The Boltzmann distribution

EstablishedPhysicsAI / MLMathematics

\htmlClass{sjt-p}{p_i} = \frac{\htmlClass{sjt-factor}{e^{-E_i / k_B T}}}{\htmlClass{sjt-Z}{Z}}

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p_i

Probability of state i

Meaning

Probability that a system in thermal equilibrium at temperature T is found in microstate i.

Its job in this equation

The output.

Read it, progressively

1. Gist

In equilibrium, low-energy states are more likely, and heat lets the system explore higher-energy ones.

What it does not establish

It applies to systems in thermal equilibrium. Living cells and training runs are generally far from equilibrium. The analogy to softmax licenses borrowed math, not borrowed physical conclusions.

Play with it

2D Ising model · a phase transition you can watch

Temperature T2.27 (T/Tc 1.00)

magnetisation m0.000

phasenear critical

Each cell is a spin (up = dark, down = light) that prefers to match its four neighbours. Temperature T sets how often a spin flips against its neighbours (Metropolis algorithm). Below the critical temperature large aligned domains form and the lattice magnetises. Above it, noise wins. Lars Onsager solved the 2D square lattice exactly in 1944: T_c = 2/ln(1+√2) ≈ 2.269 (in units where J = k_B = 1). Near T_c, clusters of all sizes coexist: that is criticality. The energy function here has the same form as a Hopfield network (1982) and other energy-based models. That is shared mathematical structure; it does not make a neural network a magnet. A 64×64 lattice is small, so the transition is smeared.

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§ Module 4: Probability, Entropy, and Information Next equation: Kleiber’s law (metabolic scaling) →