Thermo-Kinetic Uncertainty Relations

• Thermo-kinetic uncertainty relations are nonequilibrium bounds that link the precision of trajectory observables to thermodynamic costs like entropy production and kinetic measures such as dynamical activity.
• They unify classical Markov jump process analysis with quantum corrections, offering complementary limits through TURs, KURs, and their unified extensions.
• These relations provide practical insights into modeling energy transport, biological motors, and quantum thermometry by clarifying the trade-offs between dissipation, fluctuations, and coherence.

Searching arXiv for recent and foundational papers on thermo-kinetic uncertainty relations and closely related variants. Querying arXiv for "thermo-kinetic uncertainty relation Markov jump process", "kinetic uncertainty relation", and "unified thermodynamic kinetic uncertainty relation". Thermo-kinetic uncertainty relations are a family of nonequilibrium bounds that connect the precision of trajectory observables—most prominently time-integrated currents—to thermodynamic and kinetic costs. In the stochastic-thermodynamic setting, these costs are typically the total entropy production and the dynamical activity, while in broader quantum and metrological extensions analogous trade-offs involve coherence, heat fluctuations, or quantum Fisher information. In the modern Markovian formulation, the thermodynamic uncertainty relation (TUR) and the kinetic uncertainty relation (KUR) are complementary: near equilibrium the entropy-production bound is often tighter, whereas far from equilibrium the activity bound can become dominant; unified thermo-kinetic bounds combine both ingredients into a single stronger constraint (Terlizzi et al., 2018, Vo et al., 2022, Nishiyama, 2022).

1. Conceptual origins and scope

A deep antecedent of the subject is the operational link between uncertainty and thermodynamics. For two binary measurements $f=\{f_0,f_1\}$ and $g=\{g_0,g_1\}$, the fine-grained uncertainty relation

$$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$

was used to show that if the quantum bound were violated, one could construct a cyclic process with positive net work gain, $\Delta W>0$, in contradiction with the second law (Hänggi et al., 2012). In that construction, stronger-than-quantum predictability lowers the entropy cost of a work-extraction cycle enough to produce net positive work. This established an operational justification for the exact quantum form of uncertainty relations.

A second line of development concerns thermodynamic fluctuation complementarity. In a finite system where energy $U$, temperature $T$, and multiplicity $N$ may all fluctuate, the Lindhard-style relation

$$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$

was generalized to

$$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$

by allowing multiplicity fluctuations and the covariance between energy and temperature (Wilk et al., 2011). In that framework, Tsallis nonextensivity and Negative Binomial multiplicity statistics encode hidden temperature or mean-multiplicity fluctuations.

These precursors are not yet the modern current-fluctuation TKUR, but they already display the central theme: uncertainty is constrained by thermodynamic structure. A plausible implication is that the later stochastic-thermodynamic literature did not introduce an entirely new principle so much as sharpen an older one for trajectory observables, dissipation, and kinetics.

2. Classical current-precision relations in stochastic dynamics

The modern thermo-kinetic program is formulated for continuous-time Markov jump processes with transition rates $k_{ij}$, escape rates $g=\{g_0,g_1\}$0, and trajectory observables $g=\{g_0,g_1\}$1 of finite mean. In this setting, the KUR follows from the fluctuation-response inequality of Dechant and Sasa under the global rescaling

$g=\{g_0,g_1\}$2

For this perturbation, the quadratic Kullback–Leibler cost is $g=\{g_0,g_1\}$3, where

$g=\{g_0,g_1\}$4

is the integrated mean dynamical activity, i.e. the mean number of jumps up to time $g=\{g_0,g_1\}$5. The resulting KUR is

$g=\{g_0,g_1\}$6

This bound holds at all times, for transient and steady-state dynamics alike, and for broad classes of observables, including currents, counting observables, and nonlinear functions of counts (Terlizzi et al., 2018).

The corresponding time-dependent TUR can be written in the same notation,

$g=\{g_0,g_1\}$7

with $g=\{g_0,g_1\}$8 the total entropy production over $g=\{g_0,g_1\}$9. The physical distinction is sharp: the TUR is tied to dissipation and local detailed balance, whereas the KUR is frenetic and measures the sheer volume of transitions. Near equilibrium, entropy production is typically the tighter control parameter; far from equilibrium, activity can become the limiting one because dissipation may grow faster than the transition rate (Terlizzi et al., 2018).

The unified thermodynamic-kinetic uncertainty relation makes this complementarity explicit. For a time-integrated current $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$0, total entropy production $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$1, total dynamical activity $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$2, and differential operator $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$3, one form of the bound is

$$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$4

where $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$5 is the inverse of $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$6 (Vo et al., 2022). In the limits $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$7 and $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$8, this reduces to the TUR and KUR, respectively. The unified form therefore treats entropy production and activity not as competing explanations but as two asymptotic faces of a single precision bound.

3. Tightness, information theory, and variational refinements

The unified TKUR was subsequently recast in a form that makes its sharpness explicit. Writing $$\frac{1}{2}\Big(p(f_{y_1}\mid p)+p(g_{y_2}\mid p)\Big)\leq \zeta_y$$9 and letting $\Delta W>0$0 denote the inverse of $\Delta W>0$1, the bound becomes

$\Delta W>0$2

This formulation was shown to be the tightest bound within a broad class of inequalities depending only on $\Delta W>0$3, $\Delta W>0$4, and $\Delta W>0$5 (Nishiyama, 2022). The same work gave an information-theoretic interpretation by introducing forward and backward flux distributions $\Delta W>0$6 and $\Delta W>0$7 with

$\Delta W>0$8

and a binary pair $\Delta W>0$9 determined by the precision-to-activity ratio $U$0, yielding

$U$1

In this representation, the TKUR is an extremal Kullback–Leibler-divergence bound.

A distinct variational refinement appears in the Thermodynamic Uncertainty Theorem for time-symmetrically controlled computations. There, the minimal scaled variance over all time-antisymmetric currents is

$U$2

and the minimizing current is

$U$3

Because the right-hand side depends on $U$4, the bound retains the effect of all odd higher moments of entropy production, not merely $U$5 (Ray et al., 2022). This shifted attention from average dissipation to the full entropy-production distribution.

A related but structurally different development is the “Thermo-Kinetic Relations” program for overdamped Markov jump processes. In stationary settings it derives bounds on entropy production using absolute fluctuations of time-antisymmetric observables rather than mean-square fluctuations, and in non-stationary settings it becomes a generalized classical speed limit. It also separates total entropy production into non-adiabatic and housekeeping contributions and shows a trade-off: housekeeping entropy production can be increased to reduce non-adiabatic entropy production, though only to a limited extent (Delvenne et al., 2021).

A further unification was later obtained directly at the stochastic-trajectory level. Instead of using auxiliary perturbations and a Cramér–Rao argument for deterministic evolution equations, one introduces an auxiliary stochastic observable $U$6 and applies Cauchy–Schwarz,

$U$7

Different choices of $U$8 recover activity-controlled or entropy-production-controlled bounds, yielding TURs and KURs from the same stochastic-representation principle (Kwon et al., 2024). This suggests that the thermo-kinetic structure is intrinsic to stochastic path measures rather than specific to one proof technique.

4. Extensions to many-body, driven, and transient classical systems

For interacting many-body systems, a single current is generally not the optimal thermodynamic probe. The multidimensional TUR gives

$U$9

where $T$0 is the vector of particle currents and $T$1 their covariance matrix (Koyuk et al., 2022). In the thermodynamic limit of driven mixtures with short-range interactions, the optimal estimator depends not only on one-particle currents and diffusion coefficients but also on pair-current correlations. For one species,

$T$2

and if $T$3, the quality factor becomes

$T$4

The key point is that pair correlations survive collectively in the thermodynamic limit even when each individual correlation vanishes as $T$5 (Koyuk et al., 2022).

Under arbitrary control protocols, the TUR structure acquires extra kinetic and boundary terms. For observables satisfying the scaling condition $T$6, the paper on Langevin systems derives bounds of the form

$T$7

with $T$8 a kinetic term built from the dynamical generator and $T$9 a nonnegative boundary term (Vu et al., 2019). This framework applies to overdamped and underdamped dynamics, to current and noncurrent observables, and to arbitrary initial states. It makes explicit that entropy production alone is not sufficient for time-symmetric or protocol-dependent observables.

Concrete examples clarify the division of labor between thermodynamic and kinetic costs. For a biased random walk, the KUR becomes dominant far from equilibrium because entropy production grows faster than activity (Terlizzi et al., 2018). In random fully connected networks, the KUR remains valid for nonlinear observables such as

$N$0

For kinesin, the bound $N$1 is reported to be the relevant ceiling on motion precision under physiological conditions, roughly $N$2 (Terlizzi et al., 2018). For irreversible predator-prey dynamics with missing reverse rates, the usual entropy-production TUR cannot be formulated in the standard way, but the KUR still applies, illustrating the wider domain of activity-based control (Terlizzi et al., 2018).

5. Quantum thermo-kinetic relations

In Markovian open quantum dynamics, uncertainty relations survive but acquire coherence-dependent corrections. For Lindblad evolution unraveled into quantum-jump trajectories, finite-time lower bounds were derived for currents, counting observables, static observables, and first-passage times, with coherence entering through additive terms $N$3 that modify the effective entropy-production and activity denominators (Vu et al., 2021). In a generic class of dissipative processes with nondegenerate energy levels satisfying $N$4, the paper proves $N$5, so coherence tends to enhance fluctuations rather than suppress them. Under stronger nonresonance conditions, the classical TUR is exactly recovered (Vu et al., 2021).

A different quantum construction treats thermodynamic currents themselves as Hermitian operators. For a composite Hamiltonian

$N$6

the operator-based approach defines

$N$7

so that their expectation values reproduce the average power, heat rate, and internal-energy rate (Sathe et al., 2024). Applying the Robertson–Schrödinger relation

$N$8

yields a power–heat-rate uncertainty relation and, for quantum batteries, an energy–power uncertainty relation. In this formulation, the lower bound is controlled by non-commutativity and covariance rather than by entropy production (Sathe et al., 2024).

Mesoscopic coherent transport provides explicit violations of classical thermo-kinetic bounds. In a double quantum dot coupled to two thermal reservoirs, the classical indicators

$N$9

and the unified $$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$0 can exceed $$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$1, thereby violating the corresponding classical TUR, KUR, and TKUR bounds (Prech et al., 2022). The violations accompany a peak in the coherence $$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$2 generated by interdot tunneling. The same work shows that entanglement and nonlocality are complementary to, but not equivalent to, these dynamical violations: entanglement depends on the state, whereas TUR/KUR depend on two-time current correlations (Prech et al., 2022).

Quantum transport scattering theory generalizes the combined TUR/KUR logic to multi-terminal, multi-channel bosonic and fermionic devices, even without time-reversal symmetry. There, the combined bound involves entropy production $$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$3, classical activity $$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$4, and a symmetry-breaking factor

$$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$5

Bosonic bunching makes the quantum noise contribution positive, whereas fermionic antibunching makes it negative; the resulting TKUR-like bounds are therefore statistics-dependent (Palmqvist et al., 7 Apr 2025). This places quantum transport within the same general thermo-kinetic architecture, but with carrier statistics and transmission structure explicitly shaping achievable precision.

Finite-time quantum KURs are not unique. In a two-qubit entanglement engine, two plausible finite-time definitions,

$$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$6

coincide only in the steady-state limit (Bourgeois et al., 2024). This establishes that transient quantum precision bounds depend on the operational definition of current and noise, even when the steady-state KUR is unambiguous.

6. Thermometric, heat-exchange, and experimental variants

A prominent quantum-thermodynamic branch of the subject concerns estimating temperature. For a strongly coupled equilibrium subsystem described by a Hamiltonian of mean force and effective internal energy operator $$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$7, the energy-temperature uncertainty relation becomes

$$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$8

where $$\omega_U^2+\omega_T^2=\frac{1}{\langle N\rangle}, \qquad \omega_x^2=\frac{\mathrm{Var}(x)}{\langle x\rangle^2},$$9 is the average Wigner–Yanase–Dyson skew information (Miller et al., 2018). The same paper derives the modified fluctuation-dissipation relation

$$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$0

and the temperature signal-to-noise bound

$$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$1

Here, non-commutativity between the reduced state and the effective energy operator makes temperature estimation harder (Miller et al., 2018).

In nonequilibrium probe thermometry, the fundamental fluctuation is heat rather than energy alone. The temperature-heat uncertainty relation is

$$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$2

where $$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$3 is trajectory heat and $$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$4 is correlation heat induced by conditioning on the probe measurement (Zhang et al., 2023). The decomposition shows that both heat exchange and probe–sample correlations can serve as precision resources; in equilibrium, the same combination reduces to the standard temperature-energy relation $$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$5 (Zhang et al., 2023).

A more general equilibrium formulation for thermodynamically conjugate variables $$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$6, with $$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$7, derives the quantum-Fisher-information bounds

$$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$8

leading to the uncertainty relation

$$\left|\omega_N^2-\frac{1}{\langle N\rangle}\right| = \omega_U^2+\omega_T^2-2\rho\,\omega_U\omega_T$$9

(Meng et al., 7 Nov 2025). This is a metrological uncertainty principle for an intensive classical control parameter and its extensive quantum conjugate.

Experimental and model studies of transient quantum heat transport reveal an important caveat: generalized fluctuation-theorem-based TURs can remain valid even when tighter specialized versions fail. In a two-qubit NMR experiment, the generalized bounds

$k_{ij}$0

were obeyed, whereas the specialized bound

$k_{ij}$1

was violated in regimes where $k_{ij}$2 is large enough; the criterion reported near equilibrium is $k_{ij}$3, and violation is guaranteed whenever $k_{ij}$4 (Pal et al., 2019). A related transient model study found that the tighter TUR is always satisfied for coupled harmonic oscillators with Bose-Einstein statistics, but can be violated for coupled qubits and for a hybrid qubit–oscillator model, while the generalized TURs remain valid in all cases (Saryal et al., 2020).

These results correct a common oversimplification. Thermo-kinetic uncertainty relations do not form a single universal inequality governed only by average entropy production. Depending on the setting, precision can be limited by dissipation, by dynamical activity, by pair-current correlations, by higher cumulants of entropy production, by coherence corrections in quantum trajectories, or by operator non-commutativity in current observables. The unifying idea is the existence of rigorous trade-offs between precision and physically meaningful costs, but the precise form of the trade-off is model-dependent and operationally sensitive.