Stochastic Attention via Langevin Dynamics on the Modern Hopfield Energy

Attention heads retrieve: given a query, they return a softmax-weighted average of stored values. We show that this computation is one step of gradient descent on a classical energy function, and that Langevin sampling from the corresponding distribution yields stochastic attention: a training-free sampler controlled by a single temperature. Lowering the temperature gives exact retrieval; raising it gives open-ended generation. Because the energy gradient equals the attention map, no score network, training loop, or learned model is required. We derive a closed-form entropy inflection condition that identifies the retrieval-to-generation transition temperature for any memory geometry, with a scaling law $\beta^*\!\sim\!\sqrt{d}$ for random patterns. We validate on five domains (64 to 4,096 dimensions). On MNIST digit images, stochastic attention is $2.6{\times}$ more novel and $2.0{\times}$ more diverse than the best learned baseline (a VAE trained on the same patterns), while matching a Metropolis-corrected gold standard. On protein sequences from the Pfam RRM family, the generation regime achieves $6.9{\times}$ lower amino acid composition divergence than the VAE (KL $= 0.060$ vs.\ $0.416$) at matched novelty, demonstrating that the training-free score function preserves family-level fidelity that learned models lose. A denoising diffusion baseline (DDPM) fails across all memory sizes tested ($K = 100$ to $3{,}500$), producing samples indistinguishable from isotropic noise. The approach requires no architectural changes to the underlying attention mechanism.