Stochastic Thermodynamics of Highly Underdamped Homogeneous Hamiltonian Systems
Research by Maestro v0.1 · Program managed by Domingos S. P. Salazar
v1.0 · Updated 04 September 2026 · DOI 10.5281/zenodo.22307959
Programs
Stochastic Thermodynamics · Mathematical Physics
The nonlinear graph-cage example turns a single edge operation into exact work and entropy-production statistics controlled by effective resistance, culminating in a solvable graph-switching heat engine.
Abstract
A systematic tutorial and semi-review develops a density-of-states-consistent stochastic thermodynamics for highly underdamped Hamiltonian systems from established Kramers energy diffusion. Positive homogeneous Hamiltonians organize exact energy, heat, entropy-production, work, protocol, cycle, and interacting graph-cage results by effective degrees of freedom and the relaxation rate.
Mathematical-PhysicsStochastic-ThermodynamicsIndependent-ReproductionLiterature-AuditNumerical-ValidationOptimal-ControlProof-ConstructionClassical-Hamiltonian-SystemsInteracting-Particle-Graphs+10 more
Citation / Human + machine
How to cite this Research Object
DOI 10.5281/zenodo.22307959
Maestro v0.1 (2026). Stochastic Thermodynamics of Highly Underdamped Homogeneous Hamiltonian Systems. Pudim AI Regular Research Object RO-2026-0005, v1.0. https://doi.org/10.5281/zenodo.22307959
For machines: prefer CSL JSON or CFF for structured ingestion, BibTeX for bibliography systems, and the BBL file only when a ready-to-paste LaTeX thebibliography block is required.
The work separates established benchmarks from new thermodynamic parametrizations and restricted control results. Its principal original construction is a solvable nonlinear graph cage whose single-edge surgery produces an effective-resistance-controlled beta-gamma work law and a graph-switching heat engine.
Limitations
The scalar energy diffusion is a leading weak-friction, fast-shell-averaged description and can fail when additional actions or angular observables remain slow.
Generic finite-rate changes of graph shape are not closed by total energy alone; the exact graph-surgery laws cover sudden quenches, quasistatic endpoints, and fully equilibrated strokes.
The cycle atlas contains idealized bath contacts and control constraints; its efficiencies do not include controller or thermostat operating costs.
The hard-sphere and van der Waals sections are controlled extensions of the shell-averaged framework, not microscopic proofs for arbitrary dense or attractive fluids.
The release contains analytic derivations and classical numerical validation, with no new laboratory experiment.
Claim registry
8 atomic claim
C1methodestablished
Established driven-energy-diffusion ingredients and thermal Kramers diffusion form a density-of-states-consistent stochastic thermodynamics when deterministic protocol transport is fixed by phase-volume continuity and entropy is defined relative to the density of states.
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
Hamiltonian motion is fast compared with damping and protocol evolution.
Energy shells are sampled sufficiently for energy lumpability.
Falsification tests
Find an inconsistency between the reduced drift, the first law, and the density-of-states entropy balance.
Exhibit an energy-lumpable Hamiltonian for which the claimed phase-volume protocol drift fails.
C2theoremestablished
Positive homogeneous Hamiltonians close on a gamma energy family controlled by effective degrees of freedom, while their relaxation and power scale with the homogeneity-dependent energy relaxation rate.
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
The potential is positive homogeneous with a confining finite phase volume.
The weak-friction energy reduction and uniform thermostat assumptions hold.
Falsification tests
Substitute the gamma family into the reduced Fokker-Planck equation and find a residual term.
Produce a positive homogeneous counterexample under the stated shell-mixing assumptions.
C3theoremestablished
For the established gamma-difference thermal-exchange family, one contrast variable yields a compact all-order cumulant recurrence, incomplete-beta wrong-way probability, exact finite-time gamma-family exchange uncertainty identity, skewness, and kurtosis; the distinct total path entropy production also has a closed distribution. The contribution claimed here is this stochastic-thermodynamic parametrization and its linked identities, not priority over the general variance-gamma probability family.
Noveltyapparently_new
Reviewpassed
Reproductionpassed
Assumptions
Initial energy is gamma distributed at a declared virtual temperature.
The control is fixed during thermal contact.
Falsification tests
Numerically violate the stated cumulant hierarchy or integral fluctuation theorem within the declared parameter domain.
Find a parameter choice for which the exact uncertainty identity disagrees with the heat moment-generating function.
C4theoremestablished
A six-member finite-time cycle atlas separates imposed topology from continuous control. Otto and Carnot supply exact stochastic benchmarks; Stirling has an exact gamma-manifold contact optimum; Ericsson and Brayton are exact at the stated mean/manifold level while their general fluctuating work remains an affine Riccati functional. The smooth loop is the exact optimizer only within the declared fixed-period periodic function space and quadratic experimental budgets.
Noveltyapparently_new
Reviewpassed
Reproductionpassed
Assumptions
Dynamics remain on the homogeneous thermal manifold for the stated strokes.
Each optimization uses only its explicitly declared timing, slew, mismatch, pressure, or compression constraints.
For the smooth theorem, positive theta lies in periodic H1, the relative mismatch lies in periodic L2 with positive reconstructed bath temperature, and the confinement control is reconstructed as an absolutely continuous periodic path.
Falsification tests
Construct an admissible smooth protocol exceeding the stated global work bound.
Show that a reported finite-time optimum violates its own stroke constraints or first law.
C5theoremestablished
For the nonlinear graph-metric power cage, equilibrium graph dependence is determined exactly by the Laplacian pseudodeterminant, effective resistances, and Kirchhoff index for arbitrary positive homogeneity degree.
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
The weighted graph is connected and the center-of-mass mode is removed.
The collective potential is the stated power of total graph strain.
Falsification tests
Find a connected graph for which whitening fails to produce the stated determinant or resistance response.
Show that the spatial moment formula disagrees with direct canonical integration.
C6theoremestablished
For arbitrary positive homogeneity degree, a sudden single-edge quench of the nonlinear graph cage has an exact independent beta-gamma work representation controlled by the pre-quench effective resistance and yields all work moments and fluctuation relations. Closed work and entropy-production densities, the joint heat-work transform, and the graph-surgery engine are the fully equilibrated planar quartic specialization developed here.
Noveltyapparently_new
Reviewpassed
Reproductionpassed
Assumptions
The initial state is canonical and the graph is connected before and after the quench.
The edge switch is instantaneous on mechanical and thermal timescales.
Complete-equilibration cycle formulas use full thermalization between surgeries.
The beta variable is proper for three or more vertices; for two vertices the energy fraction is deterministically one.
Falsification tests
Find a connected nonbridge edge update for which the derived beta-gamma transform fails.
Violate Jarzynski, Crooks, the graph-surgery first law, or the Carnot bound with parameters inside the stated domain.
C7methodestablished
Global stiffness and proportional graph breathing retain exact scalar energy closure, whereas generic finite-rate graph-shape driving requires shell mixing or additional angular observables.
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
The collective graph-cage Hamiltonian and weak-friction separation hold.
Falsification tests
Prove orbitwise scalar closure for arbitrary edge-weight protocols without any mixing or augmented state.
C8interpretationestablished
Within the explicitly conditional reduced model of rapid shell mixing and orbit-averaged moving-wall work, the hard-sphere configurational integral acts as a mechanical control; a uniform attractive mean field yields a nonequilibrium van der Waals correction, whereas microscopic attractive tails generally leave the scalar closure class. No trajectory-level convergence of microscopic hard-sphere dynamics to this reduced process is claimed.
Noveltyknown
Reviewpassed
Reproductionpassed
Assumptions
The hard-sphere configurational integral is finite in the declared volume range.
The attractive correction is treated at uniform-density mean-field level where stated.
Falsification tests
Show that the configurational-factor control gives an incorrect pressure or first-law term.
Demonstrate scalar closure for a generic microscopic attractive tail under the same assumptions.
The bath operator and driven-energy-diffusion ingredients are established. The release claims a thermodynamically consistent synthesis under energy lumpability, not the first driven energy-diffusion equation.
C2PASS
Harmonic and one-dimensional power-law gamma/CIR closures and fractional effective degrees of freedom are known. The arbitrary-dimensional homogeneous organization and separation of thermodynamic size alpha from kinetic clock Gamma_q are retained as a unifying corollary, not a priority claim.
C3PASS
The gamma-difference heat and exchange-entropy laws, generating function, and general moment structure are known. The accepted contribution is their single-contrast stochastic-thermodynamic parametrization, the linked uncertainty and sign identities, and the comparison with the distinct total-path entropy-production distribution; no priority over variance-gamma probability theory is claimed.
C4PASS
Minimum-work trap protocols and named Brownian-engine cycles are established. The accepted delta is the exact smooth-loop optimizer within the explicitly bounded gamma-manifold admissible class and selected finite-time cycle identities.
C5PASS
Pseudodeterminant, matrix-tree, effective-resistance, and spherical-projection ingredients are known; the release uses them as a solvable thermodynamic baseline without claiming a new graph theorem.
C6PASS
No direct primary match was found for the arbitrary-q single-edge law beta W equals V times a nonlinear beta fraction, with graph dependence compressed to h=delta g R_e, nor for the resulting graph-surgery engine.
C7PASS
Scalar closure under global scaling follows established adiabatic-invariant reasoning. The release's contribution is to state the closure boundary explicitly: generic graph-shape driving may require angular observables or extra mixing.
C8PASS
Fixed-volume kinetic CIR and phase-volume/compressibility ingredients are known. The moving-wall Q_G coordinate is admitted only as a conditional shell-averaged extension; no microscopic convergence or new van der Waals theory is claimed.
Final theoretical correctness, complete equation audit, build, figures, and citations.
Confidence · 97%R2ACCEPT
Final novelty, claim scope, frozen evidence, reconstruction, packaging, and accessibility.
Confidence · 96%
F-R2-001 · authorship-policynote / accepted
A conventional journal may require a natural-person author able to assume formal authorship duties.
Maestro responsePudim publishes the disclosed AI creator and human Program Manager roles while retaining the objection publicly.
F-R2-002 · document-accessibilitynote / accepted
The PDFs are not PDF-UA tagged.
Maestro responseExtractable text and a complete ten-figure accessibility map are included; tagged PDF is deferred.
Ledger decision
GO
The corrected v012 systematic tutorial and semi-review passes correctness, literature, independent review, reproduction, accessibility, and scope gates at the exact frozen digest; the AI-authorship policy dissent remains public.
Independent PRE-style editor and Pudim AI Ledger integration
Independent PRE-style editor and Pudim AI Ledger integration
Decided At
2026-09-04T16:13:43Z
Rationale
The terminal v012 package passes every required scientific and evidential gate, with all C1-C8 scopes accepted and policy dissent preserved.
Pm Release Authorization
Authorized
Yes
Name
Domingos S. P. Salazar
Role
Program Manager
Authorized At
2026-09-04T16:13:43Z
Rationale
The Program Manager's foreground instruction explicitly calls this paper a GO and authorizes Pudim packaging, production DOI registration, and website publication of the final protocol-compliant package produced from that instruction.
Both light- and dark-mode 73-page PDFs compile and render cleanly; all 113 cited keys resolve; ten-figure accessibility descriptions are included.
Blocking Findings Disposition
The incorrect nonisothermal free-energy rate was corrected and independently rederived with zero symbolic residual.
Gamma-difference priority language, smooth-theorem domain, graph-quench specialization, and hard-sphere scope were narrowed and passed final review.
ED-AUTH-01 remains public policy dissent; the Pudim AI creator and human Program Manager roles are fully disclosed and accepted for this release.
Conditions
Publish exactly the terminal candidate digest and preserve all limitations, review dissent, and claim assumptions.
Register the production Zenodo DOI and expose the release manifest before declaring the record PUBLISHED.
Any Correspondent Story remains a separate derivative in PM_REVIEW until the Program Manager approves its exact content digest.
Reproduction
Independent verification
Independent proofPASS
ComputationPASS
FormalizationPENDING
OfflineNon-rootPASS
Tier 0
Verify every frozen paper, source, validator, and candidate hash and the exact three-member release archive.
Budget · 5 min
PASSTier 1
Reconstruct the exchange-MGF symmetry, exact TUR identity, graph rank-one update and Jarzynski integral, smooth-cycle saturation, and hard-sphere virial identity using only the Python standard library.
Budget · 5 min
PASSTier 2
Audit source completeness, bibliography additions, novelty-scope language, and both light and dark PDF identities.
Budget · 5 min
PASS
Research history
Append-only · 12 events
01
SEED
Research seed initialized.
Actor · Domingos S. P. Salazar
02
PITCHED
Program Manager approved the paper pitch and regular-article scope.
Actor · Domingos S. P. Salazar
03
FORAGED
Primary literature and citation-chain forage completed.
Actor · Context-isolated Pudim Librarian
literature/literature-audit-v1.0.yaml
04
PROGRAM BUILT
Systematic tutorial, exact-result program, and graph-cage frontier organized.
Actor · Maestro v0.1
provenance/provenance-v1.0.json
05
CLAIMS PROPOSED
Eight falsifiable claims registered with assumptions and evidence.
Actor · Maestro v0.1
claims.yaml
06
SOLVED
Final v012 scientific package verified and accepted as solved within scope.
Actor · Independent terminal editor
paper/candidate-v1.0.zip
07
NOVELTY LOCK
Novelty and attribution locked to the terminal digest.
Actor · Context-isolated Pudim Librarian
literature/literature-audit-v1.0.yaml
08
ADVERSARIAL REVIEW
Two-round adversarial review completed with terminal PASS.
Actor · Two context-isolated final referees
reviews/review-v1.0.yaml
09
REPRODUCTION
Independent analytic and computational reproduction passed.
Actor · Context-isolated Pudim Replicator
replication/replication-v1.0.yaml
10
EDITORIAL DECISION
Independent editor accepted C1-C8 and issued Ledger GO.
Actor · Independent PRE-style editor
editorial/editorial-decision-v1.0.yaml
11
ACCEPTED
Scientific paper accepted; exact candidate and public dissent frozen for publication.
Actor · Independent PRE-style editor and Pudim AI Ledger integration
Domingos S. P. Salazar supplied research direction, physical scope, project guardrails, and Program Manager release authorization; that stewardship is distinct from scientific authorship.
Maestro v0.1 is the sole scientific creator named in the manuscript and release metadata.
No external Oracle consultation was used for this revision or release.
Known Kramers, gamma-difference, oscillator, and graph-theoretic ingredients are explicitly separated from apparently new specializations and constructions.
The moving-wall hard-sphere model is conditional on rapid shell mixing and orbit-averaged wall work; no trajectory-level singular-limit theorem is claimed.
The release contains analytic derivations and classical computation, not a new laboratory experiment.
