Physic Labs

Classical statistical mechanics

Statistical ensembles: microcanonical and canonical

Study a model of a small system exchanging energy with a heat bath: once isolated constraints are relaxed, microstate probabilities follow the Boltzmann distribution Pr∝e−βErP_r\propto e^{-\beta E_r} with β=1/(kBT)\beta=1/(k_BT) instead of being equal. Vary the temperature to watch the distribution move between the two ensemble limits.

Advanced

Equipment

  • Virtual small system immersed in a heat bath on a 3D canvas
  • Temperature slider T (normalized)
  • Model-parameter slider
  • Quantitative plot of microstate probabilities

Procedure

  1. Observe energy exchange

    In figure 1, set T at a mid value: particles of the small system constantly exchange energy with the surrounding bath — the canonical-ensemble condition (fixed T, V, N), unlike the isolated microcanonical one (fixed E, V, N).

  2. Read the distribution versus temperature

    In figure 2, lower the temperature: probability piles into the lowest-energy microstate. Raise T: higher states get populated as e−βEe^{-\beta E}; as T→∞T\to\infty the distribution flattens — approaching the microcanonical limit within the allowed energy window.

  3. Estimate the partition function and predict

    Use the normalized readout: the total weight Z=∑re−βErZ=\sum_r e^{-\beta E_r} shrinks as T rises because high states add more to the sum. Move the parameter slider to shift the level spacing, predict the log-distribution slope, then check the plot.

Simulation

Experiment history

Josiah Willard Gibbs laid the formal foundations in Elementary Principles in Statistical Mechanics (1902): he defined a statistical ensemble as a conceptual collection of copies of one system — microcanonical for isolated systems, canonical for systems in contact with a bath — a tool for computing macroscopic averages, not a physical object. The probabilistic view of states came earlier: Ludwig Boltzmann in the 1870s tied entropy to the microstate count S=kBln⁡WS=k_B\ln W, and James Clerk Maxwell gave the velocity distribution in 1860. Boltzmann's ergodic hypothesis — that a long trajectory samples the energy surface uniformly — was the first attempt to justify the microcanonical ensemble's uniformity.

Related physicists

Rudolf ClausiusWilliam Thomson (Lord Kelvin)James Prescott JouleJosiah Willard GibbsLudwig Eduard Boltzmann

Related library topics

Statistical ensembles: microcanonical and canonical