Angle-Distribution Relaxation
in a Many-Body Orbital System

Jonathan R. Landers
A coarse-grained Newtonian flattening theorem

1. Setting

Consider a dominant central mass $m_1$ and orbiting masses $m_2,\dots,m_N$. For each orbiting body $k\in\{2,\dots,N\}$, let

$$ \mathbf{r}_k(t)=(x_k(t),y_k(t),z_k(t))\in\mathbb{R}^3 $$

denote its position relative to $m_1$. In the coarse-grained flattening model below, we do not evolve the instantaneous elevation angle of the particle position. Instead, we track the signed inclination of the body's orbital plane relative to a preferred horizontal plane, denoted by

$$ \theta_k(t)\in\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]. $$

The orbital phase of body $k$ evolves separately within that plane. We are interested in the evolving distribution of the plane inclinations $\{\theta_k(t)\}$. If the initial inclinations are uniformly distributed on $[-\pi/2,\pi/2]$, then the initial law is

$$ p_0(\theta)=\frac{1}{\pi}\,\mathbf{1}_{[-\pi/2,\pi/2]}(\theta), $$

and the empirical distribution at time $t$ is

$$ p_N(\theta,t)=\frac{1}{N-1}\sum_{k=2}^N \delta\!\big(\theta-\theta_k(t)\big). $$

The full gravitational dynamics are

$$ \ddot{\mathbf{r}}_k = -Gm_1\,\frac{\mathbf{r}_k}{|\mathbf{r}_k|^3} - G\!\!\sum_{\substack{j=2\\ j\neq k}}^N m_j\,\frac{\mathbf{r}_k-\mathbf{r}_j}{|\mathbf{r}_k-\mathbf{r}_j|^3}, \qquad k=2,\dots,N. $$

If $\Phi_t$ denotes the induced phase-space flow, the angle distribution at time $t$ is the pushforward of the initial phase-space law through the map $X_0\mapsto\Theta_k(\Phi_t(X_0))$.

Observation

In the pure $N$-body Hamiltonian setting, there is generally no reason for the inclination law $p_t$ to converge to a limiting distribution $p_*$. The dynamics transport and distort orbital planes, but do not by themselves provide dissipation or alignment. If one were instead to study the instantaneous elevation angle $\arcsin(z_k/|\mathbf r_k|)$, its distribution would oscillate with orbital phase even more strongly.

Nevertheless, the Newtonian many-body system is not geometrically featureless. When the total angular momentum is nonzero, it singles out a natural axis $$ \mathbf e_z := \frac{\mathbf L_{\mathrm{tot}}}{|\mathbf L_{\mathrm{tot}}|}, $$ and hence a natural reference plane $\mathbf e_z^\perp$. The key step below is to coarse-grain the Newtonian dynamics over fast orbital phases and many weak interactions, thereby extracting a smooth mean-field preference for that plane. This produces the restoring geometry used in the effective inclination model while keeping the exact Hamiltonian and the coarse-grained relaxation theorem conceptually distinct.

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2. Coarse-Grained Newtonian Midplane Preference

We now identify the Newtonian origin of the preferred plane. After averaging over the fast orbital phases of the surrounding bodies, replace the exact many-body field experienced by one tagged body by a smooth axisymmetric coarse-grained potential $\Phi_{\mathrm{cg}}(R,z)$ about the angular-momentum axis $\mathbf e_z$, with symmetry $\Phi_{\mathrm{cg}}(R,z)=\Phi_{\mathrm{cg}}(R,-z)$.

Proposition 1 Assume $\Phi_{\mathrm{cg}}$ is $C^2$ near $z=0$ and satisfies

$$ \Phi_{\mathrm{cg}}(R,z)=\Phi_{\mathrm{cg}}(R,0)+\frac12\,\nu^2(R)\,z^2+O(z^4), \qquad \nu^2(R):=\partial_{zz}\Phi_{\mathrm{cg}}(R,0)>0. $$

Let a body of mass $m$ move on a nearly circular orbit of radius $a$ whose orbital plane has small inclination $\theta$ relative to the midplane $z=0$. Then the orbit-averaged excess Newtonian potential energy satisfies

$$ \Delta \overline{U}(\theta)=\frac14\,m\,\nu^2(a)\,a^2\,\theta^2+O(\theta^4)=\frac12\,K\,\theta^2+O(\theta^4), $$

with

$$ K=\frac12\,m\,\nu^2(a)\,a^2>0. $$

Consequently, the induced restoring term in inclination space is

$$ -\partial_\theta \Delta \overline{U}(\theta)=-K\theta+O(\theta^3). $$

For a small inclination $\theta$, the vertical displacement along the orbit may be written, to leading order, as

$$ z(\psi)=a\sin\psi\,\theta+O(\theta^3), $$

where $\psi$ denotes the orbital phase. The orbit-averaged excess potential energy is

$$ \Delta \overline{U}(\theta)=m\,\Big\langle \Phi_{\mathrm{cg}}(a,z(\psi))-\Phi_{\mathrm{cg}}(a,0)\Big\rangle_{\psi}. $$

Using the even-in-$z$ expansion of $\Phi_{\mathrm{cg}}$ near the midplane gives

$$ \Delta \overline{U}(\theta)=\frac{m}{2}\,\nu^2(a)\,\big\langle z(\psi)^2\big\rangle_{\psi}+O(\theta^4). $$

Substituting $z(\psi)=a\sin\psi\,\theta+O(\theta^3)$ yields

$$ \big\langle z(\psi)^2\big\rangle_{\psi}=a^2\theta^2\big\langle \sin^2\psi\big\rangle_{\psi}+O(\theta^4)=\frac12\,a^2\theta^2+O(\theta^4). $$

Therefore

$$ \Delta \overline{U}(\theta)=\frac14\,m\,\nu^2(a)\,a^2\,\theta^2+O(\theta^4), $$

which is the claimed quadratic energy penalty. Differentiating with respect to $\theta$ gives the linear restoring term $-\partial_\theta \Delta \overline{U}(\theta)=-K\theta+O(\theta^3)$.

Remark. Proposition 1 identifies the Newtonian origin of the preferred plane and the small-angle restoring geometry. It does not yet imply irreversible convergence of the inclination law. That final step enters only after a coarse-grained closure for the slow inclination dynamics is imposed.

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3. Effective Inclination Dynamics

To pass from the conservative restoring geometry above to a relaxation law for inclination statistics, we model the slow inclination variable by the overdamped stochastic closure

$$ \dot{\theta} = -\mu\,\partial_\theta \Delta \overline{U}(\theta) + \eta(t), $$

where $\mu>0$ is an effective mobility and $\eta(t)$ is a mean-zero unresolved forcing term representing weak encounters, phase scrambling, and other perturbative effects. Linearizing Proposition 1 at small angle and writing $\eta(t)=\sqrt{2D}\,\xi(t)$ with white noise $\xi$, we obtain

$$ d\Theta_t = -\kappa\,\Theta_t\,dt + \sqrt{2D}\,dW_t, \qquad \kappa:=\mu K. $$

Thus the linear drift in the effective model is not inserted arbitrarily: it is the small-angle gradient of the orbit-averaged coarse-grained Newtonian energy, composed with a standard slow-angle relaxation closure. For analytical tractability we now pass from the bounded physical interval $[-\pi/2,\pi/2]$ to an unbounded small-angle approximation $\Theta_t\in\mathbb{R}$, valid when the mass of the distribution concentrates near the midplane and boundary effects are negligible.

The corresponding Fokker–Planck equation for the angle density $p(\theta,t)$ is

$$ \partial_t p = D\,\partial_{\theta\theta}p + \kappa\,\partial_\theta(\theta p), \qquad \theta\in\mathbb{R}. $$

This is the simplest coarse-grained model in which a final angle distribution actually exists. A bounded-domain version with reflecting boundaries at $\pm\pi/2$ could also be used, but the unbounded Ornstein–Uhlenbeck model yields the cleanest closed-form theorem.

4. Main Theorem

Theorem 1 Let $p(\theta,t)$ evolve according to the Fokker–Planck equation above with $\kappa,D>0$. Then:

(i) the equation admits the stationary distribution

$$ p_*(\theta)=\sqrt{\frac{\kappa}{2\pi D}}\,\exp\!\left(-\frac{\kappa\theta^2}{2D}\right); $$

(ii) the relative entropy decays exponentially:

$$ \mathrm{KL}(p_t\,\|\,p_*)\;\le\;e^{-2\kappa t}\,\mathrm{KL}(p_0\,\|\,p_*). $$

Define $u(\theta,t)=p(\theta,t)/p_*(\theta)$. A direct calculation shows the Fokker–Planck equation may be rewritten as

$$ \partial_t p = D\,\partial_\theta\!\left(p_*\,\partial_\theta u\right). $$

Let $H(t):=\mathrm{KL}(p_t\,\|\,p_*) = \int p_*\,u\log u\,d\theta$. Differentiating and integrating by parts,

$$ \frac{dH}{dt} = -D\int p_*(\theta)\,\frac{(\partial_\theta u)^2}{u}\,d\theta = -4D\int p_*(\theta)\left|\partial_\theta\sqrt{u}\right|^2 d\theta. $$

Since $p_*$ is Gaussian, it satisfies the Gaussian log-Sobolev inequality

$$ H(t)\;\le\;\frac{2D}{\kappa}\int p_*(\theta)\left|\partial_\theta\sqrt{u}\right|^2 d\theta. $$

Combining yields $\frac{dH}{dt}\le -2\kappa H(t)$, and Grönwall's inequality gives the claim.

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5. Corollaries

Corollary 1 (TV contraction) By Pinsker's inequality,

$$ \mathrm{TV}(p_t,p_*)\;\le\;\sqrt{\tfrac12\,\mathrm{KL}(p_t\,\|\,p_*)}\;\le\;e^{-\kappa t}\sqrt{\tfrac12\,\mathrm{KL}(p_0\,\|\,p_*)}. $$

The angle distribution therefore converges exponentially fast, in total variation, to the stationary Gaussian law.

Corollary 2 (Relaxation time) For target accuracy $\varepsilon>0$, define $T_\varepsilon:=\inf\{t\ge 0:\mathrm{TV}(p_s,p_*)\le\varepsilon \text{ for all }s\ge t\}$. Then

$$ T_\varepsilon\;\le\;\frac{1}{\kappa}\,\log\!\left(\frac{\sqrt{\mathrm{KL}(p_0\,\|\,p_*)/2}}{\varepsilon}\right). $$

This is the expected spectral-gap scaling: convergence time grows like $\kappa^{-1}\log(1/\varepsilon)$.

6. Uniform Initial Law

If the initial angle law is uniform on $[-a,a]$,

$$ p_0(\theta)=\frac{1}{2a}\,\mathbf{1}_{[-a,a]}(\theta), $$

a direct computation yields

$$ \mathrm{KL}(p_0\,\|\,p_*) = \log\!\left(\frac{\sqrt{2\pi D/\kappa}}{2a}\right) + \frac{\kappa a^2}{6D}. $$

For the natural choice $a=\pi/2$,

$$ \mathrm{KL}(p_0\,\|\,p_*) = \log\!\left(\frac{\sqrt{2\pi D/\kappa}}{\pi}\right) + \frac{\kappa\pi^2}{24D}, $$

and substituting into Corollary 2 gives an explicit, closed-form bound on $T_\varepsilon$ in terms of $\kappa$, $D$, and $\varepsilon$ alone.

Remark. One may also compare $p_t$ and $p_*$ using Jensen–Shannon or other symmetric divergences. These choices do not automatically improve the bound; the essential ingredient is contraction of the coarse-grained dynamics in some divergence. The KL route above is the cleanest, and Pinsker converts it to TV at no real cost.

7. Reduction of the Long-Time Search Space

The coarse-grained inclination law does not invert the full Newtonian microstate: many distinct phase-space configurations can share the same inclination profile or the same angle density. It nevertheless has analytical value, because once the inclination statistics concentrate near the midplane, the admissible long-time dynamics are forced into a nearly planar regime. In that sense the flattening law is a reduction principle for the asymptotic geometry of the $N$-body problem, even though it is not an exact closed-form solver for the full microscopic flow.

Proposition 2 (Asymptotic reduction, not exact inversion) Suppose that under the effective inclination dynamics one has

$$ \max_{2\le k\le N}|\theta_k(t)|\le \varepsilon \ll 1, \qquad t\ge T. $$

Then for each orbiting body, the vertical displacement and vertical velocity relative to the preferred midplane satisfy

$$ z_k(t)=O\!\big(a_k\varepsilon\big), \qquad \dot z_k(t)=O\!\big(v_k\varepsilon\big), $$

where $a_k$ and $v_k$ denote the characteristic orbital scale and speed of body $k$. Consequently the exact Newtonian Hamiltonian admits, for $t\ge T$, the decomposition

$$ H_N(t)=H_{\mathrm{planar}}(t)+O\!\big(\varepsilon^2\big), $$

where $H_{\mathrm{planar}}$ is the Hamiltonian obtained by projecting the bodies onto the preferred plane. Thus the coarse-grained flattening law identifies an asymptotically nearly planar reduced regime of the full system, but does not uniquely determine the in-plane phases, eccentricities, or other microscopic degrees of freedom.

For a small inclination angle, the geometry of an orbit with characteristic radius $a_k$ gives $z_k=O\!\big(a_k\sin\theta_k\big)=O\!\big(a_k\varepsilon\big)$. Differentiating along the orbit yields $\dot z_k=O\!\big(v_k\varepsilon\big)$, where $v_k$ is the corresponding orbital speed scale. Hence the out-of-plane kinetic contribution is

$$ \frac12 m_k \dot z_k^2 = O\!\big(m_k v_k^2\varepsilon^2\big). $$

Likewise, for pairwise separations one has

$$ |\mathbf r_k-\mathbf r_j|^2 = |\mathbf r_k^{\parallel}-\mathbf r_j^{\parallel}|^2 + (z_k-z_j)^2, $$

where $\mathbf r_k^{\parallel}$ denotes the planar projection. Since $(z_k-z_j)^2=O\!\big(\varepsilon^2\big)$, a Taylor expansion of the Newtonian potential around the planar separation shows that each pairwise potential differs from its planar counterpart by $O\!\big(\varepsilon^2\big)$. Summing the kinetic and potential corrections yields

$$ H_N(t)=H_{\mathrm{planar}}(t)+O\!\big(\varepsilon^2\big). $$

The resulting reduction is therefore geometric and asymptotic: it shrinks the long-time search space to a nearly planar manifold, but it cannot be inverted to recover a unique microscopic Newtonian trajectory from the coarse inclination law alone.

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8. Numerical Illustration

The figure below depicts a central mass $m_1$ with $N-1=60$ orbiting bodies whose orbital-plane inclinations $\theta_k$ are drawn initially from the uniform law on $[-\pi/2,\pi/2]$. For visual clarity, the animation suppresses the stochastic forcing term and illustrates only the deterministic linear restoring drift derived above, so that each body moves on a smooth Kepler-like ellipse while its orbital plane slowly flattens toward the common midplane. The running histogram at bottom shows the empirical inclination density $p_N(\theta,t)$ concentrating toward $\theta=0$.

REC · SIM-047B
N = 60 · κ = 0.35 · Dvis = 0
t = 0.00
mean |θ| = —
κvis = 0.35 · e ∈ [0.05, 0.30] · dt = 0.012

Caption. Top: 3D view of the orbital system, with the central mass rendered as a warm illuminated sphere and the orbiting bodies colored by their orbital-plane inclination $\theta_k$. Warm hues indicate $|\theta|$ close to $\pi/2$; cool hues indicate proximity to the midplane. Each body advances along a smooth Kepler-like ellipse while the inclination obeys the deterministic flattening law $\dot\theta_k=-\kappa\theta_k$, the linearized restoring drift obtained from the small-angle gradient of the orbit-averaged coarse-grained Newtonian energy. Bottom: empirical histogram of $\{\theta_k(t)\}_{k=2}^{N}$ across $32$ bins on $[-\pi/2,\pi/2]$. The theorem above still analyzes the coarse-grained noisy density model; the visualization simply isolates its geometric flattening component by setting the visual diffusion to zero.

— J. R. Landers
New York, N.Y. · April 2026
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