Hypocoercive Entropy: From Kinetic Mixing to the Full Swarm

1. What the entropy argument proves

Noise acts directly on velocity. Position moves because velocity carries it. To measure both effects, we combine relative entropy with derivatives in position and velocity, including a cross term that records their coupling. This modified entropy has an explicit exponential decay rate for the conservative kinetic dynamics, including nonconvex confining potentials.

Three quantities determine the estimate: the logarithmic Sobolev constant of the reference law, the curvature bound on the force, and the diffusion and friction coefficients. For independent kinetic particles, tensorization makes the rate independent of population size. The same conclusion holds for interacting laws when their joint inequalities have uniform constants.

Cloning and killing enter through their actual generators. A quasi-stationary distribution is stationary after conditioning on survival; its unnormalized mass decays. The normalization therefore contributes to entropy evolution. We derive that contribution below and state the precise estimate that closes the full swarm argument.

The complete static LSI proofs, nonconvex confinement argument, and extensions are in Logarithmic Sobolev inequalities and entropy convergence. Finite-particle recurrence, the discrete QSD, and the mean-field equation are developed in Convergence, Survival, and Parameter Dependence, Equilibrium Profiles and Their Analytical Characterization, and The Mean-Field Law of the Euclidean Gas. Here we work through the kinetic calculation and show how those results can be combined.

Reading the estimates

The reference law, its state space, and the evolution must be fixed together. The kinetic Gibbs law, the finite-particle QSD, a stationary mean-field law, and the invariant law of a numerical kernel each have their own defining equation. A proof transfers between them through an identified density, generator, or measure-comparison estimate.

2. Reference law, operators, and modified entropy

Start with the part of the dynamics whose equilibrium density we can calculate. Friction removes velocity fluctuations, noise replenishes them, and Hamiltonian transport preserves total energy. Their balance produces the kinetic Gibbs law. This gives us a reference against which every integration by parts can be checked.

The calculation uses nondimensional position, velocity, and time coordinates. With physical units restored, the coefficients multiplying spatial, velocity, and mixed derivatives carry the corresponding conversion factors. Entropy is measured in nats; a decay rate has units of inverse time.

2.1. Conservative kinetic dynamics

Definition 589 (Kinetic law and relative-density operator)

Let \(x,v\in\mathbb R^m\), where \(m=dN\) for a continuous \(N\)-particle state. Consider

\[ dX_t=V_tdt,\qquad dV_t=-\nabla U_N(X_t)dt-\gamma V_tdt+\sqrt{2D}\,dW_t, \qquad D>0,\quad\gamma>0. \]

Set \(\theta=D/\gamma=\sigma_v^2/(2\gamma)\). When normalizable, its invariant law is

\[ \pi_N(dx,dv)=Z_N^{-1} \exp\!\left[-\frac{U_N(x)+|v|^2/2}{\theta}\right]dx\,dv. \]

For \(f_t=h_t\pi_N\), the forward equation becomes

\[ \partial_t h=\mathcal Kh, \qquad \mathcal K=\mathcal S+\mathcal T, \qquad \mathcal S=D\Delta_v-\gamma v\cdot\nabla_v, \qquad \mathcal T=-v\cdot\nabla_x+\nabla U_N\cdot\nabla_v. \]

Proof

The density derivatives are \(\nabla_x\log\pi_N=-\nabla U_N/\theta\) and \(\nabla_v\log\pi_N=-v/\theta\). Substitution into \(\pi_N^{-1}L_{\mathrm{kin}}^*(\pi_Nh)\) gives \(\mathcal K\); the zeroth-order terms cancel because \(D=\gamma\theta\). Integration by parts yields

\[ \int g\mathcal Sh\,d\pi_N=-D\int\nabla_vg\cdot\nabla_vh\,d\pi_N, \qquad \int g\mathcal Th\,d\pi_N=-\int h\mathcal Tg\,d\pi_N. \]

The second identity follows because \(\operatorname{div}(\pi_N(-v,\nabla U_N))=0\). These identities also verify invariance. They hold first for smooth functions with justified integration by parts, then on the corresponding operator domains.

2.2. Entropy and the three Fisher terms

Definition 590 (Modified entropy)

For \(h>0\) with \(\pi_N(h)=1\), put \(u=\log h\) and define

\[\begin{split} \begin{aligned} H(h)&=\int h\log h\,d\pi_N,\\ I_x(h)&=\int h|\nabla_xu|^2d\pi_N,\qquad I_v(h)=\int h|\nabla_vu|^2d\pi_N,\\ I_{xv}(h)&=\int h\nabla_xu\cdot\nabla_vu\,d\pi_N, \qquad I(h)=I_x(h)+I_v(h). \end{aligned} \end{split}\]

Let

\[\begin{split} G=\begin{pmatrix}cI&bI\\bI&aI\end{pmatrix}, \qquad a,c>0,\quad ac>b^2, \qquad \Phi_G(h)=H(h)+cI_x(h)+2bI_{xv}(h)+aI_v(h). \end{split}\]

Writing \(q=(\nabla_xu,\nabla_vu)^{\mathsf T}\), the gradient contribution is \(I_G(h)=\int hq^{\mathsf T}Gq\,d\pi_N\). If \(G\preceq g_+I\) and

\[ \operatorname{Ent}_{\pi_N}(f^2)\leq2C_N\int|\nabla f|^2d\pi_N, \]

then

\[ H(h)\leq\Phi_G(h)\leq(C_N/2+g_+)I(h). \]

Proof

Positive definiteness of \(G\) makes \(I_G\) nonnegative. Apply the LSI to \(f=\sqrt h\) to obtain \(H(h)\leq C_NI(h)/2\), and use \(I_G\leq g_+I\).

The cross term may be negative. What must stay positive is the entire quadratic form, which is exactly what \(ac>b^2\) ensures. Its derivative will produce the spatial dissipation that ordinary entropy misses.

The LSI uses all position and velocity derivatives. The kinetic generator itself has only velocity diffusion. A nonconstant test function depending solely on position has zero velocity gradient and positive entropy, so a velocity-only LSI cannot hold for the full phase-space law. The coupling calculation is what connects velocity noise to spatial information.

INTERACTIVE EXPERIMENT · IV-01

Transport of a displaced population

How does a position displacement affect the later velocity law? The exact configured BAOAB evolution predicts mean cos(v) from the recorded displaced initial population.

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2.3. Obtaining the LSI without global convexity

Corollary 79 (Explicit kinetic reference LSI)

Suppose the spatial Gibbs law \(\pi_x\propto e^{-U/\theta}\) satisfies an LSI with constant \(C_x\). Then \(m_U=\pi_x\otimes\mathcal N(0,\theta I_d)\) satisfies a full-gradient LSI with

\[ C_0=\max\{C_x,\theta\}. \]

The same constant holds for \(m_U^{\otimes N}\). Two sufficient nonconvex regimes are:

1. \(U=W+B\), with \(\nabla^2W\succeq\kappa I\), \(\kappa>0\), and bounded \(\operatorname{osc}(B)\). One may take \(C_x=\theta e^{\operatorname{osc}(B)/\theta}/\kappa\).

2. \(U\in C^2\), \(x\cdot\nabla U(x)\geq\alpha_U|x|^2-b_U\) with \(\alpha_U>0\), and \(\nabla^2U\succeq-M_-I\). The explicit finite constant constructed in Theorem 283 applies.

Proof

In the first regime, Bakry–Émery gives constant \(\theta/\kappa\) for \(e^{-W/\theta}\); bounded density perturbation multiplies it by at most \(e^{\operatorname{osc}(B)/\theta}\). In the second, the Lyapunov function

\[ \mathcal W(x)=\exp\!\left(\frac{\alpha_U|x|^2}{4\theta}\right) \]

satisfies a negative quadratic drift for \(\Delta-\theta^{-1}\nabla U\cdot\nabla\). The weighted-energy estimate, local Poincaré inequality, entropy-transport bound, and centering argument in Corollary 92 prove the spatial LSI. Gaussian LSI and the full tensorization proof in Theorem 280 then give \(C_0\), independently of \(N\).

Remark 192 (The actual joint law)

For interacting laws, Corollary 94 supplies four proved criteria: a product reference, a uniformly bounded joint density tilt, uniform joint curvature, or a uniformly contractive additive-noise invariant flow. A QSD can use this corollary when its actual law satisfies one of these criteria. Conditional laws on continuous status strata also require the discrete entropy accounting in Proposition 176 when the strata are combined.

3. Complete kinetic dissipation calculation

Differentiating velocity information creates a position–velocity cross term. Differentiating that cross term creates negative spatial information. This is the transfer mechanism. The second derivatives generated by diffusion must be collected into one quadratic form; the mixed second-derivative term has no sign by itself.

We use a global force-Hessian bound in this calculation. A compact region containing most of the probability does not supply that bound for a Fisher integral, which weights squared density gradients. If only local curvature bounds are available, the tail terms need a weighted estimate of their own.

3.1. Exact identities

Lemma 179 (Evolution of entropy and Fisher information)

Suppose \(\|\nabla^2U_N\|_{\mathrm{op}}\leq M\) globally. For a smooth positive relative density evolving under \(\mathcal K\), with finite integrals and justified integration by parts, write \(H_U=\nabla^2U_N\). Then

\[\begin{split} \begin{aligned} \dot H&=-DI_v,\\ \dot I_v&=-2\gamma I_v-2I_{xv} -2D\int h\|\nabla_v^2u\|_{\mathrm{HS}}^2d\pi_N,\\ \dot I_{xv}&=-I_x-\gamma I_{xv} +\int h(\nabla_vu)^{\mathsf T}H_U\nabla_vu\,d\pi_N\\ &\quad-2D\int h\sum_j\partial_{v_j}\nabla_xu\cdot\partial_{v_j}\nabla_vu\,d\pi_N,\\ \dot I_x&=2\int h(\nabla_xu)^{\mathsf T}H_U\nabla_vu\,d\pi_N -2D\int h\|\nabla_v\nabla_xu\|_{\mathrm{HS}}^2d\pi_N. \end{aligned} \end{split}\]

Proof

Symmetry of \(\mathcal S\), antisymmetry of \(\mathcal T\), and the diffusion chain rule give \(\dot H=-DI_v\). The derivative commutators are

\[ [\nabla_v,\mathcal K]=-\nabla_x-\gamma\nabla_v, \qquad [\nabla_x,\mathcal K]=H_U\nabla_v. \]

Set \(B=\begin{pmatrix}0&H_U\\-I&-\gamma I\end{pmatrix}\). Since \(\partial_tu=\mathcal Ku+D|\nabla_vu|^2\) and \(\nabla\mathcal Ku=\mathcal Kq+Bq\), differentiating \(I_G\) gives

\[ \frac{d}{dt}I_G =-2D\int h\sum_j(\partial_{v_j}q)^{\mathsf T}G(\partial_{v_j}q)d\pi_N +\int hq^{\mathsf T}(GB+B^{\mathsf T}G)q\,d\pi_N. \]

Indeed, integration by parts combines the terms containing \(\mathcal K\) into the displayed second-derivative form. The derivative of \(D|\nabla_vu|^2\) cancels its accompanying first-gradient terms. Reading off the velocity, cross, and position entries gives the three identities; this is the calculation of Lemma 236 with the same conventions.

3.2. Positive coefficients and a rigorous rate

Theorem 247 (Explicit hypocoercive decay rate)

Suppose the kinetic Gibbs law \(\pi_N\) satisfies a full-gradient LSI with constant \(C_N\), and \(\|\nabla^2U_N\|_{\mathrm{op}}\leq M\). Define

\[\begin{split} L_M=2M+\gamma+2, \qquad \eta=\frac{D}{2(1+2M+L_M^2)}, \qquad G=\eta\begin{pmatrix}2I&I\\I&2I\end{pmatrix}. \end{split}\]

Then

\[ \frac{d}{dt}\Phi_G(h_t) \leq-\eta I_x(h_t)-\frac D2I_v(h_t) \leq-\eta I(h_t), \]

and

\[ \Phi_G(h_t)\leq e^{-r_Nt}\Phi_G(h_0), \qquad H(h_t)\leq e^{-r_Nt}\Phi_G(h_0), \qquad r_N=\frac{\eta}{C_N/2+3\eta}>0. \]

The estimate holds for finite-\(\Phi_G\) initial data, and from any positive time \(t_0\) at which \(\Phi_G(h_{t_0})<\infty\).

Proof

The matrix has eigenvalues \(\eta\) and \(3\eta\). Thus the second-derivative term in the preceding lemma is nonpositive. The Hessian bound and Cauchy–Schwarz give

\[ \left|I_{xv}\right|\leq\sqrt{I_xI_v},\quad \int h(\nabla_vu)^{\mathsf T}H_U\nabla_vu\,d\pi_N\leq MI_v, \]

\[ \left|\int h(\nabla_xu)^{\mathsf T}H_U\nabla_vu\,d\pi_N\right| \leq M\sqrt{I_xI_v}. \]

With \(a=c=2\eta\) and \(b=\eta\), the identities therefore imply

\[ \dot\Phi_G\leq-2\eta I_x-(D+4\eta\gamma-2\eta M)I_v +2\eta L_M\sqrt{I_xI_v}. \]

Apply \(2\eta L_M\sqrt{I_xI_v}\leq\eta I_x+\eta L_M^2I_v\):

\[ \dot\Phi_G\leq-\eta I_x- [D+4\eta\gamma-\eta(2M+L_M^2)]I_v. \]

The chosen \(\eta\) satisfies \(\eta(2M+L_M^2)\leq D/2\) and \(\eta\leq D/2\), which proves dissipation. The LSI gives \(\Phi_G\leq(C_N/2+3\eta)I\). Grönwall’s inequality proves the decay estimate. Approximation extends the calculation from smooth data to its finite-functional domain, as in Theorem 285.

There are no approximate signs in this rate. It is a sufficient lower bound, not an optimized rate. The initial factor is the modified entropy: a small coefficient in front of Fisher information does not make arbitrary gradients small.

Nonconvexity enters through two different quantities. The Hessian norm controls the dynamical mixing calculation; the LSI constant also records confinement and barriers between wells. The preceding spatial proof supplies that constant without requiring global convexity.

4. Cloning and conditioning on survival

A jump changes an entire swarm state. Its transition kernel must include the sampled companions, acceptance rule, and correlated offspring perturbations. The finite-particle kernel is a linear Markov operator on swarm laws even when each transition uses empirical fitness. A closed one-particle equation requires a separate mean-field argument.

For conservative dynamics, an invariant reference makes entropy decrease under a Markov step by data processing. For killed dynamics, we must also divide by the surviving mass. That division changes the derivative, even when the conditional law has already reached its QSD.

4.1. A common invariant target

Theorem 248 (Kinetic dynamics with a common-target cloning kernel)

Let \(P\) be a Markov kernel preserving the kinetic reference \(\pi_N\), and let \(P^\dagger\) be its action on relative densities. Suppose

\[ I_G(P^\dagger h)\leq A_JI_G(h) \]

for every density in the form domain. Add jumps at rate \(\omega\geq0\), so \(\partial_th=\mathcal Kh+\omega(P^\dagger h-h)\). If

\[ \delta=\eta-3\eta\omega(A_J-1)_+>0, \]

then

\[ H(h_t)\leq\Phi_G(h_t) \leq\exp\!\left[-\frac{\delta t}{C_N/2+3\eta}\right]\Phi_G(h_0). \]

Proof

Data processing gives \(H(P^\dagger h)\leq H(h)\) because \(\pi_NP=\pi_N\). Entropy and \(I_G=\int(\nabla h)^{\mathsf T}G\nabla h/h\,d\pi_N\) are convex in \(h\), the latter by the convexity of a quadratic perspective. Hence the directional derivative along the jump generator is bounded by

\[ D\Phi_G(h)[\omega(P^\dagger h-h)] \leq\omega(A_J-1)_+I_G(h) \leq3\eta\omega(A_J-1)_+I(h). \]

Combine this with the kinetic dissipation and the LSI closure. The full kernel and domain requirements are those of Theorem 286.

4.2. The normalized QSD entropy identity

Proposition 167 (Entropy under killed diffusion and jumps)

Let the conservative backward generator on a continuous swarm state space be

\[ LF(S)=b(S)\cdot\nabla F(S)+\operatorname{tr}(a(S)\nabla^2F(S)) +\int[F(S')-F(S)]r(S,dS'), \]

with \(a\succeq0\). Let \(A=L-\kappa\) for a bounded nonnegative killing rate, on a domain without absorbing boundary flux. Let \(\nu_N\) be a QSD satisfying

\[ L^*\nu_N=(\kappa-\lambda_N)\nu_N, \qquad\lambda_N=\nu_N(\kappa). \]

For \(f_t=h_t\nu_N\), the conditioned equation is

\[ \partial_tf_t=L^*f_t-\kappa f_t+\bar\kappa_t f_t, \qquad\bar\kappa_t=f_t(\kappa). \]

Set \(\psi(h)=h\log h-h+1\) and \(\mathcal B(s,t)=s\log(s/t)-s+t\). Then

\[\begin{split} \begin{aligned} \mathcal D_{\nu_N}(h) &=\int\frac{(\nabla h)^{\mathsf T}a\nabla h}{h}\,d\nu_N\\ &\quad+\iint\mathcal B(h(S),h(S'))\,\nu_N(dS)r(S,dS'), \end{aligned} \end{split}\]

and the exact entropy identity is

\[ \frac{d}{dt}H_{\nu_N}(h_t) =-\mathcal D_{\nu_N}(h_t) +\bar\kappa_tH_{\nu_N}(h_t)-\nu_N[\kappa\psi(h_t)]. \]

In particular, \(\dot H\leq-\mathcal D_{\nu_N}+\operatorname{osc}(\kappa)H\).

Proof

Normalize the unnormalized solution \(\partial_t\widetilde f=A^*\widetilde f\) by its mass to obtain the displayed equation. The diffusion and jump chain rules give

\[ hL\log h=Lh-\frac{(\nabla h)^{\mathsf T}a\nabla h}{h} -\int\mathcal B(h(S),h(S'))r(S,dS'). \]

Integrate against \(\nu_N\), use its eigenmeasure equation, and differentiate \(H_{\nu_N}(h)=\int h\log h\,d\nu_N\). The killing terms combine into \(\bar\kappa_tH-\nu_N(\kappa\psi(h))\). Since \(\psi\geq0\), \(\nu_N\psi(h)=H\), and \(\bar\kappa_t\leq\sup\kappa\), the upper bound follows. This is Proposition 178.

Remark 193 (Absorbing boundary flux)

For a kinetic absorbing boundary with outgoing set \(\Gamma_+=\{(x,v):v\cdot n(x)>0\}\), the total mass-loss rate is

\[ \ell_t=f_t(\kappa)+\int_{\Gamma_+}(v\cdot n)f_t. \]

Under the trace and domain hypotheses of Proposition 179, replace \(\bar\kappa_t\) by \(\ell_t\) and add \(-\int_{\Gamma_+}(v\cdot n)\nu_N\psi(h_t)\) to the entropy identity. A confinement moment estimate does not remove these boundary terms.

4.3. Closure for the full conditioned generator

Corollary 80 (Actual-law hypocoercive convergence)

Let the actual continuous-law QSD \(\nu_N\) satisfy a full-gradient LSI with constant \(C_N\). Suppose \(G_N\succ0\), \(G_N\preceq g_+I\), and the complete normalized evolution, including cloning and any killing or boundary contributions, satisfies

\[ \frac{d}{dt}\Phi_{G_N}(h_t)\leq-\delta_NI_{\nu_N}(h_t), \qquad\delta_N>0. \]

Then

\[ H_{\nu_N}(h_t)\leq\Phi_{G_N}(h_t) \leq e^{-\delta_Nt/(C_N/2+g_+)}\Phi_{G_N}(h_0). \]

Proof

The LSI gives \(\Phi_{G_N}\leq(C_N/2+g_+)I_{\nu_N}\). Substitute into the full derivative bound and apply Grönwall. The first-variation formula needed to check that derivative is Lemma 240; the complete theorem is Theorem 287.

Remark 194 (Alternative through the conditioned process)

A positive right survival eigenfunction yields a conservative Doob-transformed process with invariant law proportional to that eigenfunction times the QSD. The exact conjugacy and entropy transfer are proved in Proposition 180 and Theorem 288. Uniform transfer requires the stated survival-eigenfunction ratio bounds, or a replacement weighted comparison. This provides another route to the actual conditional law when those estimates hold.

INTERACTIVE EXPERIMENT · IV-02

Measure entropy along an executed trajectory

How does a declared spatial histogram change as the gas evolves? Entropy is computed from recorded walkers with a fixed window and bin definition.

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Rust engine experiment · Seed 7 · Poster after 7 experiment steps. Interactive view starts from the same seed.

5. Selection gradients and persistent forcing

A selection equation can make a density sharper. A derivative bound must distinguish growth proportional to existing gradients from a source that creates gradients even at the reference density. The latter produces a square-root term in Fisher information. Keeping that term determines whether the calculation proves convergence to the reference or a bound around it.

Proposition 168 (Exact derivative for a normalized multiplication model)

For this proposition only, consider the specified selection equation

\[ \partial_tf=\omega(V/\bar V_f-1)f, \qquad\bar V_f=\int V f, \]

with a fixed smooth \(V\geq v_*>0\), and a fixed positive reference \(\pi\). Let \(h=f/\pi\), \(c_f=\omega(V/\bar V_f-1)\), and \(G\) be constant positive definite. Then

\[ \dot H_\pi(h)=\frac{\omega}{\bar V_f} \operatorname{Cov}_f(V,\log h), \]

\[ \dot I_G(h)=\int c_f hq^{\mathsf T}Gq\,d\pi +\frac{2\omega}{\bar V_f}\int hq^{\mathsf T}G\nabla V\,d\pi, \qquad q=\nabla\log h. \]

If \(|V/\bar V_f-1|\leq K\) and \((\nabla V)^{\mathsf T}G\nabla V\leq S_G^2\), then

\[ \dot I_G\leq\omega K I_G+\frac{2\omega S_G}{v_*}\sqrt{I_G}. \]

Proof

Mass preservation gives \(\int c_f f=0\), so differentiating entropy gives the covariance formula. Since \(\dot h=c_fh\) and \(\partial_t\nabla\log h=\nabla c_f=\omega\nabla V/\bar V_f\),

\[ \frac{d}{dt}\int hq^{\mathsf T}Gq\,d\pi =\int c_fhq^{\mathsf T}Gq\,d\pi +2\int hq^{\mathsf T}G\nabla c_f\,d\pi. \]

Weighted Cauchy–Schwarz and the stated bounds prove the estimate.

Remark 195 (Scope of the multiplication model)

This equation is a specified normalized selection model. Identifying it with an algorithm’s mean-field limit requires deriving it from the actual sampled companion and acceptance law; see The Mean-Field Law of the Euclidean Gas. Its entropy covariance has no general negative sign. Moreover, \(f=\pi\) is stationary for this selection term only when \(V\) is constant \(\pi\)-almost everywhere. Kinetic and selection terms may balance at a full stationary law, but their separate reference-law cancellations must then be recomputed.

Lemma 180 (A square-root forcing term leaves a quantitative floor)

Suppose a nonnegative functional satisfies \(\Phi\leq C_I I\) and

\[ \dot\Phi\leq-aI+B\sqrt I+E, \qquad a,C_I>0,\quad B,E\geq0. \]

Then, with \(r=a/(2C_I)\),

\[ \Phi(t)\leq e^{-rt}\Phi(0) +\frac{B^2/(2a)+E}{r}(1-e^{-rt}). \]

Proof

Young’s inequality gives \(B\sqrt I\leq(a/2)I+B^2/(2a)\). Thus \(\dot\Phi\leq-r\Phi+B^2/(2a)+E\). Integrate this scalar differential inequality.

For the common-invariant-kernel theorem, the jump estimate is proportional to existing information, so a strict margin gives decay to zero. A persistent source instead leaves the displayed floor. An estimate for spatial Fisher alone also leaves the entropy and cross-Fisher derivatives to be controlled. These distinctions determine the actual parameter condition; a friction-versus-cloning slogan cannot replace them.

Proposition 169 (Exact cancellation at the full discrete QSD)

Let \(Q_N\) be the complete marked Euclidean Gas sub-Markov kernel and \(\nu_NQ_N=\alpha_N\nu_N\). Its backward conditional kernel is

\[ B_N(S',dS)=\frac{\nu_N(dS)Q_N(S,dS')}{\alpha_N\nu_N(dS')}, \qquad A_Ng(S')=\int g(S)B_N(S',dS). \]

The quotient denotes a regular conditional probability, not a pointwise density assumption. For a bounded \(g\) with \(\nu_Ng=0\), put \(M=\|g\|_\infty\) and \(c=\nu_N(A_Ng)\). For \(|\varepsilon|M<1\), the input law \(\mu_\varepsilon=(1+\varepsilon g)\nu_N\) has the exact conditioned output density

\[ \frac{d(\mu_\varepsilon Q_N/\mu_\varepsilon Q_N1)}{d\nu_N} =\frac{1+\varepsilon A_Ng}{1+\varepsilon c} =1+\frac{\varepsilon(A_Ng-c)}{1+\varepsilon c}. \]

For \(|\varepsilon|M\leq1/8\), its entropy increment satisfies

\[\begin{split} \begin{aligned} &D\!\left(\frac{\mu_\varepsilon Q_N}{\mu_\varepsilon Q_N1} \middle\Vert\nu_N\right)-D(\mu_\varepsilon\Vert\nu_N)\\ &\quad=\frac{\varepsilon^2}{2} \left[\operatorname{Var}_{\nu_N}(A_Ng)-\nu_N(g^2)\right] +\mathcal R_\varepsilon,\qquad |\mathcal R_\varepsilon|\leq12|\varepsilon|^3M^3. \end{aligned} \end{split}\]

Thus the full entropy balance at its actual QSD has no term of first order in \(\varepsilon\). The centering \(A_Ng-c\) includes the derivative of the survival normalization.

Proof

The joint measure \(\nu_N(dS)Q_N(S,dS')/\alpha_N\) has output marginal \(\nu_N\), so \(B_N\) is Markov and \(\|A_Ng\|_\infty\leq M\). Moreover

\[ \mu_\varepsilon Q_N =\alpha_N(1+\varepsilon A_Ng)\nu_N,\qquad \mu_\varepsilon Q_N1=\alpha_N(1+\varepsilon c), \]

which proves the exact density formula. Both \(g\) and \(A_Ng-c\) have zero \(\nu_N\)-mean. For \(\psi(1+t)=(1+t)\log(1+t)-t\), Taylor’s theorem gives

\[ \left|\psi(1+t)-\frac{t^2}{2}\right| \leq\frac{|t|^3}{6(1-|t|)^2},\qquad |t|<1. \]

Set \(e=|\varepsilon|M\leq1/8\) and \(t=\varepsilon(A_Ng-c)/(1+\varepsilon c)\); then \(|t|\leq2e/(1-e)\leq2/7\). Integrating Taylor’s formula gives the quadratic output term \(\varepsilon^2\operatorname{Var}_{\nu_N}(A_Ng)/[2(1+\varepsilon c)^2]\). Replacing its denominator by one costs at most \(2e^2(2e+e^2)/(1-e)^2\). The output Taylor remainder is at most \([2e/(1-e)]^3/[6(1-2e/(1-e))^2]\), and the input remainder is at most \(e^3/[6(1-e)^2]\). Their sum, including the denominator cost, is less than \(12e^3\) for \(e\leq1/8\). This proves the bound.

The cancellation uses the full eigenmeasure equation and its normalized kernel. In particular, it does not discard a positive square-root estimate for a separately centered cloning or kinetic term. The quadratic coefficient is the complete one-step response; its sign is not assigned from the separate pieces.

Imagine making a small dent in the equilibrium density: move a little probability from some swarm configurations to others. After a complete update, conditioning on survival changes the total weight of that dent. The subtraction of \(c\) accounts for this change. Once it is included, the disturbance still has zero total mass, and its entropy begins at second order in its size.

This is why we must put cloning, motion, and killing back together before evaluating their balance at the QSD. Each stage can move the equilibrium density; the complete conditioned step preserves it. The proposition computes the surviving quadratic term explicitly. Its sign tells us whether that particular disturbance loses information in one step. The cubic remainder bounds the error in this small-disturbance calculation; cancellation of the linear term alone does not determine the sign.

6. Discrete time and population-independent constants

A numerical step is a new transition operator. To transfer the continuous estimate, bound its error in the same modified entropy. An error bound for smooth observables does not by itself control density gradients. Once the functional error is known, the iteration is elementary and its accumulated floor is explicit.

Population size enters through the constants in the joint-law estimates. Tensorization proves those constants for a product reference while allowing correlated initial densities. For interacting swarms, the corresponding joint estimates must be uniform. Propagation of chaos identifies marginal limits; it is not a finite-particle spectral-gap comparison.

6.1. A discrete functional estimate

Theorem 249 (Entropy convergence for the canonical full-step gas)

For the canonical terminal absorbing-box gas of Theorem 225, fix \(N\geq1\) and \(h\ne2\). Let \(Q_N\), \(\nu_N\), \(\alpha_N\), and \(e_N\) be its actual full-step killed kernel, QSD, survival eigenvalue, and positive right eigenfunction. Normalize \(\nu_N(e_N)=1\). Use the constants proved there:

\[ 0<m_N\leq e_N\leq M_N,\qquad Q_N(S,\cdot)\geq\epsilon_N\theta_N(\cdot),\qquad \delta_N=\frac{\epsilon_N\theta_N(e_N)}{\alpha_N M_N}>0. \]

For every initial law \(\mu\) with finite \(D(\mu\Vert\nu_N)\) and every integer \(n\geq0\),

\[ D\!\left(\frac{\mu Q_N^n}{\mu Q_N^n1}\middle\Vert\nu_N\right) \leq\left(\frac{M_N}{m_N}\right)^2 (1-\delta_N)^nD(\mu\Vert\nu_N). \]

The full sampled-fitness component collision and revival rule, retained dead coordinates, BAOAB, final position noise, smooth cap, and terminal boundary are included in \(Q_N\). The constants in this statement are finite-population constants supplied by its full-kernel Gaussian minorization.

Proof

Define the conservative Doob kernel and its invariant law by

\[ P_N(S,dS')=\frac{e_N(S')Q_N(S,dS')}{\alpha_Ne_N(S)}, \qquad \pi_N=e_N\nu_N. \]

The minorization gives \(P_N\geq\delta_N\widehat\theta_N\), where \(\widehat\theta_N=e_N\theta_N/\theta_N(e_N)\). For \(\delta_N<1\) write \(P_N=\delta_N\widehat\theta_N+(1-\delta_N)R_N\), with \(R_N\) Markov. Joint convexity of relative entropy and data processing imply, for every law \(\eta\),

\[\begin{split} \begin{aligned} D(\eta P_N\Vert\pi_N) &=D\bigl(\delta_N\widehat\theta_N+(1-\delta_N)\eta R_N \Vert\delta_N\widehat\theta_N+(1-\delta_N)\pi_NR_N\bigr)\\ &\leq(1-\delta_N)D(\eta R_N\Vert\pi_NR_N)\\ &\leq(1-\delta_N)D(\eta\Vert\pi_N). \end{aligned} \end{split}\]

For \(\delta_N=1\), \(P_N\) is the constant kernel and the conclusion after one update is immediate. Iterate the inequality. The exact discrete conjugacy, obtained by canceling \(e_N\) between successive kernels, is

\[ \frac{\mu Q_N^n}{\mu Q_N^n1} =\mathcal R_{1/e_N}\bigl[(\mathcal R_{e_N}\mu)P_N^n\bigr], \qquad \mathcal R_w\eta=\frac{w\eta}{\eta(w)}. \]

Apply Lemma 241 to the initial and final reweightings. Each costs at most \(M_N/m_N\), giving the claimed bound. The proof of the minorization in Chapter 09 conditions on every accepted graph and shared rotation before integrating their actual probabilities; it therefore applies to the full mechanism rather than to independent collision outputs. Its constants need not be uniform in \(N\).

The function \(e_N\) lets us change how we weight swarm configurations while keeping an exact account of survival. In the proof, its factors cancel between successive steps. Reweighting at the beginning and end therefore recovers precisely the conditioned trajectories of the gas. All the shared collision rotations remain inside the transition being studied.

The minorization gives a common part of the transition law, of weight \(\delta_N\), that carries no information about the starting configuration. Repeated updates erase information through that common part. The two reweightings account for the prefactor in the entropy bound.

For a fixed swarm size this proves decay. As we add walkers, the common weight can shrink and the reweighting cost can grow. The cluster estimates have a further task: control the dependence on population size. The finite-swarm bound here supplies no such uniform estimate by itself.

Lemma 181 (Discrete entropy decay with the numerical defect retained)

Let \(\mathcal T_\tau\) be a continuous evolution satisfying \(\Phi(\mathcal T_\tau h)\leq e^{-r\tau}\Phi(h)\). Let \(P_\tau\) denote an actual numerical update, expressed relative to the same reference law. Suppose its functional defect satisfies

\[ \Phi(P_\tau h)-\Phi(\mathcal T_\tau h) \leq K\tau^{p+1}\Phi(h)+B\tau^{p+1}, \qquad p>0. \]

If \(q_\tau=e^{-r\tau}+K\tau^{p+1}<1\), then

\[ \Phi(h_n)\leq q_\tau^n\Phi(h_0) +\frac{B\tau^{p+1}}{1-q_\tau}(1-q_\tau^n). \]

For constants uniform as \(\tau\downarrow0\), the floor is \(O(\tau^p)\). If \(B=0\), the estimate gives convergence to the chosen reference.

Proof

The two assumed inequalities give \(\Phi(h_{n+1})\leq q_\tau\Phi(h_n)+B\tau^{p+1}\). Iterate the scalar recurrence and sum the geometric series. Since \(1-q_\tau=r\tau+o(\tau)\), the floor has the stated order. This is Lemma 245.

Remark 196 (Splitting order and the numerical target)

An \(O(\tau^3)\) local defect yields an \(O(\tau^2)\) functional floor when the lemma’s hypotheses hold. A Lie–Trotter local operator defect is generally \(O(\tau^2)\); integrating against a density functional does not automatically improve that order. BAOAB and cloning composition need estimates for their actual operators and domains. If the numerical invariant law or QSD differs from the continuous reference, identifying its entropy distance requires the corresponding measure comparison. The discrete QSD construction is in Equilibrium Profiles and Their Analytical Characterization.

6.2. Uniformity in the number of particles

Theorem 250 (Population-independent hypocoercive rates)

For product kinetic potentials \(U_N=\sum_{i=1}^N U(x_i)\), suppose the one-particle hypotheses of Corollary 79 hold and \(\sup_x\|\nabla^2U(x)\|_{\mathrm{op}}\leq M\). With fixed \(D,\gamma\), the rate in Theorem 247 is

\[ r_* = \frac{\eta}{C_0/2+3\eta}>0, \]

independently of \(N\), for arbitrary initial joint densities with finite modified entropy.

More generally, for actual interacting invariant laws or QSDs satisfying Corollary 80, if

\[ \sup_N C_N\leq C_*,\qquad \sup_N\|G_N\|_{\mathrm{op}}\leq g_*,\qquad \inf_N\delta_N\geq\delta_*>0, \]

then their rates are at least \(\delta_* /(C_*/2+g_*)\).

Proof

The product reference has LSI constant \(C_0\) by tensorization. The Hessian of \(U_N\) is block diagonal, so its operator norm is at most \(M\). The kinetic proof uses these constants and sums of squared derivatives; neither step introduces a factor of \(N\). For the interacting statement, substitute the three uniform bounds into the full-law rate.

Corollary 81 (Empirical variance from the joint LSI)

If an actual joint law \(\nu_N\) has the full-gradient LSI constant \(C_*\), and \(\|\nabla\varphi\|_\infty\leq L\), then

\[ \operatorname{Var}_{\nu_N}\!\left(\frac1N\sum_i\varphi(z_i)\right) \leq\frac{C_*L^2}{N}. \]

Proof

Linearizing LSI gives Poincaré with constant \(C_*\). Each particle gradient of the empirical average is \(N^{-1}\nabla\varphi(z_i)\), so the sum of squared gradients is at most \(L^2/N\). This is Corollary 98; independence is unnecessary once the joint inequality is established.

The final rate has a concrete interpretation: the numerator is the dissipation remaining after the full generator has been estimated, and the denominator converts Fisher information into modified entropy. Increasing friction affects both the mixing coefficient and the kinetic temperature, so the formula does not justify an unlimited increase in friction. Noise likewise changes both dissipation and the reference law.

For a mean-field conclusion, combine the finite-particle estimate with an identified marginal limit and the existence and uniqueness results for its equation. The LSI passes to fixed marginal limits under the conditions of Corollary 99. This keeps the finite-particle entropy theorem and the mean-field identification in one chain, with each estimate doing its stated job.