diff d837036..47d1413 hard_sphere_boltzmann_fluctuations_v2.txt
diff --git a/hard_sphere_boltzmann_fluctuations_v2.txt b/hard_sphere_boltzmann_fluctuations_v2.txt
index 896ad18..2f02531 100644
--- a/hard_sphere_boltzmann_fluctuations_v2.txt
+++ b/hard_sphere_boltzmann_fluctuations_v2.txt
@@ -41,7 +41,6 @@
\begin{document}
-% =====================================================================
\textbf{A gas of billiard balls.} A large box holds identical hard spheres,
like a three-dimensional break of pool: they fly in straight lines and
bounce elastically on contact. The diameter $\eps$ is tiny and the density
@@ -159,13 +158,8 @@ the collision rate, exactly $C_t$. The body of the argument proves the gas is
close enough to this picture for it to hold throughout the regular lifespan
and on the whole space.
-% =====================================================================
-\part*{Part 1. The exact objects and the theorem}
-% =====================================================================
-\section{The kinetic equation, the gas, and the fluctuation field}\label{sec:objects}
-\subsection{The Boltzmann equation and its norm}
Phase space is $\R_x^3\times\R_v^3$ and $z=(x,v)$. For a rate $\beta>0$
define the weighted sup norm
@@ -191,7 +185,6 @@ The Boltzmann equation reads
\end{equation}
all densities evaluated at the same $x$.
-\subsection{The microscopic gas}
Fix the sphere diameter $\eps>0$ and set the mean density
$\mu=\eps^{-2}$. The grand-canonical initial law, including $N=0$, is
@@ -204,7 +197,7 @@ with the non-overlap set
$\mathcal D_N^\eps=\{(z_i):|x_i-x_j|>\eps,\ i\ne j\}$ and
$1\le\mathcal Z_\eps\le e^\mu$. Given the configuration, spheres move by
free flight and deterministic elastic collisions, defined almost surely
-by [DHM, Proposition 1.2]. There is no random collision rule and no
+by Deng--Hani--Ma, Proposition 1.2. There is no random collision rule and no
resampling: the only randomness is the initial configuration.
\begin{assumption}\label{ass:kinetic}
@@ -219,7 +212,6 @@ $\beta>0$. There is a nonnegative classical solution $f$ of
Throughout, $[0,T]$ is \emph{one fixed regular kinetic interval}; nothing
below asserts that $T$ can be taken infinite.
-\subsection{The fluctuation field and the limiting law}
For real $\phi\in C_c^\infty(\R^6)$ put
\begin{equation}\label{eq:empirical}
@@ -244,7 +236,7 @@ Define the pair difference and the two limiting operators
\end{align}
$A_t$ is the linearized Boltzmann generator; the collision part carries
no extra factor $1/2$ because of the pre/post substitution together with
-$\omega\mapsto-\omega$ (\Cref{sec:collision-normal}). The quadratic form
+$\omega\mapsto-\omega$ (the collision normalization, below). The quadratic form
$C_t$ is the collision noise, nonnegative because $f_t\ge0$.
\begin{theorem}[Finite-dimensional fluctuation limit]\label{thm:main}
@@ -286,9 +278,8 @@ leave a nonzero deterministic drift already at time zero. This is why
the proof must compare exact expectations, not merely laws.
\end{remark}
-\subsection{Proof architecture and the two new estimates}
-We keep the layered collision-history framework of [DHM], stated as the
+The layered collision-history framework of Deng--Hani--Ma is kept here, stated as the
black box \Cref{thm:dhm-package}. Its role is to remove \emph{large
complexity}: histories with many entangled collisions. What remains must
be controlled at the fluctuation scale, and that is where the new work
@@ -296,15 +287,15 @@ lies.
\begin{enumerate}
\item \textbf{Initial ensemble and Gaussian target}
- (\Cref{sec:initial-target}). Exact initial cumulants, the initial CLT,
+ (the initial ensemble, below). Exact initial cumulants, the initial CLT,
the shift \Cref{rem:centering-shift}, and construction of the Gaussian
field of \Cref{thm:main}.
\item \textbf{Connected expansion with records}
- (\Cref{sec:cluster-expansion}). Factorial cumulants are written as a
+ (the connected expansion, below). Factorial cumulants are written as a
finite sum of globally connected diagrams, on a single pasted dynamics,
keeping finitely many observation records. Large complexity is dropped
- by the [DHM] package.
-\item \textbf{Sharp correlation bounds} (\Cref{sec:sharp-asymptotics},
+ by the Deng--Hani--Ma package.
+\item \textbf{Sharp correlation bounds} (the sharp bounds, below,
\Cref{thm:sharp-cumulants}). The $h$-th cumulant is $O(\mu^{1-h})$,
with the leading $h=1,2$ terms identified as signed sums of trees
weighted by $f_{i\tau}$, contact flux, and $\pm1$ signs. This is proved
@@ -312,35 +303,30 @@ lies.
clusters contributes $\eps^2[(v-v_*)\cdot\omega]_+$, and summing all
clusters costs $CT/\sqrt L<1$.
\item \textbf{Gaussian limit for the model dynamics}
- (\Cref{sec:gaussian-limit}). A mark that accumulates observations turns
+ (the Gaussian limit of the model dynamics, below). A mark that accumulates observations turns
every fixed multi-time linear combination into one scalar whose
ordinary cumulants are computed; second cumulant converges, higher ones
vanish, giving the Gaussian law with the correct covariance.
-\item \textbf{Accuracy of the model dynamics} (\Cref{sec:cutoff},
+\item \textbf{Accuracy of the model dynamics} (the cutoff comparison, below,
\Cref{prop:cutoff}). The pasted and true dynamics agree with
probability $O(\eps^{1+\eta})$, $\eta=1/20$. This is a positive
geometric estimate on terminal witnesses, carried out with local
gains, a common birth-energy bound, and a causal time-order sum free of
per-vertex logarithms.
-\item \textbf{Exact centering transfer} (\Cref{sec:transfer}). The
+\item \textbf{Exact centering transfer} (the centering transfer, below). The
discrepancy estimate, the fixed moments of $N/\mu$, and H\"older's
inequality show that expectations agree to $o(\mu^{-1/2})$, which is
what transfers the limit with exact centering.
\end{enumerate}
-% =====================================================================
-\part*{Part 2. The starting configuration and the Gaussian target}
-% =====================================================================
-\section{The initial ensemble and the Gaussian target}\label{sec:initial-target}
-This part is independent of the dynamics. It does two things: prove the
+This stage is independent of the dynamics. It does two things: prove the
fluctuation limit at time zero (a pure equilibrium statement about the
grand-canonical hard-sphere measure), and construct the Gaussian field
that \Cref{thm:main} claims to be the limit. It also records the initial
correlation bounds that later parts use.
-\subsection{Initial factorial measures and connected graphs}
Let $F_{h,0}^{\eps}$ be the density of the ordered factorial measure
divided by $\mu^h$:
@@ -364,7 +350,7 @@ The one-point function is $g_0=c_{1,0}^{\eps}=F_{1,0}^{\eps}$. Set
\]
and let $(N)_p=N\cdots(N-p+1)$ denote the falling factorial.
-The key geometric fact, before we can even state the initial CLT, is that
+The key geometric fact, before the initial CLT can even be stated, is that
connected correlations are \emph{order $\eps^3$ per connected pair}.
This is the static shadow of the dynamical statement that each merging
collision costs $\eps^2$; here excluded volume makes it $\eps^3$.
@@ -429,7 +415,7 @@ as a positive combination of $(N)_j$ gives the moment bounds.
\mathcal U_1=1,\qquad
\mathcal U_n(Z_n)=\sum_{H\text{ conn.\ on }[n]}\prod_{\{i,j\}\in H}(-\chi_{ij})\ (n\ge2).
\]
-We claim the tree bound
+The tree bound is
\begin{equation}\label{eq:tree-ineq}
|\mathcal U_n(Z_n)|\le\sum_{\mathcal T\text{ tree}}\prod_{\{i,j\}\in\mathcal T}\chi_{ij}.
\end{equation}
@@ -446,6 +432,12 @@ connected graphs, and summing one interval gives
\]
whose second factor lies in $[0,1]$, proving \eqref{eq:tree-ineq}.
+Penrose's scheme, with this precise tree bound, is reproduced in
+Bodineau--Gallagher--Saint-Raymond--Simonella, Proposition 2.3.3
+(\href{https://arxiv.org/pdf/2205.04110}{arXiv:2205.04110}); the original is
+O.~Penrose, \emph{Convergence of fugacity expansions for classical systems},
+in \emph{Statistical Mechanics} (Benjamin, 1967).
+
Integrating a tree from its leaves, keeping one free vertex, gives
\begin{equation}\label{eq:tree-integral}
\int f_0^{\otimes n}|\mathcal U_n|\le n^{n-2}b_\eps^{n-1}\qquad(n\ge2).
@@ -545,7 +537,6 @@ $\mu\|c_{2,0}^{\eps}\|_1=O(\eps)$, so the limiting initial covariance has
only the same-particle term $\int f_0\phi\psi$.
\end{remark}
-\subsection{Velocity moments and the backward propagator}
Write $M=\sup_{t\le T,x,v}e^{2\beta|v|^2}f_t(x,v)$. The noise form
\eqref{eq:noise} needs global velocity integrability, which does not
@@ -631,7 +622,6 @@ the simplex faces give $U(r,t)K_t\phi$) converges with \eqref{eq:dyson},
giving $\partial_tU(r,t)\phi=U(r,t)A_t\phi$ and hence \eqref{eq:upper-end}.
\end{proof}
-\subsection{Construction and uniqueness of the Gaussian field}
\begin{proposition}[Gaussian target]\label{prop:gaussian-target}
The equation \eqref{eq:fluc} has a centered Gaussian solution on each
@@ -700,20 +690,15 @@ exhaustion gives consistent local distributions. No path-space assertion
is needed here.
\end{proof}
-% =====================================================================
-\part*{Part 3. The microscopic engine: layered histories and the DHM package}
-% =====================================================================
-\section{Layered histories and the imported analytic package}\label{sec:dhm-inputs}
-We use Deng--Hani--Ma [DHM, arXiv:2408.07818v3, 18 July 2025]. Their
+Deng--Hani--Ma [arXiv:2408.07818v3, 18 July 2025] is used here. Its
diagram estimates are bounds for \emph{positive} geometric integrals; they
do not by themselves give a fluctuation theorem or the sharp connected
-bound. Everything in this part is either a definition we need or a
-statement quoted from [DHM] with its source location. This is the only
+bound. Everything here is either a definition that is needed or a
+statement quoted from Deng--Hani--Ma with its source location. This is the only
place where an unproved external input enters.
-\subsection{Parameters and the pasted dynamics}
Fix $\upsilon=3^{-4}=1/81$. Choose the excess bound $\Gamma$ as a large
dimensional constant \emph{before} the number of slabs $L$. Put
@@ -740,13 +725,12 @@ excess $\le\Gamma$; otherwise both particles continue straight. The
excess of a graph with $b$ edges, $n$ vertices, $k$ components is
$r=b-n+k$ (repeated edges counted with multiplicity); an internal contact
adds $1$, a merging contact adds the two excesses. Grazing, simultaneous,
-and tied-branch contacts are null sets. This is [DHM, Definition 4.3 and
-Eq.~(4.2)], with existence and mass preservation in [DHM, Proposition~4.5].
+and tied-branch contacts are null sets. This is Deng--Hani--Ma, Definition 4.3 and Eq.~(4.2), with existence and mass preservation in Deng--Hani--Ma, Proposition~4.5.
\textbf{Pasted dynamics.} Concatenate these restarted slab dynamics. It is
\emph{not} a semigroup (the prehistory resets at each slab), which is why
-we treat it as a separate object to be compared with the true flow in
-\Cref{sec:cutoff}.
+it is treated as a separate object to be compared with the true flow in
+the cutoff comparison, below.
\begin{lemma}[Kinetic envelopes]\label{lem:envelopes}
Under \Cref{ass:kinetic},
@@ -755,14 +739,13 @@ Under \Cref{ass:kinetic},
\end{equation}
\end{lemma}
\begin{proof}
-Apply [DHM, Proposition A.1] with decreasing exponent
+Apply Deng--Hani--Ma, Proposition A.1 with decreasing exponent
$\beta_*(t)=2\beta-\beta t/T$. Its hypothesis is \eqref{eq:kinetic-gaussian}
and its initial summable norms \eqref{eq:init-factorial}-\eqref{eq:bol-norm};
since $\beta_*(t)\ge\beta$, its conclusion gives \eqref{eq:bol-propagated}.
-This is [DHM, Eq.~(1.29)].
+This is Deng--Hani--Ma, Eq.~(1.29).
\end{proof}
-\subsection{Histories, roots, and normalization}
\begin{definition}[Layered C/O history]\label{def:history}
A history consists of finitely many oriented particle lines and binary
@@ -803,7 +786,7 @@ An \emph{admissible dependent history} satisfies:
\end{enumerate}
Connection in a slab means sharing a component of its atom graph; the last
alternative of (iv) covers a straight line with no atom there. These are
-[DHM, Definition 3.20(1)]. The \emph{complexity} is
+Deng--Hani--Ma, Definition 3.20(1). The \emph{complexity} is
\begin{equation}\label{eq:complexity}
\rho=|H_0|+\sum_i\bigl(|H_i|+r(\mathcal M_i)\bigr),
\end{equation}
@@ -843,11 +826,10 @@ roots the \emph{normalized} integral is
\mathcal J_M(Q)=\mu^{-h}I_M(Q):
\end{equation}
each birth carries activity $\mu$, each root a normalization $\mu^{-1}$,
-each contact surface a factor $\eps^2$ [DHM, Proposition 7.5, Eq.~(7.7)].
+each contact surface a factor $\eps^2$ Deng--Hani--Ma, Proposition 7.5, Eq.~(7.7).
To keep a root endpoint $(x,v)$ at time $t$ free, make its top end a fixed
end evaluated at $(x-tv,v)$, removing the root activity.
-\subsection{The imported analytic package}
The one-slab class $\mathcal T_i$ consists of connected histories with one
top root, conditions (i)--(ii) only, no continuation or initial-link
@@ -857,7 +839,7 @@ $\mathcal T_i$. The one-root signed tree sum contains every one-root
particle-line tree of size $<\Lambda_i/3$, with C/O choices and sign
$(-1)^{\#\mathrm O}$.
-\begin{theorem}[DHM analytic package]\label{thm:dhm-package}
+\begin{theorem}[Deng--Hani--Ma analytic package]\label{thm:dhm-package}
Assume \Cref{ass:kinetic}, the hierarchy \eqref{eq:cutoffs}--%
\eqref{eq:exponents}, and a common Gaussian $\Bol$ bound on all birth
inputs. For $L$ sufficiently large (fixed) and $\eps$ sufficiently small:
@@ -889,18 +871,16 @@ with target $f_{(i-1)\tau+\sigma}$; at $\sigma=0$ the map is the identity.
\end{theorem}
\begin{proof}[Source and envelope adaptations]
-For (a): [DHM, Proposition~7.2] counts root-labeled unlabeled histories by
-$C^b|\log\eps|^{C_*\rho}$; [DHM, Proposition~9.7 and Eq.~(9.51)] retain a
-positive gain $\propto\rho$; after regular cutting, [DHM, Eq.~(9.53)]
+For (a): Deng--Hani--Ma, Proposition~7.2 counts root-labeled unlabeled histories by
+$C^b|\log\eps|^{C_*\rho}$; Deng--Hani--Ma, Proposition~9.7 and Eq.~(9.51) retain a
+positive gain $\propto\rho$; after regular cutting, Deng--Hani--Ma, Eq.~(9.53)
bounds each full edge integral by
$\tau^{b/9}\eps^{-2h+2\upsilon/5+c_*\rho}|\log\eps|^{C_*\rho}$.
Multiplication by $\mu^{-h}=\eps^{2h}$ cancels the root power; weakening
$1/9\to1/10$ and discarding the extra positive power gives
-\eqref{eq:coarse}. These are the estimates inside the proof of [DHM,
-Proposition~6.2, \S9.2], not only its displayed conclusion.
+\eqref{eq:coarse}. These are the estimates inside the proof of Deng--Hani--Ma, Proposition~6.2, \S9.2, not only its displayed conclusion.
-The only use of a birth density is the envelope decomposition [DHM,
-Eqs.~(9.58)--(9.63)]: from \eqref{eq:bol-norm} choose a summable
+The only use of a birth density is the envelope decomposition Deng--Hani--Ma, Eqs.~(9.58)--(9.63): from \eqref{eq:bol-norm} choose a summable
sequence $(m_a)$ bounding the weighted supremum on unit balls; bounded
overlap and $\sum_am_a<\infty$ dominate $|g|$ by
$C\sum_am_a\ind_{|x-a|\le1}e^{-\beta'|v|^2}$, and the positive source
@@ -910,16 +890,15 @@ including absolute values of signed inputs. Multipliers in $[0,1]$ are
kept or dropped only when taking a positive upper bound.
For (b): the one-root integral is a function of a fixed endpoint
-[DHM, Eq.~(14.2)]; the C/O involution [DHM, Proposition~14.2] cancels
+Deng--Hani--Ma, Eq.~(14.2); the C/O involution Deng--Hani--Ma, Proposition~14.2 cancels
noncausal trees; the causal remainder satisfies the Gaussian $\Bol$
-estimate [DHM, Proposition~14.3, Eq.~(14.8)]; comparison to the short-time
-Boltzmann expansion, using \Cref{lem:envelopes}, is [DHM, Proposition~6.1,
-\S14]. To pass from $f_{(i-1)\tau}$ to $g$, expand each birth factor as
+estimate Deng--Hani--Ma, Proposition~14.3, Eq.~(14.8); comparison to the short-time
+Boltzmann expansion, using \Cref{lem:envelopes}, is Deng--Hani--Ma, Proposition~6.1, \S14. To pass from $f_{(i-1)\tau}$ to $g$, expand each birth factor as
$f_{(i-1)\tau}+(g-f_{(i-1)\tau})$; each residual costs
$\eps^{\theta_{i-1}}$ by hypothesis, and the multilinear tree bound sums
all choices; since $\theta_{i-1}>\theta_i$ the residuals are
$o(\eps^{\theta_i})$. For the positive subleading terms the bound is
-[DHM, Eq.~(14.28)] in $\Bol,\widetilde\beta_i$ with a time power per atom
+Deng--Hani--Ma, Eq.~(14.28) in $\Bol,\widetilde\beta_i$ with a time power per atom
and gain $\eps^{\upsilon/3+c_*\rho}$ in cyclic and extra-encounter
classes; \S14.3 Points~1--2 keep the Gaussian weight at the fixed root and
the spatial $\Bol$ sum, Point~4 regularizes an O-atom adjacent to the root
@@ -952,20 +931,19 @@ Bounded records carried by these lines are kept.
\end{lemma}
\begin{proof}[Source and construction]
-(i): use the error form of the cutting inequality [DHM, Eq.~(9.51)]
-retaining its deficit $\propto h$. The finale [DHM, \S13.5, p.~169] splits
+(i): use the error form of the cutting inequality Deng--Hani--Ma, Eq.~(9.51)
+retaining its deficit $\propto h$. The finale Deng--Hani--Ma, \S13.5, p.~169 splits
into a bounded-$\rho$ case, where the UP algorithm makes $\le h$ free-
component ``bad pieces'' paid by the root deficit, and a large-$\rho$ case
using the dependent cutting argument with the continuation condition of a
strictly lower layer still available; the resulting positive integration
-gives the loss $\eps^{-C_*h}$ of [DHM, Eq.~(9.82)]. Terminal size is used
+gives the loss $\eps^{-C_*h}$ of Deng--Hani--Ma, Eq.~(9.82). Terminal size is used
only later and absorbed by the time power.
(ii): take one line of the terminal component as root; its normalization
costs $\mu^{-1}$ and multiplication by $\mu$ restores the integral.
Splitting into $\le\Lambda_i$ lines and $\le\Gamma$ excess (ordinary
-admissible) versus the enlarged terminal class of [DHM, Definition
-3.20(4)], the same root operation is [DHM, Eq.~(15.4)]; summing over the
+admissible) versus the enlarged terminal class of Deng--Hani--Ma, Definition 3.20(4), the same root operation is Deng--Hani--Ma, Eq.~(15.4); summing over the
choice of line costs a cutoff polynomial.
(iii): fix nonnegative $q_0$ with $\int q_0=1$ and finite Gaussian $\Bol$
@@ -982,19 +960,15 @@ extension of \Cref{thm:dhm-package} admits $q_0$, and records are
untouched.
\end{proof}
-% =====================================================================
-\part*{Part 4. The connected expansion with records}
-% =====================================================================
-\section{A componentwise expansion retaining particle records}\label{sec:cluster-expansion}
-This part is pure algebra plus the [DHM] coarse bound. Its job: write the
+This stage is pure algebra plus the Deng--Hani--Ma coarse bound. Its job: write the
factorial cumulants of the pasted process as a \emph{finite} sum of
\emph{globally connected} diagrams, each with the correct label weight,
while carrying along finitely many observation \emph{records} (running
sums of test values on designated particles). The high-complexity part is
dropped with \Cref{thm:dhm-package}(a); what remains is what
-\Cref{sec:sharp-asymptotics} controls at the fluctuation scale. The
+the sharp bounds, below controls at the fluctuation scale. The
connected coefficients here are genuine factorial cumulants of the pasted
process, not a factorization assumption.
@@ -1006,7 +980,6 @@ the \emph{expansion} cluster cap. It differs from the \emph{dynamics} caps
$\Lambda_i,\Gamma$ and belongs only to the expansion; take $\eps$ small so
$M_i\ge2$ for all $i$.
-\subsection{Autonomous clusters and the finite forest identity}
\textbf{Idea.} Run each subset of particles on its own, with the truncated
collision rule. The full truncated history is then a union of ``autonomous
@@ -1096,7 +1069,6 @@ the second product lies in $[0,1]$, dropping it proves
\eqref{eq:ursell-tree-bound}.
\end{proof}
-\subsection{An exact one-slab operator on factorial measures}
\textbf{Idea.} The autonomous partition plus the forest identity become a
single linear operator on the factorial hierarchy. Its ``retained'' part
@@ -1172,7 +1144,6 @@ Indeed relevant $q\le(s+1)\Lambda_i^2/2<A_{i-1}/2$, with at most $q^q$
particle and $q^q$ cluster partitions, spanning/forest sum $\le q^{2q}$,
activity $\le\mu^q$, and at most $A_i\Lambda_i^2$ values of $q$; take logs.
-\subsection{The one-root map and its Boltzmann approximation}
Define the unmarked one-root measures recursively by
\begin{equation}\label{eq:one-root-definition}
@@ -1246,7 +1217,6 @@ an integrated $L^1$ inference, so it also gives density representatives;
restricting contact times gives the uniform partial-slab version.
\end{proof}
-\subsection{Marked retained hierarchies and exact resummation}
Fix finitely many observation pulses. Let $m$ bound both the number of
pulses and of record-carrying labels; record coordinates lie in a fixed
@@ -1334,7 +1304,6 @@ Bounded distinguished choices cost fixed powers of total size, with
exponent controlled by crossings and initial links.
\end{proof}
-\subsection{Variation errors and removal of high complexity}
\begin{lemma}[Stopped variation errors]\label{lem:stopped-variation}
For $L$ large in terms of the fixed data and $\Gamma$, any fixed record
@@ -1355,7 +1324,7 @@ positive C/O histories of the coarse package with previous-interface
continuations. The only top-slab difference, a stopped component with
$M_i\le\cdot<2M_i-2<\Lambda_i$ clusters, is in the same coarse family with
atom count $\ge M_i-1$. Bookkeeping: the lower independent inputs are
-already resummed into $g$; we do not resum the top stopped forest. Top
+already resummed into $g$; the top stopped forest is not resummed. Top
components failing continuation are paid by \Cref{lem:dummies}(i)
($\eps^{-C_*|S|}$), and bottom continuation of a record line with no
initial link uses \Cref{lem:dummies}(ii)--(iii) (add a unit-mass Gaussian
@@ -1417,9 +1386,8 @@ fixed graph counting, the $\Bol$ bounds, and $\Gamma$, so the chosen $L$
works for every subsequently fixed order.
\end{proof}
-\subsection{Terminal witnesses without observed roots}
-The comparison with the true flow (\Cref{sec:cutoff}) needs a positive
+The comparison with the true flow (the cutoff comparison, below) needs a positive
version with a connected \emph{final collision witness} instead of observed
outputs, and without any bound on the cutoff probability.
@@ -1455,14 +1423,10 @@ error and the argument starts from the initial cumulant expansion. Nothing
here compares true and pasted evolutions.
\end{proof}
-% =====================================================================
-\part*{Part 5. The sharp size of every correlation}
-% =====================================================================
-\section{Sharp bounds and the limiting tree measures}\label{sec:sharp-asymptotics}
-This is the central new estimate. The [DHM] package already removed large
-complexity (\Cref{thm:marked-expansion}(iii)); here we control what
+This is the central new estimate. The Deng--Hani--Ma package already removed large
+complexity (\Cref{thm:marked-expansion}(iii)); here what
remains at the \emph{sharp} fluctuation scale $\mu^{1-h}$ for the $h$-th
factorial cumulant. The mechanism is a change of \emph{relative
translations}: every collision that merges two clusters of particles
@@ -1470,12 +1434,11 @@ adjusts one relative position by the collision geometry, producing a
factor $\eps^2[(v-v_*)\cdot\omega]_+\,\dd t\,\dd\omega$. After accounting
for the activity $\mu$ per birth line, a connected diagram with $p$
physical initial links scales like $\mu^{1-h}\eps^p$, which is exactly the
-size we want; cycles and nontrivial initial correlations are smaller by a
+size sought; cycles and nontrivial initial correlations are smaller by a
positive power (proved via a null-set/dominated-convergence argument). The
sum over all cluster structures converges because each cluster slab has a
life time $\tau=T/L$ and the base $CT/\sqrt L<1$ for large $L$.
-\subsection{Conventions and the assertion}
Fix finitely many observation times and bounded continuous observation
functions. When observations are accumulated in a mark, update
@@ -1499,7 +1462,7 @@ $B_0<\infty$,
\sup_{0<\eps\le\eps_0,\,i}\|g_i\|_{\Bol,b}\le B_0,\qquad
\|g_i-f_{i\tau}\|_{\Bol,b}\longrightarrow0 .
\end{equation}
-We decrease $b$ a fixed number of times when reserving Gaussian weights;
+The exponent $b$ is decreased a fixed number of times when reserving Gaussian weights;
this does not change the microscopic dynamics.
\begin{theorem}[Sharp connected bounds]\label{thm:sharp-cumulants}
@@ -1530,7 +1493,6 @@ edges; continuing one line through an interface does not create a vertex,
while newly integrated particles do. This convention is essential for
counting translations and activity factors.
-\subsection{The finite graph and its numerical data}
Work with a term of complexity $\le K$. Replace each initial cumulant by
its bound from \Cref{prop:initial},
@@ -1572,7 +1534,6 @@ A birth at an interface of a $g_i$-supplied line is an independent phase
variable; the passage of a recorded line is a continuation of its old
variable.
-\subsection{Changing the relative translations}
\begin{lemma}[Chronological merger coordinates]\label{lem:merger-coordinates}
Fix a finite line graph, the C/O choices, the chronological order, and
@@ -1641,7 +1602,6 @@ $\le K$ and costs $\le\eps^{-2K}$ in \eqref{eq:sharp-scaling}; choosing
$D>2K+M$ makes such terms $O(\eps^M\mu^{1-h})$, with $M$ arbitrary and
chosen after the moment order.
-\subsection{Factorials and a summable majorant}
\begin{lemma}[Fresh-label factorials]\label{lem:fresh-label}
For complexity $\le K$, the absolute sum with slab sizes $(n_1,\dots,n_L)$
@@ -1683,7 +1643,7 @@ one observation per component, or after inserting a small Gaussian weight
in the designated velocities.
\end{lemma}
\begin{proof}
-We bound energy and time jointly to avoid a logarithmic loss per vertex.
+Energy and time are bounded jointly to avoid a logarithmic loss per vertex.
Let $E_0=\sum_{p=1}^m|v_p^{\mathrm{birth}}|^2$. Elastic collisions conserve
every colliding-pair energy, O-contacts do not change it, a line ending at
an interface discards its energy, and a birth adds energy already in
@@ -1747,9 +1707,8 @@ $(n_i)$ of total size $n$ (fixed $L$), so \eqref{eq:sharp-summable} gives
uniform summability and vanishing size tails, with polynomial degrees that
may grow with $h$ but never need another enlargement of $L$.
-\subsection{The extra-contact null sets}
-We need a qualitative fact: a limiting tree carries no extra contact. Only
+A qualitative fact is needed: a limiting tree carries no extra contact. Only
a fixed finite graph is examined. Its independent parameters are birth
velocities, contact times, merger directions, and one free translation per
component, all absolutely continuous. A claim about lines in one component
@@ -1828,7 +1787,6 @@ truncations with the Gaussian bounds and the spatial tightness from
\eqref{eq:sharp-summable}. This order needs no size-uniform rate and proves
the asserted $o$ for cycles and nontrivial initial cumulants.
-\subsection{Tree convergence and spatial disintegration}
For a fixed tree all prescribed C-clusters and O-blocks fall below the caps
once $\eps$ is small; by \Cref{lem:no-extra} almost every merger parameter
@@ -1877,7 +1835,6 @@ velocity sets conditional weak continuity gives convergence against
product tests, extended by the variation bounds.
\end{proof}
-\subsection{Completeness and splitting of the retained trees}
\begin{lemma}[Completeness of the tree class]\label{lem:tree-completeness}
For fixed roots, required records, and $L$, all global trees with trivial
@@ -1955,7 +1912,6 @@ conditional Gaussian bound controlling the joining flux with a fixed
exponent loss; records project because record forgetting commutes.
\end{proof}
-\subsection{Proof of the sharp bounds}
\begin{proof}[Proof of \Cref{thm:sharp-cumulants}]
\textbf{Weighted high-complexity tail.} Take the birth exponent $b>0$ of
@@ -1988,19 +1944,15 @@ exactly the leading trees by \Cref{lem:tree-completeness}. Their fluxes,
signs, input limits, and record tests are evaluated above, and the same
summable majorant yields the asserted Gaussian variation bounds for their
retained sums and limits, including the one-root and two-root limits of the
-next part. All exponential bases are independent of $K$ and $h$; those
+following stage. All exponential bases are independent of $K$ and $h$; those
quantities enter only through front constants and polynomial degrees, so
the $\Gamma,L$ choice remains the single choice made before the moment
order.
\end{proof}
-% =====================================================================
-\part*{Part 6. The Gaussian limit for the model dynamics}
-% =====================================================================
-\section{Identification of the covariance and the Gaussian limit}\label{sec:gaussian-limit}
-We now run the central limit theorem on the single pasted dynamics fixed by
+The central limit theorem is now run on the single pasted dynamics fixed by
\Cref{thm:marked-expansion,thm:sharp-cumulants}. Its $\Gamma,L$ do not
depend on the observation list or the moment order. The method: attach a
\emph{mark} that accumulates test values along each particle, so that any
@@ -2008,9 +1960,8 @@ fixed multitime linear combination becomes one scalar. Its ordinary
cumulants are computed from the sharp bounds; the second converges and the
higher ones vanish, giving a Gaussian law whose covariance is identified
by propagating marked tree laws backward. Comparison with the true flow is
-Part 7; exact centering is restored in Part 8.
+deferred to the next stage; exact centering is restored at the end.
-\subsection{Records and the limiting one- and two-root measures}
Give every particle a scalar mark, initially $0$, and at time $t_j$ update
\begin{equation}\label{eq:mark-update}
@@ -2058,7 +2009,6 @@ integrated one for the other, with $B\le|v|+|v_*|$, give an integrable
majorant, equivalently replacing equality by a smooth approximate identity
and passing to width zero.
-\subsection{Last-contact evolution and collision normalization}
At a collision, with $\xi=(x,v,y)$, $\xi_*=(x,v_*,y_*)$,
$\xi'=(x,v',y)$, $\xi_*'=(x,v_*',y_*)$, put
@@ -2127,7 +2077,6 @@ absolutely convergent tree sums. Updates are deterministic pushforwards,
giving \eqref{eq:pushforwards}.
\end{proof}
-\subsection{Collision normalization and the covariance source}\label{sec:collision-normal}
The pre/post substitution with $\omega\mapsto-\omega$ preserves
$B(v-v_*,\omega)\dd v\dd v_*\dd\omega$, as does particle interchange with
@@ -2161,9 +2110,8 @@ detailed balance nor equilibrium. The source is positive semidefinite
\end{equation}
and at an update $G$ pushes forward in both slots.
-\subsection{Backward pulses and all observation times}
-We need \eqref{eq:G-evolution} on the spaces
+\eqref{eq:G-evolution} is needed on the spaces
\[
\mathcal V_a=\{\Phi=c\,y+h:c\in\R,\ h\in E_a^0\},\quad
\|\Phi\|_{\mathcal V_a}=|c|+\|h\|_a .
@@ -2211,7 +2159,6 @@ whose variance is exactly \eqref{eq:variance-formula}, which by expanding
$h_r=\sum_{j:t_j\ge r}U(r,t_j)\phi_j$ equals \eqref{eq:target-cov}. This
identifies all multi-time and repeated-observation correlations.
-\subsection{Ordinary cumulants and Gaussian convergence}
\begin{proposition}[Gaussian limit on the pasted dynamics]\label{prop:cumulant-clt}
For the single pasted dynamics fixed above and every fixed observation list,
@@ -2266,22 +2213,18 @@ $\sigma=0$. This gives weak convergence with variance
\eqref{eq:variance-formula}.
\end{proof}
-% =====================================================================
-\part*{Part 7. Why the model dynamics is close to the real one}
-% =====================================================================
-\section{Accuracy of the pasted dynamics}\label{sec:cutoff}
The pasted dynamics recolors every restarted slab, resets the collision
prehistory at each interface, and truncates components above the caps
-$\Lambda_i,\Gamma$. The true hard-sphere flow does none of this. We must
-show the two trajectories coincide up to time $T$ with probability
+$\Lambda_i,\Gamma$. The true hard-sphere flow does none of this. It must be
+shown that the two trajectories coincide up to time $T$ with probability
$1-O(\eps^{1+\eta})$, $\eta>0$, \emph{better than} $O(\eps)$. The reason:
the fluctuation scale is $\sqrt\mu=\eps^{-1}$, and the mean of the pasted
process differs from the exact microscopic mean by a deterministic
$O(\eps)$ amount; an error of probability $O(\eps)$ alone would compete
with that shift after multiplication by $\eps^{-1}$, and exact centering
-would fail. We prove the sharper bound.
+would fail. The sharper bound is proved.
\begin{proposition}[Accuracy of the pasted dynamics]\label{prop:cutoff}
There is a large dimensional integer $\Gamma_0$ with the following
@@ -2298,21 +2241,20 @@ $A_i,\Lambda_i$ of \Cref{thm:dhm-package}; $C$ and the upper bound on $\eps$
may depend on the displayed fixed data.
\end{proposition}
-The rest of this part proves it. The plan:
+The rest of this argument proves it. The plan:
\begin{enumerate}[label=\textbf{Step \arabic*.},leftmargin=*]
\item Bound the probability of a first disagreement by the number of
``terminal witnesses'': a connected block of true collisions in one
slab, of bounded excess, meeting a fresh label and stopped by a rejection.
\item Separate coarse from fine. Coarse witnesses (complexity $>K$, or
terminal component larger than $\Lambda_i$) sum to less than any power of
- $\eps$ by the [DHM] package---this fixes $K$ large.
+ $\eps$ by the Deng--Hani--Ma package---this fixes $K$ large.
\item On the low-complexity fine regime, localize energy, then combine a
\emph{local gain} at every good cutting piece with a \emph{global} energy
and causal time-order bound, gaining enough powers of $\eps$ to beat
$1+\eta$.
\end{enumerate}
-\subsection{From first disagreement to terminal witnesses}
Distinguish the geometric cycle excess $r(M)$ of a graph from its
multislab complexity $\rho(M)$. For a graph with $b$ binary-contact
@@ -2362,7 +2304,6 @@ either absorbed into an independent $g$ input, or continues a required
record; the latter are boundedly many (the records through this interface,
including continued empty lines).
-\subsection{Coarse witnesses and the regularity of graphs}
Choose a complexity threshold $K$ after $L$ is fixed. By
\Cref{cor:terminal-witness,thm:dhm-package}, the sum of positive majorants
@@ -2382,7 +2323,6 @@ connected graph with $b$ atoms and excess $\le2\Gamma$ can be produced
without new crossings and within the same realization family by
$\le C^b$ choices of the initial labelled $M$ and its geometric data.
-\subsection{A first energy localization}
The number of scattered velocities in any witness is at most $B_\eps$, and
initial data carry only $\le2\beta$ Gaussian under \Cref{ass:kinetic}; a
@@ -2481,10 +2421,9 @@ then let the speed restriction tend to zero (zero-speed incoming contacts
have zero flux).
\end{proof}
-\subsection{Conditional local kernels and admissible cuts}
-This subsection imports the local (kernel) estimates and the cutting
-alternatives. We use them with conditional masses, not global probabilities.
+Here the local (kernel) estimates and the cutting alternatives are imported.
+They are used with conditional masses, not global probabilities.
A \emph{cut} exposes a set of contact variables. Its shared lines become
fixed phase-space data for the unexposed graph. The available degree of an
@@ -2496,10 +2435,15 @@ zero fixed ends. A $\{33\}$ piece has two degree-three atoms joined by one
bond; it is type A if one atom cuts as a type $\{3\}$ singleton and the
other becomes an allowed $\{2\}$ piece, type B otherwise. A $\{44\}$ piece
has two degree-four atoms joined by one bond
-\cite[Definitions 8.2, 8.11]{DHM2025}. The kernels compose
+(Deng--Hani--Ma, Definitions 8.2, 8.11). The kernels compose
$I_M=I_A\circ I_{M\setminus A}$ with repeated boundary variables identified
-\cite[Proposition 8.14]{DHM2025}; this is how a later kernel is estimated
-with earlier variables fixed.
+(Deng--Hani--Ma, Proposition 8.14); this is how a later kernel is estimated
+with earlier variables fixed. The uniform bound on the number of collisions in
+semi-dispersing billiards behind these counting estimates is due to Burago,
+Ferleger and Kononenko; Deng--Hani--Ma adapt it to the extended dynamics.
+Source PDF:
+\href{https://arxiv.org/pdf/2408.07818v3}{Deng--Hani--Ma, arXiv:2408.07818v3};
+\href{https://doi.org/10.2307/120962}{Burago--Ferleger--Kononenko, Ann.\ of Math.\ 147 (1998)}.
\begin{lemma}[Local kernels]\label{lem:dhm-local}
Let $R$ bound velocities and localized radii; the time length is $\le T$.
@@ -2517,7 +2461,7 @@ For a singleton with two fixed ends in a free kernel:
\end{enumerate}
Multiplying any such kernel by a bounded factor on its support costs its
supremum; no independent Gaussian distribution of a fixed-end velocity is
-assumed. This is \cite[Lemmas 8.15, 8.20, 8.23]{DHM2025}. A
+assumed. This is (Deng--Hani--Ma, Lemmas 8.15, 8.20, 8.23). A
\emph{uniformly good} singleton or type-B pair is one of these restricted
kernels with conditional mass $\eps^\upsilon|\log\eps|^{C_*}$; all
gain-producing restrictions are part of the kernel.
@@ -2557,8 +2501,7 @@ good, and $\ge(m_\pm-10^5S)/10$ type-A or type-B pairs.
\end{enumerate}
Neutral pieces keep their neutral kernels; partitions and operation
sequences have $\le C^b|\log\eps|^{C_*}$ subcases with $C$ independent of
-$L$. These are \cite[Definition 15.3; Propositions 15.1, 15.2, 15.4, 15.6,
-15.10]{DHM2025}.
+$L$. These are (Deng--Hani--Ma, Definition 15.3; Propositions 15.1, 15.2, 15.4, 15.6, 15.10).
\end{proposition}
\begin{lemma}[Adjacent scattering contacts]\label{lem:cutoff-pair}
@@ -2581,7 +2524,7 @@ edges of a type-A pair with contact times $t_1,t_2$ and
neutral type-two piece and a singleton with the same bound.
\end{enumerate}
The bounds are uniform in previously exposed lines and times. This is
-\cite[Lemma 15.12]{DHM2025}.
+(Deng--Hani--Ma, Lemma 15.12).
\end{lemma}
Fix $\delta=\tfrac12\min\{\upsilon,1/20\}>0$; every retained local gain is
@@ -2598,10 +2541,9 @@ improved pieces $g$ and free singletons $s_4$ satisfy
\end{equation}
There are $\le C^b|\log\eps|^{C_*}$ subcases; the number of non-ordinary
pieces is bounded in terms of $\Gamma$ and fixed choices. This is
-\cite[Proposition 15.11]{DHM2025}.
+(Deng--Hani--Ma, Proposition 15.11).
\end{lemma}
-\subsection{Simultaneous integration of the ordinary parameters}
The local powers must survive simultaneous integration without a
per-vertex logarithmic loss. Call a neutral type-three singleton
@@ -2669,10 +2611,9 @@ a polylogarithmic energy region, its integral is at most
\eps^\alpha|\log\eps|^{C_*}(C\tau\sqrt L)^q .
\end{equation}
The base $C$ depends on the fixed kinetic envelopes and dimension, not on
-$K,L$. This is \cite[Lemma 16.2]{DHM2025}.
+$K,L$. This is (Deng--Hani--Ma, Lemma 16.2).
\end{lemma}
-\subsection{Assembly: the discrepancy probability}
\begin{proof}[Proof of \Cref{prop:cutoff}]
Start from \Cref{lem:first-disagreement,cor:terminal-witness}, and fix the
@@ -2725,23 +2666,19 @@ The local exponent $\delta$ and the finite hierarchy of
choose $\Gamma$, then $L$ for the imported estimates and the common
exponential base; only afterward choose $K$, the localization powers, and the
small upper bound on $\eps$. No observation number or moment order enters the
-first two choices. This is what let us fix a single pasted dynamics in
-\Cref{sec:gaussian-limit}.
+first two choices. This is what allows a single pasted dynamics to be fixed in
+the Gaussian limit of the model dynamics, below.
\end{remark}
-% =====================================================================
-\part*{Part 8. Exact centering and the fluctuation limit}
-% =====================================================================
-\section{Transfer of exact centering and the hard-sphere limit}\label{sec:transfer}
The pasted process of \Cref{prop:cumulant-clt} is centered by its own
microscopic mean. The real gas is centered by a slightly different mean,
since the pasted and true laws charge the discrepancy event (probability
$\le C\eps^{1+\eta}$, \Cref{prop:cutoff}) with different mass. At the
fluctuation scale $\sqrt\mu=\eps^{-1}$ this difference of means must go to
-zero, and that is exactly the sharp estimate of \Cref{prop:cutoff}. We make
-this quantitative, then assemble \Cref{thm:main}.
+zero, and that is exactly the sharp estimate of \Cref{prop:cutoff}. This is made
+quantitative, then \Cref{thm:main} is assembled.
\begin{lemma}[Discrepancy and centering]\label{lem:center-transfer}
Couple the true and pasted dynamics from the same initial configuration.
@@ -2804,7 +2741,7 @@ $(\zeta_0^\eps(\phi_1),\dots,\zeta_0^\eps(\phi_m))$ to the centered Gaussian
vector with covariance $\int f_0\phi_i\phi_j$.
\textbf{Dynamical limit on the pasted dynamics.} Fix a single pasted
-dynamics as in \Cref{sec:gaussian-limit} ($\Gamma,L$ independent of the
+dynamics as in the Gaussian limit of the model dynamics, below ($\Gamma,L$ independent of the
observation list and order). \Cref{prop:cumulant-clt} gives, for every fixed
list of times and tests, convergence of the centered combination
$\eps\sum_i y_i(T)$ to $\sum_j\zeta_{t_j}(\phi_j)$, with convergence of all
@@ -2850,54 +2787,6 @@ correlation, the sharp tree limits, the covariance, the Gaussian central
limit, and the exact-centering transfer are proved here.
\end{remark}
-% =====================================================================
% Bibliography
-% =====================================================================
-
-\begin{thebibliography}{99}
-
-\bibitem{DHM2025}
-Z.~Deng, Z.~Hani, and X.~Ma,
-Long time derivation of the Boltzmann equation from hard sphere dynamics,
-arXiv:2408.07818v3 (18 July 2025).
-
-\bibitem{BGSS2023}
-T.~Bodineau, I.~Gallagher, L.~Saint-Raymond, and S.~Simonella,
-Long-time correlations for a hard-sphere gas at equilibrium,
-Comm.\ Pure Appl.\ Math.\ 76 (2023), no.~12, 3852--3911;
-arXiv:2012.03813.
-
-\bibitem{BGSS2024}
-T.~Bodineau, I.~Gallagher, L.~Saint-Raymond, and S.~Simonella,
-Long-time derivation at equilibrium of the fluctuating Boltzmann equation,
-Ann.\ Probab.\ 52 (2024), no.~1, 217--295; arXiv:2201.04514.
-
-\bibitem{BGSSCluster2022}
-T.~Bodineau, I.~Gallagher, L.~Saint-Raymond, and S.~Simonella,
-Cluster expansion for a dilute hard sphere gas dynamics,
-J.\ Math.\ Phys.\ 63 (2022), no.~7, 073301; arXiv:2205.04110.
-
-\bibitem{BGSS2020}
-T.~Bodineau, I.~Gallagher, L.~Saint-Raymond, and S.~Simonella,
-Fluctuation theory in the Boltzmann--Grad limit,
-J.\ Stat.\ Phys.\ 180 (2020), 873--895; arXiv:2004.00311.
-
-\bibitem{Lanford1975}
-O.~E.~Lanford, III,
-Time evolution of large classical systems,
-in: Dynamical Systems, Theory and Applications,
-Lecture Notes in Physics 38, Springer, 1975, 1--111.
-
-\bibitem{Penrose1967}
-O.~Penrose,
-Convergence of fugacity expansions for fluids and lattice gases,
-in: Statistical Mechanics (T.~A.~Bak, ed.), Benjamin, 1967.
-
-\bibitem{BuragoFerlegerKononenko1998}
-D.~Burago, S.~Ferleger, and A.~Kononenko,
-Uniform estimates on the number of collisions in semi-dispersing billiards,
-Ann.\ of Math.\ 147 (1998), 695--708.
-
-\end{thebibliography}
\end{document}