Thermodynamic process records and transfer conventions

For a system $s$ of constitution $c$, the process model fixes one Galilean inertial chart $g_s$ for kinetic-energy bookkeeping and transfer records and holds that chart fixed throughout each process. A process record is a tuple $P=(I_P,x_P^{-},x_P^{+},z_P,{\kappa }_P,r_P)\mathrm{.}$ For each finite closed time interval $I_P=[t_0,t_1]\subseteq \mathbb{R}$ with $t_0<t_1$ as in Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, measured in seconds, its component fields have these types: $x_P^{-},x_P^{+}\in D_c$ are equilibrium endpoint coordinates from Equilibrium state domains; $Z_s$ is the model-declared configuration space of the system, its auxiliaries and devices; $Z_s^{\mathrm{e}\mathrm{q}}\subseteq Z_s$ is the model-declared set of equilibrium configurations and ${\xi }_s:Z_s^{\mathrm{e}\mathrm{q}}\to D_c$ assigns their state coordinates; $z_P:I_P\to Z_s$ is a function (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); ${\mathcal{K}}_s$ is the model-declared set of control/contact records and ${\kappa }_P\in {\mathcal{K}}_s$; and ${\mathcal{R}}_s$ is the model-declared set of signed transfer records and $r_P\in {\mathcal{R}}_s$. These fields form a finite Cartesian product (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$). The admitted records ${\mathcal{P}}_s$ are the model-specified subset of tuples with these types that additionally satisfy $z_P(t_0),z_P(t_1)\in Z_s^{\mathrm{e}\mathrm{q}}$, ${\xi }_s(z_P(t_0))=x_P^{-}$, and ${\xi }_s(z_P(t_1))=x_P^{+}$. No continuity or differentiability is imposed on $z_P$, and its values at interior times need not represent equilibrium states.

For an endpoint bookkeeping quantity $F$, write $\mathrm{\Delta }F=F_P^{+}-F_P^{-}$. A closed record in the sense of Thermodynamic systems and permitted exchanges has no matter crossing the chosen outer boundary throughout $I_P$; zero net matter transfer alone does not suffice. The core closed-record class ${\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)\subseteq {\mathcal{P}}_s$ (Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$) consists exactly of the closed records $P\in {\mathcal{P}}_s$ for which the system has a supplied fixed total mass $M>0$ and defined center-of-mass velocities at both endpoints in the fixed chart $g_s$, and $\mathrm{\Delta }{K}_{\mathrm{c}\mathrm{m}}=\mathrm{\Delta }{U}_{\mathrm{e}\mathrm{x}\mathrm{t}}=0$. Here $K_{\mathrm{c}\mathrm{m}}^{\pm }=\frac{1}{2}M\mathrm{\parallel }{\mathbf{v}}_{\mathrm{c}\mathrm{m}}^{\pm }{\mathrm{\parallel }}_2^2$ are defined as in Galilean inertial charts and the center-of-mass rest frame, and $U_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\pm }$ are supplied endpoint values of any model-specified external-field potential energy tracked separately from the internal-energy coordinate in $g_s$; if no such term is tracked, both values are set to zero. Other process classes require an expanded energy bookkeeping rule.

Heat is the protocol-assigned thermal mode of energy transfer across the selected outer boundary, not a quantity stored in the equilibrium state. For each admitted $P$, its transfer record $r_P$ supplies cumulative incoming and outgoing thermal energies $H_P^{\mathrm{i}\mathrm{n}},H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}:I_P\to [0,\mathrm{\infty })$, with numerical values in joules as in SI quantities and unit conventions. Both maps start at zero at $t_0$ and satisfy $H_P^a(t)\le H_P^a(u)$ whenever $t\le u$ in $I_P$, for $a\in \{\mathrm{i}\mathrm{n},\mathrm{o}\mathrm{u}\mathrm{t}\}$. Their increments record the total thermal energy transferred in each direction over the corresponding time interval, across all declared outer-boundary thermal channels, before cancellation between directions or channels. No continuity or differentiability of these maps is required. Transfers between subsystems inside that boundary are internal and are excluded from these outer-boundary totals.

Define $Q:{\mathcal{P}}_s\to \mathbb{R}$ by $Q[P]:=H_P^{\mathrm{i}\mathrm{n}}(t_1)-H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t_1)$. The protocol also supplies $W_{\mathrm{o}\mathrm{n}}:{\mathcal{P}}_s\to \mathbb{R}$, with values in joules; both transfers use the fixed bookkeeping chart $g_s$. Thus $Q[P]>0$ means net thermal energy enters the system, and $W_{\mathrm{o}\mathrm{n}}[P]>0$ means net mechanical work is done on it. These are functionals of the process record, not state functions of $x_P^{-}$ and $x_P^{+}$. A record $P\in {\mathcal{P}}_s$ is adiabatic exactly when $H_P^{\mathrm{i}\mathrm{n}}(t)=H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t)=0$ for every $t\in I_P$: no thermal energy crosses the selected boundary throughout the record. In particular an adiabatic record has $Q[P]=0$, but zero net heat alone does not specify an adiabatic record. On ${\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$, use the restrictions of these same maps; defining them on the larger class supplies no energy-balance law for records outside the core class. A closed record may still exchange energy; it is not thereby isolated.

A preparation is a model-specified rule selecting the particular initial equilibrium state $x_P^{-}$. A statistical extension may instead specify a normalized probability measure on a declared measurable space, as in Probability measures and probability spaces; such a statistical preparation is outside this deterministic process class. A preparation does not assert later equilibration. An admitted record may pass through nonequilibrium configurations even though both endpoints lie in $D_c$.