Why cap at seven patterns?

The classic Hopfield capacity on random unbiased patterns scales like **P ~ 0.14 N** in this normalization; seven is a conservative teaching cap on **N = 100** before overlaps and spurious states dominate.

Does synchronous updating also lower E?

Not guaranteed. The Hopfield Lyapunov argument is standard for **random sequential** (single-unit) updates; parallel sweeps can increase energy and oscillate.

What does the overlap readout mean?

For each stored pattern μ, **m_μ = N⁻¹ Σ_i ξ_i^μ S_i** measures alignment of the current state with that memory; the sidebar highlights the largest magnitude among them as a quick “which memory wins” indicator near convergence.

Hopfield Associative Memory

10×10 binary Hopfield network: Hebbian pattern storage, energy E = −½ Σ w_ij S_i S_j, and random asynchronous updates that flow downhill to fixed-point recall after noisy cues.

About this model

A classical Hopfield network stores binary patterns ξ^μ on N = 100 spins arranged as a 10×10 grid. Symmetric Hebbian weights w_ij = (1/N) Σ_μ ξ_i^μ ξ_j^μ (zero diagonal) define the Lyapunov (energy) function E = −½ Σ_{i,j} w_ij S_i S_j. Asynchronous zero-temperature updates S_i ← sign(Σ_j w_ij S_j) never increase E, so noisy states relax toward stored attractors—the standard cartoon of associative memory and an energy landscape, with caveats about capacity and spurious minima when P grows.

Who it's for: Students in statistical mechanics, neural-networks introductions, or information theory who want a hands-on Hopfield energy picture.

Key terms

Hopfield model
Hebbian learning
Associative memory
Energy function
Attractor network
Asynchronous dynamics

How it works

Small Hopfield associative memory on a 10×10 grid: Hebbian weights, energy landscape, paint states, memorize patterns, add noise, and watch zero-temperature async dynamics retrieve a stored attractor.

Key equations

w_ij = (1/N) Σ_μ ξ_i^μ ξ_j^μ (i ≠ j), E = −½ Σ_{i,j} w_ij S_i S_j, update: pick i at random, set S_i = sign(Σ_j w_ij S_j) (tie keeps S_i).

Frequently asked questions

Why cap at seven patterns?: The classic Hopfield capacity on random unbiased patterns scales like P ~ 0.14 N in this normalization; seven is a conservative teaching cap on N = 100 before overlaps and spurious states dominate.
Does synchronous updating also lower E?: Not guaranteed. The Hopfield Lyapunov argument is standard for random sequential (single-unit) updates; parallel sweeps can increase energy and oscillate.
What does the overlap readout mean?: For each stored pattern μ, m_μ = N⁻¹ Σ_i ξ_i^μ S_i measures alignment of the current state with that memory; the sidebar highlights the largest magnitude among them as a quick “which memory wins” indicator near convergence.

Continue with similar topics in this category — or all 46 in Thermodynamics.

View category →

NewSchool

Brownian Motion

Launch Simulator

Heavy particle with random kicks and friction; trail and running ⟨r²⟩ vs time.

NewSchool

Joule–Thomson Throttling

Launch Simulator

Isenthalpic expansion: ideal gas ΔT = 0; toy μ_JT inversion for real gases.

NewSchool

Open-Channel Flow & Hydraulic Jump

Launch Simulator

Rectangular-channel Froude number, critical depth, conjugate depths, energy loss, and subcritical/supercritical regimes.

School

Phase Diagram

Launch Simulator

Temperature-pressure diagram with phase transitions.

NewSchool

Wolfram Elementary Cellular Automata

Launch Simulator

1D binary CA with Wolfram code R∈[0,255]: space–time raster, periodic or null boundaries, single-cell or random IC, rule-bit strip, and informal class hints for textbook rules (e.g. 30, 90, 110, 184).

NewSchool

Adiabatic Cloud Parcel

Launch Simulator

Lift moist air: dry Γ, LCL, toy moist lapse; RH and cartoon cloud vs height.

Hopfield Associative Memory

About this model

How it works

Key equations

Frequently asked questions

Related simulators

Brownian Motion

Joule–Thomson Throttling

Open-Channel Flow & Hydraulic Jump

Phase Diagram

Wolfram Elementary Cellular Automata

Adiabatic Cloud Parcel

Hopfield Associative Memory

About this model

How it works

Key equations

Frequently asked questions