopic dynamics and macroscopic behavior, linking Hamiltonian evolution at the particle level to emergent laws governing energy, irreversibility, and equilibrium.