Entropy Foundation — Examples

This is a conditional model calculation, not a report of an observed compression. The gas is taken to have one mole, $n=1\mathrm{m}\mathrm{o}\mathrm{l}$, of a model whose state variables are a positive temperature $T$ in kelvins, volume $V$ in cubic metres, and uniform pressure $p$ in pascals. The model is monatomic and ideal in the explicit sense that its molar constant-volume heat capacity is $C_V=3R\mathrm{/}2$, its internal energy is $U=n{C}_{V}T$, and its equation of state is $pV=nRT$. Its entropy difference is

$S(T,V)-S(T_0,V_0)=n{C}_V\ln (T\mathrm{/}{T}_0)+nR\ln (V\mathrm{/}{V}_0),$

for positive reference values $T_0$ in K and $V_0$ in ${\mathrm{m}}^3$. At initial equilibrium let $T_i>0$ and $V_i>0$. A piston protocol suddenly raises the external pressure from the initial value $p_i$ to a constant final load $p_f$, compresses the gas adiabatically to $V_f=(9\mathrm{/}10)V_i$, and dissipatively settles it to a final equilibrium state. The protocol admits no reverse record, so it is irreversible by Reversible thermodynamic process record. The idealization assumes no change in the included load's entropy; all numerical behavior is a model assumption and deduction, not an empirical measurement.

Take two independent separated one-species systems $A$ and $B$ with material constitutions $c_A,c_B$ inside an isolated compound. Write $D_A:=D_{c_A}$ and $D_B:=D_{c_B}$ for their equilibrium domains; each is $(0,\mathrm{\infty }{)}^3$ in the dimensionless coordinates $({\widehat{U}}_i,{\widehat{V}}_i,{\widehat{n}}_i)$ of Equilibrium state domains, and each full domain is designated as an open single-phase chart. Fix positive volumes and amounts ${\widehat{V}}_A,{\widehat{V}}_B,{\widehat{n}}_A,{\widehat{n}}_B$, total internal energy $U_A+U_B=1\mathrm{J}$, and a thermal-contact protocol that permits internal energy transfer while keeping volumes and amounts fixed, as in Thermal contact family and constraints. Adopt the retained product endpoint set and the separately specified total maps below:

$\varphi (x)=((x,{\widehat{V}}_A,{\widehat{n}}_A),(1-x,{\widehat{V}}_B,{\widehat{n}}_B)),\mathcal{F}=\{\varphi (x):0<x<1\}\subseteq D_A\times D_B\mathrm{.}$

Thus $x={\widehat{U}}_A=U_A\mathrm{/}(1\mathrm{J})$ and $1-x={\widehat{U}}_B=U_B\mathrm{/}(1\mathrm{J})$. With $s_{\ast }=1\mathrm{J}{\mathrm{K}}^{-1}$, adopt the toy constitutive entropies $S_A=2s_{\ast }{\widehat{U}}_A$ and $S_B=s_{\ast }{\widehat{U}}_B$. They are finite, continuous, affine (hence concave), $C^2$, and have positive $S_U$ on the stated open domains. For each $\lambda >0$, use the same formula in the own coordinates of the distinct tags $c_A^{(\lambda )}$ and $c_B^{(\lambda )}$. Then the coordinate scaling ${\delta }_{\lambda }$ of Additive separated systems and tagged bulk scaling gives $S_{c_A^{(\lambda )}}({\delta }_{\lambda }{x}_A)=\lambda S_A(x_A)$ and $S_{c_B^{(\lambda )}}({\delta }_{\lambda }{x}_B)=\lambda S_B(x_B)$. Adopt $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x_A,x_B):=S_A(x_A)+S_B(x_B)$ as additional entropy data on the retained product endpoints, independently of simple aggregate composition. Its value on $\mathcal{F}$ is $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(\varphi (x))=s_{\ast }(1+x)$. The model-declared feasible set is an open energy-partition interval, so its entropy has a finite supremum but no attained maximum. This is a conditional model example, not a claim about a real material.

For complete contact data, set $E:=(0,1)$ and take $Z_{A\bigsqcup B}=Z_{A\bigsqcup B}^{\mathrm{e}\mathrm{q}}=\mathcal{F}$, with constituent maps ${\xi }_A,{\xi }_B$ the two coordinate projections and contact family $\mathcal{T}:=\{(c_A,x_A):x_A\in D_A\}\cup \{(c_B,x_B):x_B\in D_B\}$. The two constitutions share one species label. Give the aggregate constitution the domain $D_{A\bigsqcup B}=(0,\mathrm{\infty }{)}^3$, and set ${\xi }_{A\bigsqcup B}(\varphi (x))=(1,{\widehat{V}}_A+{\widehat{V}}_B,{\widehat{n}}_A+{\widehat{n}}_B)$. This is a supplied aggregate coordinate map for process records, not an assertion of an aggregate entropy identity. For each $x,y\in E$ with $y\ge x$, admit the record $P_{x,y}$ of Thermodynamic process records and transfer conventions on $I=[0,1]$ seconds: its aggregate endpoints are both this constant coordinate tuple, $z_{x,y}(t)=\varphi (x)$ for $0\le t<1$ and $z_{x,y}(1)=\varphi (y)$, its control record is $(x,y)$, and its signed transfer record is the real number $x-y$. Declare the control-record set to be $\{(x,y)\in E^2:y\ge x\}$ and the transfer-record set to be $\mathbb{R}$. Each transfer record supplies $H_{P_{x,y}}^{\mathrm{i}\mathrm{n}},H_{P_{x,y}}^{\mathrm{o}\mathrm{u}\mathrm{t}}:[0,1]\to [0,\mathrm{\infty })$, both identically zero, with numerical values in joules; these maps are nondecreasing and start at zero. No path regularity is required by the supplier. The outer boundary admits no matter or energy transfer; fixed constituent volumes and amounts and zero contact work are retained constraints. Supply fixed positive constituent and compound masses and zero endpoint center-of-mass velocities in one Galilean chart, and no separately tracked external-field energy. Thus these are core closed records, with $Q[P_{x,y}]=W_{\mathrm{o}\mathrm{n}}[P_{x,y}]=0$. For the contact-record set ${\mathcal{P}}_T=\{P_{x,y}:x,y\in E,y\ge x\}$ supply $q_{A\to B}[P_{x,y}]=x-y$ in joule magnitudes as in Thermal contact family and constraints. The constituent energy changes are $(y-x)\mathrm{J}$ and $(x-y)\mathrm{J}$. Each feasible endpoint has its self-record; nonzero transfers occur when $y>x$. No contact-equilibrium designation or relaxation law is assumed.

To check consistency with the separately scoped aggregate postulates, take ${\mathcal{C}}_{AB}:=\{(x_A,x_B)\in D_A\times D_B:{\widehat{U}}_A={\widehat{U}}_B\}$ and let ${\kappa }_{AB}$ sum their three coordinates. Adopt $S_{A\bigsqcup B}(\widehat{U},\widehat{V},\widehat{n})=\frac{3}{2}{s}_{\ast }\widehat{U}$. On this nonempty admission set, $S_A+S_B=3s_{\ast }{\widehat{U}}_A=S_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))$, and all dimensioned coordinates add. Use the same aggregate formula on its scaled tags and all positive-coordinate scaling domains. Only the retained partition $x=1\mathrm{/}2$ is in this aggregate admission set; there the product entropy and aggregate entropy both equal $3s_{\ast }\mathrm{/}2$. All other retained partitions have separately adopted product entropy, so no aggregate entropy identity is asserted for them. Totals and entropy on these complete partitioned endpoints are additional declared maps allowed by Isolated compound and feasible equilibrium set. The maximum criterion of Stable equilibrium requires an attained global entropy maximum is used on their stated entropy sum; no aggregate-composition or accessibility theorem is claimed.

Take two independent separated one-species systems $A$ and $B$ with material constitutions $c_A,c_B$ inside an isolated compound. Write $D_A:=D_{c_A}$ and $D_B:=D_{c_B}$ for their equilibrium domains; each is $(0,\mathrm{\infty }{)}^3$ in $({\widehat{U}}_i,{\widehat{V}}_i,{\widehat{n}}_i)$, and each full domain is designated as an open single-phase chart. Fix positive ${\widehat{V}}_A,{\widehat{V}}_B,{\widehat{n}}_A,{\widehat{n}}_B$, total energy $U_A+U_B=1\mathrm{J}$, $s_{\ast }=1\mathrm{J}{\mathrm{K}}^{-1}$, and toy entropies $S_A=2s_{\ast }{\widehat{U}}_A$, $S_B=s_{\ast }{\widehat{U}}_B$, using the same formulas on the distinct tagged constitutions $c_A^{(\lambda )}$ and $c_B^{(\lambda )}$ for every $\lambda >0$. Adopt the partitioned endpoint entropy $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x_A,x_B):=S_A(x_A)+S_B(x_B)$ as additional model data and permit internal thermal energy transfer at fixed volumes and amounts by the explicit records below. This product entropy is not identified with a simple aggregate entropy after discarding the energy partition. Let

$\varphi (x)=((x,{\widehat{V}}_A,{\widehat{n}}_A),(1-x,{\widehat{V}}_B,{\widehat{n}}_B)),\mathcal{F}=\{\varphi (x):x\in (0,3\mathrm{/}10]\cup [3\mathrm{/}5,9\mathrm{/}10]\}\mathrm{.}$

The retained barrier constraint admits exactly this nonconvex subset of the energy partitions; all its pairs satisfy the same total energy, volume, and amount constraints. The additive entropy is $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(\varphi (x))=s_{\ast }(1+x)$, so $x=3\mathrm{/}10$ is a local maximum relative to $\mathcal{F}$ but lies below the feasible global maximum at $x=9\mathrm{/}10$. These toy constitutive laws and constraints make no claim about a real material; calling an actual preparation metastable would require a separate kinetic model.

For complete contact data, set $E:=(0,3\mathrm{/}10]\cup [3\mathrm{/}5,9\mathrm{/}10]$ and take $Z_{A\bigsqcup B}=Z_{A\bigsqcup B}^{\mathrm{e}\mathrm{q}}=\mathcal{F}$, with constituent maps ${\xi }_A,{\xi }_B$ the two coordinate projections and contact family $\mathcal{T}:=\{(c_A,x_A):x_A\in D_A\}\cup \{(c_B,x_B):x_B\in D_B\}$. The two constitutions share one species label. Give the aggregate constitution the domain $D_{A\bigsqcup B}=(0,\mathrm{\infty }{)}^3$, and set ${\xi }_{A\bigsqcup B}(\varphi (x))=(1,{\widehat{V}}_A+{\widehat{V}}_B,{\widehat{n}}_A+{\widehat{n}}_B)$. This is a supplied aggregate coordinate map for process records, not an assertion of an aggregate entropy identity. For each $x,y\in E$ with $y\ge x$, admit the record $P_{x,y}$ of Thermodynamic process records and transfer conventions on $I=[0,1]$ seconds: its aggregate endpoints are both this constant coordinate tuple, $z_{x,y}(t)=\varphi (x)$ for $0\le t<1$ and $z_{x,y}(1)=\varphi (y)$, its control record is $(x,y)$, and its signed transfer record is the real number $x-y$. Declare the control-record set to be $\{(x,y)\in E^2:y\ge x\}$ and the transfer-record set to be $\mathbb{R}$. Each transfer record supplies $H_{P_{x,y}}^{\mathrm{i}\mathrm{n}},H_{P_{x,y}}^{\mathrm{o}\mathrm{u}\mathrm{t}}:[0,1]\to [0,\mathrm{\infty })$, both identically zero, with numerical values in joules; these maps are nondecreasing and start at zero. No path regularity is required by the supplier. The outer boundary admits no matter or energy transfer; fixed constituent volumes and amounts and zero contact work are retained constraints. Supply fixed positive constituent and compound masses and zero endpoint center-of-mass velocities in one Galilean chart, and no separately tracked external-field energy. Thus these are core closed records, with $Q[P_{x,y}]=W_{\mathrm{o}\mathrm{n}}[P_{x,y}]=0$. For the contact-record set ${\mathcal{P}}_T=\{P_{x,y}:x,y\in E,y\ge x\}$ supply $q_{A\to B}[P_{x,y}]=x-y$ in joule magnitudes as in Thermal contact family and constraints. The constituent energy changes are $(y-x)\mathrm{J}$ and $(x-y)\mathrm{J}$. Each feasible endpoint has its self-record; nonzero transfers occur when $y>x$. No contact-equilibrium designation or relaxation law is assumed.

To check consistency with the separately scoped aggregate postulates, take ${\mathcal{C}}_{AB}:=\{(x_A,x_B)\in D_A\times D_B:{\widehat{U}}_A={\widehat{U}}_B\}$ and let ${\kappa }_{AB}$ sum their three coordinates. Adopt $S_{A\bigsqcup B}(\widehat{U},\widehat{V},\widehat{n})=\frac{3}{2}{s}_{\ast }\widehat{U}$. On this nonempty admission set, $S_A+S_B=3s_{\ast }{\widehat{U}}_A=S_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))$, and all dimensioned coordinates add. Use the same aggregate formula on its scaled tags and all positive-coordinate scaling domains. The retained partitions in $\mathcal{F}$ have $x\mathrm{\ne }1\mathrm{/}2$ and are outside this aggregate admission set, so their separately adopted product entropy does not contradict the aggregate identity. Totals and entropy on these complete partitioned endpoints are additional declared maps allowed by Isolated compound and feasible equilibrium set. The maximum criterion of Stable equilibrium requires an attained global entropy maximum is used on their stated entropy sum; no aggregate-composition or accessibility theorem is claimed.