Physic Labs

Classical statistical mechanics

Entropy and the generalized second law

Explore the statistical meaning of entropy: vary the temperature and model parameter, compare the plots with S=kBln⁡ΩS = k_B\ln\Omega and the Gibbs form S=−kB∑rprln⁡prS = -k_B\sum_r p_r\ln p_r, and see why the second law is a probabilistic statement.

Undergraduate

Equipment

  • “Temperature” slider (0.3–3 normalized units)
  • “Parameter” slider of the distribution model
  • Rotatable 3D figure of the thermodynamic relation
  • Quantitative plot of S versus Ω with readout

Procedure

  1. Vary the temperature and watch entropy

    Drag the “Temperature” slider and watch the 3D figure and the S–Ω plot: raising the temperature increases the number of accessible microstates Ω, so entropy grows as S=kBln⁡ΩS = k_B\ln\Omega. Drag the figure to rotate and inspect it from different angles.

  2. Reshape the distribution and compare with the Gibbs formula

    Sweep the “Parameter” slider to make the microstate distribution sharper or flatter. Check that a uniform distribution over Ω states maximizes entropy: substituting pr=1/Ωp_r = 1/\Omega into S=−kB∑rprln⁡prS = -k_B\sum_r p_r\ln p_r recovers S=kBln⁡ΩS = k_B\ln\Omega.

  3. Connect with the second law

    Pick a macrostate on the plot and predict the direction of evolution of an isolated system: it drifts toward regions of larger Ω. State in words the law ΔStot≥0\Delta S_{\text{tot}} \ge 0 and explain why small systems may show transient downward fluctuations of entropy.

Simulation

Experiment history

In 1824 Sadi Carnot analyzed the limiting efficiency of heat engines, laying the ground for the second law. Rudolf Clausius coined the word “entropy” in 1865 and restated the law through this state quantity: the entropy of the universe — the largest isolated system — “tends to a maximum”; William Thomson (Kelvin) independently developed the idea of energy dissipation. In 1877 Ludwig Boltzmann supplied the statistical foundation: entropy counts microstates, S=kBln⁡WS = k_B\ln W, the formula engraved on his Vienna tombstone. Josiah Willard Gibbs generalized it to arbitrary distributions, S=−kB∑rprln⁡prS = -k_B\sum_r p_r\ln p_r; together these expressions explain why equilibrium is overwhelmingly probable and why the second law tolerates small fluctuations.

Related physicists

Rudolf ClausiusWilliam Thomson (Lord Kelvin)James Prescott JouleLudwig Eduard BoltzmannJosiah Willard GibbsSadi Carnot

Related library topics

Entropy and the generalized second law