ISEGORIA / MATH ENCYCLOPEDIA
Statistical mechanics: microstates and temperature
Boltzmann weights, partition functions, and random walks.
Before you begin: Probability and energy
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Boltzmann weights
A two-level system with energies 0 and \(\Delta E\), in units with \(k_B=1\). Raise \(T\) and watch the excited share \(p_1\) climb toward ½, never beyond it. Sixty particles follow Metropolis dynamics: run it and the number in the upper level fluctuates around the Boltzmann share. The heat capacity peaks near \(T\approx0.42\,\Delta E\), where the excited share changes fastest.
Worked example. At \(k_BT=\Delta E\), \(p_1=1/(e+1)\approx0.269\).
Watch out. The two-level model omits interactions and degeneracy.
What happens as \(T\) approaches zero?
The lowest-energy state dominates.
2. Partition function
The levels form an evenly spaced ladder \(E_i=i\varepsilon\), and only the lowest \(n\) are accessible. The bars show the Boltzmann probabilities; the right panel shows the free energy \(F=-k_BT\ln Z\) for every \(n\) at the chosen temperature, approaching the value for the infinite ladder.
Worked example. Adding accessible states lowers the free energy at fixed temperature.
Watch out. The ladder is truncated at \(n\) levels for visualization; a real oscillator has infinitely many.
Why does multiplicity matter?
Every accessible microstate adds a positive term to Z, so F falls; states far above \(k_BT\) add almost nothing.
3. Random walk and diffusion
Each step moves one unit right with probability \((1+b)/2\) and left otherwise. Increase the number of steps \(N\): the walks drift by \(Nb\) and spread like \(\sqrt{N}\). The right panel is the exact binomial distribution of the final position, drawn on the same vertical axis as the walks.
Worked example. For an unbiased walk (\(b=0\)), \(\langle x_N^2\rangle=N\ell^2\), so the root-mean-square displacement grows like \(\sqrt{N}\).
Watch out. One realization fluctuates around the ensemble scaling law.
What grows linearly with \(N\)?
The variance \(N(1-b^2)\ell^2\), and for an unbiased walk the mean-square displacement itself; the typical displacement grows only like \(\sqrt{N}\).