Thermodynamic Attention: Entropy-Based Memory Eviction for Long-Context Transformers
TL;DR: What if we treated attention not as weighted retrieval but as active memory maintenance with an energy budget? This post presents a speculative architecture where tokens accumulate “entropy” when ignored and get evicted when their maintenance cost exceeds their relevance. Includes complete pseudocode and testable predictions. Looking for collaborators to implement and validate.
The Core Intuition: The Refrigerator Analogy
Think of an LLM’s context window not as a library of facts, but as a refrigerator. A refrigerator doesn’t “store” cold—it uses energy to pump heat out, creating a localized area of low entropy (order) within a high-entropy environment.
The question: What if we applied this metaphor to attention? Instead of treating all tokens in context as equally “available,” what if maintaining a token’s usefulness required continuous “cooling work” against a natural tendency toward noise?
This isn’t literal thermodynamics for digital computers (RAM doesn’t thermalize), but it’s a design metaphor that generates some interesting architectural ideas.
Motivation: What Problems Does This Address?
Current transformer attention has three pain points:
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Lost in the middle: Models fail to attend to information buried in long contexts
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Unprincipled KV-cache eviction: Heuristic methods (H2O, StreamingLLM) lack theoretical grounding
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No hallucination diagnostics: We can’t predict which tokens will cause hallucinated outputs
Standard attention assumes all tokens in context are equally retrievable. But what if retrieval cost should increase with “neglect”?
The Architecture: Carnot Attention Layer
Replace standard multi-head attention with this 8-step process:
Pseudocode
class CarnotAttention:
"""
Each token i carries an entropy register S_i
Updated every forward pass
"""
def forward(self, Q, K, V, S_prev):
# 1. Compute standard QK scores
e_i = (Q @ K.T) / sqrt(d_k)
# 2. Update entropy (tokens heat up when ignored)
S_i = S_prev + λ * (1 - α_prev) # λ: decay rate
# 3. Compute cooling cost (exponential in entropy)
W_i = W_0 * exp(β * S_i) # β: thermodynamic coupling
# 4. Compute relevance flux
R_i = softmax(e_i) * γ_i # γ_i: learned importance
# 5. Net energy balance
Ω_i = R_i - W_i
# 6. EVICT tokens where Ω_i < -ε (thermal death)
mask = (Ω_i >= -ε)
# 7. Attend over surviving tokens
α = softmax(max(Ω[mask], 0))
output = α @ V[mask]
# 8. RE-COOL attended tokens (feedback loop)
S_i[mask] *= (1 - α)
return output, S_i # carry entropy state forward
Key Differences from Standard Attention
Standard Attention:
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Stateless: same KV-cache accessibility every layer
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Eviction: when context exceeds window size (hard cutoff)
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Weights: based purely on similarity (QK^T)
Carnot Attention:
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Stateful: entropy register S_i tracks “neglect”
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Eviction: when maintenance cost exceeds value (Ω_i < 0)
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Weights: similarity minus upkeep cost
The Mathematics: Three Key Equations
1. Entropy Accumulation
S_i(t+1) = S_i(t) + λ(1 - α_i(t))
Tokens “heat up” (become noisier) when not attended. λ controls decay rate.
2. Eviction Threshold
Evict when: R_i < W_0 * exp(β * S_i)
Or equivalently: t* = (1/λβ) * ln(R_i / W_0)
Lifetime is logarithmic in relevance and inversely proportional to ambient noise (β).
3. Hallucination Risk Score
H_risk_i = S_i / S_max
Tokens with high entropy that barely survive eviction (Ω_i ≈ 0) contribute coherent-but-random outputs—the signature of hallucination.
Testable Predictions
1. Hallucination Correlation
Hypothesis: Tokens with high S_i values that remain in context (barely above eviction threshold) will correlate with factually incorrect outputs.
Test:
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Implement on GPT-2 scale
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Run TruthfulQA benchmark
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Track S_i values for tokens contributing to false statements
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Measure if S_i > threshold predicts hallucination
Expected result: High S_i tokens should show 60%+ correlation with hallucinated content.
2. Lost-in-the-Middle Recovery
Hypothesis: Middle context tokens accumulate more entropy (old enough to decay, not recent enough to be refreshed) and get evicted. Carnot Attention should fail gracefully rather than catastrophically.
Test:
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Use RULER benchmark (multi-hop reasoning over long context)
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Compare Carnot vs standard attention on retrieval from positions 25%, 50%, 75% into context
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Measure if eviction is predictable from Ω_i values
Expected result: Carnot Attention evicts based on relevance, not position—should outperform on middle-context retrieval.
3. Compute Efficiency vs H2O/PagedAttention
Hypothesis: Ω-based eviction should match or exceed heuristic methods while being theoretically principled.
Test:
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Implement on LLaMA-160M scale
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Benchmark against H2O, StreamingLLM, PagedAttention
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Measure: perplexity vs KV-cache size vs FLOPs
Expected result: Comparable perplexity at 30-50% smaller effective cache size.
What This Could Explain
Lost in the Middle
Current explanation: Positional encoding bias + attention sink
Thermodynamic explanation: Middle tokens accumulate the most entropy (not refreshed by recency, not maintained by high relevance). The model learns to evict them predictably.
Advantage: Provides per-token eviction criterion rather than positional heuristic.
Hallucination as Thermal Phenomenon
Novel insight: Hallucinations aren’t random errors—they’re structured outputs generated from high-entropy (noisy) tokens that the model attended to because Ω_i was marginally positive.
Diagnostic: Monitor S_i in real-time. When S_i > S_threshold for attended tokens, flag output as high-risk.
This is measurable and actionable in a way current hallucination detection isn’t.
KV-Cache Management
Current methods: Heuristics (keep recent + high-attention tokens)
Thermodynamic method: Keep tokens where relevance justifies maintenance cost
Advantage: Unified framework with learned parameters (λ, β, W_0) rather than hand-tuned heuristics.
Trainable Parameters
All four parameters can be learned end-to-end via backpropagation:
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λ (decay rate): How fast tokens “heat up” when ignored
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β (thermodynamic coupling): How steeply cost scales with entropy
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W_0 (base maintenance cost): Minimum energy to keep any token
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γ_i (per-token importance): Learned relevance weights
The model learns its own “refrigeration policy.”
Honest Limitations & Open Questions
1. The Analogy Isn’t Literal
Digital RAM doesn’t thermalize. The “entropy” here is a learned proxy for uncertainty/staleness, not a measured physical quantity. The exponential cost function is inspired by Landauer’s principle but doesn’t represent actual Joules.
Implication: This is a design metaphor that generates engineering insights, not a physical law.
2. Exponential Cost Function May Be Too Aggressive
W_i = W_0 * exp(β * S_i) could cause numerical instability during training.
Potential fixes:
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Polynomial approximation:
W_i = W_0 * (1 + β * S_i)^k -
Soft clipping:
W_i = W_0 * tanh(β * S_i) -
Learned nonlinearity instead of fixed exponential
Needs experimentation.
3. Statefulness Changes Parallelization
Carrying S_i registers across time steps makes this closer to RNNs/SSMs than pure transformers.
Trade-off:
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Loss: Can’t parallelize entropy updates as easily
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Gain: Natural context compression without external memory management
Similar to Mamba/RWKV—worth exploring, but different computational story.
4. Does “Use It or Lose It” Actually Help?
The feedback loop (attended tokens get re-cooled, ignored tokens heat up faster) creates reinforcement.
Risk: Could entrench attention patterns and make model overly rigid.
Test needed: Does entropy feedback improve or hurt sample efficiency during training?
Implementation Roadmap
Phase 1: Proof of Concept (1-2 weeks)
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Implement on GPT-2 Small (124M params)
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Train on WikiText-103 (manageable dataset)
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Compare perplexity vs standard attention
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Visualize S_i distributions across context
Phase 2: Diagnostic Validation (2-3 weeks)
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Run TruthfulQA to test hallucination correlation
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Run RULER for lost-in-the-middle
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Measure if high-S_i tokens predict errors
Phase 3: Scaling Test (1 month)
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If Phase 2 shows promise, scale to LLaMA-160M
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Benchmark against H2O, PagedAttention
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Measure compute/memory tradeoffs
Connection to Recent Research
This work is inspired by:
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Whitelam & Casert (Nature Communications, Jan 2026): “Nonlinear thermodynamic computing out of equilibrium” - proved thermodynamic neurons can perform universal function approximation using thermal fluctuations as computational resource [link]
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Cortical Labs / FinalSpark (2024-2025): Wetware computing with living neurons achieving high efficiency through self-organizing energy consumption—demonstrates biological computing is literally thermodynamic
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Sparse Attention Literature: Reformer, Longformer, BigBird (this extends their ideas with learned eviction)
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KV-Cache Optimization: H2O, StreamingLLM, PagedAttention (this provides theoretical foundation)
Note: The thermodynamic framing is metaphorical for digital transformers but grounded in real research on alternative computing substrates.
Why This Might Actually Work
Even though the physics isn’t literal:
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Sparse attention already works (Reformer, etc.) - this provides principled sparsity
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Relevance/cost tradeoffs are real - compute is finite, not all tokens are equally valuable
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Decay functions capture staleness - longer-unattended = less reliable is intuitive
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Entropy as uncertainty proxy - high S_i = high variance in token representation makes sense
The thermodynamic language organizes design decisions even if it doesn’t describe literal physics.
Call for Collaboration
I don’t have the resources to implement this myself, but the architecture is fully specified. Looking for:
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ML Engineers: Implement Carnot Attention in PyTorch/JAX
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Researchers: Test the three predictions (hallucination, lost-in-middle, efficiency)
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Theorists: Formalize the connection between S_i and actual uncertainty
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Skeptics: Find the failure modes—what would falsify this?
What I can provide:
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Full architecture specification (above pseudocode)
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Detailed comparison of 3 frontier LLM responses (DeepSeek, ChatGPT, Claude) that fleshed this out
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Specific benchmark protocols
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Ongoing discussion/refinement
Discussion Questions
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Has anyone tried entropy-based eviction before? (I haven’t found it in literature but could have missed it)
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Is the exponential cost function necessary? Could polynomial/logarithmic work just as well?
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Should S_i be per-token or per-head? (Different heads might have different “temperatures”)
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How to initialize S_i? (Zero? Learned? Sample from distribution?)
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Does this connect to existing uncertainty quantification methods? (Bayesian NNs, ensemble methods, etc.)
Source Materials
This hypothesis emerged from a theoretical exercise where I asked three frontier LLMs (DeepSeek-V3, ChatGPT-4, Claude 3.5 Sonnet) to apply thermodynamic principles to transformer attention. All three independently converged on similar architectures:
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DeepSeek: Most radical restructuring (renamed components: Queries→Entropy Probes, Keys→Potential Wells)
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ChatGPT: Most rigorous physics grounding (Landauer’s principle, Gibbs distributions)
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Claude: Most complete implementation (pseudocode + interactive visualization)
The synthesis above combines the best elements from all three.
Final Thoughts
This is speculative architecture, not proven science. The thermodynamic framing is a design fiction that might generate useful engineering insights.
The only way to know if this works: Build it and measure.
If you’re interested in collaborating, commenting, or just discussing—please engage below. And if you think this is nonsense, explain why—falsification is just as valuable as validation.
Tags: #long-context #sparse-attention #attention-mechanism #speculative-architecture #transformer-optimization #kv-cache #hallucination-detection
Code: (Will update this post with GitHub link if anyone implements)
Status: Seeking collaborators for proof-of-concept implementation
Disclaimer: “Thermodynamic” here is metaphorical for digital systems. The entropy registers are learned proxies, not measured physical quantities. This is exploratory research, not established theory.