Adiabatic accessibility

Fix a material constitution $c$ and its equilibrium state domain $D_c$ from Equilibrium state domains. For each admitted record $P\in {\mathcal{P}}_s$, use the nonnegative, nondecreasing cumulative incoming and outgoing thermal-energy maps $H_P^{\mathrm{i}\mathrm{n}},H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}:I_P\to [0,\mathrm{\infty })$ of Thermodynamic process records and transfer conventions, with numerical values in joules and initial values zero. For $x,y\in D_c$, write $x{\prec }_{c}y$, and say that $y$ is adiabatically accessible from $x$, exactly when the declared process model admits at least one record $P\in {\mathcal{P}}_s$ with $x_P^{-}=x$, $x_P^{+}=y$, $H_P^{\mathrm{i}\mathrm{n}}(t)=H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t)=0$ for every $t\in I_P$, and the model’s stated auxiliary and work-device endpoint conditions satisfied. The material constitution is the same at both endpoints. The process record must account for the selected outer boundary and every auxiliary included by its protocol; transfers inside that boundary are internal. These maps are supplied on all admitted records, but no energy-balance law outside the core closed-record class follows.

The sign convention is $Q>0$ for net heat into the system and $W_{\mathrm{o}\mathrm{n}}>0$ for work done on it, as fixed in Thermodynamic process records and transfer conventions. An adiabatic record has $Q[P]=0$, but zero net heat alone is insufficient: equal nonzero incoming and outgoing transfers cancel in $Q$. Adiabatic means absence of thermal transfer across the selected boundary throughout the record; it imposes neither zero work nor zero energy change, isolation, quasistatic evolution, or reversibility. The relation concerns equilibrium endpoints only; intermediate configurations may be nonequilibrium. Existence of a witness record is model input, not a consequence of this definition.

When the constitution is clear, write $x\prec y$. Define $x\sim y$ by $x\prec y$ and $y\prec x$, and $x\prec \prec y$ by $x\prec y$ and not $y\prec x$. These are respectively accessibility equivalence and strict one-way accessibility; neither is defined by equality of energy alone.

The order-theoretic relation of Lieb and Yngvason is an abstract model of this operation. Goddard’s Carathéodory formulation uses a distinct local-inaccessibility route; it is cited as context and does not supply the product comparison results used later.