From ae43eb669085fda81caded4f7bf29dd106029c46 Mon Sep 17 00:00:00 2001 From: Ben Mares Date: Wed, 15 Jul 2026 11:21:41 +0200 Subject: [PATCH 01/11] Add thermodynamic-foundations companion to the discontinuity analysis A self-contained, first-principles derivation of the moist-parcel entropy that tcpyPI holds fixed on ascent (utilities.entropy_S), plus why it is conserved and how the temperature inversion recovers T from it. Builds with `latexmk -pdf` in the same setup as discontinuity_analysis.tex. Chain, tied at every step to the source (constants.py, utilities.py, pi.py): - Statistical origin: the operator picture (Z = Tr e^{-bH}, the spectrum as the {E_i}, ensembles as Legendre transforms), the Boltzmann distribution derived from energy conservation with its assumptions stated, then Sackur-Tetrode and its polyatomic generalisation to s = c_p ln T - R ln p. - Classical scaffolding: entropy as a state function via the 1/T integrating factor. - The parcel as a Dalton mixture; Gibbs additivity; phase equilibrium giving Clausius-Clapeyron and the L_v/T vaporisation entropy. - Assembly into Emanuel (1994) eq. 4.5.9 = entropy_S, term for term. - Conservation on the reversible adiabat (adiabatic => no flux, reversible => no production), with the Liouville / adiabatic-theorem reading. - The moist adiabat and the well-posed inversion s(T,p)=s0 -> T, whose monotonicity and Newton step match solve_temperature_from_entropy. 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+\theoremstyle{definition} +\newtheorem{definition}{Definition} +\newtheorem{remark}{Remark} + +% units +\newcommand{\hPa}{\,\mathrm{hPa}} +\newcommand{\Jkg}{\,\mathrm{J\,kg^{-1}}} +\newcommand{\JkgK}{\,\mathrm{J\,kg^{-1}\,K^{-1}}} +\newcommand{\K}{\,\mathrm{K}} +\newcommand{\ms}{\,\mathrm{m\,s^{-1}}} +% thermodynamic symbols +\newcommand{\Rd}{R_d} +\newcommand{\Rv}{R_v} +\newcommand{\cpd}{c_{pd}} +\newcommand{\cpv}{c_{pv}} +\newcommand{\cl}{c_\ell} +\newcommand{\Lv}{L_v} +\newcommand{\rt}{r_t} +\newcommand{\rvv}{r_v} +\newcommand{\rl}{r_\ell} +\newcommand{\es}{e_s} +\newcommand{\pd}{p_d} +\newcommand{\eps}{\varepsilon} +\newcommand{\dd}{\mathrm{d}} +\newcommand{\Trho}{T_\rho} + +\title{Thermodynamic foundations of the moist-parcel entropy in \texttt{tcpyPI}\\ +\large A first-principles derivation of \texttt{entropy\_S}, its conservation,\\ +and the temperature inversion on the reversible adiabat} +\author{Companion to \emph{An artificial discontinuity in the tcpyPI pressure solver}} +\date{July 14, 2026} + +\begin{document} +\maketitle + +\begin{abstract} +\noindent +The potential-intensity solver in \texttt{tcpyPI} lifts an air parcel along a +\emph{reversible moist adiabat} and, at each pressure level, recovers the +parcel temperature by holding one quantity fixed: the total specific entropy +computed by \texttt{utilities.entropy\_S}. This note derives that quantity, and +everything around it, from first principles. We start from the statistical +definition of entropy (Boltzmann/Gibbs), pass through the Sackur--Tetrode +equation to the ideal-gas form $s=c_p\ln T-R\ln p+\mathrm{const}$, treat the +parcel as a multicomponent Dalton mixture with a condensed phase, derive the +Clausius--Clapeyron relation and the latent-heat entropy from equality of +chemical potentials, and assemble Emanuel's total specific entropy +(E94, eq.~4.5.9)---the exact expression in the code. We then prove it is +conserved on the ascent (adiabatic $\Rightarrow$ no entropy flux; reversible +$\Rightarrow$ no entropy production), give the microscopic Liouville/adiabatic +-theorem reading of that conservation, and finish with the moist adiabat and +the well-posed \emph{inversion} $s(T,p)=s_0\mapsto T$ that the solver performs +by Newton iteration. Every formula is tied back to the specific function and +constant it corresponds to in the source. Numerical constants are quoted as in +\texttt{tcpyPI/constants.py}; SI throughout, pressures excepted (hPa). +\end{abstract} + +\tableofcontents + +\section{Scope and the target formula}\label{sec:scope} + +The object we are after is the total specific entropy per unit mass of +\emph{dry} air, as coded in \texttt{utilities.entropy\_S(T,R,P)}: +\begin{equation}\label{eq:target} +s \;=\; (\cpd+\rt\,\cl)\,\ln T \;-\; \Rd\,\ln(p-e) \;+\; \frac{\Lv\,\rvv}{T} + \;-\; \rvv\,\Rv\,\ln H, +\end{equation} +with $T$ the temperature (K), $p$ the total pressure, $e$ the partial +pressure of water vapour, $p-e=\pd$ the partial pressure of dry air, +$\rt$ the total-water mixing ratio, $\rvv$ the vapour mixing ratio, $\Lv$ the +latent heat of vaporisation, $H=e/\es$ the relative humidity (capped at $1$), +and $\cpd,\cl,\Rd,\Rv$ material constants (Table~\ref{tab:const}). The code +evaluates \eqref{eq:target} at the parcel's launch state, where the air is +unsaturated so $\rvv=\rt=R$; that is why a single argument \texttt{R} feeds all +water terms. When the parcel later saturates, the same $s$ is held fixed and +$T$ is solved for---the \emph{inversion} of \S\ref{sec:inversion}. + +Our task is to show where \eqref{eq:target} comes from, why it is the right +thing to conserve, and why the inversion is well posed. We build it in five +movements: statistical mechanics (\S\ref{sec:statmech}), classical +thermodynamic scaffolding (\S\ref{sec:classical}), the multicomponent parcel +(\S\ref{sec:parcel}), phase equilibrium and latent heat (\S\ref{sec:phase}), +and the assembly (\S\ref{sec:assembly}). Conservation +(\S\ref{sec:conservation}), the moist adiabat (\S\ref{sec:adiabat}) and the +inversion (\S\ref{sec:inversion}) follow. + +\begin{table}[htb] +\centering +\small +\begin{tabular}{llr} +\toprule +Symbol & Meaning & Value (\texttt{constants.py})\\ +\midrule +$\cpd$ & specific heat of dry air at const.\ $p$ & $1005.7\JkgK$\\ +$\cpv$ & specific heat of water vapour at const.\ $p$ & $1870.0\JkgK$\\ +$\cl$ & (modified) specific heat of liquid water & $2500.0\JkgK$\\ +$\Rd$ & gas constant of dry air & $287.04\JkgK$\\ +$\Rv$ & gas constant of water vapour & $461.5\JkgK$\\ +$\eps=\Rd/\Rv$ & mass ratio of the two gases & $0.6220$\\ +$\Lv^{0}$ & latent heat at $0^{\circ}$C & $2.501\times10^{6}\Jkg$\\ +$\cpv-\cl$ & Kirchhoff slope $\dd\Lv/\dd T$ & $-630\JkgK$\\ +\bottomrule +\end{tabular} +\caption{Material constants, quoted from \texttt{tcpyPI/constants.py}. The +``modified'' $\cl=2500$ (rather than $4190$) is a tuning of the reference, +discussed in Remark~\ref{rem:modcl}.} +\label{tab:const} +\end{table} + +\section{The statistical origin of entropy}\label{sec:statmech} + +\subsection{The operator picture underneath}\label{ssec:operator} + +Before the classical development, it is worth stating the exact +quantum-statistical object that development approximates, because it removes +several apparent ambiguities in the symbol ``$\sum_i$'' that a careful reader +will otherwise trip on. Rigorously, energy is not a number but the Hamiltonian +\emph{operator} $\hat H$ acting on the $N$-particle Hilbert space +$\mathcal H_N$, and the partition function is a trace, +\begin{equation}\label{eq:trace} +Z=\operatorname{Tr}e^{-\beta\hat H}, +\end{equation} +whose value is independent of the basis in which it is evaluated. The microstate +energies $\{E_i\}$ are the eigenvalues of $\hat H$; the index $i$ labels an +orthonormal energy eigenbasis; and the classical sum $\sum_i e^{-\beta E_i}$ +used throughout this section is exactly \eqref{eq:trace} written in that +eigenbasis, where $\hat H$ is diagonal with entries $E_i$. Three things this +clarifies at once: +\begin{itemize} +\item \textbf{The index $i$ is discrete, and $\mathcal H_N,\hat H$ depend on +$(V,N)$.} Confinement to a finite volume discretises the spectrum---a particle +in a box of side $V^{1/3}$ has $\varepsilon\propto(n_x^2+n_y^2+n_z^2)\,V^{-2/3}$ +with $n\in\mathbb Z_+^3$---so $i$ is a genuine countable label, not a continuum. +The parameters $V$ and $N$ do \emph{not} index microstates; they select which +Hilbert space and Hamiltonian one is on. Fixing $(V,N)$, as the canonical +ensemble does, means tracing over the eigenbasis of one definite operator +$\hat H_{V,N}$. +\item \textbf{The classical sum is the $\hbar\to0$ limit.} The trace +\eqref{eq:trace} reduces to the phase-space integral +$(N!\,h^{3N})^{-1}\!\int\dd\Gamma\,e^{-\beta H_{\mathrm{cl}}}$ of +\S\ref{ssec:lambda}, with relative corrections $O(\hbar^2)$. The factors +$h^{3N}$ (one quantum state per phase-space cell of volume $h^{3N}$) and $N!$ +(indistinguishability, from the (anti)symmetrisation +$\mathcal H_N\subset\mathcal H_1^{\otimes N}$) are \emph{not} classical +inputs---they are residues of the quantum count, which is why they enter +\eqref{eq:ztrans2}--\eqref{eq:ZN} seemingly by fiat. +\item \textbf{Everything is a spectral functional of $\hat H$.} $Z(\beta)$ is +the Laplace transform of the density of states +$\rho(E)=\operatorname{Tr}\delta(E-\hat H)$, and $F,\langle E\rangle,S$ all +follow from it: thermodynamics is the spectral geometry of the Hamiltonian. For +$N$ free identical particles the canonical $Z_N$ is, exactly, a symmetric +function of the single-particle Boltzmann weights $x_k=e^{-\beta\varepsilon_k}$ +---the elementary $e_N(x)$ for fermions, the complete homogeneous $h_N(x)$ for +bosons, and the power-sum leading term $p_1(x)^N/N!$ for the Maxwell--Boltzmann +count \eqref{eq:ZN}---so the indistinguishability that forces the $N!$ is the +$S_N$-symmetry of these functions. +\end{itemize} +Finally, the various \emph{ensembles} are Legendre transforms of one another, +each holding fixed a different member of the conjugate pairs $(E,T)$, $(V,p)$, +$(N,\mu)$; ``summing over'' a mechanical variable trades it for its conjugate: +\begin{center} +\small +\begin{tabular}{lll} +\toprule +sum runs over & held fixed & potential\\ +\midrule +microstates ($E$ free) & $E,V,N$ & $S(E,V,N)$\quad(microcanonical)\\ +microstates & $T,V,N$ & $F(T,V,N)$\quad(canonical)\\ +microstates $+\;N$ & $T,V,\mu$ & $\Omega(T,V,\mu)$\quad(grand)\\ +microstates $+\;V$ & $T,p,N$ & $G(T,p,N)$\quad(isobaric)\\ +\bottomrule +\end{tabular} +\end{center} +The canonical development below fixes $(T,V,N)$ and works with $F$; the parcel +of the companion analysis ultimately wants $(T,p)$---the Gibbs potential +$G$---which is why the specific entropy is finally expressed as $s(T,p)$ and +inverted for $T$ at prescribed $p$. What follows carries out \eqref{eq:trace} in +its classical $\hbar\to0$ form; the operator identity is the exact statement +being approximated. + +\subsection{Boltzmann and Gibbs, and the system we will use}\label{ssec:BG} + +Entropy is, microscopically, a count of microstates. Fix the system once, for +the whole section: a gas of $N$ particles, each of mass $m$, confined to a +container of volume $V$ and held in equilibrium at temperature $T$. Its +\emph{macrostate} is specified by the three numbers $(T,V,N)$; a +\emph{microstate} $i$ is a complete specification of every particle's position +and momentum (classically) or of the joint quantum state (quantally), and has a +definite total energy $E_i$. This distinction---$E_i$ is the energy of the +\emph{whole system} in microstate $i$, whereas $\varepsilon$ (introduced in +\S\ref{ssec:lambda}) will denote the energy of a \emph{single particle}---is +used consistently below. + +For an \emph{isolated} system all $\Omega$ microstates consistent with the +macrostate are equally probable (the fundamental postulate of statistical +mechanics), and Boltzmann's principle is +\begin{equation}\label{eq:boltzmann} +S \;=\; k_B \ln \Omega ,\qquad k_B=1.381\times10^{-23}\,\mathrm{J\,K^{-1}} . +\end{equation} +A parcel is \emph{not} isolated---it exchanges energy with its surroundings---so +its microstates are not equiprobable; they carry probabilities $p_i$, and the +entropy is the more general Gibbs (information) form +\begin{equation}\label{eq:gibbs} +S \;=\; -k_B \sum_i p_i \ln p_i , +\end{equation} +which reduces to \eqref{eq:boltzmann} when the $p_i$ are uniform, $p_i=1/\Omega$. +To use \eqref{eq:gibbs} we must first find the $p_i$. + +\subsection{The Boltzmann distribution from energy conservation}\label{ssec:boltzdist} + +The $p_i$ are fixed by one physical input---\emph{conservation of energy} +between the system and its surroundings---plus the postulate above. Put the +system in weak thermal contact with a large reservoir (subscript $\mathrm{res}$ +throughout this subsection; not to be confused with the specific gas constant +$R$ that appears bare from \S\ref{ssec:intensive} onward), and treat the +composite (system $+$ reservoir) as isolated with a fixed total energy +\begin{equation}\label{eq:econs} +E_{\mathrm{tot}}=E_i+E_{\mathrm{res}}=\mathrm{const}. +\end{equation} +``Weak contact'' means the interaction energy is negligible, so the energies are +additive \eqref{eq:econs}; ``isolated composite'' lets us apply the equal-\emph{a +priori}-probability postulate to it. When the system sits in one definite +microstate $i$ (energy $E_i$), the reservoir may be in any of its +$\Omega_{\mathrm{res}}(E_{\mathrm{tot}}-E_i)$ microstates, all equally likely. Hence the +probability of system-microstate $i$ is proportional to that reservoir count, +\begin{equation}\label{eq:pprop} +p_i \;\propto\; \Omega_{\mathrm{res}}\!\big(E_{\mathrm{tot}}-E_i\big). +\end{equation} +Because the reservoir is large, $E_i\ll E_{\mathrm{tot}}$, so expand its +\emph{logarithm} (the extensive quantity) to first order: +\begin{equation}\label{eq:expand} +\ln\Omega_{\mathrm{res}}(E_{\mathrm{tot}}-E_i) +\;\simeq\;\ln\Omega_{\mathrm{res}}(E_{\mathrm{tot}}) +-E_i\underbrace{\frac{\partial\ln\Omega_{\mathrm{res}}}{\partial E}\Big|_{E_{\mathrm{tot}}}}_{\displaystyle\equiv\,\beta}. +\end{equation} +The coefficient $\beta$ is a property of the reservoir alone; by +\eqref{eq:boltzmann} applied to the reservoir, $S_{\mathrm{res}}=k_B\ln\Omega_{\mathrm{res}}$, and the +thermodynamic definition of temperature $1/T=\partial S_R/\partial E$, it is +\begin{equation}\label{eq:beta} +\beta=\frac{1}{k_B}\frac{\partial S_{\mathrm{res}}}{\partial E}=\frac{1}{k_BT}. +\end{equation} +Exponentiating \eqref{eq:expand} and absorbing the $E_i$-independent factor into +the normalisation gives the \emph{Boltzmann distribution} +\begin{equation}\label{eq:boltzdist} +p_i=\frac{e^{-\beta E_i}}{Z},\qquad +Z=\sum_i e^{-\beta E_i}, +\end{equation} +with the \emph{partition function} $Z$ enforcing $\sum_i p_i=1$. The +assumptions are now explicit: (i) equal \emph{a priori} probabilities for the +isolated composite; (ii) conservation of energy \eqref{eq:econs}; (iii) a +reservoir large enough that the linear term in \eqref{eq:expand} suffices; (iv) +weak coupling, so energies add. (Equivalently, by Jaynes' maximum-entropy +argument, \eqref{eq:boltzdist} is the distribution that maximises +\eqref{eq:gibbs} subject only to $\sum_i p_i=1$ and a fixed mean energy +$\langle E\rangle=\sum_i p_iE_i$, with $\beta$ the Lagrange multiplier for +energy---the two derivations agree.) + +\subsection{From \texorpdfstring{$Z$}{Z} to entropy, and indistinguishable particles} +\label{ssec:Z} + +Everything thermodynamic now follows from $Z$. Insert \eqref{eq:boltzdist} into +the Gibbs entropy \eqref{eq:gibbs}, using $\ln p_i=-\beta E_i-\ln Z$: +\begin{equation}\label{eq:Sderiv} +S=-k_B\sum_i p_i\big(-\beta E_i-\ln Z\big) + =k_B\beta\sum_i p_iE_i+k_B\ln Z + =\frac{\langle E\rangle}{T}+k_B\ln Z , +\end{equation} +where $\langle E\rangle=\sum_i p_iE_i=-\partial\ln Z/\partial\beta$ is the mean +total energy (the thermodynamic internal energy). Defining the Helmholtz free +energy $F\equiv-k_BT\ln Z$, equation \eqref{eq:Sderiv} is +\begin{equation}\label{eq:SfromF} +S=\frac{\langle E\rangle-F}{T} + =-\Big(\frac{\partial F}{\partial T}\Big)_{V,N}, +\end{equation} +the last equality being the standard thermodynamic identity (both $V$ and $N$, +the other natural variables of $F$, are held fixed). So the whole task reduces +to computing $Z$ for the parcel's constituents and differentiating. + +\begin{remark}[what depends on what]\label{rem:args} +It is worth being explicit about arguments, as the distinction drives the +conservation argument later. The microstate energies $E_i$ are \emph{mechanical}: +each is set by the Hamiltonian at a fixed volume and particle number, so +$E_i=E_i(V,N)$---a function of the microstate label $i$ and of $(V,N)$, but +\emph{not} of temperature. Consequently the partition function and free energy +carry the mechanical parameters as well, $Z=Z(\beta,V,N)$ and $F=F(T,V,N)$ +---which is why $(V,N)$ sit in the derivative subscript of \eqref{eq:SfromF} +---while the \emph{mean} energy $\langle E\rangle=\sum_i p_iE_i=U(T,V,N)$ is a +thermodynamic function of state, belonging to the whole ensemble and to no +single microstate. + +A caution on the occupations, since it is easy to over-simplify: $p_i=e^{-\beta +E_i}/Z$ is \emph{not} a function of temperature alone. It depends on $(T,V,N)$ +jointly---on $T$ through $\beta$ and on $(V,N)$ through $E_i$---so what fixing +the macrostate determines is the \emph{whole} distribution $\{p_i\}$, with no +residual freedom. This is exactly the structure the reversible-adiabatic +argument of \S\ref{sec:conservation} turns on. Under a slow (quasi-static) +change of $V$ the levels $E_i(V)$ shift, yet along an adiabat the product +$\beta E_i$ is held fixed for \emph{every} $i$ simultaneously---for a monatomic +ideal gas $E_i\propto V^{-2/3}$ and the adiabat is $TV^{2/3}=\mathrm{const}$, +so $\beta E_i\propto T^{-1}V^{-2/3}=\mathrm{const}$---and therefore each +occupation $p_i$, and with it $S=-k_B\sum_i p_i\ln p_i$, is unchanged even as +both $T$ and the $E_i$ move. ``The levels shift; the occupations are frozen'' is +this statement, and it is what makes the ascent isentropic. +\end{remark} + +One structural fact makes $Z$ tractable. For $N$ \emph{non-interacting} +particles the total energy is a sum of single-particle energies, +$E_i=\sum_{a=1}^{N}\varepsilon_{a}$, so the system sum factorises into a product +of identical single-particle sums, $z^{N}$. But the particles are +\emph{indistinguishable}: permuting them yields the same physical microstate, so +$z^{N}$ overcounts by the $N!$ permutations. The corrected result---which +resolves the Gibbs mixing paradox and is the classical limit of quantum +statistics---is +\begin{equation}\label{eq:ZN} +Z=\frac{z^{N}}{N!},\qquad +z=\sum_{\text{single-particle states}}e^{-\beta\varepsilon}, +\end{equation} +with $z$ the single-particle partition function and $\varepsilon$ a +single-particle energy (contrast $E_i$, the whole-system energy of +\S\ref{ssec:BG}). Everything now hinges on $z$. + +\subsection{The translational partition function and the thermal de~Broglie wavelength} +\label{ssec:lambda} + +For a structureless particle in a box of volume $V$ the energy is purely +kinetic, $\varepsilon=\mathbf p^{2}/2m$. In the classical (dilute, +non-degenerate) regime the sum over single-particle states becomes a +phase-space integral with one state per volume $h^{3}$ of phase space: +\begin{equation}\label{eq:ztrans1} +z_{\mathrm{tr}}=\frac{1}{h^{3}}\int_V\!\dd^{3}r\int_{\mathbb R^3}\!\dd^{3}p\; +e^{-\beta \mathbf p^{2}/2m} +=\frac{V}{h^{3}}\prod_{i=x,y,z}\int_{-\infty}^{\infty}\!\dd p_i\, +e^{-\beta p_i^{2}/2m}. +\end{equation} +Each of the three Gaussian integrals evaluates to +$\int_{-\infty}^{\infty}e^{-\beta p^{2}/2m}\dd p=\sqrt{2\pi m/\beta} +=\sqrt{2\pi m k_BT}$, so +\begin{equation}\label{eq:ztrans2} +z_{\mathrm{tr}}=\frac{V\,(2\pi m k_BT)^{3/2}}{h^{3}}\equiv\frac{V}{\Lambda^{3}}, +\qquad +\boxed{\;\Lambda\equiv\frac{h}{\sqrt{2\pi m k_BT}}\;} +\end{equation} +The length $\Lambda$ is the \emph{thermal de~Broglie wavelength}: the quantum +coherence scale of a particle with thermal momentum +$\sim\!\sqrt{m k_BT}$. It carries the entire temperature dependence of +$z_{\mathrm{tr}}$, as $\Lambda^{3}\propto T^{-3/2}$. The classical treatment +\eqref{eq:ZN}--\eqref{eq:ztrans1} is valid precisely when $n\Lambda^{3}\ll1$ +(mean spacing $\gg$ coherence length); for air at surface conditions +$n\Lambda^{3}\sim10^{-6}$, so the approximation is excellent. + +\subsection{Free energy and the Sackur--Tetrode entropy}\label{ssec:sackur} + +Take a monatomic gas first, $z=z_{\mathrm{tr}}=V/\Lambda^{3}$ (no internal +structure). From \eqref{eq:ZN}, $\ln Z_N=N\ln z-\ln N!$, and with Stirling's +approximation $\ln N!\simeq N\ln N-N$, +\begin{equation}\label{eq:Fmon} +F=-k_BT\ln Z_N +=-N k_BT\Big[\ln\frac{z}{N}+1\Big] +=-N k_BT\Big[\ln\frac{V}{N\Lambda^{3}}+1\Big]. +\end{equation} +The entropy follows from \eqref{eq:SfromF}. Writing +$\ln(V/N\Lambda^{3})=\ln(V/N)+\tfrac32\ln T+C_0$ with +$C_0=\tfrac32\ln(2\pi m k_B/h^{2})$ temperature-independent, only the +$\tfrac32\ln T$ piece and the explicit prefactor $T$ depend on temperature: +\begin{align} +S=-\frac{\partial F}{\partial T} +&=N k_B\Big[\ln\frac{V}{N\Lambda^{3}}+1\Big] + +N k_BT\cdot\frac{3}{2T} \notag\\ +&=N k_B\Big[\ln\frac{V}{N\Lambda^{3}}+\underbrace{1}_{\text{from }F} + +\underbrace{\tfrac32}_{\partial_T(\tfrac32 T\ln T)}\Big] + =N k_B\Big[\ln\frac{V}{N\Lambda^{3}}+\frac{5}{2}\Big]. +\label{eq:sackur} +\end{align} +This is the \emph{Sackur--Tetrode} equation. Its additive $\tfrac52$ has a +definite provenance: $1$ from the indistinguishability remainder (the $+1$ in +$F$, i.e.\ the $-\ln N!$ Stirling term) and $\tfrac32$ from differentiating the +thermal energy $\tfrac32 N k_BT$. Physically the two variable terms are the +split we will track to the end: $\ln(V/N)$ is \emph{configurational} (positional +microstates---more room per particle, more states), and +$-3\ln\Lambda=\tfrac32\ln T+\mathrm{const}$ is \emph{thermal} (the momentum +distribution broadens with $T$). + +\subsection{Intensive form, and the fate of the two \texorpdfstring{$5/2$}{5/2}'s} +\label{ssec:intensive} + +Reduce \eqref{eq:sackur} to a specific entropy. Per particle, in units of +$k_B$, +\begin{equation}\label{eq:sperparticle} +\tilde s\equiv\frac{S}{N k_B}=\ln\frac{V}{N\Lambda^{3}}+\frac52 . +\end{equation} +Use the ideal-gas law in the form $V/N=k_BT/p$ and +$\Lambda^{-3}=(2\pi m k_B/h^{2})^{3/2}T^{3/2}$: +\begin{align} +\ln\frac{V}{N\Lambda^{3}} +&=\ln\frac{k_BT}{p}+\frac32\ln T+C_0 + =\underbrace{\ln T}_{\text{from }V/N}\;-\;\ln p\;+\; + \underbrace{\tfrac32\ln T}_{\text{from }\Lambda^{-3}}\;+\;C_1 \notag\\ +&=\frac52\ln T-\ln p+C_1 ,\qquad C_1=\ln k_B+C_0 . +\label{eq:lnVNL} +\end{align} +Substituting into \eqref{eq:sperparticle}, +\begin{equation}\label{eq:monatomic} +\tilde s=\frac52\ln T-\ln p+\Big(C_1+\frac52\Big) + =\underbrace{\frac52}_{=\,c_p/R}\ln T-\ln p+\mathrm{const}. +\end{equation} +There are \emph{two} factors of $5/2$ here and they behave oppositely---this is +the crux: +\begin{itemize} +\item The \textbf{additive} $\tfrac52$ inherited from Sackur--Tetrode + \eqref{eq:sperparticle} is a pure constant. It is swallowed into the + reference constant in \eqref{eq:monatomic} and cancels in every entropy + \emph{difference}. So yes---it drops out of anything physical; it survives + only in the absolute, third-law-anchored entropy. +\item The \textbf{coefficient} $\tfrac52$ multiplying $\ln T$ is not a constant + and does \emph{not} drop out. Equation~\eqref{eq:lnVNL} shows it is + assembled as $1+\tfrac32$: one power of $\ln T$ from the configurational + $\ln(V/N)=\ln(k_BT/p)$, and $\tfrac32$ from the thermal + $\Lambda^{-3}\!\propto T^{3/2}$. For a monatomic gas this equals $c_p/R$. +\end{itemize} +Multiplying \eqref{eq:monatomic} by the specific gas constant $R=k_B/m$ (mass +$m$ per particle) turns $\tilde s$ into entropy per unit mass, and +$\tfrac52 R=c_p$ for a monatomic gas, giving the building block +\begin{equation}\label{eq:idealS} +\boxed{\;s \;=\; c_p\ln T - R\ln p + \mathrm{const}.\;} +\end{equation} + +\subsection{Beyond monatomic: internal degrees of freedom}\label{ssec:polyatomic} + +Sackur--Tetrode as derived assumes a structureless particle, so +\eqref{eq:sackur} does \emph{not} apply verbatim to air ($\mathrm N_2,\mathrm +O_2$) or water vapour ($\mathrm H_2\mathrm O$), which rotate and vibrate. Two +things change, both controlled. First, the single-particle energy is a sum +$\varepsilon=\varepsilon_{\mathrm{tr}}+\varepsilon_{\mathrm{int}}$ +(translation plus rotation, vibration, electronic), so the partition function +\emph{factorises}, +\begin{equation}\label{eq:zfactor} +z=z_{\mathrm{tr}}(T,V)\,z_{\mathrm{int}}(T), +\end{equation} +with the internal factor a function of $T$ alone (the internal level spacings do +not depend on the box). Second, carrying $z_{\mathrm{int}}$ through +\eqref{eq:Fmon}--\eqref{eq:sackur} adds an internal energy +$u_{\mathrm{int}}=k_BT^{2}\,\dd\ln z_{\mathrm{int}}/\dd T$ per particle and a +matching internal entropy; the extra heat capacity is +$c_{v,\mathrm{int}}=\dd u_{\mathrm{int}}/\dd T$. + +The \textbf{equipartition theorem} makes this concrete: every quadratic degree +of freedom that is thermally active contributes $\tfrac12 k_B$ to the +per-particle heat capacity. Translation always gives $3\times\tfrac12$; +rotation adds $2\times\tfrac12$ for a linear molecule ($\mathrm N_2,\mathrm +O_2,\mathrm{CO}_2$) or $3\times\tfrac12$ for a nonlinear one ($\mathrm H_2\mathrm +O$); each active vibrational mode counts twice (kinetic $+$ potential). Hence +$c_v=\tfrac{f}{2}k_B$ over $f$ active quadratic freedoms, $c_p=c_v+k_B$, and the +monatomic ratio $c_p/R=\tfrac52$ is replaced by $c_p/R=\tfrac{f}{2}+1$. + +Crucially, the \emph{functional form} of the entropy is unchanged. Whenever +$c_p$ is constant---all active modes classical, none freezing in or out over the +range of interest---integrating the thermodynamic identity +$\dd s=c_p\,\dd T/T-R\,\dd p/p$ (derived independently from the first and second +laws in \S\ref{sec:classical}) gives +\begin{equation}\label{eq:idealSgen} +s=c_p\ln T-R\ln p+\mathrm{const} +\end{equation} +for \emph{any} ideal gas, with the material $c_p$. Sackur--Tetrode is just the +monatomic special case that additionally fixes the absolute constant. For the +atmosphere ($\approx\!200$--$320\K$) the diatomic rotational modes are fully +excited while the stiff vibrational modes are largely frozen, so $\cpd$ and +$\cpv$ (Table~\ref{tab:const}) are very nearly temperature-independent and +\eqref{eq:idealSgen} holds to high accuracy; the small residual $c_p(T)$ +variation is absorbed into the effective constants and, for water, into the +$\cl$ tuning of Remark~\ref{rem:modcl}. + +\begin{remark}[what the additive constant hides, and why it does not matter] +The reference constant in \eqref{eq:idealS}/\eqref{eq:idealSgen} absorbs +$\Lambda$, the Sackur--Tetrode $\tfrac52$, the internal-mode ground states, and +each species' zero of energy. Because the ascent conserves a \emph{difference} +of entropies at fixed composition (\S\ref{sec:conservation}) and buoyancy +depends on temperature \emph{differences} (\S\ref{sec:adiabat}), these constants +never enter a physical prediction; the code is free to drop them, and does. +\end{remark} + +\subsection{The building block, and how it feeds everything below} +\label{ssec:buildingblock} + +Equation~\eqref{eq:idealSgen} is the payoff of this section and the answer to +``how does the ideal-gas entropy relate to everything that follows''. It is not +a monatomic curiosity: it is the single functional shape that each gaseous +constituent of the parcel instantiates \emph{once}. Dry air and water vapour are +both ideal gases, so each contributes +\begin{equation}\label{eq:perspecies} +s_k=c_{p,k}\ln T-R_k\ln(\text{its own partial pressure})+\mathrm{const}, +\end{equation} +differing only in which $c_p$, which $R$, and---by Dalton's law +(\S\ref{sec:parcel})---which partial pressure it sees. The total specific +entropy of the parcel is then the mixing-ratio-weighted sum of these identical +blocks plus a condensed-phase term (Gibbs additivity, \S\ref{sec:parcel}); once +phase equilibrium is used to reintroduce the latent heat (\S\ref{sec:phase}), +that sum is exactly Emanuel's \eqref{eq:E94}. In short: derive the block once +here, and ``everything below'' is bookkeeping---which partial pressure, which +$c_p$, which mixing ratio---layered on top of \eqref{eq:idealSgen}. + +\section{Classical scaffolding: why \texorpdfstring{$s$}{s} is a state function} +\label{sec:classical} + +Statistical mechanics gives $s$; classical thermodynamics tells us it is a +\emph{function of state}---indispensable, since the solver evaluates $s$ at a +point $(T,p,\rvv)$ with no reference to how the parcel got there. + +\subsection{First and second laws} + +The first law for a closed system is $\dd U=\delta Q-\delta W$, with +$\delta W=p\,\dd V$ for quasi-static volume work. Neither $\delta Q$ nor +$\delta W$ is an exact differential: $\oint \delta Q\neq 0$. The second law +supplies an \emph{integrating factor}. In Carath\'eodory's formulation, the +empirical fact that arbitrarily close to any equilibrium state there exist +states not reachable by any adiabatic path ($\delta Q=0$) is equivalent, for +the Pfaffian form $\delta Q$, to the existence of an integrating denominator; +that denominator is the absolute temperature $T$, and +\begin{equation}\label{eq:dS} +\dd s \;=\; \frac{\delta Q_{\mathrm{rev}}}{T} +\end{equation} +is an \emph{exact} differential. Its integral, $s$, is a state function. This +is the precise sense in which ``entropy exists''. + +\subsection{Recovering \texorpdfstring{$s=c_p\ln T-R\ln p$}{s=cp lnT - R lnp} thermodynamically} + +For an ideal gas, $pV=nRT$ and $U=U(T)$ with $\dd U=c_v\,\dd T$ (Joule's law). +Then $\delta Q_{\mathrm{rev}}=\dd U+p\,\dd V=c_v\,\dd T+p\,\dd V$, and with +$p/T=R/v$ (specific volume $v$), +\begin{equation} +\dd s=\frac{c_v\,\dd T}{T}+\frac{R\,\dd v}{v} + =\frac{c_p\,\dd T}{T}-\frac{R\,\dd p}{p}, +\end{equation} +using Mayer's relation $c_p-c_v=R$ and $\dd(\ln pv)=\dd\ln(RT)$ to switch +variables. Integrating returns \eqref{eq:idealS}, now with the material $c_p$. +Statistical mechanics and classical thermodynamics agree, as they must; the +exactness of $\dd s$ (equivalently the Maxwell relation +$\partial^2 s/\partial T\partial p=\partial^2 s/\partial p\partial T$) is what +lets \texttt{entropy\_S} be a pure function of its three arguments. + +\section{The parcel as a multicomponent mixture}\label{sec:parcel} + +\subsection{Composition and the per-dry-air convention} + +The parcel is dry air plus water in vapour and (once saturated) condensed form. +Masses are bookkept as \emph{mixing ratios} relative to the dry-air mass $m_d$: +\begin{equation} +\rvv=\frac{m_{\text{vapour}}}{m_d},\qquad +\rl=\frac{m_{\text{liquid}}}{m_d},\qquad +\rt=\rvv+\rl . +\end{equation} +Entropy is quoted \emph{per unit mass of dry air} because $m_d$ is the one mass +that is conserved along the ascent (water changes phase, but dry air is inert +and neither created nor removed); dividing by it gives a bookkeeping base that +does not move. This is why \eqref{eq:target} carries $\Rd$ bare but weights the +water terms by mixing ratios. + +\subsection{Dalton's law and the vapour--pressure relation} + +Each gas in an ideal mixture behaves as if it alone occupied the volume, so the +total pressure is the sum of partial pressures (Dalton): $p=\pd+e$, with $e$ +the vapour partial pressure and $\pd=p-e$ the dry-air partial pressure. The +mixing ratio and the partial pressure are two encodings of the same amount of +vapour; equating $\rvv=m_v/m_d=(\eps^{-1}e)/(p-e)$ with $\eps=\Rd/\Rv$ gives the +mutually inverse maps +\begin{equation}\label{eq:evrv} +e=\frac{\rvv\,p}{\eps+\rvv},\qquad +\rvv=\frac{\eps\,e}{p-e}, +\end{equation} +which are exactly \texttt{utilities.ev} and \texttt{utilities.rv}. The +appearance of $p-e$ (not $p$) inside the dry-air logarithm of \eqref{eq:target} +is Dalton's law at work: the dry air sees only its own partial pressure. + +\subsection{Additivity of entropy: Gibbs' theorem} + +For an ideal-gas mixture the entropy is the sum of the entropies each component +would have alone at its own partial pressure and the common temperature +(Gibbs' theorem---there is no entropy of mixing beyond what the partial +pressures already encode, because ideal gases do not interact). Hence we may +build $s$ additively: +\begin{equation}\label{eq:additive} +s \;=\; s_d \;+\; \rvv\, s_v \;+\; \rl\, \sigma_\ell, +\end{equation} +$s_d$ the specific entropy of dry air, $s_v$ that of vapour, and $\sigma_\ell$ +that of liquid water, each per unit mass of its own species. We now need the +three pieces. + +\section{Water substance and phase equilibrium}\label{sec:phase} + +The gaseous pieces use \eqref{eq:idealS}: +\begin{equation}\label{eq:sd-sv} +s_d = \cpd\ln T - \Rd\ln \pd + \mathrm{const}_d,\qquad +s_v = \cpv\ln T - \Rv\ln e + \mathrm{const}_v. +\end{equation} +Liquid water is nearly incompressible, so its entropy is essentially thermal, +\begin{equation}\label{eq:sl} +\sigma_\ell = \cl\ln T + \mathrm{const}_\ell . +\end{equation} +What ties the vapour and liquid constants together---and produces the latent +term in \eqref{eq:target}---is phase equilibrium. + +\subsection{Chemical potential and the Clausius--Clapeyron relation} + +At a flat interface, vapour and liquid coexist when their specific Gibbs free +energies (chemical potentials per unit mass) are equal, $g_v(T,\es)=g_\ell(T)$, +which \emph{defines} the saturation vapour pressure $\es(T)$. Differentiating +along the coexistence curve with $\dd g=-s\,\dd T+v\,\dd p$ gives +\begin{equation} +-s_v\,\dd T + v_v\,\dd\es \;=\; -\sigma_\ell\,\dd T + v_\ell\,\dd\es +\;\Longrightarrow\; +\frac{\dd\es}{\dd T}=\frac{s_v-\sigma_\ell}{v_v-v_\ell}. +\end{equation} +The reversible latent heat is the enthalpy of the isothermal, isobaric phase +change, $\Lv=T(s_v-\sigma_\ell)$, i.e.\ the \emph{entropy of vaporisation} is +\begin{equation}\label{eq:svap} +s_v-\sigma_\ell=\frac{\Lv}{T}. +\end{equation} +With $v_v\gg v_\ell$ and $v_v=\Rv T/\es$ (ideal vapour), this is the +Clausius--Clapeyron equation +\begin{equation}\label{eq:cc} +\frac{\dd\es}{\dd T}=\frac{\Lv\,\es}{\Rv\,T^{2}}. +\end{equation} + +\subsection{Kirchhoff's relation and the coded fits} + +The latent heat itself depends on temperature. Applying +$\dd\Lv/\dd T=\cpv-\cl$ (Kirchhoff's law: the two phases warm at different +specific heats) and integrating from $0^{\circ}$C, +\begin{equation}\label{eq:kirchhoff} +\Lv(T)=\Lv^{0}+(\cpv-\cl)\,(T-T_0), +\end{equation} +which is exactly \texttt{utilities.Lv} (Table~\ref{tab:const}: +$\Lv^0=2.501\times10^6$, slope $\cpv-\cl=-630$). Substituting +\eqref{eq:kirchhoff} into \eqref{eq:cc} and integrating yields a closed +expression for $\es(T)$; the code uses the algebraically cheaper +August--Roche--Magnus fit +\begin{equation}\label{eq:esfit} +\es(T)=6.112\,\exp\!\Big(\frac{17.67\,T_C}{243.5+T_C}\Big)\ \mathrm{hPa}, +\qquad T_C=T-273.15, +\end{equation} +i.e.\ \texttt{utilities.es\_cc}, which reproduces the Clausius--Clapeyron +integral to sub-percent accuracy over atmospheric temperatures. Equations +\eqref{eq:svap}--\eqref{eq:esfit} are all we need from phase physics. + +\begin{remark}[the ``modified'' $\cl=2500\JkgK$]\label{rem:modcl} +The true specific heat of liquid water is $\approx4190\JkgK$; \texttt{tcpyPI} +(following \texttt{pcmin}) uses $2500$. This is a deliberate effective value +that folds the ice phase and the temperature dependence of $\Lv$ into a single +linear Kirchhoff slope $\cpv-\cl=-630$, tuned so that \eqref{eq:kirchhoff} +tracks the observed $\Lv(T)$ across the mixed-phase range the parcel traverses. +It changes the reference thermodynamics slightly but not the structure of any +derivation here. +\end{remark} + +\section{Assembling the total specific entropy}\label{sec:assembly} + +Insert \eqref{eq:sd-sv}--\eqref{eq:sl} into the additive decomposition +\eqref{eq:additive} and use $\rl=\rt-\rvv$: +\begin{align} +s &= \big[\cpd\ln T-\Rd\ln\pd\big] + + \rvv\big[\cpv\ln T-\Rv\ln e\big] + + (\rt-\rvv)\,\cl\ln T + \mathrm{const} \notag\\ + &= \cpd\ln T-\Rd\ln\pd + + \rt\,\cl\ln T + + \rvv\big[(\cpv-\cl)\ln T-\Rv\ln e\big] + \mathrm{const}. +\label{eq:collected} +\end{align} +The bracket on the vapour term is where the latent heat re-enters. Evaluate +\eqref{eq:svap} using \eqref{eq:sd-sv}--\eqref{eq:sl} \emph{at saturation} +($e=\es$): +\begin{equation}\label{eq:latentconst} +\frac{\Lv}{T}=s_v-\sigma_\ell +=(\cpv-\cl)\ln T-\Rv\ln\es+(\mathrm{const}_v-\mathrm{const}_\ell). +\end{equation} +Solve \eqref{eq:latentconst} for $(\cpv-\cl)\ln T$ and substitute into the +bracket of \eqref{eq:collected}: +\begin{align} +(\cpv-\cl)\ln T-\Rv\ln e +&=\Big[\tfrac{\Lv}{T}+\Rv\ln\es-(\mathrm{const}_v-\mathrm{const}_\ell)\Big]-\Rv\ln e +\notag\\ +&=\frac{\Lv}{T}-\Rv\ln\frac{e}{\es}-(\mathrm{const}_v-\mathrm{const}_\ell) + =\frac{\Lv}{T}-\Rv\ln H+\mathrm{const}. +\end{align} +Putting this back into \eqref{eq:collected} and absorbing all reference +constants, +\begin{equation}\label{eq:E94} +\boxed{\;s=(\cpd+\rt\,\cl)\ln T-\Rd\ln(p-e)+\frac{\Lv\,\rvv}{T}-\rvv\,\Rv\ln H.\;} +\end{equation} +This is Emanuel (1994), eq.~4.5.9, and term-for-term it is \eqref{eq:target} +and \texttt{utilities.entropy\_S}. The four terms carry the four physical +meanings we tracked from the start: +\begin{center} +\begin{tabular}{ll} +\toprule +Term & Physical content \\ +\midrule +$(\cpd+\rt\cl)\ln T$ & thermal entropy of dry air $+$ all condensate carried\\ +$-\Rd\ln(p-e)$ & configurational entropy of the dry air (Dalton partial pressure)\\ +$+\Lv\rvv/T$ & latent entropy stored in the vapour fraction (Clausius--Clapeyron)\\ +$-\rvv\Rv\ln H$ & dilution entropy of sub-saturated vapour ($H<1\Rightarrow{>}0$)\\ +\bottomrule +\end{tabular} +\end{center} + +\begin{remark}[the single-argument \texttt{R}, and reduction below saturation] +The code's \eqref{eq:target} uses one mixing ratio \texttt{R} in every water +term, i.e.\ it identifies $\rvv=\rt=R$. That is exact wherever the parcel is +unsaturated, which is precisely where \texttt{entropy\_S} is called: at the +launch state, to set the conserved reference $s_0$. Below the lifting +condensation level all water is vapour, $\rl=0$, and \eqref{eq:E94} is the +familiar dry/unsaturated entropy with $H<1$. The generalisation to $\rvv<\rt$ +(condensate present) is carried implicitly by the saturated form used in the +inversion, \eqref{eq:SGcode} below, where $H=1$ retires the last term. +\end{remark} + +\section{Why \texorpdfstring{$s$}{s} is conserved on the ascent} +\label{sec:conservation} + +\subsection{The exact entropy budget} + +For the parcel as a closed system, the Clausius inequality splits any change of +entropy into a reversible flux and an irreversible production: +\begin{equation}\label{eq:budget} +\dd s \;=\; \underbrace{\frac{\delta Q}{T}}_{\text{flux}} + \;+\; \underbrace{\dd s_{\mathrm{irr}}}_{\ge 0\ \text{(production)}} . +\end{equation} +The parcel model annihilates both terms by two idealisations. + +\paragraph{Adiabatic: no flux, $\delta Q=0$.} Convective ascent is fast +compared with the time for heat to diffuse or radiate across the parcel +boundary. Formally, the ratio of the advective timescale $H_\rho/w$ (density +scale height over vertical velocity, minutes) to the thermal-diffusion and +radiative timescales (hours to days) is $\ll1$, so $\delta Q\approx0$ over the +ascent. The first term of \eqref{eq:budget} vanishes. + +\paragraph{Reversible: no production, $\dd s_{\mathrm{irr}}=0$.} The parcel is +assumed to remain in internal thermodynamic equilibrium---uniform $T$ and $p$, +no turbulent entrainment or mixing with the environment, and, crucially, phase +change occurring exactly at saturation. At saturation the chemical potentials +are equal, $g_v=g_\ell$ (\S\ref{sec:phase}), so transferring water across the +phase boundary produces \emph{no} entropy: condensation onto a parcel held at +$H=1$ is reversible. (Condensing sub- or super-saturated vapour would carry a +finite $\Delta g$ per unit mass and generate $\dd s_{\mathrm{irr}}>0$.) The +second term of \eqref{eq:budget} vanishes. + +Together, $\delta Q=0$ and $\dd s_{\mathrm{irr}}=0$ give +\begin{equation}\label{eq:isentropic} +\boxed{\;\dd s = 0 \quad\text{along the ascent}\;} +\end{equation} +---the ascent is \emph{isentropic}. This is the entire justification for +holding \texttt{entropy\_S} fixed. + +\subsection{Closed for water: reversible vs.\ pseudo-adiabatic} + +Conservation of \eqref{eq:E94} as a function of $(T,p)$ alone additionally +requires that the water inventory not change: the ``reversible'' adiabat keeps +all condensate suspended in the parcel, so $\rt=\mathrm{const}$ and the parcel +is a closed system for water. In the code this is the default +\texttt{ascent\_flag}$=0$; the density temperature (\S\ref{sec:adiabat}) then +carries the full water loading $\rt$. The alternative +\texttt{ascent\_flag}$=1$ (\emph{pseudo}-adiabatic) removes condensate the +instant it forms: the system is then \emph{open}, $\rt$ is not conserved, and +the invariant is not \eqref{eq:E94} but an approximate pseudo-entropy. The +distinction is physical, not numerical. + +\subsection{The microscopic reading} + +Statistical mechanics gives \eqref{eq:isentropic} a clean interpretation. A +reversible adiabatic change is a \emph{slow} deformation of the parcel's +Hamiltonian (volume and level spacings change as $p$ and $T$ change) with no +coupling to a heat bath. Two theorems then apply. By \textbf{Liouville's +theorem}, phase-space volume is invariant under Hamiltonian flow; by the +\textbf{adiabatic theorem}, a sufficiently slow deformation preserves the +occupation probability of each state (no transitions between energy shells). +The accessible phase-space volume $\Omega$ is therefore unchanged, and by +\eqref{eq:boltzmann} so is $S$; equivalently every $p_i$ in the Gibbs form +\eqref{eq:gibbs} is frozen, so $-k_B\sum p_i\ln p_i$ is constant even as the +individual energy levels move. Isentropic literally means \emph{the +microstate count does not change---the levels merely shift}. Note that the two +senses of ``adiabatic'' (thermodynamic: no heat; mechanical: slow) here +coincide, which is why the slow, heat-free ascent lands on a single adiabat. + +\section{The moist adiabat and buoyancy}\label{sec:adiabat} + +\subsection{From $\dd s=0$ to a path} + +Because $s$ is a state function, \eqref{eq:isentropic} is a differential +constraint that pins the path in the $(T,p)$ plane: +\begin{equation}\label{eq:pathODE} +0=\dd s=\frac{\partial s}{\partial T}\,\dd T+\frac{\partial s}{\partial p}\,\dd p +\;\Longrightarrow\; +\frac{\dd T}{\dd p}=-\frac{\partial s/\partial p}{\partial s/\partial T}. +\end{equation} +This ODE is the moist adiabat. Two regimes: + +\paragraph{Unsaturated (below the LCL): the dry adiabat.} With $\rvv=\rt$ fixed +and no condensation, the dominant balance of \eqref{eq:E94} is +$s\approx(\cpd+\rt\cl)\ln T-\Rd\ln\pd+\mathrm{const}$. Holding $s$ and the water +constant and treating the small vapour corrections as constant, $\dd s=0$ gives +$c_p\,\dd\ln T=\Rd\,\dd\ln p$ (with the appropriate moist $c_p$), i.e.\ Poisson's +relation +\begin{equation}\label{eq:poisson} +T\propto p^{\,\Rd/c_p},\qquad +\theta\equiv T\Big(\frac{p_0}{p}\Big)^{\Rd/c_p}=\mathrm{const}, +\end{equation} +the conserved (virtual) potential temperature. This is exactly the branch the +CAPE routine takes below the condensation level, where it writes +$T_G=T_P\,(p/p_P)^{\Rd/\cpd}$. + +\paragraph{The lifting condensation level.} As the unsaturated parcel cools +along \eqref{eq:poisson}, $\es(T)$ falls (Clausius--Clapeyron) while $e$ falls +only with pressure; $H=e/\es$ rises to $1$. The pressure at which $H=1$ is the +\emph{lifting condensation level} (LCL), \texttt{utilities.e\_pLCL} in the +code. Above it the parcel is saturated. + +\paragraph{Saturated (above the LCL): the moist adiabat proper.} Now +$\rvv=r_s(T,p)=\eps\es(T)/(p-\es)$ is pinned to saturation and +$\rl=\rt-r_s$ is condensate. Substituting into \eqref{eq:pathODE} (equivalently +differentiating \eqref{eq:E94} at $H=1$) gives the saturated adiabatic lapse +rate; its steepness is set by the latent-heat release, and the temperature +sensitivity is dominated by $\dd r_s/\dd T$ through Clausius--Clapeyron. Rather +than integrate this ODE, the solver conserves $s$ algebraically---the inversion +of \S\ref{sec:inversion}. + +\subsection{Density temperature and buoyancy} + +The dynamically relevant temperature is the \emph{density temperature} +$\Trho$, defined so that $p=\rho\,\Rd\,\Trho$ holds with the dry-air gas +constant despite the moisture. Accounting for vapour (which lightens the air, +$\Rv>\Rd$) and suspended condensate (which weighs it down), +\begin{equation}\label{eq:trho} +\Trho=T\,\frac{1+\rvv/\eps}{1+\rt}, +\end{equation} +exactly \texttt{utilities.Trho(T,\,$\rt$,\,$\rvv$)}. The buoyancy that drives +the parcel---and whose vertical integral is the CAPE of the companion +document---is the density-temperature excess over the environment, +\begin{equation}\label{eq:buoy} +b=\Rd\,\big(\Trho^{\text{parcel}}-\Trho^{\text{env}}\big)\big/\,\text{(a length)}, +\end{equation} +i.e.\ the parcel's self-determined adiabat (\S\ref{sec:conservation}) compared +against the sounding. The environment enters only here, in the comparison; the +parcel's own $T(p)$ came entirely from conserving $s$. + +\section{The inversion: recovering \texorpdfstring{$T$}{T} from \texorpdfstring{$s$}{s}} +\label{sec:inversion} + +\subsection{Definition and well-posedness} + +At each new pressure level the solver knows the conserved entropy $s_0$ and +must find the temperature consistent with it. This is a function +\emph{inversion}: given the forward map $T\mapsto s(T,p)$ at fixed $p$ (and +$\rt$), solve +\begin{equation}\label{eq:invert} +s(T,p)=s_0 \qquad\text{for } T=s^{-1}(s_0;p). +\end{equation} +An inverse exists and is unique wherever the forward map is strictly monotone. +Differentiate the saturated entropy---the relevant branch for the inversion, +\begin{equation}\label{eq:SGcode} +s_{\mathrm{sat}}(T)=(\cpd+\rt\,\cl)\ln T-\Rd\ln(p-\es)+\frac{\Lv\,r_s}{T}, +\end{equation} +(the $-\rvv\Rv\ln H$ term is gone because $H=1$)---with respect to $T$ at fixed +$p$. Using Clausius--Clapeyron \eqref{eq:cc} for $\dd r_s/\dd T$, the dominant +terms give +\begin{equation}\label{eq:dsdT} +\frac{\partial s_{\mathrm{sat}}}{\partial T} +=\frac{1}{T}\left(\cpd+\rt\,\cl+\frac{\Lv^{2}\,r_s}{\Rv\,T^{2}}\right) \;>\;0. +\end{equation} +Every term is manifestly positive, so $s_{\mathrm{sat}}(\cdot)$ is strictly +increasing and \eqref{eq:invert} has a unique root. By the inverse function +theorem the inverse is itself $C^1$ with derivative +$(s^{-1})'=1/(\partial s/\partial T)$. The inversion is well posed---this is the +``smooth, strictly monotone entropy relation'' invoked without proof in the +companion analysis. + +\subsection{Newton iteration, and its appearance in the code} + +There is no closed form for $s^{-1}$, so the solver inverts numerically by +Newton--Raphson on the residual $F(T)=s_{\mathrm{sat}}(T)-s_0$: +\begin{equation}\label{eq:newton} +T_{n+1}=T_n-\frac{F(T_n)}{F'(T_n)} + =T_n+\frac{s_0-s_{\mathrm{sat}}(T_n)}{\partial s_{\mathrm{sat}}/\partial T}. +\end{equation} +This is exactly \texttt{solve\_temperature\_from\_entropy}: its variable +\texttt{SG} is $s_{\mathrm{sat}}(T_n)$ of \eqref{eq:SGcode}, its \texttt{SL} is +the derivative \eqref{eq:dsdT}, +\[ +\texttt{SL}=\frac{\cpd+\rt\,\cl+\Lv^2 r_s/(\Rv T^2)}{T}, +\] +and the update \texttt{TGNEW = TG + AP*(S-SG)/SL} is \eqref{eq:newton} with a +relaxation factor \texttt{AP} (smaller for the first steps, then unity) guarding +against overshoot from a poor initial guess. Quadratic convergence follows from +$F\in C^2$ and $F'\neq0$, both guaranteed by \eqref{eq:dsdT}. The positivity of +\texttt{SL} is not a numerical convenience; it is the inverse-function-theorem +hypothesis, and \eqref{eq:dsdT} shows it holds term by term. + +\begin{remark}[why this is the ``entropy relation'', not a lapse-rate integral] +One could instead integrate the moist-adiabat ODE \eqref{eq:pathODE} in $p$. +Conserving $s$ algebraically via \eqref{eq:invert} is preferable: it carries no +accumulated integration error, it makes the conserved quantity exact at every +level by construction, and it exposes the monotonicity \eqref{eq:dsdT} that +guarantees a unique parcel temperature. The price---a root-find per level---is +cheap and quadratically convergent. +\end{remark} + +\section{Synthesis}\label{sec:synthesis} + +The chain is now complete and first-principled at every link: +\begin{enumerate} +\item Entropy is a microstate count, $S=k_B\ln\Omega=-k_B\sum p_i\ln p_i$ + (\S\ref{sec:statmech}); for an ideal gas it evaluates to + $s=c_p\ln T-R\ln p+\mathrm{const}$ (Sackur--Tetrode). +\item The second law makes $s$ an exact state function via the integrating + factor $1/T$ (\S\ref{sec:classical}), so it may be evaluated pointwise. +\item The parcel is a Dalton mixture; entropy adds over components at their + partial pressures (\S\ref{sec:parcel}), and phase equilibrium + ($g_v=g_\ell$) yields Clausius--Clapeyron and the vaporisation entropy + $\Lv/T$ (\S\ref{sec:phase}). +\item Summing dry air, vapour, and condensate gives Emanuel's total specific + entropy \eqref{eq:E94}---\texttt{utilities.entropy\_S} exactly + (\S\ref{sec:assembly}). +\item It is conserved because the ascent is adiabatic (no flux) and reversible + (no production, phase change at saturation), and reversible ascent keeps + $\rt$ fixed (\S\ref{sec:conservation}); microscopically, Liouville plus + the adiabatic theorem freeze $\Omega$. +\item Conserving $s$ defines the moist adiabat \eqref{eq:pathODE} and, via the + strictly monotone \eqref{eq:dsdT}, a well-posed inversion + $s(T,p)=s_0\mapsto T$ that \texttt{solve\_temperature\_from\_entropy} + performs by guarded Newton iteration (\S\ref{sec:adiabat}--\ref{sec:inversion}). +\end{enumerate} +Every constant, fit, and function named above appears in the \texttt{tcpyPI} +source---in \texttt{constants.py}, \texttt{utilities.py}, and \texttt{pi.py}; +the physics that justifies each is derived here rather than asserted. The one modelling choice +worth flagging for a physicist reading the code is the effective +$\cl=2500\JkgK$ (Remark~\ref{rem:modcl}); everything else is textbook +thermodynamics assembled in the meteorologist's per-dry-air bookkeeping. + +\end{document} From 5347346504dc531c178c1da6ea7cc619bb0ce15d Mon Sep 17 00:00:00 2001 From: Claude Date: Wed, 15 Jul 2026 12:16:12 +0000 Subject: [PATCH 02/11] Add canonical-typicality reading of the reservoir (SHO worked example) New subsection 2.4 derives the Boltzmann weights of 2.3 from a single pure state of one closed system, replacing the equal-a-priori postulate and the external reservoir with the Popescu-Short-Winter canonical typicality theorem. Worked out concretely on n harmonic oscillators: - the reduced density matrix rho_S = Tr_E |phi> <= sqrt(d_S/d_E^eff) = sqrt((M+1)/C(M+n-1,n-1)) gives thermality to ~1e-28 already for n=100, mu=1; - the geometric/Planck limit recovers e^{-beta hbar omega k}/Z with beta = d ln Omega/dE, meeting the reservoir derivation of 2.3 and making its Taylor truncation an explicit O(1/n) step; - a short note on why the integrable oscillator fails the dynamical (ergodicity/ETH) test: intra-shell degeneracy freezes rho_S, the commensurate spectrum gives exact revivals, and product-Fock eigenstates reduce to pure (non-thermal) subsystem states. Cross-linked with the microscopic reading in 7.3 (labelled ssec:micro) as the two faces of the same closed-system, no-reservoir stance. Builds clean with latexmk (17 pp., 0 overfull/underfull, no undefined refs); entropy_S and all physics are unchanged. Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01LZzvFfJYN7oiMWW5wZ5bhG --- .../thermodynamic_foundations.pdf | Bin 478922 -> 509427 bytes .../thermodynamic_foundations.tex | 193 +++++++++++++++++- 2 files changed, 192 insertions(+), 1 deletion(-) diff --git a/discontinuity_analysis/thermodynamic_foundations.pdf b/discontinuity_analysis/thermodynamic_foundations.pdf index 55ef5364f8a2fa6e5de94cc69cca4990fd88a605..1b2e46443e93dee28ba37d8c9bf6eef86918e2b0 100644 GIT binary patch delta 300617 zcmZ6yV{o8N)UF%bwlzs6wr$(C?c|Bgi8--t+sOnI+jb`Ae!qQAo!VzttsherJEHAD>EB+ zy6_nk4e&=tw{C+2E#Ri^RzQ7p6v2_K1j~QbW1(AXQe86gSt;IPt=mUKMsU*W_icGaHb=*^w%+Y?Mi9R zpgfv6KVBZ?{1yn}#rf&iIVqY_O*3nZ`vJZL9Yycp!&}v6|ETkwG;wkAuEq^AVby1I zf7*9e#B^gY<``tR<*(P(eS{TvYM=c1+6sd%k24s#Q5JBSCAl);bitfTR%wz4Ry2JV zYp18w^gy>YTFZ{4QQ}^+=%IP5844+BmTgaeUAuSPe+%<|GxrFoLz*U(?IyUh8bC$s zbfEVR;rL!4c`Bt>tM=cAQ)=zMAu}A5{5-dXopio_+~~~Fieu6@^9d}>$NZOc?CsD_ 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Switch to \mathbb I_R, a clean identity symbol consistent with the document's existing blackboard-bold use (R^3, Z_+^3 in 2.1) and needing no extra package. Build unchanged: 17 pp., 0 overfull/underfull, no undefined references. 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z9FPI01J(v&5p?j8lE4-Rol)2rbTVOM(0PQ70rNtWb1e!|(4nLg2lIl?nGqldF)v_V zz^yO=P6M_*Fv0_SA8;#-IX70Oi{fg))xc>GnEZh=AYg5{1aK?hR#*X87O*VHGMo;C zIuL=#MWZI6h9F)>yo|DcXTiDo^^ag}#M+3f5mzIwMqG`!8n?K@oANF4L=p2Mj}>Xt zs5ZsMI0l%pk;ji}Sgeg$8`*YkbZx}ih_w-GW4&{C)^yJe&OQ89RijH6eo@ur(zhR|YIf;=+J~xIT>8iFs#;z8 zc>TL=E?4qcRlCbocdF`exvBNi7Rw1WFVk_g@I2qjIbO}GECgdaArQk%##dre0cFfh9$J+|1w6bCF2NM@_bYXIIcyeWC3N Date: Thu, 16 Jul 2026 07:12:11 +0000 Subject: [PATCH 04/11] Define the partial trace; use a true blackboard-bold 1 (bbm) Two follow-ups on the canonical-typicality subsection: - Add \usepackage{bbm} and render the microcanonical projector as \mathbbm{1}_R, a genuine double-struck 1 (msbm/\mathbb has no digit glyph). Requires the bbm package (texlive-fonts-extra). - Define Tr_E where it first appears (eq. 10). New eq. (11) gives the partial trace on product operators, Tr_E(A (x) B) = A Tr B, contrasted with the full trace Tr(A (x) B) = Tr A Tr B, extended to all operators by linearity; and notes it returns an operator on H_S (a density matrix), not a scalar -- the point that distinguishes it from the full trace. Build clean: 17 pp., 0 overfull/underfull, no undefined references. Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01LZzvFfJYN7oiMWW5wZ5bhG --- .../thermodynamic_foundations.pdf | Bin 509304 -> 511094 bytes .../thermodynamic_foundations.tex | 17 +++++++++++++++-- 2 files changed, 15 insertions(+), 2 deletions(-) diff --git a/discontinuity_analysis/thermodynamic_foundations.pdf b/discontinuity_analysis/thermodynamic_foundations.pdf index 73faee3b66140775de45168433055e8c92b3a855..8303241001534c38853e18af2e0e22768000bc54 100644 GIT binary patch delta 113236 zcmV(;K-<6giXZlvACM*jH8Ph0L;)y&&05`$-E3yz6Z9_ zFoiTgP(%xVG7a2ZmKUjG##+#!U$>}HhVy@3FR_xQOqqf71ua|l zVb3jSuV^p3zQ83wN}e)d^_2a*-qCDQFwE|Y!x0X~CPka~r?v=-gIaC!cJ0TzizxMM zc3S|A^ECHW7|VBMexLWmg|dtb(7EDF!LMiHo91xL>tWHpA(^cl%WMsQ1Lk?uddd?d zho8LdOH}{1%qtq%mE9Hznz}FRQ`w(sxC|4BS$v?g_Rq%_XB1s`q6Zr7n(A%QE**h! z%o$LsW;zYJ{Qgv(Ndlo?km!<2gR`%^1f+ATjAcd|fB2Fq4o9OolOQF@c;EGHzU^<# zPe=1P%N#0)uf$4l#T<8kt5jQNz`XAc0bW#hWsfgs zB1ZA)IO#q5krQLy!@g33djMCHeaG%}lVfQ!FnuwdS(Ojopkdp*4!>pK1N?sL9Ucd( 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Switch those nested \binom to \dbinom so they typeset full size and stay legible; the standalone d_R binomial is unchanged. Build clean: 17 pp., 0 overfull/underfull, no undefined references. Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01LZzvFfJYN7oiMWW5wZ5bhG --- .../thermodynamic_foundations.pdf | Bin 511094 -> 511025 bytes .../thermodynamic_foundations.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/discontinuity_analysis/thermodynamic_foundations.pdf b/discontinuity_analysis/thermodynamic_foundations.pdf index 8303241001534c38853e18af2e0e22768000bc54..a1ce29d06038761f6351620afe3ad7fc8d3138b5 100644 GIT binary patch delta 9875 zcmai)WmBC!xP>?F?oM%M<5Jw+y|@=GR-6a-4HS2GDDEyTUK|P(cXuex>3hx(IJ4F@ zlS%UN&YDc_+^192r&H89S;Fw}aU=x^F#v0ZpW^?>eGT|qt40oQY9YJYVKQ_WfwnnH zgM*M28-!HX2(^U0R>&G9Txc23m$s4XRvDdE7+t)wGr@Lk+NxK5QlWX^cc4L2HH@eK zI*5Y(_V$`Z(EF}4jz`;0n_9EpWi*ow_L9|cFVSXeH~h@#>a=kD%j~Y~RLxPv!wq1Y zw7S7Dt6ZYkJaS^i{WXT+sbsTvgd@^W-BGlB<)qx`cvrAu9Nk@e=Pr=a6MXWM0RLH2 z-$8Wk_3@YJ>)WS6y8S`CQHaJ>=>1&|js9!T6@;E(5b~=T(`6?QF-BYj0 z;rk^9_2CJkLDgscPFUo~0yL>kFFg!Z#8suox&|i2+j4zO~iQlUi 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With $d_S=M+1$ accessible levels, \begin{equation}\label{eq:sho-bound} \big\langle\lVert\rho_S-\varrho_S\rVert_1\big\rangle - \le\sqrt{\frac{M+1}{\binom{M+n-1}{\,n-1\,}}} + \le\sqrt{\frac{M+1}{\dbinom{M+n-1}{\,n-1\,}}} \;\approx\;\sqrt{\mu n}\;e^{-n\sigma(\mu)/2}, \qquad \sigma(\mu)=(1{+}\mu)\ln(1{+}\mu)-\mu\ln\mu, From 83974a8472448a4b4efdbdd273030f14bb1316f3 Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 16 Jul 2026 07:51:22 +0000 Subject: [PATCH 06/11] Flesh out sec 2.1 spoilers; add a "where typicality fails" remark Section 2.1 (the operator picture): - Spoiler for the spectral functionals: give F = -kT ln Z, = -d ln Z/d beta, and S = (-F)/T = -dF/dT inline where Z is called the Laplace transform of rho(E) (full derivation deferred to 2.5, noted). - Define the three symmetric functions and the Maxwell-Boltzmann count: elementary e_N (fermions, strictly increasing indices = Pauli), complete homogeneous h_N (bosons, repeats allowed), and the leading power sum p_1^N/N! = z^N/N! (classical count, dilute limit; deferred to 2.5/2.6, noted). - Gloss the Legendre conjugate pairs: pressure p and chemical potential mu (free-energy cost of one particle). Section 2.4 (canonical typicality): - New Remark 1 "where canonical typicality fails": the exceptional set is the low-S:E-entanglement states; the product Fock state |M,0,...,0> gives rho_S = |M> Claude-Session: https://claude.ai/code/session_01LZzvFfJYN7oiMWW5wZ5bhG --- .../thermodynamic_foundations.pdf | Bin 511025 -> 517167 bytes .../thermodynamic_foundations.tex | 68 +++++++++++++++--- 2 files changed, 58 insertions(+), 10 deletions(-) diff --git a/discontinuity_analysis/thermodynamic_foundations.pdf b/discontinuity_analysis/thermodynamic_foundations.pdf index a1ce29d06038761f6351620afe3ad7fc8d3138b5..96c7e1243d0b3cbe99203fa2f90e59f299a66ad2 100644 GIT binary patch delta 213333 zcmV(}K+wOjm>;jmACM*jG&YlgJScz7T3fH%HWGgCukb!q#>Nb9BBd66*ml#R?V>1} z11zx3VxeR691E2tSCW%>fBnvINLiBQv`G)Td9p+e$(iBIHy4_(9#?$z%_safe!sl? zlk>D(mlxvZ6{pMJ-__l>NR}rDB*RW{cHb?wD(_mSz|9$rzu$7M4 z64o*+#cCr3%SA?<_1ok5$8Q62Kv9j4zSTHu5^nWwPn36 zkCj9IrA>COEnlzGH1R?aEy#Z~aC2E+q!}~Tf)4$rV7Qqg_+IcJ0y;n2cox 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SHO concretisation: Z = sum Omega(M,n) x^M = (1-x)^{-n}, the generating function whose coefficients are the shell dimensions. - Thermodynamic limit = Legendre transform: the saddle S'(E*) = 1/T gives F = E* - T S(E*), ln Z and S/k_B Legendre-dual with beta,E conjugate; the ensemble table of 2.1 is the N->infty shadow of the Laplace transform (Cramer/large-deviation duality noted). - Why Z is elegant: distribution -> analytic function; convolution -> product (so the sec-2.3 reservoir maneuver is a convolution done by saddle point, and Z dissolves it); ln Z generates the cumulants (Var E = k T^2 C_V); Z is what one actually computes. - Caveat: Legendre returns only the concave hull, so non-concave S(E) (first-order transitions, long-range/gravitating, negative T, small systems) gives ensemble inequivalence and S(E) is the faithful object; equivalence holds for short-range, N->infty -- the parcel's regime. Build clean: 20 pp., 0 overfull/underfull, no undefined references. 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a/discontinuity_analysis/thermodynamic_foundations.tex +++ b/discontinuity_analysis/thermodynamic_foundations.tex @@ -592,6 +592,109 @@ \subsection{From \texorpdfstring{$Z$}{Z} to entropy, and indistinguishable parti single-particle energy (contrast $E_i$, the whole-system energy of \S\ref{ssec:BG}). Everything now hinges on $z$. +\subsection{The bridge to \texorpdfstring{$S(E)$}{S(E)}: Laplace, Legendre, and +why \texorpdfstring{$Z$}{Z} is the elegant object}\label{ssec:bridge} + +Two objects describe the same equilibrium: the microcanonical entropy $S(E)$ +---geometric and primary, the logarithm of the phase-space volume at fixed +energy---and the canonical $Z(\beta)$ just used. Making the link exact explains +both why they agree and why the $Z$ side carried every derivation above so much +more lightly than the reservoir bookkeeping of \S\ref{ssec:boltzdist} did. + +\paragraph{The transform.} Let $\Omega(E)$ be the density of states ($\rho(E)$ +of \S\ref{ssec:operator}), so $S(E)=k_B\ln\Omega(E)$ up to a subextensive window +constant. Then $Z$ is exactly its Laplace transform, +\begin{equation}\label{eq:bridge-laplace} +Z(\beta)=\int_0^\infty\!\dd E\,\Omega(E)\,e^{-\beta E} + =\int_0^\infty\!\dd E\,\exp\!\Big[\tfrac{1}{k_B}S(E)-\beta E\Big], +\end{equation} +invertible by the Bromwich integral +$\Omega(E)=\frac{1}{2\pi i}\int Z(\beta)\,e^{\beta E}\dd\beta$: the two functions +carry identical information. The oscillator shows this concretely. The +microcanonical count $\Omega(M,n)=\binom{M+n-1}{n-1}$ of \eqref{eq:sho-dR} is a +genuine combinatorial sequence, but it is packaged whole by +\begin{equation}\label{eq:bridge-genfn} +Z(x)=\sum_{M\ge0}\Omega(M,n)\,x^{M} + =\sum_{M\ge0}\binom{M+n-1}{n-1}x^{M} + =\frac{1}{(1-x)^{n}},\qquad x=e^{-\beta\hbar\omega}, +\end{equation} +so $Z$ is nothing but the \emph{generating function} whose Taylor coefficients +are the shell dimensions; a single $\Omega(M,n)$ comes back as the coefficient +integral $\frac{1}{2\pi i}\oint Z(x)\,x^{-M-1}\dd x$, whose saddle point gives +the entropy asymptotics. ``Laplace transform of the density of states'' and +``generating function of the counts'' are one statement, continuous and discrete. + +\paragraph{The thermodynamic limit is a Legendre transform.} For a macroscopic +system the integrand of \eqref{eq:bridge-laplace} is sharply peaked, and +Laplace's method evaluates it at the saddle $E^\ast(\beta)$ where the exponent is +stationary, +\begin{equation}\label{eq:bridge-saddle} +\frac{\dd}{\dd E}\Big[\tfrac{1}{k_B}S(E)-\beta E\Big]_{E^\ast}=0 +\;\Longrightarrow\; +\frac{\dd S}{\dd E}\Big|_{E^\ast}=k_B\beta=\frac1T, +\end{equation} +which is exactly the microcanonical definition of temperature. +Back-substituting, $-k_BT\ln Z=E^\ast-T\,S(E^\ast)=F$: the free energy is the +Legendre transform of $S(E)$, with $\beta$ and $E$ conjugate, +\begin{equation}\label{eq:bridge-legendre} +\ln Z(\beta)=\sup_E\Big[\tfrac{1}{k_B}S(E)-\beta E\Big],\qquad +\tfrac{1}{k_B}S(E)=\inf_\beta\big[\ln Z(\beta)+\beta E\big], +\end{equation} +with duality relations $\beta=\tfrac1{k_B}S'(E)$ and $E=-\partial_\beta\ln Z$. +So the Legendre transform relating the ensembles (the table of +\S\ref{ssec:operator}) is just the $N\to\infty$ shadow of the exact Laplace +transform \eqref{eq:bridge-laplace}: the saddle collapses the whole spectrum onto +one energy $E^\ast$ and one temperature. (This is the thermodynamic face of the +Cram\'er/large-deviation duality between the scaled cumulant generating function +$\ln Z$ and its rate function $S/k_B$.) + +\paragraph{Why $Z$ is the elegant object.} Every awkward operation on $S(E)$ is +an easy one on $Z$, because the Laplace transform converts them: +\begin{itemize} +\item \textbf{A distribution becomes an analytic function.} +$\Omega(E)=\operatorname{Tr}\delta(E-\hat H)$ is a comb of spectral spikes, and +$S(E)$ needs a coarse-graining window to define at all; $Z(\beta)=\operatorname{Tr} +e^{-\beta\hat H}$ is a smooth, entire function of $\beta$ for any spectrum bounded +below---no delta functions, no window. +\item \textbf{Convolution becomes multiplication---and the reservoir evaporates.} +Independent systems ($\hat H=\hat H_1+\hat H_2$) \emph{convolve} their densities +of states, $\Omega=\Omega_1\!*\Omega_2$, but simply \emph{multiply} their +partition functions, $Z=Z_1Z_2$, so $\ln Z$ is manifestly extensive. The whole +``system\,$+$\,reservoir, expand $\ln\Omega_{\mathrm{res}}(E_{\mathrm{tot}}-E_i)$'' +maneuver of \S\ref{ssec:boltzdist} was a convolution evaluated by saddle point; +in the $Z$ picture one never introduces the reservoir---with the intensive +$\beta$ as primary, the Boltzmann weight $e^{-\beta E_i}/Z$ is immediate. The +reservoir's only job was to fix $\beta=S_{\mathrm{res}}'/k_B$, which $Z$ takes as +its argument. +\item \textbf{$\ln Z$ generates the cumulants.} $\langle E\rangle=-\partial_\beta +\ln Z$ and $\operatorname{Var}(E)=\partial_\beta^{2}\ln Z=k_BT^{2}C_V\ge0$; every +fluctuation is a derivative of one smooth function, and fluctuation--dissipation +is one line. Energy fluctuations are not even representable on a single +microcanonical $E$. +\item \textbf{$Z$ is what one computes.} Path integrals, transfer matrices, +perturbation theory, and Monte Carlo all act on $Z$ or $\ln Z$; counting states +at fixed energy is the hard problem. The oscillator is the epitome: +$\Omega(M,n)=\binom{M+n-1}{n-1}$ is a nontrivial count, while $Z=(1-x)^{-n}$ is +one line \eqref{eq:bridge-genfn}. +\end{itemize} + +\paragraph{When $S(E)$ is the truer object.} The elegance has a precise price: +the Legendre transform \eqref{eq:bridge-legendre} can only return the +\emph{concave hull} of $S(E)$, so wherever $S(E)$ is non-concave the $Z$ picture +silently discards information---replacing the offending region by its Maxwell +tangent. That is exactly what happens at first-order phase transitions (the flat +tangent is the latent heat), and in long-range or self-gravitating systems, at +negative temperatures, and in small systems---all cases of genuine \emph{ensemble +inequivalence}, where the microcanonical $S(E)$ is faithful and the canonical +average is not. So the honest ranking is the geometric one: $S(E)$ is primary and +information-complete; $Z(\beta)$ is its Laplace transform---smoother, factorising, +fluctuation-generating, and matched to how a system is actually held (at fixed +$T$, not fixed $E$)---and the two are fully equivalent precisely when $S(E)$ is +concave, i.e.\ short-range interactions as $N\to\infty$. That is the parcel's +regime, and the reason the intensive $s(T,p)$---not a fixed-energy count---is the +right variable for the solver. Having set $Z$ against its microcanonical shadow, +we return to the object everything hinges on: the single-particle $z$. + \subsection{The translational partition function and the thermal de~Broglie wavelength} \label{ssec:lambda} From e81090f4c5b53af89fca3de0af1e2f0f105280e1 Mon Sep 17 00:00:00 2001 From: Claude Date: Sat, 18 Jul 2026 15:36:20 +0000 Subject: [PATCH 08/11] Extend sec 2.6: hard vs soft constraint and the separation of scales Adds the inverse-transform reading of the S(E) <-> Z(beta) bridge, answering why exact energy conservation on the whole and the Boltzmann distribution on a part are the "same" only asymptotically: - Hard (microcanonical, delta(E_tot - H), zero fluctuation) vs soft (canonical, e^{-beta H}, beta a Lagrange multiplier). Linked by the operator identity delta(E_tot - H) = (1/2pi i) int e^{beta(E_tot-H)} d beta -- the hard projector is a superposition of soft weights over imaginary temperature (inverse Bromwich of the Laplace bridge). - Whole-system equivalence guarded by 1/sqrt(N): the saddle collapses to one temperature, relative energy fluctuation sqrt(Var E)/ = sqrt(k T^2 C_V)/ ~ 1/sqrt(N). - Subsystem equivalence guarded by N_S/N_B: marginalising the shell over the bath gives p(E_S) ~ Omega_S Omega_B(E_tot - E_S); linearising the bath entropy (stiff temperature, C_B ~ N_B) yields the Boltzmann weight -- this IS the sec-2.3 Taylor truncation, small parameter named. - When the guard fails: non-concave S(E) (competing saddles) or small systems -> the ensembles genuinely diverge. Build clean: 21 pp., 0 overfull/underfull, no undefined references. Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01LZzvFfJYN7oiMWW5wZ5bhG --- .../thermodynamic_foundations.pdf | Bin 530891 -> 538067 bytes .../thermodynamic_foundations.tex | 76 +++++++++++++++++- 2 files changed, 74 insertions(+), 2 deletions(-) diff --git a/discontinuity_analysis/thermodynamic_foundations.pdf b/discontinuity_analysis/thermodynamic_foundations.pdf index f544c4730141a5b54f77b229a6060ae41f941829..3b54d003a68f4c16f1702adb92b19eac2f194a14 100644 GIT binary patch delta 123390 zcmV)BK*PVwvmn#!Adn{kF_WP@D3MSEf8(|he$TIPpQ3}Vxx9)}f;=?orbrqDLG}Oz znk@<)o5xloOO7P(uKVjd!y#o!mN(h$L9$Ph$RRl#&gGk-_-eP}tG|81f93h*{U^`A z)^a5*lfvlL{o^WEOzD*soTcgNe!Kc9x!=3?(A=$MnrzQ?aVR%;YsF>q*qrKZf6)P`5oiT~u6w$(-25v6Xe+!eb)LM|CpV!s6hx5N*F0o<^Q|UnZjHWI7u;vD& z=i1Av&u|LZB{xi1J!QYBw=|jr45NE@IKsi$#I;3#YF(Hd#A=h*cfP&5Y^9!!ZXM8M zp5~znZN;`M9*Um0P?m84GFO}_czPzjY7WPu9wzM@+Ow78J=>H4^E^u3fA9p!;Zd}G ziQ?asMMW*!vfCg*Q}<JMKs#o=f)XA*Wvdwl5nw%GKy#+%W2 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a/discontinuity_analysis/thermodynamic_foundations.tex +++ b/discontinuity_analysis/thermodynamic_foundations.tex @@ -601,16 +601,42 @@ \subsection{The bridge to \texorpdfstring{$S(E)$}{S(E)}: Laplace, Legendre, and both why they agree and why the $Z$ side carried every derivation above so much more lightly than the reservoir bookkeeping of \S\ref{ssec:boltzdist} did. -\paragraph{The transform.} Let $\Omega(E)$ be the density of states ($\rho(E)$ -of \S\ref{ssec:operator}), so $S(E)=k_B\ln\Omega(E)$ up to a subextensive window -constant. Then $Z$ is exactly its Laplace transform, +\paragraph{The transform.} Two objects must be kept apart. The \emph{density of +states} +\begin{equation}\label{eq:dos} +\rho(E)=\operatorname{Tr}\,\delta(E-\hat H)=\sum_i\delta(E-E_i) +\end{equation} +of \S\ref{ssec:operator}---with $\delta(E-\hat H)=\sum_i\delta(E-E_i)\, +|i\rangle\langle i|$ the operator function of $\hat H$ (evaluated on the spectrum +and traced \emph{after}, exactly as $Z=\operatorname{Tr}e^{-\beta\hat H} +=\sum_i e^{-\beta E_i}$)---is a distribution, a comb of one spike per level; its +logarithm is meaningless. The Boltzmann entropy instead uses the dimensionless +\emph{count} +\begin{equation}\label{eq:dos-count} +\Omega(E)=\int_E^{E+\Delta E}\!\rho(E')\,\dd E'\approx\bar\rho(E)\,\Delta E, +\qquad S(E)=k_B\ln\Omega(E), +\end{equation} +the number of levels in a macroscopically thin shell $[E,E+\Delta E]$, with +$\bar\rho$ the smoothed envelope of the comb. Because the spectrum is spaced like +$e^{-N}$, any such shell holds astronomically many levels, so $\Omega$ is a huge +\emph{smooth} number and $\ln\Omega$ is well defined and extensive; the window +enters only through the subextensive $k_B\ln\Delta E$, negligible beside +$S\sim k_BN$. ($\Omega$ and $\bar\rho$ thus agree \emph{inside} the logarithm but +are not the same object---they differ by the units-carrying $\Delta E$, which is +why $\rho$ and $\Omega$ get different symbols.) The partition function is the +Laplace transform of the \emph{density} $\rho$, \begin{equation}\label{eq:bridge-laplace} -Z(\beta)=\int_0^\infty\!\dd E\,\Omega(E)\,e^{-\beta E} - =\int_0^\infty\!\dd E\,\exp\!\Big[\tfrac{1}{k_B}S(E)-\beta E\Big], +Z(\beta)=\int_0^\infty\!\dd E\,\rho(E)\,e^{-\beta E}=\sum_i e^{-\beta E_i} + \approx\int_0^\infty\!\dd E\,\exp\!\Big[\tfrac{1}{k_B}S(E)-\beta E\Big], \end{equation} -invertible by the Bromwich integral -$\Omega(E)=\frac{1}{2\pi i}\int Z(\beta)\,e^{\beta E}\dd\beta$: the two functions -carry identical information. The oscillator shows this concretely. The +and the middle equality is the crux: the exponential kernel integrates straight +over the comb, so $Z$ does \emph{for free} the coarse-graining that $S(E)$ needed +a window for---the ``distribution\,$\to$\,analytic function'' elegance, met again +below. It is invertible by the Bromwich integral +$\rho(E)=\frac{1}{2\pi i}\int Z(\beta)\,e^{\beta E}\dd\beta$: $\rho$ and $Z$ carry +identical information. The oscillator shows this concretely---and more cleanly +still, since its spectrum is genuinely discrete, so the ``count'' is an honest +integer degeneracy needing no window at all. The microcanonical count $\Omega(M,n)=\binom{M+n-1}{n-1}$ of \eqref{eq:sho-dR} is a genuine combinatorial sequence, but it is packaged whole by \begin{equation}\label{eq:bridge-genfn} @@ -652,13 +678,13 @@ \subsection{The bridge to \texorpdfstring{$S(E)$}{S(E)}: Laplace, Legendre, and an easy one on $Z$, because the Laplace transform converts them: \begin{itemize} \item \textbf{A distribution becomes an analytic function.} -$\Omega(E)=\operatorname{Tr}\delta(E-\hat H)$ is a comb of spectral spikes, and +$\rho(E)=\operatorname{Tr}\delta(E-\hat H)$ is a comb of spectral spikes, and $S(E)$ needs a coarse-graining window to define at all; $Z(\beta)=\operatorname{Tr} e^{-\beta\hat H}$ is a smooth, entire function of $\beta$ for any spectrum bounded below---no delta functions, no window. \item \textbf{Convolution becomes multiplication---and the reservoir evaporates.} Independent systems ($\hat H=\hat H_1+\hat H_2$) \emph{convolve} their densities -of states, $\Omega=\Omega_1\!*\Omega_2$, but simply \emph{multiply} their +of states, $\rho=\rho_1\!*\rho_2$, but simply \emph{multiply} their partition functions, $Z=Z_1Z_2$, so $\ln Z$ is manifestly extensive. The whole ``system\,$+$\,reservoir, expand $\ln\Omega_{\mathrm{res}}(E_{\mathrm{tot}}-E_i)$'' maneuver of \S\ref{ssec:boltzdist} was a convolution evaluated by saddle point; From 323743b151aa7b90e71a4016f1494a3a81067688 Mon Sep 17 00:00:00 2001 From: Claude Date: Mon, 20 Jul 2026 21:12:24 +0000 Subject: [PATCH 10/11] Full sweep: figures throughout, definitions at first use, consistency Exposition sweep extrapolating the review principles applied so far (never conflate objects; define or explicitly defer everything at first use; spoil payoffs; exact worked examples with counterexamples; tie every claim to code or a named theorem): New "notational hazards" remark (Remark 1) collecting every symbol collision: Omega count vs grand potential, rho(E) vs density matrices vs T_rho, single-particle epsilon vs eps=Rd/Rv, mu chemical potential vs mean occupancy, H vs Hhat vs mathcal-H, R gas constant vs code argument. Definitions added where first used: trace norm (with the Helstrom operational meaning), Haar measure (with the Gaussian sampling recipe), Levy's lemma (explicit concentration inequality, footnote), effective dimension as inverse purity/participation number, stars-and-bars (footnote). Physics fix: the ergodicity "cure" was stated as a bilinear coupling lambda x_i x_{i+1}, which is still quadratic hence integrable; corrected to the cubic FPUT term, with the bilinear trap called out explicitly. Sloppiness fix: buoyancy "(a length)" replaced by the real relations b = g dTrho/Trho and CAPE = int Rd dTrho dlnp, matching the PFAC trapezoid accumulated in pi.py. Eight figures, all generated by the new scripts/foundations_figs.py (numpy/matplotlib; colorblind-validated palette; the inversion figure transcribes es_cc/Lv/ev/rv/entropy_S and the solver loop and asserts against the source docstring examples): - fig_sf_classical_marginal: hard constraint -> Boltzmann as n grows - fig_sf_quantum_marginal: exact shell marginal vs Planck limit - fig_sf_typicality: exact Dirichlet sampling of ||rho_S - varrho_S||_1 vs the PSW bound vs the atypical Fock state - fig_sf_comb: DOS delta comb vs envelope vs windowed count - fig_sf_bridge: Legendre tangent geometry + saddle sharpening - fig_sf_nonconcave: convex intruder, Maxwell tangent, what Z loses - fig_sf_fput: integrable freeze vs alpha-FPUT mixing + recurrence - fig_sf_inversion: monotone s_sat per level + guarded Newton error Abstract and synthesis updated to cover the typicality and bridge material and the figure provenance; date bumped. Build clean: 26 pp., 0 overfull/underfull, no undefined references. 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figures for thermodynamic_foundations.tex. + +Every figure is computed from the formulas derived in the document; the +entropy-inversion figure (fig_sf_inversion) additionally reproduces, line for +line, the algebra of ``utilities.entropy_S`` / ``pi.solve_temperature_from_entropy`` +(transcribed below and cross-checked against the docstring examples of the +source, so a numba-free interpreter can run this script). + +Usage: python3 foundations_figs.py (writes ../figures/fig_sf_*.{pdf,png}) +""" + +import math +import numpy as np +import matplotlib + +matplotlib.use("Agg") +import matplotlib.pyplot as plt +from pathlib import Path + +FIGDIR = Path(__file__).resolve().parent.parent / "figures" +FIGDIR.mkdir(exist_ok=True) + +RNG = np.random.default_rng(20260720) + +# ---------------------------------------------------------------------------- +# Style: match the lmodern/Computer-Modern look of the document; recessive +# axes/grids, thin marks, direct labels. Categorical palette (fixed order, +# validated): blue, green, magenta, yellow. Sequential families (increasing n) +# use a single blue hue, light -> dark. +# ---------------------------------------------------------------------------- +C_BLUE, C_GREEN, C_MAGENTA, C_YELLOW = "#2a78d6", "#008300", "#e87ba4", "#eda100" +INK, INK2 = "#0b0b0b", "#52514e" +BLUES = ["#b9d1ef", "#7fa9e0", "#3e7fd0", "#134f96"] # light -> dark + +plt.rcParams.update({ + "font.family": "serif", + "mathtext.fontset": "cm", + "font.size": 9, + "axes.labelsize": 9, + "axes.titlesize": 9, + "xtick.labelsize": 8, + "ytick.labelsize": 8, + "legend.fontsize": 8, + "axes.spines.top": False, + "axes.spines.right": False, + "axes.linewidth": 0.7, + "axes.edgecolor": INK2, + "xtick.color": INK2, + "ytick.color": INK2, + "axes.labelcolor": INK, + "grid.color": INK2, + "grid.alpha": 0.18, + "grid.linewidth": 0.5, + "axes.grid": True, + "axes.axisbelow": True, + "legend.frameon": False, + "lines.linewidth": 1.6, + "figure.dpi": 110, +}) + + +def save(fig, name): + for ext in ("pdf", "png"): + fig.savefig(FIGDIR / f"{name}.{ext}", bbox_inches="tight", + dpi=200 if ext == "png" else None) + plt.close(fig) + print(f"wrote {name}") + + +def lbinom(a, b): + """log C(a, b) via lgamma (floats fine, a,b >= 0).""" + return math.lgamma(a + 1) - math.lgamma(b + 1) - math.lgamma(a - b + 1) + + +def binom(a, b): + return math.exp(lbinom(a, b)) + + +# ============================================================================ +# Fig 1 -- classical shell marginal: hard constraint on the whole becomes the +# Boltzmann weight on one mode as n grows. P(eps) = ((n-1)/E)(1-eps/E)^(n-2), +# E = n kT (so the mean energy per mode is kT throughout). +# ============================================================================ +def fig_classical_marginal(): + fig, ax = plt.subplots(figsize=(4.9, 3.0)) + eps = np.linspace(0, 5, 400) + ns = [3, 10, 30, 100] + for n, c in zip(ns, BLUES): + E = float(n) # kT = 1 + P = np.where(eps < E, (n - 1) / E * np.maximum(1 - eps / E, 0) ** (n - 2), 0.0) + ax.plot(eps, P, color=c, label=f"$n={n}$") + ax.plot(eps, np.exp(-eps), color=INK, ls="--", lw=1.3, + label=r"$e^{-\varepsilon/k_BT}$ (limit)") + ax.annotate(r"$P(\varepsilon)\propto\Omega_{\mathrm{bath}}(E-\varepsilon)" + r"\propto(1-\varepsilon/E)^{\,n-2}$", + xy=(1.95, 0.40), fontsize=8.5, color=INK2) + ax.set_xlabel(r"single-mode energy $\varepsilon/k_BT$") + ax.set_ylabel(r"$P(\varepsilon)\,k_BT$") + ax.set_xlim(0, 5) + ax.set_ylim(0, 1.05) + ax.legend(loc="upper right", handlelength=1.6) + save(fig, "fig_sf_classical_marginal") + + +# ============================================================================ +# Fig 2 -- quantum marginal: exact rho_S(k) of the microcanonical shell for +# finite n vs the geometric (Planck) limit, mu = M/n = 1. +# ============================================================================ +def fig_quantum_marginal(): + fig, ax = plt.subplots(figsize=(4.9, 3.0)) + ks = np.arange(0, 13) + for n, c in zip([4, 16, 64], BLUES[1:]): + M = n # mu = 1 + dR = lbinom(M + n - 1, n - 1) + p = np.array([math.exp(lbinom(M - k + n - 2, n - 2) - dR) + if k <= M else np.nan for k in ks]) + ax.plot(ks, p, "o-", ms=3.6, color=c, lw=1.2, label=f"$n={n}$") + geo = 0.5 ** (ks + 1) # (1-x) x^k, x = mu/(1+mu) = 1/2 + ax.plot(ks, geo, "s", ms=4.5, mfc="none", mec=INK, mew=1.0, ls="--", + color=INK, lw=1.0, label=r"$(1-x)\,x^{k}$ (limit)") + ax.set_yscale("log") + ax.set_xlabel(r"system occupation $k$") + ax.set_ylabel(r"$\varrho_S(k)$") + ax.set_xlim(-0.4, 12.4) + ax.set_ylim(1e-5, 1) + ax.annotate(r"$\varrho_S(k)=\binom{M-k+n-2}{n-2}/\binom{M+n-1}{n-1}$," + r" $\mu=1$", xy=(0.03, 0.05), xycoords="axes fraction", + fontsize=8.5, color=INK2) + ax.legend(loc="upper right", handlelength=1.6) + save(fig, "fig_sf_quantum_marginal") + + +# ============================================================================ +# Fig 3 -- canonical typicality Monte Carlo. For a Haar-random state of the +# shell, the vector of diagonal entries rho_S(k) is exactly Dirichlet +# distributed with parameters (N_0, ..., N_M), N_k = Omega(M-k, n-1); sample it +# and compare the trace distance to the PSW bound and to the atypical Fock +# state |M,0,...,0>. +# ============================================================================ +def fig_typicality(): + fig, ax = plt.subplots(figsize=(4.9, 3.1)) + ns = np.arange(2, 31) + means, lo, hi, bound, atyp = [], [], [], [], [] + NSAMP = 400 + for n in ns: + M = int(n) # mu = 1 + ldR = lbinom(M + n - 1, n - 1) + Nk = np.array([math.exp(lbinom(M - k + n - 2, n - 2)) for k in range(M + 1)]) + pk = np.array([math.exp(lbinom(M - k + n - 2, n - 2) - ldR) + for k in range(M + 1)]) + samp = RNG.dirichlet(Nk, size=NSAMP) # exact law of diag(rho_S) + d1 = np.abs(samp - pk).sum(axis=1) # ||rho_S - varrho_S||_1 + means.append(d1.mean()) + lo.append(np.percentile(d1, 5)) + hi.append(np.percentile(d1, 95)) + bound.append(math.exp(0.5 * (math.log(M + 1) - ldR))) # sqrt(d_S/d_R) + atyp.append(2 * (1 - math.exp(-ldR))) # 2(1 - 1/d_R) + ax.fill_between(ns, lo, hi, color=C_BLUE, alpha=0.18, lw=0) + ax.plot(ns, means, color=C_BLUE, label="Haar mean (5–95% band)") + ax.plot(ns, bound, color=INK, ls="--", lw=1.2, + label=r"PSW bound $\sqrt{d_S/d_R}$") + ax.plot(ns, atyp, color=C_MAGENTA, ls=":", lw=1.6, + label=r"atypical $|M,0,\dots,0\rangle$") + ax.annotate("extrapolates to 28 orders\nof magnitude by $n=100$", + xy=(0.80, 0.72), xycoords="axes fraction", fontsize=8, + color=INK2, ha="center") + ax.set_yscale("log") + ax.set_ylim(top=30) + ax.set_xlabel(r"number of oscillators $n$ ($\mu=1$)") + ax.set_ylabel(r"$\|\rho_S-\varrho_S\|_1$") + ax.set_xlim(2, 30) + ax.legend(loc="lower left", handlelength=1.7) + save(fig, "fig_sf_typicality") + + +# ============================================================================ +# Fig 4 -- density of states rho(E) (delta comb, weights Omega(M,3)) vs the +# smoothed envelope rho-bar and the windowed count Omega(E). +# ============================================================================ +def fig_comb(): + fig, ax = plt.subplots(figsize=(4.9, 3.0)) + n = 3 + Ms = np.arange(0, 15) + w = np.array([binom(M + n - 1, n - 1) for M in Ms]) # C(M+2,2) + Ee = np.linspace(0, 14.4, 300) + env = (Ee + 1) * (Ee + 2) / 2 # smooth envelope + win = (8.5, 11.5) + inwin = (Ms >= win[0]) & (Ms <= win[1]) + count = int(w[inwin].sum()) + ax.axvspan(*win, color=C_YELLOW, alpha=0.22, lw=0) + ml, sl, bl = ax.stem(Ms, w, basefmt=" ") + plt.setp(sl, color=C_BLUE, lw=1.3) + plt.setp(ml, color=C_BLUE, ms=3.5) + ax.plot(Ee, env, color=INK, ls="--", lw=1.2) + ax.annotate(r"$\bar\rho(E)$ (smoothed envelope)", xy=(9.5, 87), + fontsize=8.5, color=INK, ha="right") + ax.annotate(r"$\rho(E)=\sum_M\Omega(M,n)\,\delta(E-M\hbar\omega)$", + xy=(0.4, 100), fontsize=8.5, color=C_BLUE) + ax.annotate(rf"window $\Delta E$: $\Omega={count}$ states," + r" $S=k_B\ln\Omega$", + xy=(10.0, 128), fontsize=8.5, color=INK2, ha="center", + annotation_clip=False) + ax.set_xlabel(r"$E/\hbar\omega$ ($n=3$ oscillators)") + ax.set_ylabel(r"level degeneracy $\Omega(M,n)$") + ax.set_xlim(-0.5, 14.5) + ax.set_ylim(0, 125) + save(fig, "fig_sf_comb") + + +# ============================================================================ +# Fig 5 -- the bridge: (a) Legendre geometry on the exact SHO entropy +# sigma(mu); (b) the saddle integrand rho(E)e^{-beta E} sharpening as 1/sqrt(n). +# ============================================================================ +def fig_bridge(): + fig, (a, b) = plt.subplots(1, 2, figsize=(6.4, 2.9)) + mu = np.linspace(1e-3, 3.2, 400) + sig = (1 + mu) * np.log(1 + mu) - mu * np.log(mu) + a.plot(mu, sig, color=C_BLUE) + mus, slope = 1.0, math.log(2) # beta* hbar omega = ln 2 at mu=1 + sigs = 2 * math.log(2) + tang = sigs + slope * (mu - mus) + a.plot(mu, tang, color=INK, ls="--", lw=1.1) + a.plot([mus], [sigs], "o", ms=4, color=C_MAGENTA, zorder=5) + a.plot([0], [sigs - slope * mus], "o", ms=4, mfc="none", mec=C_GREEN, mew=1.2, + zorder=5, clip_on=False) + a.annotate(r"$S(E)/nk_B=\sigma(\mu)$", xy=(2.28, 1.80), color=C_BLUE, + fontsize=8.5, ha="left", va="top") + a.annotate(r"slope $=\beta^{*}\hbar\omega=\ln 2$", xy=(1.95, 2.25), + fontsize=8.5, color=INK2, rotation=20, ha="center") + a.annotate(r"intercept $=-F/nk_BT=\ln 2$", xy=(0.06, 0.48), + fontsize=8.5, color=C_GREEN) + a.set_xlabel(r"$\mu=E/n\hbar\omega$") + a.set_ylabel(r"entropy per oscillator $/k_B$") + a.set_xlim(0, 3.2) + a.set_ylim(0, 3.0) + a.set_title("(a) Legendre geometry", fontsize=9, color=INK2) + + x = np.linspace(0.4, 1.8, 600) + for n, c in zip([10, 100, 1000], BLUES[1:]): + expo = n * ((1 + x) * np.log(1 + x) - x * np.log(x) - x * math.log(2)) + expo -= expo.max() + b.plot(x, np.exp(expo), color=c, label=f"$n={n}$") + b.axvline(1.0, color=INK2, lw=0.7, ls=":") + b.annotate(r"width $\propto 1/\sqrt{n}$", xy=(1.28, 0.55), fontsize=8.5, + color=INK2) + b.set_xlabel(r"$E/E^{*}$") + b.set_ylabel(r"$\rho(E)\,e^{-\beta E}$ (norm.)") + b.set_title("(b) the saddle sharpens", fontsize=9, color=INK2) + b.legend(loc="upper left", handlelength=1.5) + fig.tight_layout(w_pad=2.0) + save(fig, "fig_sf_bridge") + + +# ============================================================================ +# Fig 6 -- non-concave S(E): the Legendre/canonical picture sees only the +# concave hull (Maxwell double tangent); the convex intruder is invisible to Z. +# ============================================================================ +def fig_nonconcave(): + fig, ax = plt.subplots(figsize=(4.9, 3.0)) + E = np.linspace(0.02, 3.0, 900) + S = 1.9 * np.sqrt(E) - 0.42 * np.exp(-((E - 1.15) / 0.30) ** 2) + # upper concave hull via monotone scan of slopes + pts = list(zip(E, S)) + hull = [pts[0]] + for p in pts[1:]: + while len(hull) >= 2: + (x1, y1), (x2, y2) = hull[-2], hull[-1] + if (y2 - y1) * (p[0] - x2) <= (p[1] - y2) * (x2 - x1): + hull.pop() + else: + break + hull.append(p) + hx = np.array([p[0] for p in hull]) + hy = np.array([p[1] for p in hull]) + Hy = np.interp(E, hx, hy) + gap = Hy - S > 1e-4 + i1, i2 = np.argmax(gap), len(gap) - 1 - np.argmax(gap[::-1]) + E1, E2 = E[i1], E[i2] + ax.fill_between(E, S, Hy, where=gap, color=C_MAGENTA, alpha=0.20, lw=0) + ax.plot(E, S, color=C_BLUE, label=r"$S(E)$ (microcanonical)") + ax.plot(E, Hy, color=INK, ls="--", lw=1.2, label=r"concave hull $=$ what $Z$ sees") + for Ep in (E1, E2): + ax.plot([Ep], [np.interp(Ep, E, Hy)], "o", ms=4, color=C_GREEN, zorder=5) + ax.annotate("", xy=(E2, 0.55), xytext=(E1, 0.55), + arrowprops=dict(arrowstyle="<->", color=INK2, lw=0.9)) + ax.annotate(r"latent heat $\Delta E$", xy=((E1 + E2) / 2, 0.60), + ha="center", fontsize=8.5, color=INK2) + ax.annotate(r"convex intruder" "\n" r"(lost by Legendre)", + xy=((E1 + E2) / 2, 1.28), ha="center", fontsize=8.5, + color="#b0447a") + ax.annotate(r"double tangent, slope $1/T_c$", xy=(2.0, 2.36), fontsize=8.5, + color=INK2, rotation=13) + ax.set_xlabel(r"$E$ (arb.)") + ax.set_ylabel(r"$S(E)$ (arb.)") + ax.set_xlim(0, 3.0) + ax.set_ylim(0, 3.4) + ax.legend(loc="upper left", handlelength=1.7) + save(fig, "fig_sf_nonconcave") + + +# ============================================================================ +# Fig 7 -- ergodicity and its failure: mode energies of a 32-oscillator chain, +# harmonic (frozen) vs alpha-FPUT (mixing, with the famous near-recurrence). +# ============================================================================ +def fput_modes(alpha, tmax, dt=0.05, N=32, nrec=1200): + """Velocity-Verlet integration of the fixed-end alpha-FPUT chain, energy + started in mode 1; returns times (in mode-1 periods) and E_k(t), k=1..4.""" + i = np.arange(1, N + 1) + k = np.arange(1, 5)[:, None] + modes = np.sqrt(2.0 / (N + 1)) * np.sin(np.pi * k * i / (N + 1)) # (4, N) + om = 2 * np.sin(np.pi * np.arange(1, 5) / (2 * (N + 1))) + x = np.sin(np.pi * i / (N + 1)) # amplitude 1 in mode 1 + v = np.zeros(N) + + def force(x): + xp = np.concatenate(([0.0], x, [0.0])) + d = np.diff(xp) # d_j = x_j - x_{j-1} + f = d[1:] - d[:-1] + alpha * (d[1:] ** 2 - d[:-1] ** 2) + return f + + nst = int(tmax / dt) + every = max(1, nst // nrec) + ts, Es = [], [] + f = force(x) + for s in range(nst): + v += 0.5 * dt * f + x += dt * v + f = force(x) + v += 0.5 * dt * f + if s % every == 0: + Q = modes @ x + P = modes @ v + Es.append(0.5 * (P ** 2 + (om * Q[: 4].T) ** 2)) + ts.append((s + 1) * dt) + T1 = 2 * np.pi / om[0] + return np.array(ts) / T1, np.array(Es) + + +def fig_fput(): + fig, (a, b) = plt.subplots(1, 2, figsize=(6.4, 2.9), sharey=True) + cols = [C_BLUE, C_GREEN, C_MAGENTA, C_YELLOW] + t0, E0 = fput_modes(alpha=0.0, tmax=30000) + t1, E1 = fput_modes(alpha=0.25, tmax=30000) + for k in range(4): + a.plot(t0, E0[:, k], color=cols[k], lw=1.1) + b.plot(t1, E1[:, k], color=cols[k], lw=1.1, + label=f"mode {k+1}") + a.annotate("mode 1", xy=(0.5, 0.86), xycoords="axes fraction", + color=C_BLUE, fontsize=8.5) + a.annotate("modes 2–4 (exactly zero forever)", xy=(0.5, 0.10), + xycoords="axes fraction", color=INK2, fontsize=8.5, ha="center") + a.set_ylim(0, 0.088) + a.set_title(r"(a) harmonic ($\alpha=0$): integrable, frozen", + fontsize=9, color=INK2) + b.set_title(r"(b) $\alpha$-FPUT ($\alpha=1/4$): mixing $+$ recurrence", + fontsize=9, color=INK2) + a.set_xlabel(r"$t$ (mode-1 periods)") + b.set_xlabel(r"$t$ (mode-1 periods)") + a.set_ylabel(r"mode energy $E_k$") + b.legend(loc="upper right", ncols=2, columnspacing=0.9, handlelength=1.3) + fig.tight_layout(w_pad=1.6) + save(fig, "fig_sf_fput") + + +# ============================================================================ +# Fig 8 -- the entropy inversion, with the code's own formulas. +# Transcribed from src/tcpyPI/utilities.py and pi.py (constants.py values); +# cross-checked below against the source docstring examples. +# ============================================================================ +CPD, CPV, CL = 1005.7, 1870.0, 2500.0 +CPVMCL = CPV - CL +RV, RD = 461.5, 287.04 +EPS = RD / RV +ALV0 = 2.501e6 + + +def es_cc(TC): + return 6.112 * math.exp(17.67 * TC / (243.5 + TC)) + + +def Lv(TC): + return ALV0 + CPVMCL * TC + + +def ev(R, P): + return R * P / (EPS + R) + + +def rv_(E, P): + return EPS * E / (P - E) + + +def entropy_S(T, R, P): + EV = ev(R, P) + ES = es_cc(T - 273.15) + RH = min(EV / ES, 1.0) + ALV = Lv(T - 273.15) + return (CPD + R * CL) * math.log(T) - RD * math.log(P - EV) \ + + ALV * R / T - R * RV * math.log(RH) + + +def s_sat(T, RP, P): + """Saturated entropy SG exactly as in solve_temperature_from_entropy.""" + TC = T - 273.15 + ES = es_cc(TC) + RG = rv_(ES, P) + ALV = Lv(TC) + EM = ev(RG, P) + return (CPD + RP * CL) * math.log(T) - RD * math.log(P - EM) + ALV * RG / T + + +def newton_track(S, P, RP, T_initial): + """The code's guarded Newton iteration, returning the iterate track.""" + TGNEW, TG, NC, track = T_initial, 0.0, 0, [T_initial] + while abs(TGNEW - TG) > 0.001: + TG = TGNEW + TC = TG - 273.15 + ENEW = es_cc(TC) + RG = rv_(ENEW, P) + NC += 1 + ALV = Lv(TC) + SL = (CPD + RP * CL + ALV * ALV * RG / (RV * TG * TG)) / TG + EM = ev(RG, P) + SG = (CPD + RP * CL) * math.log(TG) - RD * math.log(P - EM) + ALV * RG / TG + AP = 0.3 if NC < 3 else 1.0 + TGNEW = TG + AP * (S - SG) / SL + track.append(TGNEW) + if NC > 500 or ENEW > P - 1: + break + return track + + +def check_transcription(): + assert abs(es_cc(20) - 23.369) < 1e-2, es_cc(20) + assert abs(es_cc(0) - 6.112) < 1e-9 + assert abs(Lv(20) - 2488400.0) < 1e-6 + assert abs(ev(0.01, 1000) - 15.823) < 1e-2 + assert abs(rv_(15.942, 1000) - 0.010076) < 1e-5 + assert abs(entropy_S(300, 0.01, 1000) - 3987.17) < 1e-1 + tr = newton_track(4000.0, 1000.0, 0.01, 300.0) + assert abs(tr[-1] - 292.676) < 1e-2, tr[-1] + print("transcription cross-checks vs source docstrings: OK") + + +def fig_inversion(): + check_transcription() + RP, s0 = 0.018, entropy_S(300.0, 0.018, 1000.0) + fig, (a, b) = plt.subplots(1, 2, figsize=(6.4, 2.9), + gridspec_kw={"width_ratios": [1.35, 1]}) + T = np.linspace(232, 312, 500) + press = [1000, 850, 700, 500, 300] + blues5 = ["#c4d8f1", "#8fb3e5", "#5a8dd8", "#2a6ac0", "#123f7e"] + for P, c in zip(press, blues5): + sv = np.array([s_sat(t, RP, P) for t in T]) + a.plot(T, sv, color=c) + if P < 900: # mark roots only above the LCL (~950 hPa here) + Tstar = newton_track(s0, P, RP, 300.0)[-1] + a.plot([Tstar], [s0], "o", ms=3.8, color=c, zorder=5) + a.annotate(f"{P}", xy=(T[-1] + 1, sv[-1]), fontsize=7.5, color=c, + va="center", annotation_clip=False) + a.axhline(s0, color=INK2, lw=0.9, ls="--") + a.annotate(r"$s_0$", xy=(234, s0 + 22), fontsize=9, color=INK2) + a.annotate("hPa", xy=(313, s_sat(312, RP, 300)), fontsize=7.5, color=INK2, + annotation_clip=False, va="bottom") + a.set_xlabel(r"$T$ (K)") + a.set_ylabel(r"$s_{\mathrm{sat}}(T,p)$ (J kg$^{-1}$K$^{-1}$)") + a.set_title(r"(a) monotone $s_{\mathrm{sat}}$, one root per level", + fontsize=9, color=INK2) + a.set_xlim(232, 312) + + for P, T0, c, lab in [(500, 300.0, C_BLUE, r"start $300\,$K"), + (500, 240.0, C_GREEN, r"start $240\,$K")]: + tr = np.array(newton_track(s0, P, RP, T0)) + Tstar = tr[-1] + err = np.abs(tr - Tstar)[:-1] + err = np.where(err > 0, err, 1e-14) + b.semilogy(range(len(err)), err, "o-", ms=3.2, lw=1.1, color=c, + label=lab) + b.axvspan(-0.4, 2.0, color=C_YELLOW, alpha=0.18, lw=0) + b.annotate(r"guarded" "\n" r"$AP=0.3$", xy=(0.8, 3e-4), fontsize=8, + color=INK2, ha="center") + b.set_xlabel(r"Newton iteration") + b.set_ylabel(r"$|T_n-T^{*}|$ (K)") + b.set_title("(b) guarded Newton, $p=500$ hPa", fontsize=9, color=INK2) + b.set_ylim(top=2000) + b.legend(loc="upper right", handlelength=1.4) + fig.tight_layout(w_pad=2.2) + save(fig, "fig_sf_inversion") + + +if __name__ == "__main__": + fig_classical_marginal() + fig_quantum_marginal() + fig_typicality() + fig_comb() + fig_bridge() + fig_nonconcave() + fig_fput() + fig_inversion() + print("all figures written to", FIGDIR) diff --git a/discontinuity_analysis/thermodynamic_foundations.pdf b/discontinuity_analysis/thermodynamic_foundations.pdf index 3480501ef5bb35992070769a8e5636dad1c0abdc..9d6c52a6449d829ba6eac92a6d770e94616595f3 100644 GIT binary patch delta 586836 zcmY(qV{j&1)TJHUwr$(C?T(X+H3z z6aPbUoX<`}sUji8$jZbCM>)SdvJS_}$wR_I;%I6MCm;aFtYG0_M)=8Sx+hQrDfoNBlL$XN<(6Q229^3%xsHo6bnCR@Ao@?VSx>& z*W#A`zxqrVQyNf+u*$;U%=O@Qk5s9Zl2MzDOq1#s3H}LE{9CDxj?I4v)H$$#o_OWB zX(yZY6pEr@4u<;V$jW`)M&y#esgi!3GBWBv`_F+DTWvY7=^KEx$8a80WG3aU@nCc*=I)y;j<I{=nMY2yoOrv9G5Wp4~U^T+HwC4^QU)(`>U#nAM6vry>dLwwzN>^m#y-H^edbIH*Q|M{-k-L6 zBSU*#eFvbF4gfeH^DOX*NF2B~a;r+x8OgXMWs~%zwd+7#kYT^wg$wZ|BH2i`mclTC z+mI@^Uaga-=b%+*l!9d!p_3~5EEtU_ezu!U1- z6O4fh1_=(IT0T~h)0?^N&iJ7cz{hlOs?!VaL@XU=0nFD@LN>9tA|={hxHyXkJg07= zi@4tGHViLtYx-YLx`u~6BX-7Y^QV?^*==spr2{h%P>$2u&7ELFau(r{sC**UiLx(? z+1Tf02(8OOD>RN7y6^}w9@`H=a*=S5o=@wio;)+c6iFt};%#aKD(YGKW8KF@9EiP* zh+vE6THm{im|QH!Q9Iu$r8CW_b$in0_s_HI%Fx{rECyi7;d-Z+8ABPKxutfZY*I-H 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[PATCH 09/11] Fix rho/Omega conflation in the bridge (density vs count) The 2.6 "transform" paragraph wrongly equated the density of states rho(E) with the count Omega(E) and then took log of "the density of states," which is ill-defined. Rewritten to keep them distinct: - rho(E) = Tr delta(E - H) = sum_i delta(E - E_i): a distribution (comb of spikes); note delta(E - H) is the operator function of H traced AFTER (same rule as Z = Tr e^{-beta H}), and its log is meaningless. - Omega(E) = int_E^{E+dE} rho ~ rhobar(E) dE: the dimensionless count of levels in a thin shell, and S = k_B ln Omega. Omega and rhobar agree inside the log but differ by the units-carrying window dE -- hence the two symbols. - Laplace transform is of the density: Z = int rho e^{-beta E} = sum_i e^{-beta E_i}, the kernel coarse-graining the comb for free (why Z needs no window). Bromwich inverse recovers rho, not Omega. - SHO noted as the clean case: discrete spectrum, Omega(M,n) an honest integer degeneracy, no window needed. - Consistency fixes in the elegance bullets: rho(E) = Tr delta(E - H) and rho = rho_1 * rho_2. Build clean: 21 pp., 0 overfull/underfull, no undefined references. Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01LZzvFfJYN7oiMWW5wZ5bhG --- .../thermodynamic_foundations.pdf | Bin 538067 -> 540560 bytes .../thermodynamic_foundations.tex | 46 ++++++++++++++---- 2 files changed, 36 insertions(+), 10 deletions(-) diff --git a/discontinuity_analysis/thermodynamic_foundations.pdf b/discontinuity_analysis/thermodynamic_foundations.pdf index 3b54d003a68f4c16f1702adb92b19eac2f194a14..3480501ef5bb35992070769a8e5636dad1c0abdc 100644 GIT binary patch delta 149411 zcmV(;K-<65>mZQ-Adn^jHDxe_t^{-$Ba!@0NX>Gc!;6Avp#K}J01Lg~O1H>STh^QMSUKch+GOk6^5xo?#BYjd zVNU}$m+6JcSZXcE(9i2?+{5|b&zD#+e}<`aAbmp9mVH=r1JZNtWz{D*1?-X=Caj*a zU({O~O#+6|y*nJ?U~J;rqCd4ROb%kT$;-8G?=D-ZXQNvOG?}M)s6tz@EsKYuCoYs_ zT!73KX9}L4iLaW&v8abh`-b*x<#^9FWxza-Qa3z7a(EPNU!wRoWl>Sfw(K@Yf6&x@ zS)a=OOwDB&K+NI`owa{Iwm8Cd-HBePwQH)Eu3b8UG-Fu`Le)&0Aj|Jg)tPo6^b^{; zy?&OViRL)tJPnUty z2B-1zqOhp~A5(Tcy?Wh664Kv=O=NSiD+6vWrP;x}P!p8J@&4(G&06Nkf1bXMt=Bk5 zUsW)k%8yCGm^KYIM3W*aQePYl-VdA~31TqZY-G65`Y;@)P4-O%1GXi}%YuCbqTfzM zP>0LulTUl?wx4;!6^zUIPNia2#Dhkzv;?DSdbC3=@}vunWJC5z{my$5a?pM$Y}|kc zIQ4Ez;~tRW{3cb)G=r82e*>q!tjfN09hm6Z=)bRI&;ae9jfBI#2ahW>8c67IQ!AUM{B?U05f-u7X9~yG<8H1~WH{1xAcXbI;&`53vKrgoK z-NE}W-!n!!xkoykcQht=T#m=mxdRTQ&<`!o4ipJbuKKj2rV!9@$}Wgu2xy+I!1c)4 zT}1@RQb>uk5hww_S9FEXa>R*RgVX=b2OgPdW>}6se-Y%KhM1}n_Zo3vcty;B5y%T4 zo67JE@33~KKS{GD3d^nEp-L0T!?oXz$=$}1P44O z>oy_c(>LxrBn0JV!L1Bt$^i;zYC0~M$*v?^h;;c7;?Ty@`N_zP z@pNz)o3`mT&C%!FF!G18IlU~a%00cDi|@FDIh)p@^=S^Gfo^mqJNov9=NI;#PxDLO zX=C2$o5|XMk`P0;r)Kh1W7d~+N~BDPAg2h+f2|o(O8C$bK$(bTO!CSfVpZJ*=ZfC% zIyD%pje+bEW9)dnvVIwI-P^%gaR7O0I_?Gn7<7QAY_<{_#hVs^AgCbioQ%0&-(@NR zy%njw6&HskB`gUcYrd$!ELXk}eJE`Jxu6r#;o1SXJ^# zGURsj?%_-^xLsWFW&~EbWRmNMgIb6#e<5<#TOR?_0HaeXpzsKB;H7I$fN^ls;E#Sh z9RN&(@S9$E00R&66m1yy+o|!%o^R}i$!{lon7P|jO_` z$YhewC*W`pN(RwKhf{x!B_gF_R??h&5~kFUy?6BKUT@ry_61*P)BB(B2nw_`e@3hc zP%r^N9Zd>riqbL({sTNBQ&z@uSiM#vLEwUqjtW428%!wVlxem4b#>MBJ(ZQfTjV?! 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