Irreversible Boundaries — Examples

Let $\mathrm{\Omega }=\{{\omega }_1,{\omega }_2\}$ with ${\omega }_1\mathrm{\ne }{\omega }_2$, and declare the finite numerical energy map $ϵ:\mathrm{\Omega }\to \mathbb{R}$ by $ϵ({\omega }_1)=ϵ({\omega }_2)=0$, so the assigned physical energies are $E({\omega }_i)=0\mathrm{J}$. This supplies the finite microstate pair $(\mathrm{\Omega },ϵ)$ required in Macroscopic and microscopic entropy quantities; no mechanical internal-energy or equilibrium-macrostate identification is assumed. Choose $\mathcal{M}=\{m\}$ and the constant coarse-graining map $\pi :\mathrm{\Omega }\to \mathcal{M}$, so both states belong to the macrocell $M_m=\mathrm{\Omega }$, and assign $p_1=1,p_2=0$. The probability vector is deliberately nonuniform. The example shows that the macrocell-count definition and the Gibbs–Shannon definition can take different values on separately specified inputs.

Let $\mathrm{\Omega }=\{{\omega }_{+},{\omega }_{-}\}$ with weights $w({\omega }_{+})=3\mathrm{/}4$ and $w({\omega }_{-})=1\mathrm{/}4$. Both weights are nonnegative and sum to $1$, so $(\mathrm{\Omega },w)$ is a finite probability space as defined in Finite probability spaces, outcome weights, events, and event probabilities. For every event $A\subseteq \mathrm{\Omega }$, write $P(A)={\sum }_{\omega \in A}w(\omega )$ for its event probability. Let $Y:\mathrm{\Omega }\to \mathbb{R}$ be given by $Y({\omega }_{+})=2a$ and $Y({\omega }_{-})=-2a$, where $a>0$ and the values are numerical values in the unit $\mathrm{J}{\mathrm{K}}^{-1}$. By Real random variables on finite probability spaces and their finite distributions, $Y$ is a finite-valued real random variable on this finite probability space, so its expectation is a finite sum. The label $Y$ is only an illustrative entropy-valued variable; this example adopts no physical path law.

This arithmetic example illustrates the distinction in Hypotheses for microscopic fluctuation claims.

Use the one-coordinate convention of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ explicitly: $\iota :\mathbb{R}\to {\mathbb{R}}^1$ sends $a$ to the function on $\{0\}$ with value $\iota (a)(0)=a$, and the inverse takes the coordinate at index zero. In every scalar display below, a one-coordinate parameter or field is written through this bijection. Thus $\mathcal{Z}$ means $\iota ((0,\mathrm{\infty }))$, the scalar formula for $z$ denotes its $\iota$ image, $C(e,y)=e$ means $C(e,y)=\iota (e)$, $c(a)=a$ is the identity on $\mathcal{Z}$, and $q(\iota (a))=(a,0)$; temperature and inequalities use the scalar coordinate $a$. This is a notation convention, not a set-theoretic identity between a real and a function.

Take $\mathcal{O}=(0,L{)}^3$ in metre coordinates and $I=(0,\tau )$ in second coordinates, with $L>0\mathrm{m}$ and $\tau >0\mathrm{s}$. Choose ${\mathrm{\Theta }}_0>0\mathrm{K}$ and dimensionless $0<\alpha <1$. Instantiate $m=1$ and the local-state parameter set $\mathcal{Z}=(0,\mathrm{\infty })\subseteq {\mathbb{R}}^1$, and define its one-coordinate field $z:\mathcal{O}\times I\to \mathcal{Z}$ by

$z(\mathbf{x},t)=\frac{{\mathrm{\Theta }}_0}{1\mathrm{K}}(1+\alpha \frac{x_1}{L}),T(\mathbf{x},t)=(1\mathrm{K})z(\mathbf{x},t)={\mathrm{\Theta }}_0(1+\alpha \frac{x_1}{L})\mathrm{.}$

For the homogeneous comparison cell, supply $\mathcal{X}=(0,\mathrm{\infty })\times \mathbb{R}$ with states $(e,y)$, physical energy $(1\mathrm{J})e$, and numerical entropy $s(e,y)=\ln e-y^2$, where $\ln :(0,\mathrm{\infty })\to \mathbb{R}$ is the natural logarithm of The natural logarithm as the inverse of the exponential function and physical entropy is $(1\mathrm{J}{\mathrm{K}}^{-1})s(e,y)$. The coordinate $y$ is dimensionless. Set $r=1$, $C(e,y)=e$, $c(a)=a$, and $B_a=\mathcal{X}$ for every $a\in \mathcal{Z}$, so the fixed-energy admissible set is $A_a=\{(a,y):y\in \mathbb{R}\}$. Its entropy-maximizing set is $E_a=\{(a,0)\}$. Define ${\mathcal{E}}_{\mathrm{e}\mathrm{q}}={\bigcup }_{a>0}{E}_a$ and supply $q(a)=(a,0)$; assign this comparison state temperature $(1\mathrm{K})a$. Stipulate the local-equilibrium approximation with this map, so the local constitutive state at $(\mathbf{x},t)$ is $q(z(\mathbf{x},t))$. The field is static and nonconstant; this stipulation is a model assumption, not an assertion that a physical sample realizes the profile.

On $\mathcal{O}=(0,L{)}^3$ in metre coordinates and $I=(0,\tau )$ in second coordinates, with $L>0\mathrm{m}$ and $\tau >0\mathrm{s}$, prescribe $T(\mathbf{x},t)=T_0+\mathrm{\Delta }T{x}_1\mathrm{/}L$, where $T_0>0\mathrm{K}$ and $\mathrm{\Delta }T>0\mathrm{K}$. In the chosen Cartesian coordinates, define ${\mathbf{e}}_1:=(1,0,0)\in {\mathbb{R}}^3$. Compare two isotropic model choices ${\kappa }_1,{\kappa }_2>0$, ${\kappa }_1\mathrm{\ne }{\kappa }_2$, each using ${\mathbf{j}}_Q=-{\kappa }_i\mathrm{\nabla }T$. The coefficients have units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$.

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