Cycles And Low Temperature

Fix a working system $s$ of constitution $c$ and a closed process record $P\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$ with equilibrium endpoints as in Thermodynamic process records and transfer conventions. A cycle of the working system is such a record with $x_P^{-}=x_P^{+}$; the reservoirs and work store need not return to their initial energy states.

An ideal thermal reservoir $b$ has an energy interval $J_b\subseteq \mathbb{R}$ measured in joules and a finite entropy function $S_b:J_b\to \mathbb{R}$ measured in $\mathrm{J}{\mathrm{K}}^{-1}$. On the energy interval admitted for a cycle, assume the exact affine law $S_b(E_b)=s_{b,0}+\frac{E_b}{T_b}$ for a constant $s_{b,0}$ and fixed $T_b>0$ kelvins. Thus $S_b^{\mathrm{\prime }}(E_b)=1\mathrm{/}{T}_b$ in the interior and every admitted endpoint change is exactly $\mathrm{\Delta }{S}_b=\mathrm{\Delta }{E}_b\mathrm{/}{T}_b$. This constant is the absolute thermodynamic temperature under Entropic temperature, pressure, and molar chemical potentials. The affine entropy law on the admitted interval is the ideal-bath model; a finite reservoir whose temperature changes is outside this definition.

An ideal work store is a named auxiliary with an energy coordinate $E_w$ measured in joules and an assigned entropy $S_w(E_w)=s_w$ constant on the energy interval used by the cycle. It exchanges only mechanical work and no matter or heat. Its energy may change, while its entropy change is zero by this model definition.

A two-reservoir cycle consists of one working-system cycle, two labelled baths $h,c$ at fixed temperatures $T_h,T_c>0$, and the ideal work store. The labels specify the transfer roles below; the general bookkeeping permits equal or reversed positive bath temperatures. In the strictly ordered subfamily $0<T_c<T_h$, call $h$ the hot bath and $c$ the cold bath. The engine-efficiency and refrigerator bounds use this subfamily. No existence of a cycle with prescribed transfer signs is asserted. They lie inside one isolated outer boundary in the sense of Thermodynamic systems and permitted exchanges, so the full compound has no external matter or energy transfer. There are no unlisted heat, work, matter, kinetic-energy, or external-potential-energy changes. The admitted process and transfer signs are those of Thermodynamic process records and transfer conventions.

For use of Macroscopic second law for isolated compounds, the model supplies an admitted core closed process record $P_{\mathrm{t}\mathrm{o}\mathrm{t}}\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s_{\mathrm{t}\mathrm{o}\mathrm{t}})$ for this full compound $s_{\mathrm{t}\mathrm{o}\mathrm{t}}$. In particular, it supplies the compound's fixed positive mass and defined endpoint center-of-mass velocities in its fixed bookkeeping chart, as required by Thermodynamic process records and transfer conventions. Its two endpoint configurations lie in the declared equilibrium set of that process model, its separately tracked center-of-mass kinetic and external-field potential energies have zero endpoint change, and its endpoint association ${\rho }_{P_{\mathrm{t}\mathrm{o}\mathrm{t}}}:\{-,+\}\to \mathcal{F}$ has both values in the same feasible set $\mathcal{F}$ from that postulate. In the constituent ordering of ${\rho }_{P_{\mathrm{t}\mathrm{o}\mathrm{t}}}(\pm )$, the working-system state is $x_P^{\pm }$ and the bath and work-store components have the corresponding energy coordinates $E_h^{\pm },E_c^{\pm },E_w^{\pm }$. The total-compound record and the working-system record therefore have matching working-system endpoints; all bath and work-store energy differences below refer to these same total-compound endpoint tuples. The former record is the input to the second-law postulate, and $P$ is the working-system record to which the first-law balance applies.

For each endpoint $\pm$, the model declares the additive entropy split $S_{\mathrm{t}\mathrm{o}\mathrm{t}}\text{\hspace{0.17em}⁣}({\rho }_{P_{\mathrm{t}\mathrm{o}\mathrm{t}}}(\pm ))=S_{\mathrm{w}\mathrm{s}}(x_P^{\pm })+S_h(E_h^{\pm })+S_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{d}}(E_c^{\pm })+S_w(E_w^{\pm }),$ where $S_{\mathrm{w}\mathrm{s}}$ is the working-system entropy and the bath and work-store functions are those defined here. This constituent split is assumed to be admitted by Concave macroscopic fundamental entropy and is the entropy supplied to Macroscopic second law for isolated compounds; $S_w$ contributes zero change. When reversibility is asserted, $P_{\mathrm{t}\mathrm{o}\mathrm{t}}$ is reversible relative to its declared outer boundary and protocol as in Reversible thermodynamic process record.

In the heat-engine mode, the positive magnitudes $Q_h,Q_c$ in joules mean respectively heat taken from bath $h$ into the working system and heat rejected by the working system to bath $c$; accordingly $\mathrm{\Delta }{E}_h=-Q_h$, $\mathrm{\Delta }{E}_c=Q_c$, $Q[P]=Q_h-Q_c$, and work output is $W_{\mathrm{o}\mathrm{u}\mathrm{t}}=-W_{\mathrm{o}\mathrm{n}}[P]>0$. The work store gains exactly this transferred work, so $\mathrm{\Delta }{E}_w=W_{\mathrm{o}\mathrm{u}\mathrm{t}}$. In the refrigerator mode, $Q_c>0$ is heat taken from bath $c$ into the working system, $Q_h>0$ is heat rejected to bath $h$, so $\mathrm{\Delta }{E}_c=-Q_c$, $\mathrm{\Delta }{E}_h=Q_h$, $Q[P]=Q_c-Q_h$, and work input is $W_{\mathrm{i}\mathrm{n}}=W_{\mathrm{o}\mathrm{n}}[P]>0$. The store supplies that work, so $\mathrm{\Delta }{E}_w=-W_{\mathrm{i}\mathrm{n}}$. These are idealized model conventions, not claims that a particular apparatus realizes the cycle.

Let an admitted heat-engine cycle in the sense of Ideal-reservoir thermodynamic cycles operate between fixed ideal-bath temperatures $0<T_c<T_h$, absorb $Q_h>0$, reject $Q_c>0$, and deliver $W_{\mathrm{o}\mathrm{u}\mathrm{t}}>0$. Define its efficiency by $\eta =W_{\mathrm{o}\mathrm{u}\mathrm{t}}\mathrm{/}{Q}_h$. Then $\eta \le 1-\frac{T_c}{T_h}\mathrm{.}$ Equality holds exactly when the total entropy change of the isolated compound is zero. In particular, a reversible cycle in the stated boundary model saturates the bound; zero entropy change is not here asserted to imply full reversibility.

Let an admitted refrigerator cycle in the sense of Ideal-reservoir thermodynamic cycles operate between fixed ideal-bath temperatures $0<T_c<T_h$, extract $Q_c>0$ from the cold bath, reject $Q_h>0$ to the hot bath, and consume $W_{\mathrm{i}\mathrm{n}}>0$. Define its cooling coefficient of performance by ${\mathrm{C}\mathrm{O}\mathrm{P}}_R=Q_c\mathrm{/}{W}_{\mathrm{i}\mathrm{n}}$. Then ${\mathrm{C}\mathrm{O}\mathrm{P}}_R\le \frac{T_c}{T_h-T_c}\mathrm{.}$ Equality holds exactly when the total entropy change of the isolated compound is zero. A reversible cycle in the stated boundary model saturates the bound; no converse about reversibility is asserted.

Fix a material constitution $c$, a positive temperature $T_0$ in kelvins, and a nonempty set $K$ equipped with a specified topology ${\tau }_K$ such that the topological space $(K,{\tau }_K)$ is compact. Compact means that every cover of $K$ by ${\tau }_K$-open subsets has a finite subcover; the label $a\in K$ holds its listed external constraints fixed while temperature varies. Let $D_{c,+}\subseteq D_c$ be the designated open equilibrium charts on which the finite $C^2$ entropy and positive absolute temperature $T_c=1\mathrm{/}{S}_{c,U}$ are defined, as in Concave macroscopic fundamental entropy and Entropic temperature, pressure, and molar chemical potentials. A model-specified family map $\varphi :(0,T_0)\times K⟶D_{c,+}$ assigns an equilibrium state $\varphi (t,a)$ at temperature $t$, so $T_c(\varphi (t,a))=t$. The map need not be continuous in $a$; the postulate below itself requires the displayed supremum to be finite for all sufficiently small $t$. Define $S_K(t,a):=S_c(\varphi (t,a))$, measured in $\mathrm{J}{\mathrm{K}}^{-1}$.

The model adopts the following uniform compact-family Nernst heat-theorem postulate: ${\lim }_{t\downarrow 0}{\sup }_{a,b\in K}\mid S_K(t,a)-S_K(t,b)\mid =0,$ where the supremum is the least upper bound of the displayed real differences. This is an additional low-temperature physical postulate, not a consequence of the first or second law. It is a deliberately scoped uniform formulation; it is not asserted equivalent to every statement called the third law or Nernst's theorem.

The postulate constrains entropy differences within the declared family. It does not assert that any individual finite limit $S_K(0,a):={\lim }_{t\downarrow 0}{S}_K(t,a)$ exists, that such a limit is zero, or that the family includes every material or constraint. If the indicated individual limit is to be used, its existence and finiteness are an additional premise.

For a declared material constitution $c$, let $D_{c,+}\subseteq D_c$ be its designated equilibrium states with finite positive absolute temperature, and require $D_{c,+}\mathrm{\ne }\mathrm{\varnothing }$ for a nonvacuous application. Let $D_0$ be a separately declared nonempty set of zero-temperature boundary preparations, disjoint from $D_{c,+}$, and let ${\bar{D}}_c:=D_{c,+}\bigsqcup D_0,{\bar{T}}_c:{\bar{D}}_c\to [0,\mathrm{\infty })$ extend the thermodynamic temperature from Entropic temperature, pressure, and molar chemical potentials by ${\bar{T}}_c{\mathrm{\mid }}_{D_{c,+}}=T_c$ and ${\bar{T}}_c(y)=0$ for $y\in D_0$.

Let ${\mathcal{U}}_c$ be the model-declared set of cooling controls, and let $\mathcal{C}\subseteq {\bar{D}}_c\times {\bar{D}}_c\times {\mathcal{U}}_c\times (0,\mathrm{\infty })$ be the model's full declared family of admitted cooling operation records. Each record $\gamma =(x^{-},x^{+},u,\tau )$ has initial-endpoint map $i(\gamma )=x^{-}$, final-endpoint map $f(\gamma )=x^{+}$, control $u\in {\mathcal{U}}_c$, and finite positive duration $\tau (\gamma )=\tau$ seconds. Its positive-temperature portions obey the process conventions of Thermodynamic process records and transfer conventions, while the model separately declares how a boundary endpoint in $D_0$ is represented. Define two records to be compatible exactly when $f(\gamma )=i({\gamma }^{\mathrm{\prime }})$. A finite protocol is a tuple $({\gamma }_1,\dots ,{\gamma }_m)$ for $m\in \mathbb{N}$, $m\ge 1$ (The natural numbers $\mathbb{N}$ (von Neumann)), with each successive pair compatible; its initial and final endpoints are $i({\gamma }_1)$ and $f({\gamma }_m)$, and its duration is ${\sum }_{j=1}^m\tau ({\gamma }_j)$. Let ${\mathcal{P}}_{\mathcal{C}}$ be the set of all such finite protocols and let ${\mathcal{P}}_{\mathcal{C}}^{+}:=\{p\in {\mathcal{P}}_{\mathcal{C}}:i(p)\in D_{c,+}\}$. For each $p=({\gamma }_1,\dots ,{\gamma }_m)$, define $i(p):=i({\gamma }_1)$, $f(p):=f({\gamma }_m)$, and $\tau (p):={\sum }_{j=1}^m\tau ({\gamma }_j)<\mathrm{\infty }$. For a nonvacuous application, require ${\mathcal{P}}_{\mathcal{C}}^{+}\mathrm{\ne }\mathrm{\varnothing }$ and declare $\mathcal{C}$ as the full operation family rather than defining it to exclude boundary endpoints. Define ${\mathrm{Reach}}_{\mathcal{C}}:=\{(i(p),f(p)):p\in {\mathcal{P}}_{\mathcal{C}}\}\subseteq {\bar{D}}_c\times {\bar{D}}_c$.

The adopted operational unattainability postulate is ${\mathrm{Reach}}_{\mathcal{C}}\cap (D_{c,+}\times D_0)=\mathrm{\varnothing }\mathrm{.}$ This is a separate physical assumption from the Nernst heat-theorem postulate; neither is deduced here from the first or second law. If $D_{c,+}$, $D_0$, or ${\mathcal{P}}_{\mathcal{C}}^{+}$ is empty, or if no boundary endpoint is available in the declared operation-record space because the model defines every operation to stay in $D_{c,+}$, the assertion is absent or vacuous and is not evidence for unattainability. The postulate does not remove the need to state the actual state space, controls, and admitted operations.

Masanes and Oppenheim, Introduction, formulate unattainability in terms of finitely many operations in finite time. The reachability notation above is this framework's explicit formulation of that operational principle. Their quantum cooling bounds require additional dynamical, bath, and work-resource assumptions, listed in their Fundamental Assumptions; those bounds are not a derivation of the postulate for the arbitrary declared operation family here. Tong §4.5 and Arovas §2.6.2 provide context for the distinct entropy form of the third law, not this operational exclusion.

Residual entropy here means a finite entropy limit at low temperature under a model's specified entropy convention. In a finite canonical model, let a finite set of microstates have energies $E_i$ in joules, assign $p_i(T)\propto \exp [-E_i\mathrm{/}(k_{\mathrm{B}}T)]$ for $T>0$, and use the Gibbs entropy $-k_{\mathrm{B}}{\sum }_i{p}_i(T)\log p_i(T)$. If the ground energy has finite degeneracy $g$ and there is a positive gap to all excited states, this model-specific entropy tends to $k_{\mathrm{B}}\log g$; Residual entropy in a finite canonical model ↗ supplies the full finite-state calculation. Here $g$ is the number of ground states and $\log$ is the natural logarithm of The natural logarithm as the inverse of the exponential function.

This formula belongs to that specified microscopic preparation and entropy convention. It does not identify every Gibbs entropy with the macroscopic $S_c$, and it does not establish a universal value $S_c(0)=0$. The uniform Nernst postulate Uniform Nernst heat-theorem postulate on a compact family constrains differences across its declared family, not the common entropy offset: adding one fixed constant to every $S_K(t,a)$ leaves every displayed difference unchanged. Thus that postulate alone neither selects an absolute zero nor forces a unique ground state. The Nernst heat theorem, zero-entropy convention for a unique ground state, and operational unattainability remain distinct model statements.