Irreversible Boundaries

Fix a material constitution $c$ and its equilibrium state domain $D_c$ from Equilibrium state domains. The adopted macroscopic entropy is the finite-valued state function $S_c:D_c\to \mathbb{R}$ of Concave macroscopic fundamental entropy: for every $x\in D_c$, $S_c(x)$ is a finite real value measured in $\mathrm{J}{\mathrm{K}}^{-1}$ under SI quantities and unit conventions. Its argument is an equilibrium macrostate. The definition does not assign $S_c$ to an arbitrary nonequilibrium configuration or to a microscopic trajectory.

For a finite microstate model $(\mathrm{\Omega },ϵ)$ of Finite statistical microstate model, $\mathrm{\Omega }$ is finite and nonempty and $ϵ:\mathrm{\Omega }\to \mathbb{R}$ assigns a finite numerical energy. The model's physical energy is $E(x)=ϵ(x)\mathrm{J}$ for $x\in \mathrm{\Omega }$, with its zero and any retained internal offsets fixed by the declared material model. By The cardinality $\mid A\mid$ of a finite set, $N:=\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$ is the unique natural number equinumerous with $\mathrm{\Omega }$, and $N\ge 1$. Choose an enumeration $\mathrm{\Omega }=\{{\omega }_1,\dots ,{\omega }_N\}$. A chosen coarse-graining is a finite label set $\mathcal{M}$ and a function $\pi :\mathrm{\Omega }\to \mathcal{M}$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain). For $m\in \mathcal{M}$, let $M_m=\{x\in \mathrm{\Omega }:\pi (x)=m\}\subseteq \mathrm{\Omega }$ (Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$). Each $M_m$ is finite: choose a bijection $b:N\to \mathrm{\Omega }$ and set $J_m=b^{-1}(M_m)\subseteq N$. Induction on $n$ (The principle of mathematical induction) proves every subset $J\subseteq n$ finite. For $n=0$, $J=\mathrm{\varnothing }$; for $n=\sigma (k)$, if $k\mathrm{\notin }J$, then $J\subseteq k$; if $k\in J$, the set $J_0:=\{j\in J:j\mathrm{\ne }k\}\subseteq k$ is finite by induction, say via a bijection $f:n_0\to J_0$, and extending $f$ by $f(n_0)=k$ gives a bijection $\sigma (n_0)\to J$. Thus $J_m$ is finite, and the restriction $b{\mathrm{\mid }}_{J_m}:J_m\to M_m$ is a bijection. By The cardinality $\mid A\mid$ of a finite set, $N_m:=\mathrm{\mid }{M}_m\mathrm{\mid }$ is therefore defined. Only labels with $N_m\ge 1$ are admitted here. The Boltzmann macrocell entropy of this specified coarse-graining is

$S_{\mathrm{B}}(m):=k_{\mathrm{B}}\log N_m\mathrm{.}$

Under this enumeration, a probability vector $p=(p_1,\dots ,p_N)$ assigns probability $p_i$ to microstate ${\omega }_i$, where $p_i\ge 0$ and ${\sum }_{i=1}^N{p}_i=1$. Define its Gibbs–Shannon entropy by

$S_{\mathrm{G}}(p):=-k_{\mathrm{B}}{\sum }_{i=1}^N{p}_i\log p_i,$

with the summand $p_i\log p_i$ defined to be $0$ when $p_i=0$. Both expressions have units $\mathrm{J}{\mathrm{K}}^{-1}$, because their logarithms are dimensionless and $k_{\mathrm{B}}$ has those units. The finite sum and logarithm have the meanings of Finite sums and finite products, by recursion and The natural logarithm as the inverse of the exponential function.

For a density operator $\rho$ in the declared finite-dimensional sector class of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, its von Neumann physical entropy is $S_{\mathrm{v}\mathrm{N}}(\rho )=k_{\mathrm{B}}s(\rho )$, with $s(\rho )$ the dimensionless entropy defined there. That supplier defines no such quantity for its infinite-dimensional spatial sectors.

These are distinct definitions on different domains: $S_c$ acts on an equilibrium macrostate, $S_{\mathrm{B}}$ on a chosen macrocell, $S_{\mathrm{G}}$ on a specified probability vector, and $S_{\mathrm{v}\mathrm{N}}$ on a finite-sector density operator. A numerical equality among them requires an additional, explicit identification of model, coarse-graining, preparation and applicable limit. This item supplies no such identification and adopts no canonical or microcanonical sampling rule.

The finite microstate model and its numerical-to-physical energy assignment are those of Finite statistical microstate model. That definition supplies neither a sampling rule nor an identification of microscopic entropy with $S_c$; any such bridge requires separate assumptions and results.

Evidence record.

An empirical claim in a later item must identify the source report and exact locator, the material and sample, apparatus and geometry, boundary and control conditions, operational definitions of measured quantities, calibration, sampling procedure, and the source's reported uncertainty and systematic limitations. Preserve the units, resolution, sample size, and selection conditions actually given by the source. If the source omits one of these details, record the omission and limit the claim; do not supply a plausible value or precision.

Keep four records separate: source-reported observations; derived estimates from those observations; model predictions under stated assumptions; and statistical comparisons between estimates and predictions. State the sampling and measurement model and the inference method when a statistical conclusion is made. Agreement supports the specified model-and-apparatus combination; it does not prove a postulate. Evidence belongs in an experiment item or a nonlogical evidence relation unless a theorem actually assumes a qualified empirical result, in which case that qualification must be carried as an empirical premise.

The source treatments cited here discuss modelled fluctuations and transport. Their illustrative numerical values are not measurements made by this library. No experiment item or observation is supplied by this remark.

Scope.

The adopted Macroscopic second law for isolated compounds compares the total macroscopic entropies of specified equilibrium endpoints of an admitted isolated process record. It assigns no entropy function to nonequilibrium intermediate configurations and does not assert that every microscopic realization has nonnegative entropy change. The macrostate and finite-model quantities remain distinct as defined in Macroscopic and microscopic entropy quantities.

A stochastic fluctuation claim needs its own state space or path space, probability law, preparation, driving protocol, time interval, measured or modelled random variable, and averaging convention. A trajectory observable has the units stated by its model (for physical entropy production, $\mathrm{J}{\mathrm{K}}^{-1}$; for dimensionless entropy production, no units). If a trajectory variable is called entropy production, define it and state the hypotheses of the relation being used, including the forward and reverse protocols and the support and integrability conditions needed for the stated averages. Positivity of an ensemble mean does not imply positivity in every realization.

For a precise model, let $X$ be a specified set of states and let $\tau \in (0,\mathrm{\infty })\subseteq \mathbb{R}$ be the dimensionless numerical duration in seconds. The physical duration is $(1\mathrm{s})\tau$. The numerical time domain $[0,\tau ]$ is the closed real interval of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length; its coordinate $t$ represents physical elapsed time $(1\mathrm{s})t$. Let $\mathrm{\Gamma }$ be a specified set of allowed sample paths $\gamma :[0,\tau ]\to X$, where each $\gamma$ is a function in the sense of A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain. Any further path regularity or admissibility conditions are part of that model; none is imposed universally here. Equip $\mathrm{\Gamma }$ with an event $\sigma$-algebra $\mathcal{F}$ and choose a probability measure $P$ on $(\mathrm{\Gamma },\mathcal{F})$. A trajectory observable is a measurable map $\mathrm{\Sigma }:(\mathrm{\Gamma },\mathcal{F})\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$, where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra; if it represents physical entropy production, the model assigns its values units $\mathrm{J}{\mathrm{K}}^{-1}$, while a dimensionless entropy production has no units. When $\mathrm{\Sigma }$ is $P$-integrable, meaning ${\int }_{\mathrm{\Gamma }}\mathrm{\mid }\mathrm{\Sigma }\mathrm{\mid }dP<\mathrm{\infty }$, its expectation is the integral ${\mathbb{E}}_P[\mathrm{\Sigma }]:={\int }_{\mathrm{\Gamma }}\mathrm{\Sigma }dP$ by Expectation of a nonnegative or integrable random variable. A driving protocol is a declared control map $\lambda :[0,\tau ]\to \mathrm{\Lambda }$, with its parameter space $\mathrm{\Lambda }$, regularity and endpoint controls specified by the model; $\lambda$ is also a function as in A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain. Any reverse-protocol comparison needs its own path measure and support conditions.

The fluctuation–dissipation theorem discussed in the cited treatments relates equilibrium fluctuations to a specified linear response of a coupled model. It is distinct from a trajectory fluctuation relation and does not by itself prove the macroscopic second-law postulate. No universal microscopic fluctuation formula or trajectory-wise entropy inequality is asserted here.

Scope and objects.

For a declared continuum model, let $m\in \mathbb{N}$ with $m\ge 1$ (The natural numbers $\mathbb{N}$ (von Neumann)) and let $\mathcal{O}\subseteq {\mathbb{R}}^3$ be open in the metric topology induced by the Euclidean norm (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$). Its spatial coordinates are in metres. In the conventional physical display $\mathbf{x}=(x_1,x_2,x_3)$, $x_i$ denotes the function coordinate at index $i-1$ of the library's ${\mathbb{R}}^3$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). Let $I=(t_0,t_1)\subseteq \mathbb{R}$ be a nonempty open interval of time (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) with coordinates in seconds. A field is a function on the stated space-time domain, with its domain and codomain shown, as in A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain. The model chooses a local-state parameter set $\mathcal{Z}\subseteq {\mathbb{R}}^m$ and a field $z=(z_1,\dots ,z_m):\mathcal{O}\times I\to \mathcal{Z}$ whose component functions have continuous first partial derivatives in the three spatial coordinates and time; the partial derivatives are those of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. Each $z_j$ is the dimensionless numerical value of its local quantity divided by its model-declared reference unit $u_j$; the model specifies $u_j$ and the physical meaning of each component. In particular, if temperature is the component $z_{j_T}$, the model declares that index and reads $T(\mathbf{x},t)=(1\mathrm{K})z_{j_T}(\mathbf{x},t)$, so $T:\mathcal{O}\times I\to (0,\mathrm{\infty })$ has units kelvins. If the model includes mass density and fluid velocity, their types are $\rho :\mathcal{O}\times I\to [0,\mathrm{\infty })$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ and $\mathbf{u}:\mathcal{O}\times I\to {\mathbb{R}}^3$ in $\mathrm{m}{\mathrm{s}}^{-1}$; these fields may also be required to have continuous first partial derivatives. Each model must specify its remaining coordinates and equation of state; $\mathcal{Z}$ is not universal.

A local-equilibrium approximation requires an explicit equilibrium criterion for a homogeneous comparison cell at each parameter value $a\in \mathcal{Z}$. One entropy-based specification consists of a set $\mathcal{X}$ of cell states, a finite-valued numerical entropy $s:\mathcal{X}\to \mathbb{R}$ (physical entropy $(1\mathrm{J}{\mathrm{K}}^{-1})s$), a map $C:\mathcal{X}\to {\mathbb{R}}^r$ of retained conserved quantities in declared reference units, a map $c:\mathcal{Z}\to {\mathbb{R}}^r$ of their fixed values, and a specified set $B_a\subseteq \mathcal{X}$ of states allowed by the retained cell constraints. Here $r\ge 1$ is a specified integer, and the physical meanings of the states, conserved quantities and constraints are model data. Define

$A_a:=\{e\in B_a:C(e)=c(a)\},E_a:=\{e\in A_a:s(e)\ge s(e^{\mathrm{\prime }})\text{ for every }{e}^{\mathrm{\prime }}\in A_a\},{\mathcal{E}}_{\mathrm{e}\mathrm{q}}:={\bigcup }_{a\in \mathcal{Z}}{E}_a\mathrm{.}$

Thus equilibrium means entropy maximization under the stated constraints, not an arbitrary label. The model must supply a map $q:\mathcal{Z}\to {\mathcal{E}}_{\mathrm{e}\mathrm{q}}$ satisfying $q(a)\in E_a$ for every $a$; existence, attainment and a selection in the case of multiple maximizers are requirements on the supplied model, not conclusions here. No choice axiom or equilibration theorem is being invoked to construct $q$. A different equilibrium formulation must instead give its full mathematical condition, its physical constraints and parameter matching, and verify that its supplied $q(a)$ satisfies that condition; naming a target set equilibrium does not suffice. For example, a kinetic collision-equilibrium condition must specify the distribution class, collision operator and retained moments, and require the selected distribution to have those moments and zero collision term. This condition alone does not assert a maximum-entropy characterization or solve the full transport equation.

The local-equilibrium assumption assigns the comparison-cell state $q(z(\mathbf{x},t))$ at each point. It adopts the cell's equilibrium constitutive values locally; it does not claim that different cells are in mutual equilibrium. A model using finite coarse-grained cells must separately specify its cell partition and the map from cells to local parameters; this remark adopts no such partition. A kinetic model may declare a mean free path $\mathrm{\ell }>0\mathrm{m}$ from its intercollision-distance law, a characteristic field-variation length $L>0\mathrm{m}$, a collision-relaxation time ${\tau }_{\mathrm{c}}>0\mathrm{s}$, and a protocol-variation time ${\tau }_{\mathrm{m}}>0\mathrm{s}$. The model must give the mathematical construction of each scale (including any distribution and finite-mean condition when it uses an expectation) and its criterion for scale separation. Such a model may require $\mathrm{\ell }\mathrm{/}L\ll 1$ and ${\tau }_{\mathrm{c}}\mathrm{/}{\tau }_{\mathrm{m}}\ll 1$; the meaning of $\ll 1$ is set by that model and has no universal numerical threshold. Where a kinetic model has a distribution $f$ on a declared phase space and a local-equilibrium distribution $f_0(z)$, write its deviation as $\delta f:=f-f_0(z)$; $f$, $f_0$ and their domains are supplied by that model, and the local-equilibrium distribution alone need not solve the kinetic evolution equation.

Nonconstant fields $T(\mathbf{x},t)$, $\rho (\mathbf{x},t)$, or $\mathbf{u}(\mathbf{x},t)$ are permitted in this continuum model. The pointwise comparison-cell criterion alone neither establishes nor excludes equilibrium of the whole system: that requires a separately specified whole-system equilibrium criterion, including its external forces, boundary conditions and retained constraints. In particular, spatial nonconstancy alone is not a universal test for absence of global equilibrium. Initial data, boundary conditions, conservation equations, constitutive assumptions and their empirical scope must be supplied before any time evolution or transport conclusion is claimed.

Adopt exactly the state, observable, autonomous-dynamics and canonical-ensemble assumptions, readings, finite-dimensional sector class, and normalized canonical preparation of Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one. The sector ${\mathfrak{H}}_N^{\sigma }$ and its self-adjoint energy operator $H_N^{\sigma }$ are those of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable. Model time $t$ is in seconds, $H_N^{\sigma }$ is in joules, and $\mathrm{\hslash }=h\mathrm{/}(2\pi )$ is the reduced Planck constant in $\mathrm{J}\mathrm{s}$, as in Dimensions, SI unit conventions and quantity types of the framework. The evolution definition Assigned energy observable, unitary flow and the two pictures of autonomous evolution gives $U(t)=\exp (-it{H}_N^{\sigma }\mathrm{/}\mathrm{\hslash })$ and $\rho (t)=U(t)\rho (0)U(t{)}^{\ast }$. Both the Axiom of Countable Choice and the Axiom of Choice are assumed (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Define $S_{\mathrm{v}\mathrm{N}}(\rho )=k_{\mathrm{B}}s(\rho )$ as in Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model. Then, for every declared sector preparation $\rho (0)$ and every $t\in \mathbb{R}$:

1. $S_{\mathrm{v}\mathrm{N}}(\rho (t))=S_{\mathrm{v}\mathrm{N}}(\rho (0))$;
2. the adopted canonical preparation is stationary; and
3. a preparation different from the canonical one is never mapped to it by this unitary flow.

These are statements about the exact closed finite-sector model. They do not identify $S_{\mathrm{v}\mathrm{N}}$ with the macroscopic equilibrium entropy $S_c$ of Macroscopic and microscopic entropy quantities, derive the canonical preparation, or establish bath relaxation or thermalization.

Scope.

Use the fields and local-equilibrium regime of Scope of a local-equilibrium continuum branch. For this constitutive relation, assume $T$ has continuous first spatial partial derivatives, defined as in Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, and write

$\mathrm{\nabla }T:=({\mathrm{\partial }}_{x_1}T,{\mathrm{\partial }}_{x_2}T,{\mathrm{\partial }}_{x_3}T)\in {\mathbb{R}}^3\mathrm{.}$

Here the physical coordinate indices $i=1,2,3$ correspond to function indices $i-1$ in The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$.

The components of $\mathrm{\nabla }T$ have units $\mathrm{K}{\mathrm{m}}^{-1}$.

Matrices are elements of $M_{3\times 3}(\mathbb{R})$ as in The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations. In the physical one-based display, $K_{ij}$ denotes the library matrix entry with indices $(i-1,j-1)$, and $v_i$ denotes vector coordinate $i-1$, for $i,j=1,2,3$. The action is $(Kv{)}_i:={\sum }_{j=1}^3{K}_{ij}{v}_j$; $I_3$ is the matrix with entries $1$ on the diagonal and $0$ off it as in Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes. A Fourier heat-transport model may postulate

${\mathbf{j}}_Q(\mathbf{x},t)=-K(\mathbf{x},t)\mathrm{\nabla }T(\mathbf{x},t),$

where ${\mathbf{j}}_Q:\mathcal{O}\times I\to {\mathbb{R}}^3$ is a continuous heat-energy flux field in $\mathrm{W}{\mathrm{m}}^{-2}$ and $K:\mathcal{O}\times I\to M_{3\times 3}(\mathbb{R})$ is a declared conductivity tensor field with entries in $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. In an isotropic model $K(\mathbf{x},t)=\kappa (\mathbf{x},t)I_3$, where $\kappa :\mathcal{O}\times I\to [0,\mathrm{\infty })$ is the declared scalar conductivity field. A diffusion model may separately postulate ${\mathbf{j}}_n=-D\mathrm{\nabla }n$, where number density $n:\mathcal{O}\times I\to [0,\mathrm{\infty })$ is in ${\mathrm{m}}^{-3}$, number flux ${\mathbf{j}}_n:\mathcal{O}\times I\to {\mathbb{R}}^3$ is in ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, and $D$ is either a scalar field $D:\mathcal{O}\times I\to \mathbb{R}$ or a matrix field $D:\mathcal{O}\times I\to M_{3\times 3}(\mathbb{R})$ with coefficient units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$; its regularity and positivity conditions are model data.

These are constitutive choices for specified material, geometry, regime, field regularity and boundary conditions. In a kinetic model with microscopic state space $\mathrm{\Xi }$ and allowed collision transitions $\mathcal{C}\subseteq \mathrm{\Xi }\times \mathrm{\Xi }$, a collision invariant is a real-valued function $Q:\mathrm{\Xi }\to \mathbb{R}$ satisfying $Q({\xi }^{\mathrm{\prime }})=Q(\xi )$ for every $(\xi ,{\xi }^{\mathrm{\prime }})\in \mathcal{C}$. If the model has a distribution $f$ on a declared phase space and a collision operator $C[f]$ giving the collision contribution to its evolution, a single-relaxation-time replacement may be $C_{\tau }[f]=-(f-f_0)\mathrm{/}\tau$, with model-declared reference distribution $f_0$ and $\tau >0$. This replacement need not preserve collision invariants unless it is constrained to preserve their integrals against the model's state distribution. Different kinetic, linear-response and quantum transport models use different hypotheses and coefficient definitions; a coefficient derived in one is not thereby universal. No kinetic operator or reciprocity law is defined or asserted here. Microscopic calculations and measurements may determine or test coefficients for a specified model and material, but neither the equilibrium entropy postulates nor dimensional consistency alone determine them.

Any heat equation or transport conclusion additionally needs the relevant conservation balance, sources or work terms, constitutive relation, initial data and boundary conditions. No such evolution equation, Onsager relation, or transport coefficient is proved or adopted by this remark.