Flow Lab · Helmholtz–Hodge & NESS

01 / Helmholtz–Hodge decomposition

One field. Three different kinds of motion.

The arrows describe a velocity at every point. Split a random smooth field into a gradient part, a solenoidal part and a harmonic part. Adjust their strengths to rebuild the total.

Seed 42 · 12 Fourier modes

Gradient 1.00

Solenoidal 1.00

Harmonic 0.35

Total field f

The pointwise sum of all three fields.

Gradient −∇φ

Flows downhill in φ. Curl is zero.

Solenoidal R∇ψ

Follows contours of ψ. Divergence is zero.

Harmonic h

Uniform transport. Curl and divergence are zero.

Arrows share one linear scale. Tracers follow each field independently; their paths do not add. All coordinates and times are dimensionless.

Gradient share of squared field norm

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Solenoidal / harmonic shares

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Reconstruction error · relative L²

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RMS curl(gradient) / div(solenoidal)

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A complicated flow can be a simple sum.

The blue part organises motion down a potential; the green part moves along its own stream-function contours; the amber part carries everything across the periodic domain.

Try thisTurn off the gradient. The flow can still move and circulate, but it no longer locally compresses area. Then leave only the harmonic part: motion survives even though both curl and divergence vanish.

THE INTUITION

Downhill, around, and across

A landscape gives you a downhill direction. A stream function gives you contour-following motion. A periodic domain also permits a constant current around the whole space. The decomposition separates these geometries.

WHAT “RANDOM” MEANS HERE

A random field, fixed in time

Each seed creates smooth, arbitrary vector-valued Fourier coefficients. The field is then projected into its components. These tracers have no noise; randomness enters when the field is created. Experiment 02 adds ongoing stochastic forcing.

THE BOUNDARY MATTERS

This square is a flat torus

Leaving one edge brings a tracer back at the opposite edge. Under these periodic conditions, the harmonic field is the spatial mean. Other domains and boundary conditions can give a different harmonic part and a different split.

The decomposition, step by step Equations & definitions

f(x) = −∇φ(x) + R∇ψ(x) + h

f, x: The velocity field and a point in the two-dimensional state space. A tracer obeys dx/dt = f(x).
φ: A scalar potential. Its gradient points uphill; the minus sign makes this part point downhill.
ψ, R: A scalar stream function and a 90° anticlockwise rotation: R(a, b) = (−b, a). Rotating ∇ψ makes the flow tangent to contours of ψ.
h: The constant mean field on this flat periodic domain. It cannot be absorbed into a single-valued periodic potential.

In two dimensions, divergence is ∂_xf_x + ∂_yf_y: local expansion or compression. Scalar curl is ∂_xf_y − ∂_yf_x: local rotation. Mixed partial derivatives cancel, so

curl(−∇φ) = 0, ∇·(R∇ψ) = 0, curl h = ∇·h = 0.

How the random field is actually decomposed

For every nonzero wavevector k, project the field’s Fourier coefficient f̂(k) parallel to k and perpendicular to k. The zero-frequency coefficient is the harmonic part. This finite Fourier projection is exact up to floating-point arithmetic; the animation uses a grid only for drawing arrows.

ĝ(k) = kk^T|k|²f̂(k), ŝ(k) = f̂(k) − ĝ(k), h = f̂(0)

The sliders multiply the projected components before summing them. Because the components are orthogonal under the spatial L² inner product on this domain, their squared norms add: ‖f‖² = ‖g‖² + ‖s‖² + ‖h‖². The displayed percentages use this mathematical norm; they are not a claim about physical energy.

Two different potentials. The gradient part follows φ; the solenoidal part follows contours of ψ. In a general Helmholtz–Hodge decomposition, ψ need not equal φ, and the two vector components need not be perpendicular at each point. Their orthogonality here is an integral over the domain.

Why “downhill” and “incompressible” are precise claims

For dx/dt = −∇φ: dφdt = −|∇φ|² ≤ 0.

The potential cannot increase along pure gradient flow. This does not require divergence to be negative everywhere: a gradient field may locally expand near a maximum while still descending in φ.

For dx/dt = R∇ψ: dψdt = ∇ψ·R∇ψ = 0.

Antisymmetry gives contour-following motion. Zero divergence also preserves infinitesimal area under the deterministic flow. Neither statement means that every divergence-free field conserves every physical energy you might assign to it.

Numerical notes

The domain is [−π, π)². Twelve independent integer wavevectors carry seeded random sine and cosine vector coefficients, with high frequencies attenuated for smoothness. The mean flow points along (1, 0.35). Diagnostics use analytic mode derivatives and a 32 × 32 quadrature grid. Tracers use fourth-order Runge–Kutta steps with periodic wrapping; their finite trails are illustrative, not a density estimator.

02 / A nonequilibrium steady state

The pattern is steady. The particles are not.

Inward drift confines the cloud; noise spreads it out. Add rotation and probability circulates without changing the stationary density. Here, “steady” describes the distribution, not each trajectory.

Confinement κ 1.00

Circulation ω 1.50

Diffusion D 0.50

Noisy trajectories dX = f dt + √(2D) dW

Blue arrows: deterministic drift f. Dots: 450 stochastic trajectories. The bright dot has a short trail.

Stationary probability current j* = ω R x p*

Green arrows: net current j*. Background: stationary density p*, coloured relative to its peak. Rings: 1, 2 and 3 coordinate standard deviations.

Two even, overdamped state coordinates; constant isotropic diffusion. The square is a cropped view of ℝ², with no wrapping or reflecting walls. Parameter changes restart the stationary cloud.

Stationary variance · each coordinate

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Observed / stationary mean r²

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Steady entropy production · kB = 1

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Current state

NESS

The same density can hide a different dynamics.

Change ω while keeping κ and D fixed. The Gaussian density stays the same, but the circulating probability current changes direction and strength.

Try thisSet ω to zero. The current disappears, yet the dots keep moving. Now restore rotation: a steady circulating current appears. Reverse ω to reverse the arrow of circulation.

CONFINEMENT + NOISE

A cloud instead of a point

Without noise, the inward drift would pull every trajectory towards the origin. Noise continually scatters particles. Their balance produces a Gaussian cloud whose width is set by D/κ.

STEADY ≠ EQUILIBRIUM

Traffic can flow through a steady pattern

The number of particles expected in each region stays constant when inflow balances outflow. At equilibrium the net probability current is zero. A NESS has persistent current even though its density does not change.

A CONTROLLED EXAMPLE

Rotation leaves this density unchanged

The rotational drift is tangent to the circular density contours. This is deliberately an exactly solvable example. In a general noisy system, altering circulation can also alter the stationary density.

From a noisy equation to a stationary current Equations & definitions

dX_t = [−κX_t + ωRX_t] dt + √(2D) dW_t

X, W: A random two-dimensional state and two independent standard Wiener processes. Noise increments over Δt have variance Δt.
κ > 0: Inward relaxation rate. A larger κ gives stronger confinement.
ω: Angular circulation rate. Positive means anticlockwise; negative reverses it.
D > 0: Diffusion coefficient. Each coordinate receives noise with variance 2D Δt.

The probability density p(x, t) obeys a continuity equation—the Fokker–Planck equation for this constant diffusion:

∂_tp = −∇·j, j = f p − D∇p.

f p transports probability with the deterministic drift; −D∇p spreads probability from dense regions to sparse regions. Stationarity asks for ∇·j_* = 0, not necessarily j_* = 0.

p_*(x) = κ2πD exp(−κ|x|²2D), j_*(x) = ωRx p_*(x).

To check this, use ∇p_* = −(κ/D)x p_*. Diffusion cancels the inward drift in the probability current. Only circulation remains. Since R x is orthogonal to x, the circulation follows density contours and has zero divergence.

These arrows represent different things. Drift f is a velocity; current j_* is probability transported per unit boundary length and time. Their arrow scales are separate and can adjust with the parameters for readability. Within each panel, arrow lengths are proportional to field magnitude. Current is strongest away from the origin, but becomes small far out where little probability remains.

The link to surprisal, gradient flow and solenoidal flow

Define the stationary surprisal Φ(x) = −log p_*(x). A likely state has low surprisal. For constant scalar diffusion, rearranging the current equation gives

f = −D∇Φ + u, u = j_*p_*, ∇·(p_*u) = 0.

The first term restores probability towards more likely regions. The second is the steady current velocity. Its defining constraint is weighted by the stationary density.

This is not automatically the ordinary Helmholtz–Hodge split. In general ∇·(p_*u) = 0 does not imply ∇·u = 0, or that u is tangent to surprisal contours. The divergence-free object guaranteed by stationarity is the probability current. Those stronger properties hold in this particular example.

Here, Φ = κ|x|²/(2D) + constant. With the constant antisymmetric matrix Q = (ωD/κ)R, the drift can also be written

f = (Q − D I)∇Φ, Q^T = −Q, ∇Φ·Q∇Φ = 0.

This is the connection to the gradient/solenoidal notation often used for stochastic systems and in Bayesian mechanics. Our sign convention puts +Q in the drift; conventions for the name Q vary. The simple expression above assumes constant coefficients. State-dependent diffusion or antisymmetric matrices require derivative terms; boundaries and topology also matter.

A stochastic trajectory need not go downhill

d E[Φ]dt = E[−D|∇Φ|² + DΔΦ] = 0 at stationarity here.

The deterministic drift reduces Φ, but the Itô diffusion correction offsets that decrease in the stationary ensemble. Individual noisy trajectories can move to higher surprisal. A steady state is not a continual collapse of the distribution onto its mode.

Irreversibility, entropy production and the simulation

σ = ∫ |j_*|²D p_* dx = ω²D E_*[|x|²] = 2ω²κ.

σ is the steady-state entropy production rate in units with Boltzmann’s constant k_B = 1, using the standard overdamped, all-even-coordinate time reversal. It vanishes at ω = 0 and is positive otherwise. In this model a circulating drift can preserve the quadratic surprisal along its own deterministic paths while maintaining positive entropy production in the noisy steady state.

This interpretation must not be transferred unchanged to a position–momentum system. Momentum reverses sign under time reversal, and equilibrium Hamiltonian motion can have reversible phase-space current. Experiment 03 uses that different mechanical setting.

Exact Gaussian transitions

The simulation uses the exact finite-time transition for this linear Ornstein–Uhlenbeck process, rather than Euler–Maruyama. For a step Δt, rotate by ωΔt, contract by exp(−κΔt), and add independent Gaussian noise with variance (D/κ)[1 − exp(−2κΔt)] per coordinate. The initial cloud is sampled from the exact stationary Gaussian, except when “Release from one point” is chosen.

The density and current panels show stationary analytic quantities even during a transient release. During that release, the observed cloud is not yet stationary and its instantaneous current need not match j_*. The 450-particle moment estimate fluctuates; it is not expected to equal the theoretical moment on every frame.

03 / Conservative and dissipative dynamics

Going round is different from running down.

Compare the same oscillator with and without friction. Without friction, energy stays constant. With friction, trajectories lose energy and spiral towards rest—or return without oscillating if damping is strong enough.

Friction γ · right-hand oscillator 0.35

Click either phase plot to choose a new starting state. Both systems start together.

Conservative γ = 0

The trajectory stays on one energy contour. The small patch of nearby initial states preserves its area.

With friction γ = 0.35

Energy decreases whenever momentum is nonzero and γ > 0. The patch of nearby states contracts.

Energy remaining · H(t) / H(0)

ConservativeWith friction

Conservative energy remaining

100%

Damped energy remaining

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Damped phase-space area remaining

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Damped flow divergence

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Friction removes energy, not just speed at one point.

The undamped system exchanges potential and kinetic energy. Friction makes their sum decrease. It also compresses a cloud of nearby states in phase space.

Try thisSet γ = 0 and the two motions coincide. Add weak friction to make a spiral. Set γ above 2 to see an overdamped return. Scrub time to inspect the energy and area changes.

CONSERVATIVE

Energy changes form

A mass on an ideal spring trades kinetic energy for potential energy and back. It can keep moving without losing total mechanical energy. The phase portrait uses position and momentum, not two positions.

DISSIPATIVE

Energy leaves the oscillator

Friction transfers mechanical energy to degrees of freedom outside this model, such as heat in the environment. The oscillator’s energy falls; energy conservation for a larger closed system need not fail.

THE LANGUAGE TRAP

A gradient force can be conservative

The spring force is minus the gradient of a potential. In Newton’s second-order equation it supports conservative oscillation. The first-order equation “velocity equals minus a gradient” instead describes relaxation.

Hamiltonian flow plus friction Equations & definitions

Use unit mass and spring stiffness. Let q be position and p momentum. The Hamiltonian is the total mechanical energy:

H(q, p) = ½q² + ½p², q̇ = p, ṗ = −q − γp.

q, p: Position and momentum. The two axes are phase-space coordinates.
½q², ½p²: Potential energy in the spring and kinetic energy in the moving mass.
γ ≥ 0: Friction coefficient. γ = 0 is conservative; γ > 0 removes mechanical energy.

Differentiate H along a trajectory. The exchange terms cancel:

dHdt = qq̇ + pṗ = qp + p(−q − γp) = −γp² ≤ 0.

The conservative part (p, −q) is a rotated energy gradient. The friction part (0, −γp) is a negative gradient in phase space:

f_H = J∇H = (p, −q), f_γ = −∇(½γp²).

J(a, b) = (b, −a) is the conventional Hamiltonian rotation, so J = −R from the earlier experiments. Its antisymmetry ensures ∇H·J∇H = 0. This is an explicit geometric split, not a unique periodic Hodge projection of this unbounded linear field.

Why the nearby-state patch contracts

∇·f = −γ, A(t)A(0) = exp(−γt).

A is the area of a small set of initial conditions carried by the deterministic flow. Liouville’s formula gives the exponential area change. With γ = 0 the area stays fixed; with friction it contracts. This exact area ratio is distinct from the energy ratio, which also depends on the state.

Numerics. Both trajectories and the nearby-state patch use the exact matrix exponential of this linear oscillator, including the critically damped case γ = 2. The time window is 20 dimensionless units. The phase plots are cropped views of the plane; there are no boundary forces. The energy curve shows the history up to the selected time.

Keep these three meanings of “conservative” separate

Statement	What it means	What it does not establish
A force is conservative	F = −∇U; its work is path independent when the potential is globally defined.	It does not mean first-order motion ẋ = F conserves U.
A flow is solenoidal	∇·f = 0; deterministic flow preserves local volume.	It does not by itself specify a conserved physical energy.
A dynamical system conserves energy	A specified H satisfies ∇H·f = 0 along trajectories.	It does not, by itself, imply zero stochastic entropy production under every noise model.
A distribution is stationary	∂_tp = 0, or ∇·j_* = 0.	It does not imply that individual states stop moving or that detailed balance holds.

The useful question is always: which field, which space, which conserved quantity, which boundary conditions, and which definition of time reversal? The three experiments choose these explicitly so the visual parallels do not become false equivalences.