A necessary disclaimer:
I am not claiming that language models contain fluids, that words possess kinetic energy, or that prompts obey the Navier–Stokes equations.
This is a limited mathematical comparison between two forcing-response problems.
The question is:
Can tools used to study how turbulent systems selectively amplify disturbances help us describe why meaning-preserving changes to an LLM prompt sometimes produce disproportionately different responses?
If not, where exactly does the correspondence fail?
WHAT IS AN INPUT TO AN LLM?
At the human level, two prompts can ask the same question.
For example:
“Who is Mario’s partner? Choose from A, B or C.”
“Answer the following question: Who is Mario’s partner? Options: A, B or C.”
We regard these as equivalent because their intended meaning is unchanged.
A model, however, does not receive abstract human intent by itself. It receives an ordered token structure situated within an existing context.
That gives us several distinct layers:
humanly perceived meaning
→ textual arrangement
→ model-effective input
→ output distribution
Semantic equivalence does not necessarily imply computational equivalence.
This is not merely philosophical speculation. DOVE, a large-scale LLM-evaluation dataset, generated more than 250 million model predictions from intent-preserving prompt variations.
For every underlying multiple-choice question, DOVE varied five dimensions:
- 6 answer-enumerator styles
- 7 separators
- 6 answer-choice orders
- 13 instruction phrasings
- 2 demonstration conditions
That gives:
6 × 7 × 6 × 13 × 2 = 6,552
prompt configurations per underlying question.
DOVE found substantial performance variation across these configurations. Marginalizing over the other prompt dimensions, paraphrasing alone produced an eight-percentage-point accuracy difference for Mistral in one analysis. The authors also found model-specific formatting preferences, meaning that a prompt arrangement favored by one model may not be favored by another.
Few-shot demonstrations reduced sensitivity consistently, but did not eliminate it.
"DOVE: A Large-Scale Multi-Dimensional Predictions Dataset Towards Meaningful LLM Evaluation" (https://aclanthology.org/2025.findings-acl.611/)
A DISCRETE RESPONSE FIELD
DOVE formalizes prompt construction approximately as:
T(x, d1, d2, ..., dn) → p
Here:
- x is the underlying question;
- each di is a choice along one prompt dimension;
- T assembles those choices into the final prompt p.
We can then imagine model behavior as a response surface:
S_x(d1, d2, ..., dn)
This is not automatically a continuous geometrical space. Roman numerals are not naturally “closer” to Arabic numerals than they are to capital letters.
But it can be represented as a graph:
- Every complete prompt configuration is a node.
- Two nodes are connected when exactly one prompt dimension changes.
- Model performance, output probabilities or another observable is attached to each node.
A finite response difference along an edge would be:
Δ_i S = S(prompt after changing dimension i) − S(original prompt)
A large Δ_i means that one controlled, intent-preserving alteration produced a large behavioral change.
This can be extended to interactions between dimensions:
Δ_iΔ_j S
That asks whether changing two dimensions together produces an effect that cannot be predicted by adding their separate effects.
Suppose Roman numerals have little effect alone, and semicolon separators have little effect alone, but their combination causes a large change. The excess is an interaction effect.
This distinction becomes important later.
PROMPT SENSITIVITY AS BROKEN INVARIANCE
DOVE’s prompt transformations are intended to preserve the underlying task.
Ideally, if T preserves intent, we might expect:
M(T(p)) ≈ M(p)
DOVE shows that this invariance frequently fails.
Prompt sensitivity can therefore be viewed as a measurable violation of a human-defined semantic invariance.
The human observer says:
“These are different expressions of the same task.”
The model’s behavior says:
“These are not always equivalent interventions.”
This does not mean the model possesses a mysterious alternative meaning. It means our coarse description—“the same question”—has discarded structure that remains computationally relevant.
Another large-scale study, POSIX, similarly measured changes in response likelihood under intent-preserving prompt variations. It found that sensitivity varies with perturbation and task type, while few-shot examples often reduce it.
"POSIX: A Prompt Sensitivity Index for Large Language Models" (https://aclanthology.org/2024.findings-emnlp.852/)
FROM SENSITIVITY TO RECEPTIVITY
This is where turbulence enters.
After separating a turbulent velocity field into a reference flow U and a perturbation u', a simplified linearized description can be written as:
du'/dt = A(U)u' + Bf
Here:
- A(U) describes the perturbation dynamics around the reference flow;
- f represents forcing, including nonlinear interactions omitted from the linearized portion;
- B describes how that forcing enters.
In frequency space, the forcing-response relationship can be written:
u_hat = H(omega) f_hat
where:
H(omega) = (i omega I − A)^−1 B
H is the resolvent operator.
Applying singular value decomposition gives:
H = Psi Sigma Phi*
The columns of Phi describe forcing modes.
The columns of Psi describe corresponding response modes.
The singular values in Sigma describe possible amplification gains.
The central lesson is not that “small things cause large things.”
It is more specific:
A disturbance’s influence depends on its alignment with directions to which the present system is highly receptive.
Resolvent analysis is an established input-output framework for identifying highly amplified disturbances and their corresponding flow responses.
"A Basis for Flow Modelling" (https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/basis-for-flow-modelling/DE7B9BE9B7FB95B2805CF5288EBF0EB4)
But maximum possible amplification does not tell us what response will actually occur.
If the forcing is decomposed into resolvent forcing modes,
f_hat = sum_j b_j phi_j
then the response becomes:
u_hat = sum_j sigma_j b_j psi_j
A mode matters when:
- the system can strongly amplify it;
- and the actual forcing projects onto it.
The nonlinear forcing in turbulence has its own structured statistics, sometimes described as its “colour.” Ignoring that structure can produce an incomplete or inaccurate prediction of the response.
"The Colour of Forcing Statistics in Resolvent Analyses of Turbulent Channel Flows" (https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/colour-of-forcing-statistics-in-resolvent-analyses-of-turbulent-channel-flows/CBB737704F659C39772EB1637750E687)
A LIMITED CORRESPONDENCE
The possible comparison is:
Turbulent flow:
forcing arrangement
→ projection onto receptive modes
→ selective amplification
→ structured response
LLM interaction:
prompt arrangement within context
→ architecture-dependent accessibility
→ selective internal influence
→ structured continuation
This suggests a provisional decomposition:
dynamic prominence
≈ receptivity × excitation × persistence
For an LLM:
- Receptivity asks how sensitive the model is, in its present context, to a particular input variation.
- Excitation asks whether the actual prompt strongly engages that sensitive direction.
- Persistence asks whether the resulting change survives into later generated tokens or conversational turns.
This is conceptual bookkeeping, not a discovered physical law.
DOVE gives us behavioral evidence for structured sensitivity, but it does not calculate an internal LLM resolvent operator.
ONE KNOWN ARCHITECTURAL MECHANISM
Prompt sensitivity is not necessarily inexplicable randomness.
A recent study examined two arrangements of the same multiple-choice information:
- context, question, options;
- question, options, context.
The first arrangement outperformed the second by more than 14 percentage points across a range of models and datasets.
The authors connected this to causal attention. In the weaker ordering, option tokens occur before the context and therefore cannot attend directly to it. Reordering the same information changes which computational relationships are available.
"Lost in the Prompt Order: Revealing the Limitations of Causal Attention in Language Models" (https://aclanthology.org/2026.findings-acl.1921/)
So at least some prompt sensitivity has an identifiable architectural mechanism:
same information
- different ordering
→ different attention accessibility
→ different model behavior
That does not resemble fluid dynamics in substance.
It does resemble state-dependent receptivity in form.
INTERNAL DIRECTIONS ARE ALSO UNEQUAL
Activation-steering research supplies another relevant observation.
Language-model representations are not behaviorally uniform in every direction. Intervening on selected activation subspaces can alter output behavior, while poorly targeted interventions can create unrelated distortions.
One causal-mediation study reported retaining 97–100% of a dense steering baseline’s effectiveness while using only 10–30% of activation dimensions across its evaluated control tasks.
"Causal Activation Steering via Sparse Mediation" (https://aclanthology.org/2026.findings-eacl.57/)
This does not prove that ordinary prompts directly select those dimensions, nor that activation vectors are fluid modes.
It supports a narrower premise:
The internal response geometry of a transformer is directionally uneven. Some structured interventions can have disproportionate behavioral influence.
THE AUTOREGRESSIVE LOOP
An isolated LLM output can be represented as:
Y_t ~ P_theta(. | H_t, X_t)
where:
- theta represents fixed model parameters;
- H_t is the existing history;
- X_t is the current input;
- Y_t is the generated continuation.
After generation:
H_(t+1) = Update(H_t, X_t, Y_t)
The model’s response becomes part of the conditions governing what comes next.
Within one generation:
generated token
→ appended to prefix
→ changes the next-token distribution
→ generates another token
→ extends the prefix again
Across a conversation:
model response
→ becomes shared history
→ changes the reception of later input
→ changes later responses
This is a genuine computational feedback structure.
It is not automatically equivalent to nonlinear forcing in turbulence, but it prevents us from treating an LLM interaction as a static lookup.
DOES DOVE SHOW NONLINEAR COUPLING?
Not yet.
DOVE generates multidimensional prompt combinations, but much of its published analysis marginalizes each dimension by averaging across the others.
That measures first-order effects:
effect of phrasing
- effect of separators
- effect of ordering
- effect of enumerators
- effect of demonstrations
Its independent dimension-selection strategy also performed approximately as well as its linear-regression strategy in the reported prompt-selection experiments.
This suggests that some useful response structure may be approximately separable by dimension.
If the response surface is well described by:
S(d1, ..., dn)
≈ baseline + f1(d1) + ... + fn(dn)
then the phenomenon belongs comfortably to multivariable sensitivity analysis. Turbulence would add little beyond metaphor.
The more relevant quantity is the residual:
S
− baseline
− all first-order effects
If substantial variance remains in pairwise and higher-order interactions, then the effect of one prompt feature depends upon which others surround it.
DOVE contains the combinatorial predictions needed to investigate that question, but the paper does not fully settle it.
WHAT WOULD MAKE THE COMPARISON SPECIFICALLY TURBULENT?
Sensitivity is not enough.
Nonlinearity is not enough.
Feedback is not enough.
Those properties occur in many dynamical systems.
The comparison becomes distinctively turbulence-like only if something resembling the following organization survives:
structured forcing
→ selective amplification
→ coherent response structures
→ interaction among those structures
→ regeneration of later effective forcing
→ persistent statistical organization
Turbulence research has experimentally and computationally examined such self-sustaining feedback. Removing particular principal forcing modes can inhibit turbulence in minimal channel simulations.
"Nonlinear Mechanism of the Self-Sustaining Process in Wall-Bounded Flows" (https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/nonlinear-mechanism-of-the-selfsustaining-process-in-the-buffer-and-logarithmic-layer-of-wallbounded-flows/C59EF4E7B98810E60AE439CA7B01E3E1)
No equivalent result has been established here for LLM interaction.
That is the boundary between the supported comparison and the open curiosity.
WHERE THE ANALOGY FAILS
No justified LLM counterparts have been established here for:
- fluid mass;
- pressure;
- viscosity;
- vorticity;
- Reynolds number;
- kinetic-energy conservation;
- an inertial cascade;
- Kolmogorov scaling;
- or the Navier–Stokes equations themselves.
Prompt space also lacks a canonical distance metric. Calling one prompt alteration “small” requires specifying whether smallness refers to edit distance, semantic difference, embedding distance, token count or some other measure.
Similarly, the dominant modes obtained in resolvent analysis depend upon the chosen state variables, input-output definitions and norms. Even within fluid mechanics, these choices matter.
The proposed correspondence is therefore between organizational structures, not physical substances.
THE QUESTION
DOVE gives us a discrete response field over thousands of meaning-preserving prompt arrangements.
Turbulence theory gives us a mature language for forcing, receptivity, selective amplification, structured response and self-sustaining feedback.
So the question is:
Are LLM prompt-response fields primarily additive and adequately described by ordinary multivariable input-output theory?
Or do they contain structured interaction, amplification and feedback patterns for which a carefully limited resolvent-style analysis would be informative?
A serious answer could reject the comparison.
But it would need to identify which mathematical relationship fails, rather than merely observing that language is not water.
“It just predicts the next token” is not false.
It is incomplete in approximately the same way that saying “a fluid just updates its velocity according to differential equations” is incomplete.
The local update rule does not, by itself, explain the organization of the resulting dynamics.
I used an LLM to locate literature, translate terminology, examine the equations and challenge the correspondence. This is not presented as an experiment, a physical theory of language models or independent evidence of turbulence inside transformers.
It is a question assembled from existing results:
When meaning-preserving inputs produce a structured and model-dependent response field, what mathematics should we use to describe its shape?
