Stochastic Protocols

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.

A finite rate model consists of a finite nonempty state set $S$ and rates $q_{ij}\ge 0$ in seconds${}^{-1}$ for $i\mathrm{\ne }j$. Set $q_{ii}=-{\sum }_{j\mathrm{\ne }i}{q}_{ij}$. A row probability vector is $p=(p_i{)}_{i\in S}$ with $p_i\ge 0$ and ${\sum }_i{p}_i=1$; write $\mathrm{\Delta }(S)=\{p\in [0,1{]}^S:{\sum }_i{p}_i=1\}$. A time-dependent law is a coordinatewise continuously differentiable map $p:[0,\mathrm{\infty })\to \mathrm{\Delta }(S)$, with a right derivative at $t=0$, that satisfies the continuous-time master equation ${\dot{p}}_i(t)={\sum }_{j\mathrm{\ne }i}[p_j(t)q_{ji}-p_i(t)q_{ij}]$ for every $t\ge 0$. A strictly positive normalized vector $\pi$ is a detailed-balance law when ${\pi }_i{q}_{ij}={\pi }_j{q}_{ji}$ for every distinct pair. Its undirected rate graph has an edge $\{i,j\}$ exactly when ${\pi }_i{q}_{ij}>0$; detailed balance makes this equivalent to ${\pi }_j{q}_{ji}>0$.

On a finite nonempty state set $S$, a discrete transition kernel is a matrix $K:S\times S\to [0,1]$ satisfying ${\sum }_{j\in S}K(i,j)=1$ for every $i\in S$. For a strictly positive row probability vector $\pi \in \mathrm{\Delta }(S)$, the kernel satisfies $\pi$-detailed balance when ${\pi }_{i}K(i,j)={\pi }_{j}K(j,i)$ for every $i,j\in S$. This identity pairs the supports of forward and reversed transitions. A rate model or a reversible kernel is a mathematical/physical model class; no observed trajectory is asserted by the definition.

For a protocol of Finite driven protocol and path work, fix a bath temperature $T_b>0$ kelvin and set $\beta =(k_{\mathrm{B}}{T}_b{)}^{-1}$ in joules${}^{-1}$, using the exact constant and SI conventions of SI quantities and unit conventions. The finite-state clause of the adopted Canonical sampling assumption for finite and normalized statistical models then gives the stage-zero law ${\pi }_0(i)=e^{-\beta E_0(i)}\mathrm{/}{Z}_0$; adopt that the initial microstate is drawn from this law, so ${\mu }_0={\pi }_0$. Here $\mathrm{\Omega }$ is finite and nonempty and $E_0(i)$ is finite, so $0<Z_0<\mathrm{\infty }$ as required by that clause. This is a model premise about the preparation distribution. It does not follow from the finite partition-sum definition, does not assert that an arbitrary system equilibrates in finite time, and does not report a measured preparation. The countable-state and measure-integral branches of Canonical sampling assumption for finite and normalized statistical models are not used.

For the protocol in Finite driven protocol and path work, adopt, at each stage $k=1,\dots ,m$, the stationarity condition ${\sum }_{i\in \mathrm{\Omega }}{\pi }_k(i)K_k(i,j)={\pi }_k(j)$ for every $j\in \mathrm{\Omega }$. This says that if the bath transition begins with law ${\pi }_k$ at fixed energy $E_k$, it ends with the same law. It is a physical modeling assumption for this protocol family, not a consequence of the transition-kernel definition and not an assertion that the kernel mixes every initial law to ${\pi }_k$.

Choose a Galilean inertial chart $\mathcal{F}$ as defined in Galilean inertial charts and the center-of-mass rest frame for the specified fixed closed trap-and-bath body, assumed to have positive total mass and the center-of-mass map required there, and supply its componentwise differentiable center-of-mass trajectory on the whole modeled time interval. Assume that its center-of-mass velocity is one constant vector on that interval, so the single constant boost of that supplier selects $\mathcal{F}$ with zero center-of-mass velocity throughout. Take $\mathcal{F}$ to be this common rest chart. The trap-and-bath background is held fixed in this idealization; center-of-mass recoil or accelerating-frame dynamics are not part of the scalar process. In this chart let ${\mathbf{r}}_c\in {\mathbb{R}}^3$ be the fixed trap-center position and let $\mathbf{e}\in {\mathbb{R}}^3$ be a fixed unit vector along the modeled trap axis. For a particle position $\mathbf{r}\in {\mathbb{R}}^3$, define the signed displacement $x=⟨\mathbf{r}-{\mathbf{r}}_c,\mathbf{e}⟩$ in metres; the process $X_t$ in Ornstein–Uhlenbeck integral equation is this coordinate of the particle position at time $t$. Thus $U(x)=k{x}^2\mathrm{/}2$ is the harmonic potential relative to the trap center, not an absolute spatial coordinate. Using the SI conventions and exact $k_{\mathrm{B}}$ of SI quantities and unit conventions, take stiffness $k>0$ in joules per square metre, constant mobility $\mu >0$ in square metres per joule-second, and bath temperature $T_b>0$ kelvin. Adopt the one-dimensional overdamped model $d{X}_t=-\gamma X_{t}dt+\sigma d{B}_t$, interpreted through the integral equation of Ornstein–Uhlenbeck integral equation, with $\gamma =\mu k$ in seconds${}^{-1}$ and ${\sigma }^2=2\mu k_{\mathrm{B}}{T}_b$ in square metres per second. Here $B$ is standard Brownian motion with its stated filtration and increment hypotheses, and the bath noise is independent of the initial preparation. This Einstein noise relation and the linear Markov model are physical assumptions. They do not follow from equilibrium thermodynamics alone and do not report a measured spectrum or calibration.

Let $M$ be a finite memory-state set and $R$ a finite retained-register set. A joint preparation is a probability law $p(m,r)\ge 0$ with ${\sum }_{m,r}p(m,r)=1$. Write $a_r={\sum }_{m}p(m,r)$ and, for $a_r>0$, $q(m\mathrm{\mid }r)=p(m,r)\mathrm{/}{a}_r$. Use the finite Shannon entropy in nats and the conditional entropy $H(M\mathrm{\mid }R)$ exactly as defined by Conditional entropy of finite memory given side information. For a fixed target memory state $m_0$, a reset is the endpoint map $p^{\mathrm{\prime }}(m,r)={\mathbf{1}}_{\{m=m_0\}}{a}_r$: it places the memory at $m_0$ while leaving the register marginal unchanged. The definition alone assigns no physical entropy, work cost, heat transfer, or implementation.

For every path $\omega =(i_0,\dots ,i_m)$ of Finite driven protocol and path work, the endpoint energy change is $E_m(i_m)-E_0(i_0)=W(\omega )+Q_{\mathrm{i}\mathrm{n}}(\omega )$, with the stated sign convention.

Let $S$ have $N\ge 2$ states, positive detailed-balance law $\pi$, and connected undirected rate graph as in Finite detailed-balance rate and transition models. For every strictly positive initial probability vector $p(0)$, the master equation has a unique strictly positive probability solution $p(t)$ for all $t\ge 0$. The relative entropy $D(p(t)\mathrm{\parallel }\pi )={\sum }_i{p}_i(t)\log (p_i(t)\mathrm{/}{\pi }_i)$ is nonincreasing, and $p_i(t)\to {\pi }_i$ for every state. No monotonicity of Shannon entropy $-{\sum }_i{p}_i\log p_i$ is asserted.

For the forward path law of Finite driven protocol and path work, under Canonical initial preparation for a finite protocol and Finite bath kernels preserve the stage canonical law, the finite expectation satisfies ${\mathbb{E}}_F[e^{-\beta W}]=Z_m\mathrm{/}{Z}_0=e^{-\beta \mathrm{\Delta }F}$. This is an ensemble identity; it does not say $W=\mathrm{\Delta }F$ on each path.

Assume the setup of Finite driven protocol and path work, the canonical initial law of Canonical initial preparation for a finite protocol, the stationarity model of Finite bath kernels preserve the stage canonical law, and, additionally, ${\pi }_k(i)K_k(i,j)={\pi }_k(j)K_k(j,i)$ for every $k,i,j$. For every forward path with positive probability, define its reverse probability by $P_R({\omega }^R)={\pi }_m(i_m){\prod }_{k=1}^m{K}_k(i_k,i_{k-1})$. Then $P_R({\omega }^R)>0$ and $P_F(\omega )\mathrm{/}{P}_R({\omega }^R)=e^{\beta (W(\omega )-\mathrm{\Delta }F)}$. Consequently, at each work value $w$, $P_F(W=w)=e^{\beta (w-\mathrm{\Delta }F)}{P}_R(W_R=-w)$. The reverse protocol first applies the stage-$k$ reverse transition at fixed $E_k$, then reverses the quench $E_k\to E_{k-1}$; its work is $W_R({\omega }^R)=-W(\omega )$.

Assume the Axiom of Choice The Axiom of Choice: every family of nonempty sets has a choice function. It is inherited from the Brownian and Itô-integral suppliers used for the stationary Gaussian model.

Under Linear Ornstein-Uhlenbeck bath model, prepare $X_0\sim N(0,V)$ independently of future Brownian increments, where $V=k_{\mathrm{B}}{T}_b\mathrm{/}k$. Then the conditional linear model has stationary covariance $C(\tau )=V{e}^{-\mu k\mathrm{\mid }\tau \mathrm{\mid }}$. With angular-frequency convention $S(\omega )=(2\pi {)}^{-1}{\int }_{\mathbb{R}}C(\tau )e^{i\omega \tau }d\tau$, its spectrum is $S(\omega )=\mu k_{\mathrm{B}}{T}_b\mathrm{/}(\pi [(\mu k{)}^2+{\omega }^2])$. For every averaging time $T>0$, $\mathrm{Var}(T^{-1}{\int }_0^T{X}_{t}dt)\le 2k_{\mathrm{B}}{T}_b\mathrm{/}(\mu k^{2}T)$. These are model predictions, not reported observations.

For the reset model of Finite memory and retained-register reset model, use the exact SI value and conventions of SI quantities and unit conventions. Assume (i) the dimensionless Shannon entropies in nats from Conditional entropy of finite memory given side information give memory-register thermodynamic entropy $S_{\mathrm{m}\mathrm{e}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{y}+\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}=k_{\mathrm{B}}H(M,R)$ in joules per kelvin at both endpoints, with $k_{\mathrm{B}}$ in joules per kelvin, (ii) a single ideal reservoir remains at fixed temperature $T_b>0$ kelvin and receives heat $Q_b$ in joules, (iii) the memory-register endpoint energy is unchanged so the adopted first-law convention gives work on the system $W=Q_b$ in joules, and (iv) the total entropy change obeys $\mathrm{\Delta }{S}_{\mathrm{m}\mathrm{e}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{y}+\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}+Q_b\mathrm{/}{T}_b\ge 0$ in joules per kelvin. Then $W\ge k_{\mathrm{B}}{T}_{b}H(M\mathrm{\mid }R)$, with both sides in joules. Equality is not asserted without a separately specified reversible implementation.

Under the hypotheses of Jarzynski equality for a finite protocol with stationary kernels, the mean work obeys ${\mathbb{E}}_{F}W\ge \mathrm{\Delta }F$. The conclusion concerns the path ensemble only; individual paths may have $W<\mathrm{\Delta }F$.