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1 Introduction↩︎

Markovian master equations are a standard language for transport through mesoscopic conductors, quantum dots, thermal machines, and monitored open systems [1]–[8]. A master equation for the system state determines relaxation rates, stationary occupations, and state response functions. Transport experiments often measure reservoir-resolved records instead, including particle, energy, heat, spin, photon, or other transfer statistics [9]–[23]. The central question of this paper is whether complete knowledge of the Markovian state dynamics is enough to determine those thermodynamic records.

The issue arises because a state generator contains total transition rates between system states, whereas a thermodynamic transport model also specifies which reservoir or energy-filter channel realizes each transition and which increment is recorded in that channel. We call a state-only Markovian description thermodynamically incomplete for a given record when it fixes the state probabilities and their response, but does not fix that record because different reservoir-channel assignments remain compatible with the same generator. This mechanism is distinct from hidden-state inference or coarse-grained non-Markovian dynamics, where missing entropy production is caused by eliminating part of the state space [24]–[32]. Here the state equation itself is assumed to be known exactly. No transition rate in the state generator is missing. The missing information is the assignment of those transitions to reservoirs, filters, and measurement records.

The practical issue is that state tomography is often used to validate a Markovian transport model before the model is used to predict thermodynamic performance [9], [10], [24], [27]. We show when that inference is justified and when it is underdetermined: all state-level tests can agree with the same generator, while heat currents, entropy production, and noise measurements distinguish the underlying reservoir-channel assignments.

We first formulate a reconstruction criterion: a record observable is determined by the state generator only if it is unchanged under every redistribution of transition rates among reservoir channels that keeps the generator fixed. We then apply the criterion to a multi-terminal Coulomb-blockaded quantum dot with energy-filtered contacts [33]–[35], where identical occupation dynamics can coexist with different heat currents, entropy production, and current noise. Finally, we derive the geometric and topological form of the criterion. The projection from channel currents to state dynamics gives a quotient fluctuation geometry, while the connectivity of the transport channels gives exact bounds on the heat, particle, and spin records still compatible with the same state generator.

Section 2 defines the transport data and thermodynamic records. Section 3 states the reconstruction criterion. Section 4 constructs state-identical quantum-dot devices with different thermodynamic records. Section 5 derives the geometric and topological form of the criterion. Section 6 summarizes the experimental implication.

2 Markovian transport data↩︎

We consider a finite set of system states \(n=1,\ldots,N\). The probability vector \(p(t)\) obeys the Markovian master equation \[\begin{align} \frac{\mathsf{d}}{\mathsf{d}t}p_{n} = \sum_{m\neq n} \left[ W_{nm}p_{m} - W_{mn}p_{n} \right] \equiv \left(Lp\right)_{n}. \label{Eq-state-master} \end{align}\tag{1}\] The generator \(L\) contains the total transition rate \(W_{nm}\) from state \(m\) to state \(n\). A transport experiment generally resolves this total rate into reservoir or measurement channels, \[\begin{align} W_{nm} = \sum_{\alpha\in{\cal A}_{nm}} w_{nm}^{\alpha}. \label{Eq-channel-resolution} \end{align}\tag{2}\] The label \(\alpha\) specifies the physical channel that realizes the transition. In an electronic conductor it can include the reservoir index, an energy-filter index, or a spin-selective contact. Each channel also carries a record increment \(d_{nm}^{\alpha,\mu}\) for the measured quantity \(\mu\), such as transferred charge or reservoir heat.

For a probability vector \(p\), the mean rate of the record \(\mu\) is \[\begin{align} J_{\mu}(p) = \sum_{m,n,\alpha} d_{nm}^{\alpha,\mu}w_{nm}^{\alpha}p_{m}. \label{Eq-mean-current} \end{align}\tag{3}\] The full counting statistics are generated by the tilted Markov generator \[\begin{align} \left[L(\boldsymbol{\chi})\right]_{nm} = \sum_{\alpha\in{\cal A}_{nm}} w_{nm}^{\alpha} \exp\left( \sum_{\mu}\chi_{\mu}d_{nm}^{\alpha,\mu} \right), \qquad n\neq m, \label{Eq-tilted-offdiag} \end{align}\tag{4}\] with diagonal entries chosen so that \(L(0)=L\). The long-time cumulants are obtained from the dominant eigenvalue of \(L(\boldsymbol{\chi})\) [5], [9], [10], [33], [34]. Eqs. (1 4 ) show the basic separation. The state generator depends on \(W_{nm}\). The thermodynamic records depend on the channel rates \(w_{nm}^{\alpha}\) and on the increments \(d_{nm}^{\alpha,\mu}\).

3 Thermodynamic completeness criterion↩︎

We now ask when a thermodynamic record is determined by the state generator. Fix the total rates \(W_{nm}\). A generator-preserving perturbation of the channel rates is a collection \(\delta w_{nm}^{\alpha}\) satisfying \[\begin{align} \sum_{\alpha\in{\cal A}_{nm}} \delta w_{nm}^{\alpha} = 0 \quad \text{for every transition }m\rightarrow n . \label{Eq-generator-preserving} \end{align}\tag{5}\] For sufficiently small amplitude, such a perturbation changes the physical channel assignment but leaves the Markovian state equation (1 ) unchanged.

For a generator-preserving channel redistribution, if Eq. (?? ) holds, changing the channel assignment at fixed \(L\) does not change the record rate. The record is then a function of the state generator alone. If Eq. (?? ) fails, one can choose a small parameter \(\epsilon\) such that \(w_{nm}^{\alpha}+\epsilon\delta w_{nm}^{\alpha}\geq0\). The perturbed and unperturbed devices have the same total rates \(W_{nm}\), but their record rates differ by the nonzero first-order change in Eq. (?? ). Thus the state generator \(L\) does not contain enough information to choose a unique reservoir-channel assignment.

The same criterion applies to current noise and higher cumulants. The tilted generator \(L(\boldsymbol{\chi})\) is determined by \(L\) only if every generator-preserving perturbation leaves the off-diagonal tilted rates in Eq. (4 ) unchanged for the chosen counting fields. If two channel assignments have the same \(L\) but different \(L(\boldsymbol{\chi})\), then state tomography cannot determine the corresponding counting statistics.

The first two cumulants make the separation between state dynamics and record data explicit. Let \(\langle{\boldsymbol{1}}|\) denote the row vector with all entries equal to one, let \(|p^{\mathrm{ss}}\rangle\) be the stationary state of \(L\), and define \[\begin{align} L_{\mu} = \left. \frac{\partial L(\boldsymbol{\chi})}{\partial\chi_{\mu}} \right|_{\boldsymbol{\chi}=0}, \qquad L_{\mu\nu} = \left. \frac{\partial^{2} L(\boldsymbol{\chi})}{\partial\chi_{\mu}\partial\chi_{\nu}} \right|_{\boldsymbol{\chi}=0}. \label{Eq-tilt-derivatives} \end{align}\tag{6}\] The mean current is \[\begin{align} J_{\mu}^{\mathrm{ss}} = \langle{\boldsymbol{1}}| L_{\mu} |p^{\mathrm{ss}}\rangle . \label{Eq-fcs-mean} \end{align}\tag{7}\] The zero-frequency current-noise matrix can be written as \[\begin{align} S_{\mu\nu} = \langle{\boldsymbol{1}}|L_{\mu\nu}|p^{\mathrm{ss}}\rangle - \langle{\boldsymbol{1}}| \left( L_{\mu}{\cal R}L_{\nu} + L_{\nu}{\cal R}L_{\mu} \right) |p^{\mathrm{ss}}\rangle , \label{Eq-fcs-noise} \end{align}\tag{8}\] where \({\cal R}\) is the Drazin inverse of \(L\) on the subspace orthogonal to the stationary state [5], [9], [10]. The state generator fixes \(|p^{\mathrm{ss}}\rangle\) and \({\cal R}\), but it does not fix \(L_{\mu}\) or \(L_{\mu\nu}\). Thus a state-only model may correctly predict relaxation and stationary occupations while failing to predict current means, current noise, or cross correlations.

4 Multi-terminal energy-filtered quantum dot↩︎

We apply the criterion to a Coulomb-blockaded quantum dot with \(N\) single particle levels \(\varepsilon_{i}\) [33]–[35]. The dot is either empty, denoted by \(0\), or occupied by one electron in level \(i\). Reservoir \(r\) has chemical potential \(\mu_{r}\) and temperature \(T_{r}\). Energy-filtered contacts produce channel rates \[\begin{align} w_{i0}^{r\lambda} = \gamma_{i}^{r\lambda} f_{r}\left(\varepsilon_{i}\right), \qquad w_{0i}^{r\lambda} = \gamma_{i}^{r\lambda} \left[ 1-f_{r}\left(\varepsilon_{i}\right) \right], \label{Eq-dot-rates} \end{align}\tag{9}\] where \(f_{r}\) is the Fermi function and \(\lambda\) labels an energy-filter channel. The state dynamics depends only on \[\begin{align} \Gamma_{i}^{+} = \sum_{r,\lambda}w_{i0}^{r\lambda}, \qquad \Gamma_{i}^{-} = \sum_{r,\lambda}w_{0i}^{r\lambda}. \label{Eq-total-dot-rates} \end{align}\tag{10}\] The occupation equation is \[\begin{align} \frac{\mathsf{d}}{\mathsf{d}t}p_{i} = \Gamma_{i}^{+}p_{0} - \Gamma_{i}^{-}p_{i}, \qquad p_{0}=1-\sum_{i=1}^{N}p_{i}. \label{Eq-dot-master} \end{align}\tag{11}\] Thus the entire occupation dynamics, including stationary occupations and linear response of the dot probabilities, is fixed by \(\Gamma_{i}^{+}\) and \(\Gamma_{i}^{-}\). The stationary solution is \[\begin{align} p_{0}^{\mathrm{ss}} = \left( 1+\sum_{i=1}^{N} \frac{\Gamma_{i}^{+}}{\Gamma_{i}^{-}} \right)^{-1}, \qquad p_{i}^{\mathrm{ss}} = \frac{\Gamma_{i}^{+}}{\Gamma_{i}^{-}} p_{0}^{\mathrm{ss}} . \label{Eq-dot-ss} \end{align}\tag{12}\] The linearized relaxation matrix around this stationary state is \[\begin{align} A_{ij} = -\Gamma_{i}^{-}\delta_{ij} - \Gamma_{i}^{+}. \label{Eq-dot-response} \end{align}\tag{13}\] The notation means that the second term is independent of the column index \(j\). Hence every state-only relaxation measurement is determined by the same set of total rates \(\{\Gamma_{i}^{+},\Gamma_{i}^{-}\}\).

The reservoir heat record is not fixed by these totals. For an electron entering the dot from reservoir \(r\), the reservoir heat increment is \(-(\varepsilon_{i}-\mu_{r})\). For an electron leaving the dot into reservoir \(r\), it is \(+(\varepsilon_{i}-\mu_{r})\). The heat current from reservoir \(r\) is therefore \[\begin{align} \dot{Q}_{r} = \sum_{i,\lambda} \left(\varepsilon_{i}-\mu_{r}\right) \left[ w_{0i}^{r\lambda}p_{i} - w_{i0}^{r\lambda}p_{0} \right]. \label{Eq-dot-heat} \end{align}\tag{14}\] The total entropy-production rate of the resolved jump process is \[\begin{align} \dot{S}_{\mathrm{env}} = \sum_{i,r,\lambda} \left[ w_{i0}^{r\lambda}p_{0} - w_{0i}^{r\lambda}p_{i} \right] \ln \frac{ w_{i0}^{r\lambda}p_{0} }{ w_{0i}^{r\lambda}p_{i} } . \label{Eq-dot-entropy} \end{align}\tag{15}\] This expression depends on the reservoir-channel rates before they are summed into \(\Gamma_{i}^{+}\) and \(\Gamma_{i}^{-}\). It is therefore not a function of the occupation generator alone. At stationarity, where the system entropy does not change on average, this rate equals the entropy flow into the reservoirs.

State data can nevertheless give bounds when the set of possible reservoirs is known. Let \(q_{ir}=\varepsilon_{i}-\mu_{r}\). Suppose first that an electron-entering transition \(0\rightarrow i\) can be supplied by reservoirs in a set \(R_{i}^{+}\), while the total entering rate \(\Gamma_{i}^{+}\) is known from state tomography. Since the channel rates form a nonnegative decomposition of \(\Gamma_{i}^{+}\), the entering contribution to the heat current lies in the interval \[\begin{align} -p_{0}\Gamma_{i}^{+} \max_{r\in R_{i}^{+}}q_{ir} \leq -p_{0} \sum_{r,\lambda}q_{ir}w_{i0}^{r\lambda} \leq -p_{0}\Gamma_{i}^{+} \min_{r\in R_{i}^{+}}q_{ir}. \label{Eq-heat-bound-in} \end{align}\tag{16}\] Similarly, if the electron-leaving transition \(i\rightarrow0\) can empty into reservoirs in \(R_{i}^{-}\), then \[\begin{align} p_{i}\Gamma_{i}^{-} \min_{r\in R_{i}^{-}}q_{ir} \leq p_{i} \sum_{r,\lambda}q_{ir}w_{0i}^{r\lambda} \leq p_{i}\Gamma_{i}^{-} \max_{r\in R_{i}^{-}}q_{ir}. \label{Eq-heat-bound-out} \end{align}\tag{17}\] The intervals collapse to single values only when all allowed reservoirs have the same heat increment for that transition, or when the channel assignment is specified independently. Thus state dynamics can bound heat currents, but it does not determine them except in thermodynamically complete cases.

We now give an explicit pair of state-identical devices. They have the same occupation master equation, but they differ in heat currents and heat-current noise because the same total dot transition rate is assigned to different reservoirs.

To prove the statement, choose a transition \(0\rightarrow i\) and shift a small rate \(\eta\) from reservoir \(s\) to reservoir \(r\): \(\delta w_{i0}^{r}=\eta\), \(\delta w_{i0}^{s}=-\eta\), with all other channels fixed. Eq. (10 ) is unchanged, so the occupation dynamics is identical. Eq. (14 ) changes by \(-\eta(\mu_{s}-\mu_{r})p_{0}\). The off-diagonal tilted rate for this transition changes from \(\sum_{a=r,s}w_{i0}^{a}e^{-\chi(\varepsilon_{i}-\mu_{a})}\) to the same expression plus \(\eta[e^{-\chi(\varepsilon_{i}-\mu_{r})}-e^{-\chi(\varepsilon_{i}-\mu_{s})}]\). Thus the heat counting statistics differ for generic \(\chi\).

The same conclusion can be expressed directly in the full-counting-statistics generator. If \(\chi_{r}^{Q}\) counts heat entering reservoir \(r\), the off-diagonal entries of the tilted dot generator are \[\begin{align} \left[L(\boldsymbol{\chi}^{Q})\right]_{i0} = \sum_{r,\lambda} w_{i0}^{r\lambda} e^{-\chi_{r}^{Q}\left(\varepsilon_{i}-\mu_{r}\right)}, \label{Eq-dot-tilt-in} \end{align}\tag{18}\] \[\begin{align} \left[L(\boldsymbol{\chi}^{Q})\right]_{0i} = \sum_{r,\lambda} w_{0i}^{r\lambda} e^{+\chi_{r}^{Q}\left(\varepsilon_{i}-\mu_{r}\right)} . \label{Eq-dot-tilt-out} \end{align}\tag{19}\] The state generator is recovered at \(\boldsymbol{\chi}^{Q}=0\), where only the totals in Eq. (10 ) remain. Derivatives with respect to \(\boldsymbol{\chi}^{Q}\) restore the reservoir-resolved heat increments. The dominant eigenvalue \(\lambda_{\max}(\boldsymbol{\chi}^{Q})\) therefore contains information that is absent from \(L\). In particular, the heat-noise matrix \[\begin{align} S_{rs}^{Q} = \left. \frac{\partial^{2}\lambda_{\max}}{\partial\chi_{r}^{Q}\partial\chi_{s}^{Q}} \right|_{\boldsymbol{\chi}^{Q}=0} \label{Eq-dot-heat-noise} \end{align}\tag{20}\] is not fixed by occupation dynamics when different reservoir assignments give different tilted generators.

The prediction is experimentally concrete. Two samples can be tuned to have the same occupation relaxation rates and the same stationary dot populations. If the tunnel couplings distribute the same total rates among reservoirs with different chemical potentials or different energy filters, heat-noise and cross-correlation measurements distinguish the samples even though state tomography does not.

The channel-space mechanism can be seen explicitly. Consider the two entering channels \(0\rightarrow i\) from reservoirs \(r\) and \(s\). In the channel-current basis \((j_{i0}^{r},j_{i0}^{s})\), the projection \(P\) from reservoir-resolved channels to the total ordered transition current has the local block \[\begin{align} P_{i0}^{(r,s)} = \begin{pmatrix} 1 & 1 \end{pmatrix}, \label{Eq-dot-P-block} \end{align}\tag{21}\] where \(P_{i0}^{(r,s)}j=j_{i0}^{r}+j_{i0}^{s}\). The vector \[\begin{align} c_{i0}^{rs} = \begin{pmatrix} 1\\ -1 \end{pmatrix} \label{Eq-dot-kernel-vector} \end{align}\tag{22}\] therefore satisfies \(P_{i0}^{(r,s)}c_{i0}^{rs}=0\). It redistributes current between two reservoir channels without changing the total transition rate, and therefore without changing the occupation generator. The heat-record row for these two channels is \[\begin{align} D_{Q}^{(r,s)} = \begin{pmatrix} -\left(\varepsilon_{i}-\mu_{r}\right) & -\left(\varepsilon_{i}-\mu_{s}\right) \end{pmatrix}, \label{Eq-dot-D-row} \end{align}\tag{23}\] and hence \[\begin{align} D_{Q}^{(r,s)}c_{i0}^{rs} = \mu_{r}-\mu_{s}. \label{Eq-dot-Dc} \end{align}\tag{24}\] Whenever the two reservoirs have different chemical potentials, this generator-invisible channel direction carries a heat record. This is the finite-dimensional geometric origin of the incompleteness in the dot.

The construction also shows that the effect appears directly in transport statistics. The matrices \(L_{\mu}\) and \(L_{\mu\nu}\) entering Eqs. (7 8 ) are changed at first order by a redistribution of rates between reservoirs with different heat increments. The matrix \(L\), its stationary state, and its relaxation spectrum are unchanged. Therefore the discrepancy appears directly in the transport statistics rather than through an error in the inferred state dynamics. In this regime, state measurements are internally consistent but thermodynamically incomplete.

5 Geometric and topological form of the criterion↩︎

The criterion above can be expressed in geometric form. We use a directed transition-channel representation. The nodes are system states, and the directed channels are reservoir- or filter-resolved transitions. Several distinct channels may connect the same ordered pair of states. There are two linear projections. The first projection \(P\) maps channel currents \(j\) to the total ordered-transition currents \(u=Pj\), with \(u_{nm}=\sum_{\alpha}j_{nm}^{\alpha}\). The Markovian state generator fixes this first projection. The second projection \(B\) maps ordered-transition currents to state velocities, so that \(\dot{p}=Bu=BPj\). We assume that the full state generator is known. The primary hidden space is therefore \(\ker P\), rather than the larger nullspace of \(BP\).

Near a typical current \(\bar j\), the channel-current action has the quadratic expansion \[\begin{align} I(j) = \frac{1}{2} \left(j-\bar j\right)^{T} R \left(j-\bar j\right) +o\left(\|j-\bar j\|^{2}\right), \label{Eq-current-geometry} \end{align}\tag{25}\] where \(R\) is the local inverse covariance of channel currents. If only the generator-visible ordered-transition current \(u=P(j-\bar j)\) is retained, the effective cost is the minimum channel-current cost over all reservoir-channel perturbations producing the same \(u\): \[\begin{align} I_{\mathrm{gen}}(u) = \inf_{P\delta j=u} \frac{1}{2} \delta j^{T}R\delta j = \frac{1}{2}u^{T} \left(PR^{-1}P^{T}\right)^{\dagger}u . \label{Eq-quotient} \end{align}\tag{26}\] The dagger denotes the Moore-Penrose inverse on the range of \(P\). This quotient geometry is the part of channel-current fluctuations visible in the state generator. It removes the nullspace \(\ker P\), which consists of reservoir-channel redistributions that leave all total transition rates unchanged [36]–[47].

If \(Dc=0\) for all \(c\in\ker P\), then \(Dj\) is constant on every affine set of channel currents with the same total ordered-transition currents \(Pj\). Therefore the record \(Dj\) factors through the state generator. The row-space condition is the finite-dimensional identity \((\ker P)^{\perp}=\operatorname{ran}P^{T}\). If there is \(c\in\ker P\) with \(Dc\neq0\), then \(j\) and \(j+\epsilon c\) give the same state generator and different transport records. Eq. (?? ) is the dimension of the image of the generator-invisible channel space under the record map \(D\). This gives the measurable count \(d_{\mathrm{lost}}\) of thermodynamic record directions that cannot be reconstructed from the state generator.

The criterion can also be used quantitatively. It gives the complete set of thermodynamic records compatible with a known state generator. Fix the total ordered-transition currents \(u_{nm}\). For each ordered transition, the compatible channel currents form a simplex: \[\begin{align} j_{nm}^{\alpha} = u_{nm}q_{nm}^{\alpha}, \qquad q_{nm}^{\alpha}\geq0, \qquad \sum_{\alpha\in{\cal A}_{nm}}q_{nm}^{\alpha}=1 . \label{Eq-channel-simplex} \end{align}\tag{27}\] This simplex is the set of all reservoir or filter assignments that give the same total transition current \(u_{nm}\).

The proof uses the simplex variables in Eq. (27 ). The mean record associated with one ordered transition is \(u_{nm}\sum_{\alpha}q_{nm}^{\alpha}d_{nm}^{\alpha}\), which is exactly \(u_{nm}H_{nm}\). Summing over ordered transitions gives Eq. (?? ). A linear function on a convex hull reaches its extrema at the listed channel increments, giving Eqs. (?? ?? ). The geometry therefore supplies an operational reconstruction bound. From the state generator and the allowed transport increments one can determine the full range of heat, particle, or spin records that remain compatible with the same state dynamics.

The same nullspace controls current noise and full counting statistics, but the record map must then be understood at the level of the tilted generator. Let \(\mathcal{L}_{\boldsymbol{\chi}}\) denote the off-diagonal channel-resolved tilted rates in Eq. (4 ). For a generator-preserving perturbation \(c\in\ker P\), the first-order change of the tilted generator is \[\begin{align} \delta_{c}\mathcal{L}_{\boldsymbol{\chi}} = \sum_{e} c_{e} e^{\boldsymbol{\chi}\cdot d_{e}} |n(e)\rangle\langle m(e)|, \label{Eq-FCS-null-variation} \end{align}\tag{28}\] where \(e\) runs over directed channels, \(m(e)\) and \(n(e)\) are its initial and final states, and \(d_{e}\) is the vector of recorded increments. The channel-resolved tilted generator is fixed by the state generator only if \(\delta_{c}\mathcal{L}_{\boldsymbol{\chi}}=0\) for every \(c\in\ker P\) and for the counting fields being used. Equivalently, the channel-resolved exponential weights must be insensitive to all generator-invisible channel redistributions. If this condition fails and the dominant eigenvalue of the tilted generator changes, then at least one current cumulant is not determined by state dynamics. The quantum-dot construction in Sec. 4 gives this change explicitly.

The geometric and topological interpretation is therefore operational. The quotient in Eq. (26 ) gives the fluctuation cost seen after reservoir-channel assignments have been eliminated, while \(C(u)\) gives the remaining heat, particle, spin, or photon records compatible with the same state generator. The discrete connectivity pattern of the transport channels sets the number of independent redistributions. If there are \(E\) reservoir-resolved channels and \(E_{0}\) ordered state transitions after reservoir labels are forgotten, then \[\begin{align} \dim\ker P = E-E_{0}. \label{Eq-loop-dimension} \end{align}\tag{29}\] The number \(\dim\ker P\) is unchanged by continuous changes of the positive transition rates. It changes only when a transport channel is added, removed, or attached to a different ordered transition. Eq. (?? ) refines this connectivity count for thermodynamics: not every generator-invisible redistribution is visible in a chosen transport measurement, and only the image of \(\ker P\) under the record map \(D\) produces missing thermodynamic data. Thus tuning tunnel rates can change the size of the compatible-record set \(C(u)\), but it cannot remove a nonzero ambiguity unless the channel connectivity or the measured records are changed. If one knows only an instantaneous state velocity rather than the full state generator, the larger nullspace \(\ker(BP)\) also contains closed paths through the reduced state network. With the full Markovian state generator as input, the relevant hidden directions are the reservoir-channel redistributions counted by Eq. (29 ).

For the quantum dot of Sec. 4, a rate shift between two reservoir channels for the same dot transition is an element of \(\ker P\). If the two reservoirs have different \(\varepsilon_{i}-\mu_{r}\), the heat record matrix \(D\) does not annihilate that null direction. This gives precisely the incompleteness shown in Proposition 1. It also gives the measurement prescription: state tomography must be supplemented by enough independent reservoir-current or heat-current measurements to remove the nonzero directions counted by \(d_{\mathrm{lost}}\).

This prescription can be used without reconstructing every microscopic channel. Suppose that a set of measured records is represented by the rows of a matrix \(D_{\mathrm{meas}}\). The remaining ambiguity after state tomography and these transport measurements is \[\begin{align} {\cal K}_{\mathrm{rem}} = \ker P \cap \ker D_{\mathrm{meas}} . \label{Eq-remaining-kernel} \end{align}\tag{30}\] An additional target observable \(D_{\mathrm{tar}}j\) is predictable from the available data if and only if \[\begin{align} D_{\mathrm{tar}}c=0 \quad \text{for every }c\in{\cal K}_{\mathrm{rem}} . \label{Eq-diagnostic} \end{align}\tag{31}\] Eqs. (30 31 ) are the practical diagnostic. They say which extra transport records must be measured before a state-only model can be used for thermodynamic prediction. In the energy-filtered dot, measuring only the total occupation dynamics leaves rate redistributions among reservoirs in \({\cal K}_{\mathrm{rem}}\). Measuring the independent reservoir heat currents removes exactly the null directions that would otherwise change heat noise and heat cross correlations.

6 Conclusion↩︎

We have shown that a Markovian state master equation can be thermodynamically incomplete. The failure does not require memory effects, non-Markovian reservoirs, or an approximate state description. It can occur inside an ordinary Markovian transport model when the state generator fixes only total transition rates and not their reservoir-channel assignments.

The main result is a reconstruction criterion. A thermodynamic record is determined by state dynamics only if it is invariant under every change of channel assignment that leaves the state generator unchanged. In a multi-terminal energy-filtered quantum dot, this criterion gives a measurable prediction: devices with identical occupation dynamics can have different heat currents, entropy production, and heat-current noise [33], [34]. The geometric and topological analysis explains why. State dynamics is a projection of channel-current dynamics. The quotient geometry gives the state fluctuations, while generator-invisible reservoir-channel redistributions carry the missing thermodynamic records. The compatible-record set in Eq. (?? ) turns this statement into a practical bound on which thermodynamic records remain possible after the state generator has been measured.

The practical limitation is direct: a master equation fitted from state tomography may be sufficient for relaxation and stationary occupations, but insufficient for thermodynamic performance and noise. To make thermodynamic predictions complete, one must specify not only the state generator but also the reservoir and measurement records associated with the transitions [9], [24], [27].

References↩︎