Jul 28, 2025
Momentum–Point-Perplexity Mechanics in Large Language Models
Lorenzo Tomaz, Thomas Jones
This work analyzes the hidden states of twenty different open-source transformer language models, ranging from small to medium size and covering five major architectures. The key discovery is that these models show signs of "energy conservation" during inference—meaning a certain measure combining changes in hidden states and token unpredictability stays almost constant as the model processes text.
The authors developed a new framework inspired by physics to jointly analyze how hidden states and prediction confidence evolve over time. They propose that transformers' behavior can be understood as following certain mechanical principles, much like how physical systems follow rules like conservation of energy.
Their experiments show that this conserved quantity varies very little between tokens, especially in untrained (random-weight) models, where it's extremely stable. In pre-trained models, the average energy drops more due to training, but there are larger relative fluctuations from token to token.
They also introduce a new method, based on this framework, for controlling transformer outputs by "steering" the hidden states. This method achieves good results—producing completions rated as higher in semantic quality, while still maintaining the same kind of energy stability.
Overall, the findings suggest that viewing transformer models through the lens of physical mechanics gives new, principled ways to interpret and control their behavior. It also highlights a key difference: random models behave more like balanced systems, while trained models make quicker, more decisive state changes at the cost of less precise energy conservation.
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Cite this work
@misc {
title={
@misc {
},
author={
Lorenzo Tomaz, Thomas Jones
},
date={
7/28/25
},
organization={Apart Research},
note={Research submission to the research sprint hosted by Apart.},
howpublished={https://apartresearch.com}
}
