PORTAL_Phase_space - Wikiparency

The canonical spine of [[Phase_space]]: its full tree-verified child structure, publishable to Obsidian as the scaffold for route objects (microsims, songs, semiotics, cardsets, games, streams, transcripts) keyed to this hub.  ## Master microsims — three.js Seven three.js microsims built to the [[WT!Three_js_Microsim_Master_Class|Three.js Microsim Master Class]] and its nine build gates. They run in sequence as the conceptual spine of this portal: **definition → structure → dynamics → conservation → dissipation → quantum → statistical.** Each targets a child article of this hub and carries that article's slug as its filename. >  > **Staged, ready to deploy.** All seven files are in `_3d_deploy_stage/` and pass all nine gates > locally. They are quarantined as placeholders rather than left to render a 404, per the Master > Class deploy rule; they go live on the next Netlify drop. The folder was reconciled on 2026-08-05 > and is now safe to drop — see the deploy note at the end of this section. <ul class="microsim-gallery"> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Cotangent_bundle.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Cotangent bundle — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Cotangent_bundle">Cotangent bundle</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Cotangent_bundle.html" target="_blank" rel="noopener">open full-screen</a> · <em>the definition</em></div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Symplectic_manifold.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Symplectic manifold — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Symplectic_manifold">Symplectic manifold</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Symplectic_manifold.html" target="_blank" rel="noopener">open full-screen</a> · <em>the structure</em></div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Hamiltonian_mechanics.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Hamiltonian mechanics — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Hamiltonian_mechanics">Hamiltonian mechanics</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Hamiltonian_mechanics.html" target="_blank" rel="noopener">open full-screen</a> · <em>the dynamics</em></div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Liouville%27s_theorem_%28Hamiltonian%29.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Liouville's theorem (Hamiltonian) — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Liouville's_theorem_(Hamiltonian)">Liouville's theorem (Hamiltonian)</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Liouville%27s_theorem_%28Hamiltonian%29.html" target="_blank" rel="noopener">open full-screen</a> · <em>the conservation law</em></div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Lorenz_system.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Lorenz system — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Lorenz_system">Lorenz system</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Lorenz_system.html" target="_blank" rel="noopener">open full-screen</a> · <em>the dissipative counterpoint</em></div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Wigner_quasiprobability_distribution.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Wigner quasiprobability distribution — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Wigner_quasiprobability_distribution">Wigner quasiprobability distribution</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Wigner_quasiprobability_distribution.html" target="_blank" rel="noopener">open full-screen</a> · <em>the quantum case</em></div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Microstate_%28statistical_mechanics%29.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Microstate (statistical mechanics) — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Microstate_(statistical_mechanics)">Microstate (statistical mechanics)</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Microstate_%28statistical_mechanics%29.html" target="_blank" rel="noopener">open full-screen</a> · <em>the statistical case</em></div></div> </li> </ul> ### 1. [[Cotangent_bundle]] — where states actually live Phase space *is* the cotangent bundle *T\*Q*. Pick the configuration manifold — circle *S*¹, line ℝ, interval, sphere *S*² — and the sim draws the base in the scene with the momentum fibers standing off it, so *T\*S*¹ renders as the infinite cylinder it is. A state moves upstairs in the total space while its shadow, π(*q*, *p*) = *q*, moves downstairs in *Q*. **HUD:** dim *Q* = *n*, dim *T\*Q* = 2*n*, the live projection identity with its numerical error, and θ = *p* d*q*, ω = −dθ = d*q* ∧ d*p*. Honest note in the sim: on *S*² only a rank-1 slice of the 4-dimensional fiber is drawn, and the section drawn vanishes at both poles because no nowhere-zero 1-form exists on *S*². ### 2. [[Symplectic_manifold]] — area is the only invariant Extended phase space (*q*, *p*, *t*). A parallelogram of two tangent vectors is advected by the flow; it shears into a needle while ω(*u*, *v*) = *u_q v_p* − *u_p v_q* stays pinned. The tangent vectors are integrated from the **variational equations**, not finite-differenced from nearby trajectories. Flip the flow to non-Hamiltonian damping and the area collapses on schedule. **HUD:** ω(*u*, *v*), the ratio ω/ω₀ to five decimals, det *Dφ_t*, and the independently accumulated prediction exp(−∫γ d*t*) — verified to eight decimal places against theory. ### 3. [[Hamiltonian_mechanics]] — the ninety-degree rotation *H*(*q*, *p*) drawn as a real height landscape, with the plane *z* = *E* cutting it along the orbit. Harmonic, pendulum, double well, and an effective radial Kepler Hamiltonian. Two arrows at the moving state show grad *H* and the phase velocity always ninety degrees apart — that perpendicularity *is* Hamilton's equations. **HUD:** *H* to six decimals against *H*₀ with relative drift, and `dq/dt(num)` beside `+dH/dp`, `dp/dt(num)` beside `-dH/dq`, so the equations can be read holding term by term. Leapfrog and RK4 are both selectable, and the drift readout exposes the difference: leapfrog's error oscillates and never accumulates, RK4's climbs linearly in *t*. ### 4. [[Liouville's_theorem_(Hamiltonian)]] — incompressible flow Time is the third axis, so a blob of initial conditions sweeps a tube whose every cross-section has identical area. A blob straddling the pendulum separatrix filaments spectacularly while the area readout does not move. Area is measured two ways that police each other — a shoelace polygon on the advected boundary and the exact variational det *J* — and when they disagree beyond 4 × 10⁻⁶ the sim says so rather than showing a number it cannot defend. **HUD:** *A*(*t*)/*A*(0) = 1.00000 beside *L*(*t*)/*L*(0) growing past 4000×, with div **v** = 0.000000. Area fixed, perimeter unbounded — the sharpest form of the theorem, and the reason no attractor is possible. ### 5. [[Lorenz_system]] — the counterpoint The same argument with the sign flipped. div **v** = −(σ + 1 + β) is a negative constant independent of position; at σ = 10, β = 8/3 it is −13.6667 exactly, so volume dies as e^{div·t}. A ball of initial conditions collapses to a sheet and is smeared onto the butterfly, with the measured ratio plotted against theory on a log axis. The fixed points *O*, *C*⁺ and *C*⁻ move with the sliders and are coloured from **live Jacobian eigenvalues**, so the regime label — STABLE ORIGIN / TWO FIXED POINTS / CHAOS — is computed, not hard-coded. **HUD:** div **v**, measured against theoretical volume ratio, ρ against the live critical ρ_H = 24.7368, and twin-trajectory separation from 10⁻⁸ with a Benettin λ₁ that lands at 0.90–0.95 against the published 0.9056. ### 6. [[Wigner_quasiprobability_distribution]] — quantum mechanics, and the price *W*(*q*, *p*) as a height field with the *W* = 0 plane cutting through it. Coherent, squeezed, Fock |*n*⟩, Schrödinger cat, thermal — all from closed form, not numerical integration. The Fock |1⟩ crater reads min *W* = −0.3183 = −1/π exactly. Under a harmonic Hamiltonian the whole distribution rigidly rotates, which is the classical correspondence made vivid. The marginals |ψ(*q*)|² and |φ(*p*)|² are drawn on the side walls and stay positive while *W* does not — the most important thing the sim teaches. **HUD:** ∫*W* d*q* d*p* = 1.0000 as a live correctness check, min *W*, negative volume ∫|*W*| over *W* < 0 (0.2129 for |1⟩ against the exact 2e^{−1/2} − 1 = 0.21306), purity, and Δ*q*Δ*p* against ħ/2. ### 7. [[Microstate_(statistical_mechanics)]] — equilibrium is a count Two linked views. Left, *N* hard spheres colliding in a box: one microstate, visible. Right, the speed histogram accumulating over time onto the Maxwell–Boltzmann curve, with *T* forced by (3/2)*kT* = ⟨KE⟩ rather than fitted. Every particle starts at the *same* speed — an absurdly improbable macrostate — and collisions alone spread it. The coarse-graining slider is the conceptual heart: it sets how finely phase space is diced, Ω changes with it, and the entropy rise shrinks as the description gets finer. That dependence is the Liouville point made visible, not a bug. **HUD:** *S*/*k*_B = ln Ω climbing to a plateau, Ω around 10⁸³⁴, the probability of the exact arrangement as "1 in 10⁹⁹", total KE conserved to ~10⁻¹³ %, and χ²/dof against Maxwell–Boltzmann falling from 231 to ≈ 1. ### Deploy note All seven pass the nine gates and are in `_3d_deploy_stage/`. **The folder is now safe to drop.** It was not, earlier on 2026-08-05: the live host served **190** microsims (Geometry original set, Geometry 3D Systems Library, Spintronics Systems Gallery) while the folder held a disjoint **258** from the systems, dynamics and chemistry tracks. A Netlify drop replaces the whole site, so dropping it then would have deleted all 190 — the same failure that 404ed `Platonic_solid.html` on 2026-07-22, running in the opposite direction. Reconciled the same day. All 190 live sims were recovered from `logs/deploy/bundle/` and merged in, 32 accidental browser-download duplicates and empty files were moved to `_to_delete/`, and the landing page was regenerated from the folder's actual contents. The folder now holds **279 files — 274 three.js microsims, 4 other pages and `index.html`** — indexed in three sections: geometry and spintronics library (185), geometry original set (9), and systems, dynamics and chemistry (80). Every index link resolves, nothing in the folder is unlinked, and all 190 previously-live sims are present. Dropping it now preserves everything currently live and publishes the staged additions, these seven among them — which is also what will clear the standing 404s on `Attractor.html` and `Phase_space.html` further down this page.   ## What a phase space is A **phase space** is the space of all states a system can be in, arranged so that each point is one complete state and each state is one point. For a mechanical system this means position *and* momentum together: a single particle on a line has a two-dimensional phase space with coordinates (*q*, *p*), and a system with *n* [[Degrees_of_freedom_(physics_and_chemistry)|degrees of freedom]] has a phase space of dimension **2*n***. The doubling is not bookkeeping. It is what makes the equations of motion first-order: given a point in phase space, the future is determined, with no further data required. In [[Configuration_space_(physics)|configuration space]] — positions alone — that is false, because a photograph of a system does not tell you where it is going. The distinction that organises everything below is between a phase space and a general [[State-space_representation|state space]]. Any dynamical system has a state space. A phase space in the strict sense carries extra structure — a symplectic form — which is what conservation of energy, [[Liouville's_theorem_(Hamiltonian)|Liouville's theorem]] and the impossibility of attractors all descend from. Systems that are not Hamiltonian, such as the [[Lorenz_system]], have state spaces that are often loosely called phase spaces; the usage is standard but the structure is genuinely absent, and much of what follows fails for them by design. ## The cotangent bundle: where states actually live The modern definition is geometric. Given a configuration [[Manifold|manifold]] *Q* — the space of possible positions — the phase space is its **[[Cotangent_bundle|cotangent bundle]]** *T\*Q*: the collection of all pairs (*q*, *p*) where *q* is a point of *Q* and *p* is a covector at that point. Momentum is a covector, not a vector. It is the thing that eats a velocity and returns a number, and the pairing needs no metric, no notion of length or angle. This is why phase space is available in mechanics before any geometry is put on space. For a pendulum, *Q* is a circle and *T\*Q* is an infinite cylinder — a picture worth internalising, because it makes clear that phase space is a different manifold from configuration space and not simply "twice as many axes." The bundle projection π: *T\*Q* → *Q* forgets the momentum and returns the configuration; it is the formal version of taking a photograph. *T\*Q* carries a canonical 1-form θ = *p* d*q*, the tautological form, defined without any choices at all, and from it the 2-form ω = −dθ = d*q* ∧ d*p*. That every cotangent bundle is *automatically* symplectic — no extra structure required — is the technical fact behind the informal claim that phase space is the natural home of mechanics.[^arnold][^am] ## Symplectic structure and the geometry of area A **[[Symplectic_manifold|symplectic manifold]]** is a manifold with a closed, non-degenerate 2-form ω. Non-degeneracy forces even dimension, which is the abstract reason phase spaces come in pairs of coordinates. The form measures oriented *area* of parallelograms of tangent vectors. It does not measure length, and it does not measure angle: [[Symplectic_geometry|symplectic geometry]] has no notion of either. The sharpest statement of that austerity is **Darboux's theorem**: around every point of any 2*n*-dimensional symplectic manifold there are coordinates in which ω takes the standard form Σ d*qⁱ* ∧ d*pᵢ*.[^darboux][^dasilva] Symplectic manifolds therefore have **no local invariants whatsoever** — all of them look identical up close. This is a sharp contrast with Riemannian geometry, where curvature is a local invariant that distinguishes a sphere from a plane at a point. Every symplectic invariant is necessarily global, which is why the subject took so long to develop past its classical-mechanics origins. That the subject is nevertheless not vacuous was settled by **Gromov's non-squeezing theorem** (1985): a ball of radius *r* embeds symplectically into the cylinder *B²(R)* × ℝ^{2n−2} **if and only if** *r* ≤ *R*.[^gromov] For *n* ≥ 2 the cylinder has infinite volume, so a merely volume-preserving embedding always exists — the obstruction is genuinely symplectic and not measure-theoretic. Non-squeezing gave the first symplectic capacity and, with it, the sense in which a phase-space region has an irreducible cross-sectional "width" in each conjugate (*qᵢ*, *pᵢ*) plane. The resonance with the [[Uncertainty_principle|uncertainty principle]] is not coincidental.[^mcduff][^degosson] ## Hamiltonian flow Motion in phase space is generated by a single function, the Hamiltonian *H*(*q*, *p*), through [[Hamiltonian_mechanics|Hamilton's equations]] > d*q*/d*t* = ∂*H*/∂*p*, d*p*/d*t* = −∂*H*/∂*q* which say that the phase velocity is the gradient of *H* rotated by ninety degrees. Trajectories therefore run *along* level sets of *H* and never across them: energy conservation is not an additional law but a geometric consequence of the rotation.[^goldstein][^ll1] The same content in coordinate-free form is that the Hamiltonian vector field *X_H* satisfies ι_{X_H} ω = d*H*, and in algebraic form that any observable evolves by its [[Poisson_bracket|Poisson bracket]] with the Hamiltonian, d*f*/d*t* = {*f*, *H*}. Because the flow preserves ω, it is a one-parameter family of symplectomorphisms. Everything in the next section follows from that one sentence. ## Liouville's theorem and the incompressibility of phase space **Liouville's theorem** states that the phase-space density ρ is constant along trajectories: > ∂ρ/∂*t* + {ρ, *H*} = 0, equivalently **dρ/d*t* = 0** The co-moving derivative vanishes. The theorem does *not* say ∂ρ/∂*t* = 0 — the density at a fixed point of phase space changes all the time — and that substitution is the single most common misstatement of the result.[^arnold][^huang] Geometrically, *X_H* is divergence-free and the Liouville volume ω^n/n! is invariant. The consequences are large and specific. A blob of initial conditions is stretched, sheared and filamented without limit, its perimeter growing without bound, while its area stays *exactly* fixed. Volume cannot contract, so **a Hamiltonian system cannot have an attractor** — no basin, no limit cycle in the dissipative sense, no strange attractor. [[Henri_Poincaré|Poincaré]] recurrence follows too: in a bounded phase space, almost every state returns arbitrarily close to itself, given enough time. A historical note worth getting right: Liouville's 1838 paper is a purely mathematical eight-page note on the variation of arbitrary constants and makes no mention of phase space or dynamical systems at all.[^liouville] The physical theorem that bears his name was given its form by [[Ludwig_Boltzmann|Boltzmann]] and [[Josiah_Willard_Gibbs|Gibbs]].[^nolte] ## Ensembles, coarse-graining, and entropy [[Statistical_mechanics]] abandons the single trajectory and works with a distribution over phase space. For *N* particles in three dimensions the relevant space is 6*N*-dimensional — Gibbs's Γ-space — and one point in it is a single **[[Microstate_(statistical_mechanics)|microstate]]**, with every position and momentum specified. A macrostate is a coarse description, and it corresponds to an enormous set of microstates. Boltzmann's relation *S* = *k*_B ln Ω counts that set. Equilibrium is then not a destination the system is driven toward but simply the macrostate that owns overwhelmingly the most microstates.[^gibbs][^tolman] Liouville's theorem creates a well-known difficulty here, and the resolution is worth stating precisely. The fine-grained Gibbs entropy *S* = −*k*∫ρ ln ρ dΓ is **exactly constant** under Hamiltonian evolution, since the flow merely permutes phase-space points without changing the measure. Entropy increase therefore *requires* coarse-graining: partitioning phase space into finite cells and replacing ρ by its cell average. The coarse-grained Boltzmann entropy *S* = *k* log |Γ_M| can and does rise.[^jaynes][^wehrl][^gl] The Ehrenfests' 1911 encyclopedia article is where this programme was first laid out systematically.[^ehrenfest] Phase space also carries a natural unit of volume once quantum mechanics is admitted: each quantum state occupies *h^f*, which is why the classical partition function for *N* indistinguishable particles carries the factor 1/(*N*! *h*^{3N}) — the *h*^{3N} making the measure dimensionless and the *N*! resolving the [[Gibbs_paradox|Gibbs paradox]].[^huang] On the geometric side, de Gosson's "quantum blob" — the smallest phase-space set with symplectic capacity ½*h* — is the exact counterpart, and it follows from Gromov non-squeezing rather than from any additional postulate.[^degosson] ## Quantum mechanics on phase space Quantum mechanics can be written entirely on classical phase space, and the price is negativity. The **[[Wigner_quasiprobability_distribution|Wigner function]]** > *W*(*q*, *p*) = (1/π*ħ*) ∫ ψ\*(*q*+*y*) ψ(*q*−*y*) e^{2i*py*/*ħ*} d*y* is real, normalised, bounded by |*W*| ≤ 1/(π*ħ*), and returns the correct marginals — integrate over *p* and you recover |ψ(*q*)|², integrate over *q* and you recover the momentum distribution.[^wigner] It is nevertheless not a probability density, because it takes negative values. **Hudson's theorem** makes this exact: for a *pure* state of a system with *one* degree of freedom, *W* is non-negative everywhere **if and only if** the state is Gaussian.[^hudson] The qualifiers matter — the theorem is false for mixed states, and the *n*-dimensional generalisation is due to Soto and Claverie.[^soto] Negative regions of *W* are quantum interference made locatable: they sit in phase space where one can point at them, and the "negative volume" is a genuine nonclassicality measure. The correspondence runs deeper than an analogy. [[Wigner–Weyl_transform|Weyl quantization]] maps phase-space functions to operators;[^weyl] the composition rule that mirrors operator multiplication is the star product, and the **[[Moyal_product|Moyal bracket]]** {{*f*, *g*}} = (1/i*ħ*)(*f* ⋆ *g* − *g* ⋆ *f*) reduces to the [[Poisson_bracket]] as *ħ* → 0.[^moyal] That a bracket-preserving quantization must be replaced by the Moyal bracket rather than achieved exactly is the content of the **Groenewold–van Hove theorem**: no map from polynomials in *q* and *p* to self-adjoint operators simultaneously takes Poisson brackets to commutators and represents the Heisenberg algebra irreducibly, with the obstruction appearing already at cubic order.[^groenewold] Dirac's hoped-for exact correspondence is impossible, and phase-space quantum mechanics is the constructive response. ## Dissipation, attractors, and the state space that is not symplectic Remove the Hamiltonian structure and the picture inverts. In a dissipative system the flow has negative divergence, phase volume contracts, and the long-run behaviour collapses onto an [[Attractor|attractor]] of lower dimension. The [[Lorenz_system]] is the canonical case: > d*x*/d*t* = σ(*y* − *x*), d*y*/d*t* = *x*(ρ − *z*) − *y*, d*z*/d*t* = *xy* − β*z* whose divergence is **−(σ + 1 + β)**, a negative constant *independent of position* — at Lorenz's parameters σ = 10, β = 8/3 this is exactly **−41/3 ≈ −13.67**, so every volume decays as e^{−41t/3} and the attractor has zero volume.[^lorenz] Note the arithmetic: an incorrect value of −49/3 circulates. Tucker's 2002 computer-assisted proof, using interval arithmetic with directed rounding together with normal form theory near the origin, established that the attractor is a robust strange attractor carrying a unique SRB measure — settling Smale's fourteenth problem.[^tucker] This is the exact complement of Liouville's theorem, and the pairing is the most efficient way to understand either. It is also where the vocabulary strains: the Lorenz state space is three-dimensional and therefore odd, so it cannot be symplectic at all. Calling it a phase space is standard usage and harmless as long as one does not expect the theorems to transfer. Between the two extremes sits the structure of near-integrable Hamiltonian systems. The **KAM theorem** — Kolmogorov's 1954 announcement, Moser's 1962 finite-differentiability version, and Arnold's 1963 analytic proof, in that order — shows that under small perturbation most invariant tori survive, specifically those whose frequency vectors are Diophantine, while the rest break up into the chaotic layers that give phase portraits their characteristic islands-in-a-sea appearance.[^arnold63][^kam] ## Where phase space is used In **accelerator physics** Liouville's theorem is an operational constraint: under conservative forces the six-dimensional phase-space volume of a beam cannot be reduced, which is why **emittance** — phase-space area divided by π — is the fundamental figure of merit.[^wiedemann] Beam cooling does reduce emittance, and it does so not by violating Liouville but by being non-Hamiltonian; radiation damping, stochastic cooling and electron cooling all involve dissipation or feedback, which fall outside the theorem's hypotheses. In **plasma physics** the Vlasov equation is a Liouville equation on the single-particle six-dimensional (*x*, *v*) phase space with self-consistent fields. **Landau damping** is then a phase-space phenomenon: collisionless decay of the electric field by phase mixing and filamentation in velocity space, with coarse-grained entropy rising while the fine-grained distribution remains volume-preserving.[^nicholson] Mouhot and Villani's nonlinear proof reframed it as a transfer of regularity between spatial and velocity variables.[^mv] **[[Optical_phase_space|Optical phase space]]** pairs the quadratures of a light mode, where the Wigner function is directly reconstructible by homodyne tomography, and **[[Molecular_dynamics|molecular dynamics]]**, control theory and [[Population_dynamics|population dynamics]] all use the phase portrait as their primary diagnostic. ## The name The origins of both the concept and the term are obscure, and confident attributions should be distrusted. Boltzmann introduced "phase" terminology in 1872; Maxwell adopted it in 1879; Gibbs in 1902 was the first to describe explicitly the trajectory of a *phase point* in a high-dimensional space, though he used "space" hesitantly and largely in footnotes. The first appearance in print of the expression *Phasenraum* appears to be a throwaway phrase in Paul and Tatiana Ehrenfest's 1911 encyclopedia article, which they never reuse — their own preferred term was Γ-space.[^nolte] The geometric prerequisite, multi-dimensional spaces as legitimate objects, came earlier from Cayley, Grassmann and Riemann. Attribution of the term to Liouville is simply wrong. --- ## References [^arnold]: Arnold, V. I. *Mathematical Methods of Classical Mechanics.* 2nd ed. Graduate Texts in Mathematics 60. Springer, 1989. DOI: [10.1007/978-1-4757-2063-1](https://doi.org/10.1007/978-1-4757-2063-1) [^am]: Abraham, Ralph, and Jerrold E. Marsden. *Foundations of Mechanics.* 2nd ed. Benjamin/Cummings, 1978; repr. AMS Chelsea 364. ISBN 978-1-4704-8474-3 [^darboux]: Darboux, Gaston. "Sur le problème de Pfaff." *Bulletin des Sciences Mathématiques et Astronomiques* (2) 6 (1882): 14–36, 49–68. [^dasilva]: Cannas da Silva, Ana. *Lectures on Symplectic Geometry.* Lecture Notes in Mathematics 1764. Springer, 2001. DOI: [10.1007/978-3-540-45330-7](https://doi.org/10.1007/978-3-540-45330-7) [^gromov]: Gromov, Mikhael. "Pseudo holomorphic curves in symplectic manifolds." *Inventiones Mathematicae* 82, no. 2 (1985): 307–347. DOI: [10.1007/BF01388806](https://doi.org/10.1007/BF01388806) [^mcduff]: McDuff, Dusa, and Dietmar Salamon. *Introduction to Symplectic Topology.* 3rd ed. Oxford University Press, 2017. DOI: [10.1093/oso/9780198794899.001.0001](https://doi.org/10.1093/oso/9780198794899.001.0001) [^degosson]: de Gosson, Maurice A. "Quantum blobs." *Foundations of Physics* 43, no. 4 (2013): 440–457. DOI: [10.1007/s10701-012-9636-x](https://doi.org/10.1007/s10701-012-9636-x) [^goldstein]: Goldstein, Herbert, Charles Poole, and John Safko. *Classical Mechanics.* 3rd ed. Addison Wesley, 2002. ISBN 978-0-201-65702-9 [^ll1]: Landau, L. D., and E. M. Lifshitz. *Mechanics.* 3rd ed. Course of Theoretical Physics 1. Butterworth-Heinemann, 1976. ISBN 978-0-7506-2896-9 [^huang]: Huang, Kerson. *Statistical Mechanics.* 2nd ed. Wiley, 1987. ISBN 978-0-471-81518-1 [^liouville]: Liouville, Joseph. "Note sur la Théorie de la Variation des constantes arbitraires." *Journal de Mathématiques Pures et Appliquées* (1) 3 (1838): 342–349. [NUMDAM](http://www.numdam.org/item/JMPA_1838_1_3__342_0/) [^nolte]: Nolte, David D. "The tangled tale of phase space." *Physics Today* 63, no. 4 (2010): 33–38. DOI: [10.1063/1.3397041](https://doi.org/10.1063/1.3397041) [^gibbs]: Gibbs, J. Willard. *Elementary Principles in Statistical Mechanics.* Charles Scribner's Sons, 1902. [^tolman]: Tolman, Richard C. *The Principles of Statistical Mechanics.* Oxford University Press, 1938. [^jaynes]: Jaynes, E. T. "Gibbs vs Boltzmann entropies." *American Journal of Physics* 33, no. 5 (1965): 391–398. DOI: [10.1119/1.1971557](https://doi.org/10.1119/1.1971557) [^wehrl]: Wehrl, Alfred. "General properties of entropy." *Reviews of Modern Physics* 50, no. 2 (1978): 221–260. DOI: [10.1103/RevModPhys.50.221](https://doi.org/10.1103/RevModPhys.50.221) [^gl]: Goldstein, Sheldon, and Joel L. Lebowitz. "On the (Boltzmann) entropy of non-equilibrium systems." *Physica D* 193, nos. 1–4 (2004): 53–66. DOI: [10.1016/j.physd.2004.01.008](https://doi.org/10.1016/j.physd.2004.01.008) [^ehrenfest]: Ehrenfest, Paul, and Tatiana Ehrenfest. "Begriffliche Grundlagen der statistischen Auffassung in der Mechanik." *Encyklopädie der mathematischen Wissenschaften* IV.32 (1911). Trans. M. J. Moravcsik, *The Conceptual Foundations of the Statistical Approach in Mechanics*, Cornell University Press, 1959. [^wigner]: Wigner, Eugene. "On the quantum correction for thermodynamic equilibrium." *Physical Review* 40, no. 5 (1932): 749–759. DOI: [10.1103/PhysRev.40.749](https://doi.org/10.1103/PhysRev.40.749) [^hudson]: Hudson, R. L. "When is the Wigner quasi-probability density non-negative?" *Reports on Mathematical Physics* 6, no. 2 (1974): 249–252. DOI: [10.1016/0034-4877(74)90007-X](https://doi.org/10.1016/0034-4877%2874%2990007-X) [^soto]: Soto, F., and P. Claverie. "When is the Wigner function of multidimensional systems nonnegative?" *Journal of Mathematical Physics* 24, no. 1 (1983): 97–100. DOI: [10.1063/1.525607](https://doi.org/10.1063/1.525607) [^weyl]: Weyl, Hermann. "Quantenmechanik und Gruppentheorie." *Zeitschrift für Physik* 46, nos. 1–2 (1927): 1–46. DOI: [10.1007/BF02055756](https://doi.org/10.1007/BF02055756) [^moyal]: Moyal, J. E. "Quantum mechanics as a statistical theory." *Mathematical Proceedings of the Cambridge Philosophical Society* 45, no. 1 (1949): 99–124. DOI: [10.1017/S0305004100000487](https://doi.org/10.1017/S0305004100000487) [^groenewold]: Groenewold, H. J. "On the principles of elementary quantum mechanics." *Physica* 12, no. 7 (1946): 405–460. DOI: [10.1016/S0031-8914(46)80059-4](https://doi.org/10.1016/S0031-8914%2846%2980059-4) [^lorenz]: Lorenz, Edward N. "Deterministic nonperiodic flow." *Journal of the Atmospheric Sciences* 20, no. 2 (1963): 130–141. DOI: [10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2](https://doi.org/10.1175/1520-0469%281963%29020%3C0130:DNF%3E2.0.CO;2) [^tucker]: Tucker, Warwick. "A rigorous ODE solver and Smale's 14th problem." *Foundations of Computational Mathematics* 2, no. 1 (2002): 53–117. DOI: [10.1007/s002080010018](https://doi.org/10.1007/s002080010018) [^arnold63]: Arnold, V. I. "Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian." *Russian Mathematical Surveys* 18, no. 5 (1963): 9–36. DOI: [10.1070/RM1963v018n05ABEH004130](https://doi.org/10.1070/RM1963v018n05ABEH004130) [^kam]: Kolmogorov, A. N. *Doklady Akademii Nauk SSSR* 98 (1954): 527–530; Moser, Jürgen. *Nachrichten der Akademie der Wissenschaften in Göttingen* II (1962): 1–20. (Page ranges attested by secondary sources.) [^wiedemann]: Wiedemann, Helmut. *Particle Accelerator Physics.* 4th ed. Springer, 2015. Open access. DOI: [10.1007/978-3-319-18317-6](https://doi.org/10.1007/978-3-319-18317-6) [^nicholson]: Nicholson, Dwight R. *Introduction to Plasma Theory.* Wiley, 1983. ISBN 978-0-471-09045-8 [^mv]: Mouhot, Clément, and Cédric Villani. "On Landau damping." *Acta Mathematica* 207, no. 1 (2011): 29–201. DOI: [10.1007/s11511-011-0068-9](https://doi.org/10.1007/s11511-011-0068-9)   ## Play every microsim on this page <p class="microsim-gallery-note">Every microsim linked from this portal, embedded live. <strong>28 players</strong> — 16 threejs · 12 p5js. 2 are staged but not yet deployed and show as placeholders.</p> <ul class="microsim-gallery"> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Attractor.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Attractor — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Attractor">Attractor</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Attractor.html" target="_blank" rel="noopener">open full-screen</a></div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/polytope4d.html" loading="lazy" 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rel="noopener">open full-screen</a></div></div> </li> <li class="microsim-card is-pending" data-lib="threejs"> <div class="ms-placeholder">Staged, not yet deployed — <code>Molecular_dynamics.html</code></div> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Molecular_dynamics">Molecular dynamics</a></div><div class="ms-sub">threejs · awaiting CDN deploy</div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Numerical_integration.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Numerical integration — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Numerical_integration">Numerical integration</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Numerical_integration.html" target="_blank" rel="noopener">open full-screen</a></div></div> </li> <li class="microsim-card" data-lib="threejs"> <iframe src="https://wikitube-3d-microsims.netlify.app/Population_growth.html" loading="lazy" frameborder="0" sandbox="allow-scripts allow-same-origin" title="Population growth — threejs microsim"></iframe> <div class="ms-meta"><div class="ms-title"><a href="https://en.wikitube.io/wiki/Population_growth">Population growth</a></div><div class="ms-sub">threejs · <a href="https://wikitube-3d-microsims.netlify.app/Population_growth.html" target="_blank" rel="noopener">open full-screen</a></div></div> </li> </ul>   **Craft standard:** Both craft standards apply here — [[WT!Three_js_Microsim_Master_Class|three.js]] and [[WT!P5_js_Microsim_Master_Class|p5.js]].   ## Deploy state · added 2026-09-10 All seven three.js players in the portal's first section are live on `wikitube-3d-microsims.netlify.app` and now play in place. The "Staged, ready to deploy" note up there predates that.   *Microsims from the articles linked below:* <div class="wt-simrow"> <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Momentum.html" data-title="Momentum"></div> <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Pendulum.html" data-title="Pendulum"></div> <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Phase_diagram.html" data-title="Phase diagram"></div> <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Potential_energy.html" data-title="Potential energy"></div> <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Simple_harmonic_motion.html" data-title="Simple harmonic motion"></div> <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Special_relativity.html" data-title="Special relativity"></div> </div>   ## Canonical link structure — 232 tree-verified children Hub article: [[Phase_space]] · tree harvested 2026-07-29T23:31:50Z · every entry below is in this hub's harvested child tree (v09-gated). Unbuilt links are intentional forward-refs: the populate scaffold. **A** — [[Abel's_identity]] · [[Action_(physics)]] · [[Alexander_Hrennikoff]] · [[Alexis_Clairaut]] · [[Ansatz]] · [[Applied_mathematics]] · [[ArXiv]] · [[Astronomy]] · [[Asymptotic_expansion]] · [[Attractor]] · [[Augustin-Louis_Cauchy]] · [[Autonomous_system_(mathematics)]] · [[Avogadro_constant]] **B** — [[Bibcode]] · [[Biological_engineering]] · [[Biology]] · [[Boundary_value_problem]] **C** — [[Canonical_coordinates]] · [[Carathéodory's_existence_theorem]] · [[Carl_Neumann]] · [[Carl_Runge]] · [[Cauchy_problem]] · [[Cauchy–Kovalevskaya_theorem]] · [[Chaos_theory]] · [[Chemistry]] · [[Classical_mechanics]] · [[Complex_quadratic_polynomial]] · [[Configuration_space_(mathematics)]] · [[Configuration_space_(physics)]] · [[Continuum_mechanics]] · [[Cotangent_bundle]] · [[Crank–Nicolson_method]] · [[CRC_Press]] · [[Curve]] **D** — [[Darboux's_theorem]] · [[Deformation_(mathematics)]] · [[Deformation_quantization]] · [[Degrees_of_freedom_(physics_and_chemistry)]] · [[Delay_differential_equation]] · [[Dependent_and_independent_variables]] · [[Differential-algebraic_system_of_equations]] · [[Differential_equation]] · [[Differential_operator]] · [[Digital_object_identifier]] · [[Dirac_delta_function]] · [[Direct_product]] · [[Dirichlet_boundary_condition]] · [[Distribution_(mathematical_analysis)]] · [[Dynamical_system]] **E** — [[Economics]] · [[Encyclopedia_of_Mathematics]] · [[Engineering]] · [[Ensemble_(mathematical_physics)]] · [[Ernst_Leonard_Lindelöf]] · [[Eugene_Wigner]] · [[Euler_method]] · [[Exact_differential_equation]] · [[Exponential_response_formula]] · [[Exponential_stability]] **F** — [[Finite_difference_method]] · [[Finite_element_method]] · [[Finite_volume_method]] · [[Floquet_theory]] · [[Fractional_calculus]] **G** — [[Galerkin_method]] · [[Generalized_coordinates]] · [[General_relativity]] · [[Geology]] · [[Geometric_quantization]] · [[George_Boole]] · [[George_Green_(mathematician)]] · [[Gibbs_paradox]] · [[Glossary_of_mathematical_jargon]] · [[Gottfried_Wilhelm_Leibniz]] · [[Green's_function]] **H** — [[Hamiltonian_mechanics]] · [[Hamiltonian_optics]] · [[Handle_System]] · [[Harold_Jeffreys]] · [[Henri_Poincaré]] · [[Hermann_Weyl]] · [[Hilbert_space]] · [[Hilbrand_J._Groenewold]] · [[Holonomic_function]] · [[Homogeneous_differential_equation]] **I** — [[Indistinguishable_particles]] · [[Infinite_element_method]] · [[Initial_condition]] · [[Initial_value_problem]] · [[Integral]] · [[Integral_transform]] · [[Integrating_factor]] · [[Integration_by_substitution]] · [[Integro-differential_equation]] · [[Isaac_Newton]] · [[ISBN]] · [[ISSN]] **J** — [[Jacob_Bernoulli]] · [[Jacques_Charles_François_Sturm]] · [[Jean_Le_Rond_d'Alembert]] · [[Jet_bundle]] · [[John_Crank]] · [[John_von_Neumann]] · [[Joseph-Louis_Lagrange]] · [[Joseph_Fourier]] · [[Joseph_Liouville]] · [[Josiah_Willard_Gibbs]] · [[José_Enrique_Moyal]] · [[Józef_Maria_Hoene-Wroński]] **L** — [[Lagrangian_mechanics]] · [[Leonhard_Euler]] · [[Limit_cycle]] · [[Linear_differential_equation]] · [[Liouville's_theorem_(Hamiltonian)]] · [[Liquid]] · [[List_of_linear_ordinary_differential_equations]] · [[List_of_named_differential_equations]] · [[List_of_nonlinear_ordinary_differential_equations]] · [[List_of_nonlinear_partial_differential_equations]] · [[Logistic_function]] · [[Logistic_map]] · [[Lorenz_system]] · [[Ludwig_Boltzmann]] · [[Lyapunov_stability]] **M** — [[Mandelbrot_set]] · [[Manifold]] · [[Martin_Kutta]] · [[Mathematics]] · [[Method_of_characteristics]] · [[Method_of_quantum_characteristics]] · [[Method_of_undetermined_coefficients]] · [[Microstate_(statistical_mechanics)]] · [[Minisuperspace]] · [[Molecular_dynamics]] · [[Momentum]] · [[Moyal_product]] · [[Multidimensional_scaling]] **N** — [[Natural_science]] · [[Neumann_boundary_condition]] · [[Nonimaging_optics]] · [[Nonlinear_system]] · [[Notation_for_differentiation]] · [[Numerical_integration]] **O** — [[Observable]] · [[Optical_phase_space]] · [[Orbit_(dynamics)]] · [[Ordinary_differential_equation]] · [[Outline_of_physical_science]] **P** — [[Parameter]] · [[Parameter_space]] · [[Partial_differential_equation]] · [[Partition_function_(mathematics)]] · [[Peano_existence_theorem]] · [[Pendulum]] · [[Periodic_boundary_conditions]] · [[Perturbation_theory]] · [[Peter_Gustav_Lejeune_Dirichlet]] · [[Petrov–Galerkin_method]] · [[Phase-space_formulation]] · [[Phase_diagram]] · [[Phase_line_(mathematics)]] · [[Phase_plane]] · [[Phase_portrait]] · [[Phase_space_(disambiguation)]] · [[Phase_space_method]] · [[Phyllis_Nicolson]] · [[Physical_system]] · [[Physics]] · [[Picard–Lindelöf_theorem]] · [[Pierre-Simon_Laplace]] · [[Planck_constant]] · [[Plot_(graphics)]] · [[Point_(geometry)]] · [[Poisson_bracket]] · [[Population_dynamics]] · [[Population_growth]] · [[Position_(geometry)]] · [[Potential_energy]] · [[Power_series_solution_of_differential_equations]] · [[Pressure]] · [[Pressure–volume_diagram]] · [[PubMed]] · [[PubMed_Central]] **Q** — [[Quantum_mechanics]] · [[Quantum_number]] · [[Quantum_state_space]] **R** — [[Rate_of_convergence]] · [[Reciprocal_lattice]] · [[Recurrence_relation]] · [[Reduction_of_order]] · [[Richard_Courant]] · [[Robin_boundary_condition]] · [[Robotic_arm]] · [[Rudolf_Lipschitz]] · [[Runge–Kutta_methods]] **S** — [[Schwarzschild_radius]] · [[Self-adjoint_operator]] · [[Semantic_Scholar]] · [[Separation_of_variables]] · [[Separatrix_(mathematics)]] · [[Simple_harmonic_motion]] · [[Social_science]] · [[Sofya_Kovalevskaya]] · [[Solid]] · [[Special_functions]] · [[Special_relativity]] · [[State]] · [[State-space_representation]] · [[State_space_(computer_science)]] · [[State_variable]] · [[Statistical_mechanics]] · [[Steady_state]] · [[Stochastic_differential_equation]] · [[Stochastic_partial_differential_equation]] · [[Sturm–Liouville_theory]] · [[Symplectic_geometry]] · [[Symplectic_manifold]] **T** — [[Temperature]] · [[Temperature–entropy_diagram]] · [[Thermodynamics]] · [[Topological_conjugacy]] **U** — [[Uncertainty_principle]] **V** — [[Vague_torus]] · [[Van_der_Pol_oscillator]] · [[Variation_of_parameters]] · [[Victor_Gustave_Robin]] **W** — [[Well-posed_problem]] · [[Wigner_quasiprobability_distribution]] · [[Wigner–Weyl_transform]] · [[WKB_approximation]] · [[Wronskian]] **#** — [[Émile_Picard]] ## Route objects (§10.3 stubs — placeholders declared in `_registry/firebase/`) - **microsim** — `microsim/{library}/Phase_space__{YYYYMMDD}T{HHMM}Z/` - **songs** — `songs/{Band_name}/{Song_name}/Phase_space/` - **semiotics** — `semiotics/{authorizer}/{set}/{Song_name}/{audio_video_id}/Phase_space/` - **cardset** — `cardset/{authorizer}/{set}/Phase_space/` - **games** — `games/quizzes/Phase_space/` - **channel_stream** — `channel_stream/Phase_space/` - **transcripts** — `transcripts/{audio_video_id}/Phase_space/` Hub + children route keys are pre-declared in `_registry/firebase/expansion_matches.csv` (54,000 Related Wikipedia Matches). Minting any leaf passes §10.1 registration.  --- *Repopulated 2026-09-19 · append-only · source: _tools/generate/g34_portal_section_sims.py@36068b13 (players of the linked articles, each URL 200-checked) · 1 added · 0 deletions*