Equilibrium Profiles and Their Analytical Characterization#
Stationary Equations and Conditional Laws#
An equilibrium profile is a distribution whose shape survives the evolution. We must specify which evolution, and what happens to lost mass. A conservative process preserves probability directly. A killed process loses probability, and its quasi-stationary distribution preserves its shape only after we divide by the surviving mass. A mean-field equation describes a third object: a population law evolving under its own interaction field.
These distinctions matter when we draw a proposed equilibrium curve. Its shape may solve a kinetic equation while failing the cloning balance, or it may describe the positions of alive walkers without accounting for the fraction that are alive. This chapter derives several useful reference profiles and then gives a quantitative test for a proposed profile of the full stationary model.
Definition 587 (Three equilibrium objects)
For a killed finite-swarm kernel \(Q_N\), a QSD is a probability law satisfying \(\pi_NQ_N=\alpha_N\pi_N\), \(0<\alpha_N\leq1\). Its one-walker marginal is a probability measure on the walker’s marked state space. A stationary solution of the continuous mean-field equation satisfies \(Au+\mathcal R(u)=0\). A kinetic reference law instead satisfies \(A\rho=0\).
The QSD convergence theorem in Convergence, Survival, and Parameter Dependence, the mean-field existence, uniqueness, and attraction theorems in The Discrete Population Limit and Propagation of Chaos, and the kinetic estimates in Logarithmic Sobolev inequalities and entropy convergence refer to these specified objects. Identification of a QSD marginal with a stationary mean-field law uses Theorem 236 and its consistency and concentration hypotheses.
Proposition 163 (Alive equilibrium profile and its mass)
For the normalized-kernel continuous model of
prop-chaos-alive-stationary-reconstruction, let \(\rho\) be a
probability density solving
Then its stationary alive measure is \(m_a\rho\), where \(m_a=\lambda_{\rm rev}/(\lambda_{\rm rev}+\bar c[\rho])\). The remaining mass is \(m_d=1-m_a\). Boundary-loss proposals are treated with the full marked-state equations of The Mean-Field Law of the Euclidean Gas.
Proof. Integrating the alive equation gives the stationary balance \(m_a\bar c[\rho]=\lambda_{\rm rev}m_d\). Solve this equation with \(m_a+m_d=1\), and substitute into the normalized stationary equation.
The alive fraction has a simple flow balance. The rate at which alive mass is lost equals the rate at which dead mass returns. The normalized density \(\rho\) tells us where the alive walkers are; multiplying by \(m_a\) tells us how much of the whole population they represent. A plot normalized to unit area can hide this distinction.
Companion Distances and Local Density#
In a dense cloud, the nearest particle is usually closer. To estimate the distance, expand a ball until it contains roughly one particle. Density times ball volume is then of order one, giving the familiar inverse-density scale.
Kernel selection asks a different question. It assigns a positive weight to every eligible companion and samples from those weights. A nearby candidate is favored, but it need not be chosen. The expected distance must therefore use the actual sampling law. The two calculations below show exactly where their different scales enter.
Proposition 164 (Nearest-neighbor scaling and companion selection)
For a homogeneous Poisson point process of intensity \(\lambda>0\) in \(\mathbb R^D\), the distance \(R\) from a fixed test point to the nearest point satisfies, with \(\omega_D=\pi^{D/2}/\Gamma(1+D/2)\),
The local Poisson approximation with intensity \(N\rho(z)\) therefore has scale \([N\rho(z)]^{-1/D}\). Its use requires a local Poisson approximation; exchangeability alone does not supply it.
For companion selection with a fixed kernel \(K_\epsilon(z,y)\), the continuum expected distance is instead
provided the denominator is positive and the numerator finite. This is the conditional law of one kernel-selected companion. Its regularity follows from the normalized-moment estimates in the regularity chapters.
Proof
The nearest-neighbor event is precisely that the ball of radius \(r\) contains zero Poisson points. Its count has mean \(\lambda\omega_Dr^D\), giving the survival function. Integrating it over \(r\geq0\) and substituting \(s=\lambda\omega_Dr^D\) gives the gamma factor. For kernel selection, normalize the measure \(K_\epsilon(z,y)\rho(y)dy\) and integrate \(d(z,y)\). A nearest-neighbor rule and a fixed-width kernel rule define different sampling laws, so their expectations require their respective formulas.
The Iso-Fitness Reference Model#
Consider a local model in which a region gains population whenever its fitness exceeds the population average. At a positive stationary density, every occupied region must have the same fitness; otherwise some region would still gain or lose mass.
If crowding reduces the diversity contribution, that equality determines how much density a high-reward region can support. The power law below follows directly from this balance. It is a useful reference calculation because every step is explicit. Applying its profile to the Fractal Gas requires checking the nonlocal cloning and kinetic terms as well.
Theorem 238 (Power-law equilibrium for the local replicator model)
Consider the specified local model
where \(a,\beta>0\), \(R>0\), and the displayed quantities are integrable. Among strictly positive probability densities, its stationary density is
For \(R=e^{-U}\), this is the reference Gibbs law with inverse temperature \(\alpha D/\beta\). The theorem concerns this local replicator equation; the nonlocal sampled-fitness cloning operator is the operator in The Mean-Field Law of the Euclidean Gas.
Proof
Stationarity and strict positivity give \(V[\rho]=\overline V[\rho]\) almost everywhere. Solving the resulting algebraic equation gives the displayed power law; normalization fixes its constant. Conversely, substitution makes \(V[\rho_*]\) constant, so the right side vanishes. This also proves uniqueness in the stated class. If \(Z=\infty\), the displayed profile is not a probability density. On an unbounded space the limit \(\alpha/\beta\to0\) cannot be identified with a uniform probability distribution.
Writing the reward as an exponential turns the power law into a Gibbs-shaped curve. This is an algebraic consequence of the chosen reward and local fitness formula. The name “Gibbs” does not by itself identify the stationary law of a different dynamics. The residual estimate at the end of the chapter provides a way to test that identification quantitatively.
Kinetic Equilibria and Absorption#
A boundary can change an equilibrium profile even when the motion inside is simple diffusion. With absorption and no source, mass continually disappears; a quasi-stationary profile preserves its shape while its amplitude decays. With a source, the incoming mass can instead balance the boundary loss. The sine and parabola below are the two solutions of these two balance equations.
Proposition 165 (Absorbing diffusion profile on an interval)
For \(\partial_t f=D_0\partial_{xx}f\) on \((0,L)\) with \(D_0>0\) and absorbing Dirichlet endpoints, the normalized positive eigenfunction
is a QSD density with decay rate \(\lambda_0=D_0\pi^2/L^2\). For a prescribed source \(s\), a stationary forced density instead solves \(D_0f''+s=0\). In particular, a constant source gives \(f(x)=s x(L-x)/(2D_0)\).
Proof
The sine has integral \(2L/\pi\), vanishes at both endpoints, and satisfies \(D_0\phi''=-\lambda_0\phi\). Thus the killed evolution starting from it is \(e^{-\lambda_0t}\phi\) and normalization leaves \(\phi\). Twice integrating the constant-source equation and imposing the endpoints gives the parabola. For a density with killing and revival, use its full source balance rather than the source-free eigenvalue equation. The kinetic transport boundary condition also distinguishes incoming and outgoing velocities; the scalar Dirichlet calculation applies to the specified spatial diffusion model.
Velocity has another useful reference calculation. Friction damps the initial velocity, while noise continually adds fresh fluctuations. In the Ornstein–Uhlenbeck equation these effects can be solved exactly. Their balance gives a Gaussian velocity law and an explicit rate at which the initial velocity is forgotten.
That calculation isolates the thermal part of the dynamics. A collision, reset, or selection event can change the velocity statistics, so it must also appear in the balance equation before we claim that the full velocity marginal is Maxwellian.
Theorem 239 (Ornstein–Uhlenbeck thermalization and its reference law)
For \(dV_t=-\gamma V_tdt+\sigma_vdB_t\), \(\gamma>0\), set \(T=\sigma_v^2/(2\gamma)\). The invariant law is \(M_T=\mathcal N(0,TI_d)\) and
For conservative underdamped Langevin dynamics with force \(-\nabla U\), the normalized density \(\rho_U(x,v)=Z^{-1}\exp[-(U(x)+|v|^2/2)/T]\) is invariant when \(Z<\infty\) and the generator domain has zero boundary flux. Cloning and killing require checking their contributions to the stationary equation.
Proof
Variation of constants gives \(V_t=e^{-\gamma t}V_0+\sigma_v\int_0^te^{-\gamma(t-s)}dB_s\). The Gaussian integral has covariance \(T(1-e^{-2\gamma t})I_d\), proving invariance. Couple two solutions with the same Brownian motion; their difference is \(e^{-\gamma t}(V_0-W_0)\). Optimize the initial coupling to obtain the Wasserstein estimate. For \(\rho_U\), the transport term \(-v\cdot\nabla_x\rho_U\) cancels \(\nabla U\cdot\nabla_v\rho_U\); friction cancels velocity diffusion by \(\sigma_v^2=2\gamma T\).
Remark 190 (A finite jump rate retains a velocity correction)
For the preceding OU velocity with independent resets to zero at rate \(r>0\), its generator applied to \(|v|^2\) gives
Its stationary second moment is \(d\sigma_v^2/(2\gamma+r)\), rather than \(dT\). Thus a finite jump rate alone does not imply an exactly Maxwellian stationary velocity marginal. In the Fractal Gas the actual collision, selection, killing, and revival terms determine the corresponding correction.
Quantitative Validation of Equilibrium Profiles#
Suppose we have drawn a plausible equilibrium curve. How can we tell whether it is close to the true one? Insert it into the stationary equation and measure the leftover term. A zero residual gives an exact stationary solution. A small residual gives a useful error bound when the stationary solution map contracts.
The factor below explains the role of contraction. If the map contracts strongly, it cannot move a point only slightly while that point remains far from its fixed point. If the contraction factor is close to one, a small residual is less informative. This turns a proposed decorated Gibbs profile into a testable approximation.
Theorem 240 (Error bound for a proposed decorated Gibbs profile)
Under thm-uniqueness-contraction-solution-operator, let
\(\mathcal T_C\) have contraction factor \(q_C<1\) and stationary density \(u_*\).
For any proposed normalized density
in the same complete space,
If \(g\) belongs to the generator domain, its stationary residual \(r=Ag+\mathcal R(g)\) has zero mass. Consequently
with the zero-mass semigroup constants \(K,a\) from
lem-uniqueness-scaling-hypoelliptic-constant.
Proof
Insert \(\mathcal T_Cg\) between \(g\) and \(u_*=\mathcal T_Cu_*\), then use contraction and move \(q_C\|g-u_*\|_1\) to the left. The resolvent identity \(R_C(C-A)g=g\) gives \(\mathcal T_Cg-g=R_C[Ag+\mathcal R(g)]\). Conservation makes the residual zero-mass, so the sharper resolvent bound \(K/(C+a)\) applies.
Corollary 77 (Equilibrium observables and particle fluctuations)
For a bounded observable \(\psi\), the preceding profile error implies \(|g\psi-u_*\psi|\leq\|\psi\|_\infty\|g-u_*\|_1\). For an exchangeable particle law, its empirical-observable error also contains marginal bias and sampling variance, as quantified in Theorem 313. The uniform-in-\(N\) LSI and total-relative- entropy routes in Exchangeability, Empirical Laws, and Functional Inequalities and Logarithmic Sobolev inequalities and entropy convergence supply the corresponding \(O(N^{-1})\) variance bounds under their analytical hypotheses.
Proof. Integrate \(\psi(g-u_*)\) and apply the \(L^\infty\)–\(L^1\) inequality. For an empirical average split the error into its centered fluctuation and its expectation; Cauchy–Schwarz bounds the former by its standard deviation.
A simulation adds fluctuations to the profile comparison. Averaging more walkers can reduce the random part of an observable’s error under the stated concentration hypotheses. It does not automatically reduce an error in the proposed equilibrium shape. The stationary residual controls that shape error; the finite-particle estimates control the remaining fluctuations and marginal bias.