{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.gross-pitaevskii-equation","code":"thing-q910667","url":"https://ver.cy/models/thing/q910667/","name":"Gross–Pitaevskii equation","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Information and virtual systems","entryKind":"thing","plane":"INF","domain":["INF.MED"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"The Gross-Pitaevskii equation is a nonlinear Schrödinger equation describing the mean-field evolution of the condensate order parameter in a dilute, weakly interacting Bose gas, usually with interactions represented by a local cubic term.","purpose":"Enable an agent to recognize a Gross-Pitaevskii formulation, assess its physical applicability, and select defensible calculations and interpretations of condensate behavior.","scope":{"in":["Time-dependent and stationary scalar Gross-Pitaevskii formulations","Condensate order parameter, normalization, interaction coefficient, and units","Mean-field assumptions and limits of applicability","External potentials, geometry, initial conditions, and boundary conditions","Stationary states, dynamics, stability, and numerical verification","Boundaries between the baseline equation, dimensional reductions, and named extensions"],"out":["Bose-Einstein condensation as a physical phenomenon in general","Complete many-body quantum states and microscopic interaction models","The nonlinear Schrödinger equation family beyond its condensate interpretation","Experimental apparatus design and atomic species catalogues","Software implementation details belonging to individual solvers","Full finite-temperature, strongly correlated, and quantum fluctuation theories"],"boundaries":[]},"distinguishingFeatures":["The dependent field represents a condensate order parameter; its normalization determines whether its squared magnitude is a number density or a probability density.","The baseline scalar equation contains a local cubic term g|ψ|²ψ representing contact interactions, alongside kinetic and external-potential terms.","Its condensate interpretation and physically specified coupling distinguish it from a generic cubic nonlinear Schrödinger equation with arbitrary coefficients.","The stationary equation is a nonlinear eigenvalue problem with chemical potential constrained by normalization; chemical potential must not automatically be identified with total energy per particle.","Bogoliubov-de Gennes equations describe perturbations around a condensate solution, whereas the Gross-Pitaevskii equation governs the underlying mean field."],"structure":{"bundles":[{"id":"equation-identity-and-conventions","name":"Equation identity and conventions","description":"Establish which Gross-Pitaevskii equation is being represented and how its symbols map to physical quantities.","layers":[{"id":"canonical-formulation","name":"Canonical formulation","description":"Identify the baseline differential equation and its stationary specialization.","findings":[{"id":"scalar-contact-form","name":"Scalar contact form","description":"For a particle-number-normalized field, the baseline form is iℏ∂tψ = [−ℏ²∇²/(2m) + V + g|ψ|²]ψ. Record departures term by term. See [Pitaevskii and Stringari, Nonuniform Bose Gases at Zero Temperature](https://academic.oup.com/book/26678/chapter-abstract/195458863).","questions":[{"text":"What is the exact operator equation, including every sign, coefficient, and additional term?","id":"scalar-contact-form-q01","kind":"definition"},{"text":"Which reference supports this formulation and its interpretation of the field?","id":"scalar-contact-form-q02","kind":"provenance"}]}]},{"id":"normalization-and-scaling","name":"Normalization and scaling","description":"Make physical and dimensionless representations interchangeable without changing the problem.","findings":[{"id":"field-and-coefficient-conventions","name":"Field and coefficient conventions","description":"Require an explicit normalization integral and length, time, energy, and field scales before comparing coefficients or observables.","questions":[{"text":"Does the field integrate to unity, condensate number, or another declared quantity, and where is the particle-number factor placed?","id":"field-and-coefficient-conventions-q01","kind":"definition"},{"text":"What transformation recovers physical density, time, and energy from the dimensionless variables?","id":"field-and-coefficient-conventions-q02","kind":"measurement"}]}]}]},{"id":"mean-field-applicability","name":"Mean-field applicability","description":"Record why the equation is an adequate approximation for the proposed system.","layers":[{"id":"dilute-condensate-regime","name":"Dilute condensate regime","description":"Assess interaction, condensation, and temperature assumptions separately.","findings":[{"id":"regime-evidence","name":"Regime evidence","description":"The standard treatment concerns weakly interacting, dilute condensates near the zero-temperature limit; applicability needs evidence for the actual system. See [Pethick and Smith, Theory of the condensed state](https://www.cambridge.org/core/books/abs/boseeinstein-condensation-in-dilute-gases/theory-of-the-condensed-state/19D43707671194C5704F4B1FE10055DA).","questions":[{"text":"What density and scattering-length estimates determine the relevant diluteness parameter across the cloud?","id":"regime-evidence-q01","kind":"measurement"},{"text":"What evidence bounds thermal population, quantum depletion, and correlations sufficiently for the intended observable?","id":"regime-evidence-q02","kind":"boundary"}]}]},{"id":"dimensional-and-physical-extensions","name":"Dimensional and physical extensions","description":"Distinguish justified reductions from additions that change the governing model.","findings":[{"id":"reduction-and-extension-contract","name":"Reduction and extension contract","description":"Require a derivation or cited justification for effective dimensionality and identify nonlocal interactions, multiple components, dissipation, noise, or fluctuation corrections explicitly.","questions":[{"text":"Which confinement and energy-scale conditions justify eliminating spatial directions, and how was the effective coupling obtained?","id":"reduction-and-extension-contract-q01","kind":"boundary"},{"text":"Which omitted physical effect would invalidate the baseline equation for this task, and which linked formulation addresses it?","id":"reduction-and-extension-contract-q02","kind":"action"}]}]}]},{"id":"physical-problem-specification","name":"Physical problem specification","description":"Supply the information needed to turn an equation into a determinate condensate problem.","layers":[{"id":"interactions-and-confinement","name":"Interactions and confinement","description":"Record physical coefficients and the environment acting on the condensate.","findings":[{"id":"parameter-and-potential-definition","name":"Parameter and potential definition","description":"Specify mass, interaction coefficient, particle number, potential, and reference frame with their provenance and uncertainty.","questions":[{"text":"What values and uncertainties define mass, scattering length, condensate number, and any effective coupling?","id":"parameter-and-potential-definition-q01","kind":"measurement"},{"text":"What function defines confinement or driving, and is any rotational term expressed in a declared rotating frame?","id":"parameter-and-potential-definition-q02","kind":"definition"}]}]},{"id":"domain-and-state-data","name":"Domain and state data","description":"Specify boundary behavior and either initial data or stationary-state selection constraints.","findings":[{"id":"problem-closure","name":"Problem closure","description":"Treat phase information, topology, normalization, and boundary conditions as substantive inputs rather than solver defaults.","questions":[{"text":"What initial complex field or stationary-state constraints select the intended physical state?","id":"problem-closure-q01","kind":"definition"},{"text":"Which boundary conditions represent the physical system, and which boundaries merely truncate the computational domain?","id":"problem-closure-q02","kind":"boundary"}]}]}]},{"id":"states-observables-and-stability","name":"States, observables, and stability","description":"Connect solutions to condensate quantities and distinguish existence from stability.","layers":[{"id":"stationary-state-selection","name":"Stationary state selection","description":"Interpret the nonlinear stationary equation and its normalization constraint.","findings":[{"id":"chemical-potential-and-energy","name":"Chemical potential and energy","description":"Record the stationary ansatz ψ(r,t) = φ(r)exp(−iμt/ℏ), the associated energy functional, and the constraints used to select a solution.","questions":[{"text":"Is the target a ground state, a constrained minimum, or an excited stationary branch?","id":"chemical-potential-and-energy-q01","kind":"definition"},{"text":"How are chemical potential and total energy computed separately under the chosen normalization?","id":"chemical-potential-and-energy-q02","kind":"measurement"}]}]},{"id":"observable-and-perturbation-interpretation","name":"Observable and perturbation interpretation","description":"Define measurable outputs and evidence for stability or instability.","findings":[{"id":"condensate-observables","name":"Condensate observables","description":"Attach explicit extraction rules to density, phase, current, collective motion, and vortices. These are established mean-field applications discussed in [Dalfovo et al., Theory of Bose-Einstein condensation in trapped gases](https://arxiv.org/abs/cond-mat/9806038).","questions":[{"text":"How are density, current, and phase-derived quantities calculated, including behavior near field zeros?","id":"condensate-observables-q01","kind":"measurement"},{"text":"What phase-winding and density evidence distinguishes a vortex from a numerical artifact?","id":"condensate-observables-q02","kind":"measurement"}]},{"id":"stability-evidence","name":"Stability evidence","description":"Keep energetic stability, linear dynamical stability, and finite-time persistence as separate assessments.","questions":[{"text":"What constrained energy comparison or linearized perturbation analysis supports the stated stability classification?","id":"stability-evidence-q01","kind":"measurement"},{"text":"Which perturbations and observation times have been tested before using this solution to predict persistent behavior?","id":"stability-evidence-q02","kind":"action"}]}]}]},{"id":"calculation-validation-and-use","name":"Calculation validation and use","description":"Determine whether a computed result is reproducible and accurate enough for its proposed use.","layers":[{"id":"numerical-adequacy","name":"Numerical adequacy","description":"Match solution methods and resolution to the requested state or trajectory.","findings":[{"id":"method-and-resolution-evidence","name":"Method and resolution evidence","description":"Require convergence evidence for relevant spatial and temporal scales; imaginary-time relaxation must be identified as a state-finding method.","questions":[{"text":"Does the selected method find the requested branch or evolution, and what initialization or constraints could bias its result?","id":"method-and-resolution-evidence-q01","kind":"action"},{"text":"How do the target observables change under mesh, time-step, solver-tolerance, and domain-size refinement?","id":"method-and-resolution-evidence-q02","kind":"measurement"}]}]},{"id":"physical-and-mathematical-checks","name":"Physical and mathematical checks","description":"Assess residuals, conservation or balance laws, and agreement with appropriate reference cases.","findings":[{"id":"acceptance-and-escalation","name":"Acceptance and escalation","description":"Define task-specific acceptance criteria using equation residuals, applicable invariants, benchmark limits, and independent physical evidence.","questions":[{"text":"Which norm and energy conservation or balance laws apply to this potential, boundary treatment, and equation variant?","id":"acceptance-and-escalation-q01","kind":"boundary"},{"text":"What residual, benchmark agreement, and observable uncertainty must be achieved before accepting the result or escalating to another theory?","id":"acceptance-and-escalation-q02","kind":"action"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-kinds","name":"Kinds and varieties","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these kinds and varieties hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence?","id":"evidence-kinds-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of Gross-Pitaevskii equation this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Explain the Gross-Pitaevskii equation and its uses in physics.","Solve or simulate it in models the agent is authorized to work on."],"mustNot":["Present mean-field results outside their validity range as reliable.","Fabricate numerical or experimental results.","Misattribute derivations."],"requiresHuman":["Publishing new results."]},"ethics":{"considerations":["Accurate science communication matters for trust.","Credit for scientific work matters."],"affectedParties":["Researchers","Students"]},"owners":{"steward":"Nobody owns the equation; researchers answer for their use of it.","roles":[],"masterSystems":[]},"relations":[{"target":"Q36338801","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.gross-pitaevskii-equation","Wikidata: Q910667 (https://www.wikidata.org/wiki/Q910667)"]},"properties":{"applicability":"not-applicable","items":["Equation formulation: time-dependent; stationary; explicitly identified extension","Order-parameter normalization: integral |ψ|² dV = N₀; integral |φ|² dV = 1; other explicitly defined convention","Spatial dimensionality: 3D; effective 2D; effective 1D","Particle mass: kg or a declared nondimensional mass scale","Interaction specification: s-wave scattering length aₛ in m; baseline 3D coupling g = 4πℏ²aₛ/m in J·m³","Diluteness parameter: n|aₛ|³, dimensionless for a 3D gas","Confinement and driving: external potential V(r,t), domain geometry, and any declared rotating reference frame","Solution assessment: unverified; converged; stationary; dynamically stable; unstable; applicability unresolved"]},"recognition":{"applicability":"optional","items":["Appears in physics texts as a nonlinear Schrodinger equation with a cubic interaction term."]},"capabilities":{"applicability":"required","items":["Recognize equivalent formulations by reconciling units, normalization, and reference frames.","Assess whether a specified condensate regime supports a scalar mean-field treatment.","Construct a stationary or initial-boundary-value problem from physical inputs.","Compute candidate condensate profiles and dynamics with recorded approximation limits.","Extract density, phase, current, energy, and defect observables under explicit conventions.","Reject unsupported interpretations or route the problem to an appropriate extended theory."]},"hazards":{"applicability":"required","items":["Misapplication outside the dilute, weakly interacting regime.","Numerical instability in simulations."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["The Gross-Pitaevskii equation is used in theoretical and computational studies of dilute, weakly interacting Bose-Einstein condensates.","It describes condensate density and phase and supports investigations of trapped states, collective motion and vortices within a mean-field approximation."]}},"sources":[{"title":"Wikidata item Q910667: Gross–Pitaevskii equation","url":"https://www.wikidata.org/wiki/Q910667","note":"identity and sense of the item"},{"title":"Wikipedia: Gross–Pitaevskii equation","url":"https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation","note":"general description of the item"}],"openQuestions":["Does an existing Vercy world-model publication already cover this equation, requiring the registry entry to link to that publication?","Which named extensions should be represented within this entry, and which should be linked as distinct registered concepts?","Does the assigned INF.MED domain intentionally include mathematical equations, or does this entry require a classification correction?","Which primary derivations should anchor the publication's historical attribution, normalization conventions, and reduced-dimensional formulations?","Which application regimes and target observables will determine acceptable depletion, thermal contribution, numerical error, and validation thresholds?"],"resources":{"spec":"/models/things/publications/thing-q910667/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q910667/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T10:07:19Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:33+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane INF: no invented physical properties.","interaction.interfaces":"No regulatory requirement or technical standard is intrinsic to this theoretical physics equation."},"score":1.0}}