Microcanonical Ensemble StatMech by Huang, Kerson

Chapter 6: Classical Statistical Mechanics

6.1 The Postulate of Classical Statistical Mechanics

• Statistical Mechanics Overview

  • Concerned with equilibrium properties of macroscopic molecular systems.

  • Aims to derive equilibrium properties from molecular dynamics.

  • Does not describe how systems approach equilibrium; only outlines what equilibrium is.

• Kinetic Theory of Gases

  • Approach to equilibrium is complex, but equilibrium state (Maxwell-Boltzmann distribution) is simpler.

  • Generalizes the method to discuss any macroscopic system's equilibrium.

• Characterizing Classical Systems

  • A system contains a large number (N) of molecules in a large volume (V): typical values are:

    • N ≈ 10^23

    • V ≈ 10^2 (m³)

  • Limiting case considered: N → ∞ and V → ∞, where specific volume (V/N) remains finite.

6.2 Isolated Systems and Phase Space

• Isolated System Definition

  • System's energy is constant.

  • Real-world systems interact, but weak interactions approximate isolation.

  • Dynamics determined by Hamiltonian H(p, q).

• Canonical Coordinates and Phase Space

  • State defined by 3N coordinates (q) and 3N momenta (p).

  • Combined notation: (p, q).

  • Phase space is 6N-dimensional; a point in phase space represents a system's state.

  • Energy surfaces defined by H(p, q) = E.

• Gibbsian Ensemble

  • No single system; visualize an ensemble of many systems.

  • Density function p(p,q,t) represents probability across states in phase space.

• Liouville's Theorem

  • States that the distribution in phase space is conserved over time, behaving like an incompressible fluid.

6.3 Postulate of Equal A Priori Probability

• Foundational Postulate

  • In thermodynamic equilibrium, all states satisfying macroscopic conditions are equally likely.

  • Forms the basis for the microcanonical ensemble:

    • p(p,q) = Const. if E < H(p,q) < E + Δ and 0 otherwise.

• Measurable Properties and Averages

  • For a property f(p,q), its observed value is averaged over the ensemble.

  • Two types of averages: most probable value and ensemble average:

    • Most probable value: most common f(p,q).

    • Ensemble average defined through integration: <f> =[ \frac{1}{W(E)} \int f(p, q) p(p,q) d^{3N}p d^{3N}q ]

6.4 Microcanonical Ensemble

• Entropy Definition

  • Fundamental link to thermodynamics; defined as:

    • S(E,V) = k log Γ(E) (where Γ(E) relates to volume in phase space).

  • Properties include:

    • Extensive: S₁ + S₂ = S for two subsystems.

    • S increases with V (second law of thermodynamics).

6.5 Derivation of Thermodynamics

• Connecting with Thermodynamics

  • Quasistatic transformations maintain microcanonical ensemble states at each stage.

  • Entropy change during transformation described by:

    • dS = 1/T dE + P dV/T.

  • Leads to the first law of thermodynamics:

    • dE = T dS - P dV.

6.6 Classical Ideal Gas

• Hamiltonian Structure of Ideal Gas

  • Statistical properties derived from Hamiltonian and density of states calculations.

  • Entropy expression:

    • S(E,V) is functionally related to log of phase space dimensions.

• Gibbs Paradox

  • Observes contradictions when mixing identical gases.

  • Solution requires adjusting for indistinguishable particles: S = Nk log(V/u^3/2) - log(N!).

6.7 Conclusion

• Correct Boltzmann Counting

  • Essential for accurate thermodynamic descriptions of systems, leads to consistent entropy formulations.