Mixture Dynamics And Shells

Let $K$ be a nonempty compact subset of ${\mathbb{R}}^d$, $d\ge 1$, with its Borel sigma-algebra, and let $a:K\to \mathbb{R}$ be continuous. The convex hull $\mathrm{conv}K$ is the set of finite convex combinations of points of $K$. A phase generator is a point $y\in K$ with declared energy $a(y)$. A mixture is a Borel probability measure $P$ on $K$. Its barycenter and mean energy are the coordinatewise Lebesgue integrals $\bar{y}(P):={\int }_{K}ydP(y)\in {\mathbb{R}}^d,\bar{a}(P):={\int }_{K}a(y)dP(y)\in \mathbb{R}\mathrm{.}$ These integrals are finite because each coordinate and $a$ are bounded on compact $K$. For prescribed mean $x\in \mathrm{conv}K$, the mixture energy is $a_{\mathrm{m}\mathrm{i}\mathrm{x}}(x):=\inf \{\bar{a}(P):P\text{ is a Borel probability on }K,\bar{y}(P)=x\}\mathrm{.}$ The admissible set is nonempty since every finite convex combination of points of $K$ is the barycenter of a finite atomic probability. The definition does not assert that the infimum is attained; the theorem on this page gives a compact continuous sufficient condition.

The definition concerns a mathematical probability mixture and an assigned scalar energy. It does not assert that physical phases can be prepared, exchanged, or brought to equilibrium.

Fix $d\ge 2$. The elastic collision set is $Q_{\mathrm{e}\mathrm{l}}:=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in {\mathbb{R}}^{4d}:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\},$ with its relative Borel sigma-algebra. A collision measure is a nonnegative countably additive measure $\nu$ on this measurable space. It is symmetric if it is invariant under the exchanges $(v,w)\leftrightarrow (w,v)$, $(v^{\mathrm{\prime }},w^{\mathrm{\prime }})\leftrightarrow (w^{\mathrm{\prime }},v^{\mathrm{\prime }})$, and $(v,w)\leftrightarrow (v^{\mathrm{\prime }},w^{\mathrm{\prime }})$. These invariances are hypotheses on $\nu$, not consequences of energy and momentum conservation.

Fix the velocity measure space $({\mathbb{R}}^d,\mathcal{L}({\mathbb{R}}^d),dv)$, where $dv$ is the Lebesgue set function of Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$ and its countably additive measure-space structure is assumed as part of the data. This definition makes no existence assertion for that structure or for the collision operator. An input density is a finite-valued positive Lebesgue-measurable function $f:{\mathbb{R}}^d\to (0,\mathrm{\infty })$ whose four coordinate evaluations $f(v),f(w),f(v^{\mathrm{\prime }}),f(w^{\mathrm{\prime }})$ on $Q_{\mathrm{e}\mathrm{l}}$ are $\nu$-measurable. Let ${\mathcal{D}}_{\nu }$ be a declared set of such inputs. Put ${\mathcal{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^1:=\{q:{\mathbb{R}}^d\to \mathbb{R}:q\text{ is Lebesgue measurable and }{\int }_K\mathrm{\mid }q\mathrm{\mid }dv<\mathrm{\infty }\text{ for every compact }K\subset {\mathbb{R}}^d\}$. Its elements here are chosen function representatives, rather than equivalence classes. A collision model includes a specified map $Q_{\nu }:{\mathcal{D}}_{\nu }\to {\mathcal{L}}_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$; write $Q(f):=Q_{\nu }(f)$.

For $f\in {\mathcal{D}}_{\nu }$, put $A=f(v)f(w)$ and $B=f(v^{\mathrm{\prime }})f(w^{\mathrm{\prime }})$. For a finite-valued complex Lebesgue-measurable function $\phi$ on velocity space, put $\delta \phi :=\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }})$. Define the complete admissible test class ${\mathcal{T}}_f$ to consist of exactly those $\phi$ for which $\delta \phi$ is $\nu$-measurable and ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }\phi (v)Q_{\nu }(f)(v)\mathrm{\mid }dv<\mathrm{\infty },{\int }_{Q_{\mathrm{e}\mathrm{l}}}\mathrm{\mid }(B-A)\delta \phi \mathrm{\mid }d\nu <\mathrm{\infty }\mathrm{.}$ Absolute integrability is in the sense of Integrable real and complex functions, and their integrals. The model also declares, for each $f\in {\mathcal{D}}_{\nu }$, a complex linear testing subspace ${\mathcal{S}}_f\subseteq {\mathcal{T}}_f$ containing at least one nonzero test. Require the following weak collision identity for every $f\in {\mathcal{D}}_{\nu }$ and every $\phi \in {\mathcal{S}}_f$. A formulation adopting all admissible tests specifies ${\mathcal{S}}_f={\mathcal{T}}_f$; a result using only particular tests names those tests and their linear span explicitly: ${\int }_{{\mathbb{R}}^d}\phi (v)Q(f)(v)dv=\frac{1}{4}{\int }_{Q_{\mathrm{e}\mathrm{l}}}(f(v^{\mathrm{\prime }})f(w^{\mathrm{\prime }})-f(v)f(w))(\phi (v)+\phi (w)-\phi (v^{\mathrm{\prime }})-\phi (w^{\mathrm{\prime }}))d\nu ,$ Both sides are absolutely convergent by the test-class definition. Admissible tests need not be bounded or compactly supported: $1$, velocity coordinates, squared speed, and $\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$ for a positive constant $f_{\mathrm{r}\mathrm{e}\mathrm{f}}$ belong to ${\mathcal{T}}_f$ whenever they meet the displayed measurability and integrability conditions. A consumer invoking one of these tests must establish or assume those conditions and require it to belong to the declared testing domain ${\mathcal{S}}_f$. In particular, the entropy branch uses the span of $1$ and $\log (f\mathrm{/}{f}_{\mathrm{r}\mathrm{e}\mathrm{f}})$, with absolute convergence explicitly assumed. The map $Q_{\nu }$, its input domain and its testing domains are adopted model data; the weak identity is their admissibility condition, not an assertion that every positive density and collision measure admit such a map or that these tests alone uniquely determine it. Molecular chaos or a microscopic derivation of the identity is not part of the definition.

The measure's normalization is chosen so $Q(f)$ has units of $f$ per unit time in an application. No specific cross-section or measured collision rate is asserted.

Fix a finite alphabet $A=\{1,\dots ,q\}$, $q\ge 1$, and real single-site energies $e_1,\dots ,e_q$ in a fixed energy unit. At inverse parameter $\beta \in \mathbb{R}$, define $Z_1(\beta ):={\sum }_{j=1}^q{e}^{-\beta e_j},p_j(\beta ):=e^{-\beta e_j}\mathrm{/}{Z}_1(\beta ),E(\beta ):={\sum }_{j=1}^q{p}_j(\beta )e_j\mathrm{.}$ For $N\ge 1$, a configuration $i=(i_1,\dots ,i_N)\in A^N$ has additive energy $H_N(i):={\sum }_{r=1}^N{e}_{i_r}$. Given a width ${\delta }_N\ge 0$, the shell is ${\mathrm{\Gamma }}_N(\beta ,{\delta }_N):=\{i\in A^N:\mathrm{\mid }{H}_N(i)-NE(\beta )\mathrm{\mid }\le {\delta }_N\}\mathrm{.}$ When this finite set is nonempty, its microcanonical law means the uniform probability on ${\mathrm{\Gamma }}_N$, and its dimensionless entropy is $\log \mathrm{\mid }{\mathrm{\Gamma }}_N\mathrm{\mid }$. A type is a vector $t=(n_1\mathrm{/}N,\dots ,n_q\mathrm{/}N)$, where the nonnegative integer counts sum to $N$; its multiplicity is $N!\mathrm{/}{\prod }_j{n}_j!$.

The convention fixes both the shell center and the law within the shell. It makes no physical claim that an apparatus has resolution ${\delta }_N$.

Let $(\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t{)}_{t\ge 0},P)$ carry a standard Brownian motion $B$ adapted to the filtration, with future increments independent of the present sigma-algebra. Fix constants $\gamma >0$ and $\sigma >0$, and an ${\mathcal{F}}_0$-measurable real random variable $X_0$ with finite second moment. A continuous adapted solution of the Ornstein–Uhlenbeck equation on $[0,T]$ is a continuous adapted process $X$ satisfying, almost surely for every $t\in [0,T]$, $X_t=X_0-\gamma {\int }_0^t{X}_{s}ds+\sigma B_t,$ where the time integral is the ordinary Lebesgue integral along the continuous path. In differential notation this is $d{X}_t=-\gamma X_{t}dt+\sigma d{B}_t$, with the stochastic differential interpreted through the displayed integral equation.

The definition asserts neither existence nor uniqueness. The Itô integral in the explicit representation is the square-integrable predictable-process integral of Ito integral for square-integrable predictable processes.

Let $O\subset {\mathbb{R}}^m$, $m\ge 2$, be open, let $H\in C^2(O,\mathbb{R})$, and let $J\subset \mathbb{R}$ be open. An energy $E\in J$ is regular if $\mathrm{\nabla }H(x)\mathrm{\ne }0$ for every $x\in {\mathrm{\Sigma }}_E:=H^{-1}(\{E\})$. By the implicit-function theorem, a neighborhood of each point of a nonempty regular level is a graph $x_i=h(z)$.

In a graph chart, use the local area density $\sqrt{1+\mathrm{\mid }\mathrm{\nabla }h(z){\mathrm{\mid }}^2}dz$, where $dz$ is $(m-1)$-dimensional Lebesgue measure. A graph partition on ${\mathrm{\Sigma }}_E$ is a locally finite graph-chart cover with nonnegative smooth functions ${\chi }_a$ supported in their respective charts and summing to one on the level. Whenever such a partition is supplied, define the candidate weighted shell integral by ${\mu }_E(g):={\sum }_a\int g(z,h_a(z)){\chi }_a(z,h_a(z))\mathrm{\mid }{\mathrm{\partial }}_{i}H(z,h_a(z)){\mathrm{\mid }}^{-1}dz$ for nonnegative Borel $g$. Its independence from the chosen chart partition, and its equality with ${\int }_{{\mathrm{\Sigma }}_E}g\mathrm{\mid }\mathrm{\nabla }H{\mathrm{\mid }}^{-1}d{A}_E$, are proved in the theorem on this page. If $0<{\mu }_E(1)<\mathrm{\infty }$, the microcanonical shell probability is ${\widehat{\mu }}_E:={\mu }_E\mathrm{/}{\mu }_E(1)$.

This is an explicit chart-based definition, not a formal delta distribution $\delta (H-E)$. Critical levels are excluded.

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix dimension $d\ge 1$, particle mass $m>0$, action reference $h>0$, inverse energy $\beta >0$, and positive cube volumes $V_N$ with $N\mathrm{/}{V}_N\to \rho \in (0,\mathrm{\infty })$. For $N$ indistinguishable classical noninteracting particles in a cube of volume $V_N$, use the phase-space reference $dqdp\mathrm{/}(N!h^{dN})$ and Hamiltonian $H_N(q,p)={\sum }_{i=1}^N\mathrm{\mid }{p}_i{\mathrm{\mid }}^2\mathrm{/}(2m)$. Define ${\lambda }_{\beta }=h\sqrt{\beta \mathrm{/}(2\pi m)}$. Then the dimensionless partition integral and free-energy limits are $Z_N=\frac{V_N^N}{N!{\lambda }_{\beta }^{dN}},\frac{-{\beta }^{-1}\log Z_N}{N}⟶{\beta }^{-1}[\log (\rho {\lambda }_{\beta }^d)-1],\frac{-{\beta }^{-1}\log Z_N}{V_N}⟶{\beta }^{-1}\rho [\log (\rho {\lambda }_{\beta }^d)-1]\mathrm{.}$ There is also the following finite-momentum-block limit. Let $M=dN$, let $G_1,\dots ,G_M$ be independent standard normal variables, set $A_N=M^{-1}{\sum }_{a=1}^M{G}_a^2$, and define canonical momenta $P_a^{\mathrm{c}\mathrm{a}\mathrm{n}}=\sqrt{m\mathrm{/}\beta }{G}_a$ and shell-prepared momenta $P_a^{\mathrm{s}\mathrm{h}}=\sqrt{m\mathrm{/}\beta }{G}_a\mathrm{/}\sqrt{A_N}$. On the null event $A_N=0$, set the shell-prepared vector to $(\sqrt{mM\mathrm{/}\beta },0,\dots ,0)$. The shell-prepared vector has exactly energy $dN\mathrm{/}(2\beta )$, and for every fixed $r$ and bounded continuous $\psi :{\mathbb{R}}^r\to \mathbb{R}$, $\mathbb{E}\psi (P_1^{\mathrm{s}\mathrm{h}},\dots ,P_r^{\mathrm{s}\mathrm{h}})⟶\mathbb{E}\psi (P_1^{\mathrm{c}\mathrm{a}\mathrm{n}},\dots ,P_r^{\mathrm{c}\mathrm{a}\mathrm{n}})\mathrm{.}$ This is weak convergence for each fixed block in this free-particle model. The shell preparation is the stated normalized-Gaussian construction, and no finite-width shell or interacting-gas equivalence is asserted.

Assume the Axiom of Choice. Let $(M,\omega )$ be a smooth $2n$-dimensional symplectic manifold and $H\in C^{\mathrm{\infty }}(M,\mathbb{R})$, with Hamiltonian field defined by ${\iota }_{X_H}\omega =dH$. Wherever its local flow ${\varphi }_t$ is defined, it preserves the symplectic form and the volume measure $m$ determined by ${\omega }^n\mathrm{/}n!$. No completeness is asserted. Let $R$ be a measurable region on which a nonzero-time map $T={\varphi }_{\tau }$ is defined and is forward invariant, $T(R)\subseteq R$. The restriction $T:R\to R$ preserves $m{\mathrm{\mid }}_R$ if $T(R)=R$, or more generally if $m(R\setminus T(R))=0$. This latter condition holds automatically when $m(R)<\mathrm{\infty }$. Thus for every finite-measure forward-invariant $R$ and every measurable $E\subseteq R$, almost every point of $E$ returns to $E$ infinitely often under forward iteration. Mere forward invariance of an infinite-volume region does not imply relative measure preservation.

If $m(R)=1$ and $T$ is ergodic, then for every $f\in L^1(R,m)$ the averages $N^{-1}{\sum }_{j=0}^{N-1}f(T^{j}x)$ converge almost everywhere and in $L^1$ to ${\int }_{R}fdm$. Without ergodicity, the limit is the invariant conditional expectation supplied by Birkhoff's theorem, not necessarily the space mean. Finally, fix any real scalar $k_B$. If each stated time map ${\varphi }_t$ is a diffeomorphism of a finite invariant region $R$, a density $p_0\ge 0$ with ${\int }_R{p}_{0}dm=1$ and ${\int }_R\mathrm{\mid }{p}_0\log p_0\mathrm{\mid }dm<\mathrm{\infty }$ transports by $p_t=p_0\circ {\varphi }_{-t}$ with conserved fine-grained entropy $-k_B{\int }_R{p}_t\log p_{t}dm$, using $0\log 0=0$. Volume preservation alone does not imply ergodicity or a time-average/ensemble identity.

Let $K\subset {\mathbb{R}}^d$ be nonempty compact, $d\ge 1$, and $a:K\to \mathbb{R}$ continuous. Define $a_{\mathrm{m}\mathrm{i}\mathrm{x}}$ as in Compact phase-generator mixture. Then for every $x\in \mathrm{conv}K$ the infimum defining $a_{\mathrm{m}\mathrm{i}\mathrm{x}}(x)$ is attained by a Borel probability measure supported on at most $d+1$ points. The function $a_{\mathrm{m}\mathrm{i}\mathrm{x}}$ is lower semicontinuous and convex on $\mathrm{conv}K$, and is the greatest convex function on $\mathrm{conv}K$ bounded above by $a$ on $K$. Minimizing over all Borel probabilities on $K$ gives the same attained value as minimizing over finite mixtures. This is a mathematical compactness result; it makes no phase-preparation or equilibrium postulate.

Let $A=\{1,\dots ,q\}$, $q\ge 1$, have real energies $e_j$, and fix finite $\beta \in \mathbb{R}$. Set $Z_1={\sum }_j{e}^{-\beta e_j}$, $p_j=e^{-\beta e_j}\mathrm{/}{Z}_1>0$, and $E={\sum }_j{p}_j{e}_j$. For each $N\ge 1$, let ${\delta }_N\ge q{\sum }_{j=1}^q\mathrm{\mid }{e}_j\mathrm{\mid }$ and ${\delta }_N\mathrm{/}N\to 0$. Define the shell ${\mathrm{\Gamma }}_N=\{i\in A^N:\mathrm{\mid }{\sum }_{r=1}^N{e}_{i_r}-NE\mathrm{\mid }\le {\delta }_N\}$ and the uniform law on that shell. For probability laws $\mu ,\nu$ on the finite set $A^k$, total variation distance means $\frac{1}{2}{\sum }_{x\in A^k}\mathrm{\mid }\mu (x)-\nu (x)\mathrm{\mid }$. Then ${\mathrm{\Gamma }}_N$ is nonempty for every $N$; for each fixed $k\ge 1$, the first $k$-site marginal (for $N\ge k$) converges in that distance to $p^{\otimes k}$; and $\frac{1}{N}\log \mathrm{\mid }{\mathrm{\Gamma }}_N\mathrm{\mid }⟶H(p):=-{\sum }_{j=1}^q{p}_j\log p_j\mathrm{.}$ The conclusion is for this finite alphabet, additive energy, exact shell width and uniform shell law. It does not assert finite-size equality or equivalence for interacting or continuous models.