# Answer-Distribution Trajectories: A Stochastic-Dynamics View of LLM Reasoning

arXiv: https://arxiv.org/abs/2609.09030 · HTML: https://mlsanity.com/paper/2609.09030/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 27 of its 49 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 27 of 27)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [Yi: Open Foundation Models by 01.AI](https://mlsanity.com/paper/2403.04652/) (2024)
- [The Geometry of Truth: Emergent Linear Structure in Large Language Model Representations of True/False Datasets](https://mlsanity.com/paper/2310.06824/) (2023)
- [Measuring Faithfulness in Chain-of-Thought Reasoning](https://mlsanity.com/paper/2307.13702/) (2023)
- [DeepSeek LLM: Scaling Open-Source Language Models with Longtermism](https://mlsanity.com/paper/2401.02954/) (2024)
- [Monitoring Reasoning Models for Misbehavior and the Risks of Promoting Obfuscation](https://mlsanity.com/paper/2503.11926/) (2025)
- [INSIDE: LLMs' Internal States Retain the Power of Hallucination Detection](https://mlsanity.com/paper/2402.03744/) (2024)
- [Chain of Thought Monitorability: A New and Fragile Opportunity for AI Safety](https://mlsanity.com/paper/2507.11473/) (2025)
- [When Chain of Thought is Necessary, Language Models Struggle to Evade Monitors](https://mlsanity.com/paper/2507.05246/) (2025)
- [Demystifying Reasoning Dynamics with Mutual Information: Thinking Tokens are Information Peaks in LLM Reasoning](https://mlsanity.com/paper/2506.02867/) (2025)
- [Understanding Chain-of-Thought in LLMs through Information Theory](https://mlsanity.com/paper/2411.11984/) (2024)
- [Think Just Enough: Sequence-Level Entropy as a Confidence Signal for LLM Reasoning](https://mlsanity.com/paper/2510.08146/) (2025)
- [Trace Length is a Simple Uncertainty Signal in Reasoning Models](https://mlsanity.com/paper/2510.10409/) (2025)
- [Early Stopping for Large Reasoning Models via Confidence Dynamics](https://mlsanity.com/paper/2604.04930/) (2026)
- [The Stepwise Informativeness Assumption: Why are Entropy Dynamics and Reasoning Correlated in LLMs?](https://mlsanity.com/paper/2604.06192/) (2026)
- [Thinking Past the Answer: Evaluating Harmful Overthinking in Large Reasoning Models](https://mlsanity.com/paper/2606.02835/) (2026)
- [When Do LLMs Reason? A Dynamical Systems View via Entropy Phase Transitions](https://mlsanity.com/paper/2605.22873/) (2026)

## Built on by (most cited first, 1 of 1)

- [The belief-action gap in language model agents](https://mlsanity.com/paper/2610.17444/) (2026)

## Abstract (arXiv)

Chain-of-thought reasoning provides a structured computation between a model's input and final answer. Yet it is often evaluated through endpoint accuracy, which ignores the path taken to reach that answer. An emerging line of work addresses this limitation using entropy profiles, which track how uncertainty evolves over the reasoning process but do not reveal which competing hypotheses account for that uncertainty. We introduce answer-distribution trajectories, a stochastic-dynamics-inspired representation that tracks the model's full predictive distribution over answers as reasoning unfolds. As a strictly finer representation than endpoint and entropy summaries, answer-distribution trajectories enable us to characterize a trace through a dynamical reasoning profile spanning exploration, revision, motion, and commitment, and to distinguish different dynamical mechanisms of reasoning success and failure. Across sixteen open-weight language models and four reasoning benchmarks, we show that traces with the same endpoint and similar entropy profiles can exhibit substantially different reasoning dynamics. We further find substantial variation in these dynamics both within and across models and tasks, with different objectives favoring different dynamical profiles. Additionally, we show that training and inference choices systematically reshape these profiles. Our results suggest that answer-distribution trajectories provide a rich framework for analysing and evaluating the dynamics of LLM reasoning.