Finite driven protocol and path work

Take $\mathrm{\Omega }$ to be a finite nonempty microstate set as in Finite statistical microstate model: distinct elements count as distinct microstates, and the physical energy values are finite and measured in joules in the declared rest frame. For the physical internal-energy interpretation, use one declared fixed Galilean chart for every stage, with the bulk center-of-mass and separately tracked external-energy sectors suppressed from this finite protocol. The companion mechanical model supplies the center-of-mass data required by the microstate supplier; when the common chart is called a center-of-mass rest chart throughout the protocol, its center-of-mass velocity is constant over the modeled interval, and one constant boost selects that chart. This fixed-chart idealization does not infer a rest chart from the finite set or transition kernels, and does not describe an accelerating bulk body. Fix an integer $m\ge 1$ and stage energy functions $E_k:\mathrm{\Omega }\to \mathbb{R}\mathrm{J}$ for $k=0,\dots ,m$, with the energy offsets fixed by the declared protocol model. Fix inverse temperature $\beta >0$ in joules${}^{-1}$. The finite partition sum and canonical probability vector at stage $k$ are $Z_k={\sum }_{i\in \mathrm{\Omega }}{e}^{-\beta E_k(i)}$ and ${\pi }_k(i)=e^{-\beta E_k(i)}\mathrm{/}{Z}_k$. Since $\mathrm{\Omega }$ is nonempty and finite and each energy is finite, $0<Z_k<\mathrm{\infty }$ and ${\pi }_k$ is a strictly positive probability vector.

A transition kernel at stage $k$ is a matrix $K_k(i,j)\ge 0$ with ${\sum }_{j\in \mathrm{\Omega }}{K}_k(i,j)=1$. The general class permits every such row-stochastic kernel; normalization alone imposes no thermal consistency. For the subclass interpreted as transitions due only to one equilibrium reservoir at inverse temperature $\beta$, adopt the discrete detailed-balance condition ${\pi }_k(i)K_k(i,j)={\pi }_k(j)K_k(j,i)$ for every $k,i,j$. Since ${\pi }_k$ is strictly positive, $K_k(i,j)>0$ if and only if $K_k(j,i)>0$; on positive pairs this gives $K_k(i,j)\mathrm{/}{K}_k(j,i)=e^{-\beta (E_k(j)-E_k(i))}$. In this physical subclass, also adopt that the fixed-stage energy change is entirely reservoir heat, with no untracked work, particle exchange, or intrinsic state-entropy contribution. These are model assumptions, not a deduction that an apparatus realizes this reservoir. Arbitrary row-stochastic kernels remain admissible for the formal path formulas; any other physical transfer interpretation requires a separately specified model. In the forward stage order, the energy changes instantaneously from $E_{k-1}$ to $E_k$ while the state is still $i_{k-1}$; the subsequent transition samples $i_k$ using $K_k(i_{k-1},i_k)$ at fixed energy $E_k$. A path is $\omega =(i_0,\dots ,i_m)\in {\mathrm{\Omega }}^{m+1}$. The initial law is a probability vector ${\mu }_0:\mathrm{\Omega }\to [0,1]$ with ${\sum }_{i\in \mathrm{\Omega }}{\mu }_0(i)=1$. Its path probability is $P_F(\omega )={\mu }_0(i_0){\prod }_{k=1}^m{K}_k(i_{k-1},i_k)$; summing over all paths gives $1$ by the normalization of ${\mu }_0$ and the row sums of the kernels.

The quench-work sign is positive when the quench raises the energy. The fixed-stage energy increment $Q_{\mathrm{i}\mathrm{n}}$ is positive when the subsequent transition raises the energy; it is interpreted as heat into the system under the single-reservoir subclass assumptions just stated. On the general formal class it denotes only this energy increment. Define $W(\omega )={\sum }_{k=1}^m[E_k(i_{k-1})-E_{k-1}(i_{k-1})]$ and $Q_{\mathrm{i}\mathrm{n}}(\omega )={\sum }_{k=1}^m[E_k(i_k)-E_k(i_{k-1})]$, both in joules. Define $F_k=-{\beta }^{-1}\log Z_k$ and $\mathrm{\Delta }F=F_m-F_0=-{\beta }^{-1}\log (Z_m\mathrm{/}{Z}_0)$. These are finite-model quantities; their physical interpretation is conditional on the adopted preparation and transition assumptions.