Nonadiabatic Entropy Production

• Nonadiabatic entropy production is the transient dissipative contribution that arises when a system deviates from its instantaneous steady state under time-dependent driving.
• It decomposes total entropy production into adiabatic 'housekeeping' and nonadiabatic (excess) components, thereby refining the second-law inequality.
• It underpins fluctuation theorems and informs control strategies, speed limits, and efficiency analyses in driven stochastic and quantum systems.

Nonadiabatic entropy production quantifies the contribution to total entropy production that arises from the system’s instantaneous deviation from its reference stationary—or adiabatic—state, typically in the context of time-dependent driving or environmental changes. This decomposition is central to modern stochastic, quantum, and nonequilibrium thermodynamics, providing both a refined second-law inequality and operational fluctuation theorems for systems far from equilibrium. Nonadiabatic entropy production links geometric, information-theoretic, and dynamical features of irreversibility at both trajectory and ensemble levels, and has direct implications for the thermodynamic efficiency, control, and speed limits of driven systems.

1. Definitions and Formalism

Nonadiabatic entropy production (EP), often labeled “excess” or “transient” EP, can be formally defined for a broad class of dynamics—discrete Markov jump processes, continuous Langevin/Fokker–Planck evolution, quantum CPTP maps, and Lindblad dynamics—by decomposing the total EP into adiabatic (“housekeeping”) and nonadiabatic contributions (Broeck et al., 2010, Manzano et al., 2017, Horowitz et al., 2013).

Consider a general Markovian process for the state probability vector $p(t)$ with an instantaneous steady state $p^{\mathrm{st}}(\lambda(t))$ under parameter $\lambda(t)$. The total (trajectory) entropy production $\Delta S_{\mathrm{tot}}[\gamma]$ admits the decomposition

$$\Delta S_{\mathrm{tot}} = \Delta S_{\mathrm{na}} + \Delta S_{\mathrm{a}},$$

where

• $\Delta S_{\mathrm{na}}$ is the nonadiabatic entropy production, associated with system relaxation and time-dependent driving,
• $\Delta S_{\mathrm{a}}$ is the adiabatic part, associated with steady-state “housekeeping” dissipation required to maintain nonequilibrium currents.

For continuous-state processes under Fokker–Planck dynamics, the nonadiabatic EP rate is given by

$$\dot S_{\mathrm{na}} = \int dx\, J(x,t) \partial_x \ln \frac{p(x,t)}{p^{\mathrm{ss}}(x,t)} = -\frac{d}{dt} D(p(x,t) \| p^{\mathrm{ss}}(x,t)),$$

where $p^{\mathrm{ss}}(x,t)$ is the instantaneous steady-state solution (Broeck et al., 2010).

In quantum contexts, given a CPTP map $\mathcal{E}$ with invariant state $\pi$ ($\mathcal{E}(\pi) = \pi$), and nonequilibrium potential $\Phi = -\ln \pi$, the mean nonadiabatic EP from $\rho \rightarrow \rho' = \mathcal{E}(\rho)$ reads

$$\Delta_i S_{\mathrm{na}} = S(\rho' \| \pi) - S(\rho \| \pi) \geq 0,$$

with $$S(\rho \| \pi) = \operatorname{Tr}[ \rho ( \ln \rho - \ln \pi ) ]$$, signifying the contraction of quantum relative entropy to the invariant state (Manzano et al., 2017, Horowitz et al., 2014).

Pathwise or trajectory-based formulations consistently express the nonadiabatic entropy as the log-ratio of a forward and a specifically defined dual- or accompanying-process probability (Horowitz et al., 2013, Manzano et al., 2017, García-García, 2012, Sohn, 2014): $$\Delta S_{\mathrm{na}}[\gamma] = \ln \frac{P[\gamma]}{P^{\mathrm{dual}}[\tilde{\gamma}]}$$ where $P^{\mathrm{dual}}$ corresponds to a process with reversed or dual dynamics ensuring invariance of the reference steady state.

2. Physical Interpretation and Significance

Nonadiabatic entropy production characterizes the irreversibility tied to the deviation of an instantaneous distribution from the contemporaneous steady state. It quantifies dissipation resulting from system relaxation and from explicit time-dependent driving (protocol variations) rather than from the maintenance of steady-state nonequilibrium currents (Broeck et al., 2010, Horowitz et al., 2014, Horowitz et al., 2013, Manzano et al., 2017).

Key interpretations include:

• Distance to steady state: Nonadiabatic EP is the rate at which the relative entropy to the instantaneous NESS contracts; it vanishes if $p(t) = p^{\mathrm{ss}}(\lambda(t))$ at all times, even if $\lambda(t)$ varies.
• Relaxational dissipation: It encodes all transient entropy generated as $p(t)$ chases $p^{\mathrm{ss}}(t)$ under dynamically changing environments or parameters.
• Refined second law: The average nonadiabatic term satisfies a strong second-law–type bound, $\langle \Delta S_{\mathrm{na}} \rangle \geq 0$, holding for all (Markovian and certain non-Markovian) systems with well-defined steady states (García-García, 2012, Broeck et al., 2010).

3. Fluctuation Theorems and Mathematical Properties

Nonadiabatic entropy production possesses a robust fluctuation theorem (FT) structure central to nonequilibrium statistical mechanics:

• Integral FT: $\langle e^{-\Delta S_{\mathrm{na}}} \rangle = 1$,
• Detailed FT: $$P(\Delta S_{\mathrm{na}}=x)/P(\Delta S_{\mathrm{na}}=-x) = e^{x}$$, when microreversibility (local detailed balance) holds at the trajectory level (Manzano et al., 2017, Broeck et al., 2010, Horowitz et al., 2014, García-García, 2012).

This FT validates its role as an independent measure of irreversibility and implies strict non-negativity for its ensemble average via Jensen’s inequality,

$\langle \Delta S_{\mathrm{na}} \rangle \geq 0,$

generalizing the second law to transitions between nonequilibrium steady states even in non-Markovian regimes with appropriate stability (García-García, 2012).

4. Splitting: Adiabatic Versus Nonadiabatic Contributions

The total entropy production is additively split: $$\Delta S_{\mathrm{tot}} = \Delta S_{\mathrm{a}} + \Delta S_{\mathrm{na}},$$ with precise trajectory and ensemble counterparts in both classical and quantum settings (Manzano et al., 2017, Broeck et al., 2010, Horowitz et al., 2013, Matsuo et al., 2022). The adiabatic contribution $\Delta S_{\mathrm{a}}$ quantifies the “housekeeping” dissipation required to maintain the NESS—for instance, to sustain steady-state current in the presence of driving forces or gradients—even if the system is not relaxing toward equilibrium.

Tables, such as below, organize contributions:

Quantity	Physical Meaning	Vanishing Condition
$\Delta S_{\mathrm{na}}$	Relaxational/excess, transient EP	$p(t) = p^{\mathrm{ss}}(t)$
$\Delta S_{\mathrm{a}}$	Housekeeping, adiabatic/steady EP	Equilibrium/detailed balance
$\Delta S_{\mathrm{tot}}$	Total entropy production	Equilibrium (all vanish)

This splitting is universally non-negative under local detailed balance conditions and underlies refined performance bounds and control design in nonequilibrium devices (Manzano et al., 2017, Kolchinsky et al., 11 Dec 2024).

5. Information Geometry and Trade-Off Relations

Recent advances connect nonadiabatic entropy production directly to information-geometric quantities, with the quantum or classical relative entropy playing the role of a thermodynamic “distance” from steady state (Takahashi, 2017, Kolchinsky et al., 11 Dec 2024). The decomposition in the space of probability measures or density operators reveals a Pythagorean relation: $D(P \| R) = D(P \| Q) + D(Q \| R)$ where $D$ denotes the Kullback-Leibler divergence and $Q$ is a projection of $P$ orthogonal to $R$ in the information-geometric manifold (Takahashi, 2017). The nonadiabatic part corresponds to the “dynamical” leg—distance accrued by nonadiabatic transitions induced by driving.

Thermodynamic speed limits result, constraining the excess (nonadiabatic) entropy production in finite-time protocols by the information-geometric length of the driving path and the available dynamical activity (transition rates or fluxes) (Kolchinsky et al., 11 Dec 2024): $$\dot S_{\mathrm{ex}} \geq \frac{\dot{F}^2}{\mathcal{I}},$$ with $\mathcal{I}$ a Fisher-information–like activity, and similar Wasserstein-speed limits in discrete-state systems.

6. Quantum Nonadiabatic Entropy Production

In open quantum systems described by Lindblad dynamics or general CPTP maps, nonadiabatic entropy production is constructed via:

• Quantum relative entropy: $S(\rho \| \pi) - S(\rho' \| \pi)$,
• Quantum jump trajectories: log-ratio of the path probabilities between forward and dual-reverse processes, operationally linked to experimentally accessible measurement and control protocols (Manzano et al., 2017, Horowitz et al., 2014, Horowitz et al., 2013).

Key structural features:

• The dual (reverse) process must be defined with respect to the nonequilibrium potential operator $\Phi = -\ln \pi$.
• Exact splitting into adiabatic and nonadiabatic parts—and the validity of separate fluctuation theorems for each—requires each Kraus (or jump) operator to effect a well-defined change in the nonequilibrium potential: $$[\Phi, M_{\mu \nu}] = \Delta\phi_{\mu \nu} M_{\mu \nu}$$ (Manzano et al., 2017).

Failure of this commutation condition (e.g., for coherent maps) can compromise the positivity and splitting, leading to cases where the “adiabatic” EP can become negative transiently.

7. Applications and Examples

Nonadiabatic entropy production plays a pivotal role in diverse physical contexts:

• Driven open quantum systems: Exemplified by three-level thermal machines or harmonic oscillators coupled to multiple reservoirs, where nonadiabatic EP quantifies dissipation from rapid driving and guides the design of protocols minimizing excess EP (Manzano et al., 2017, Horowitz et al., 2013).
• Classical transport and relaxation: In stochastic thermal transport, the relaxational component of the entropy production (nonadiabatic EP) encodes transient heat and density redistribution under time-dependent temperature fields (Ford et al., 2015).
• Many-body and hydrodynamic systems: In quasi-adiabatic parameter sweeps, nonadiabatic EP controls the minimal unattainable entropy for finite-rate protocols, with hydrodynamic fluctuations fundamentally limiting the approach to adiabaticity at rates dictated by diffusive modes (Weiß et al., 2020).
• Thermodynamic control and speed limits: The geometric structure of nonadiabatic EP underpins universal lower bounds on entropy production for rapid state transformations, relevant to optimal thermodynamic control, shortcuts to adiabaticity, and precision quantum operations (Takahashi, 2017, Kolchinsky et al., 11 Dec 2024, Zhelnin et al., 19 Jun 2025).

Experimental protocols that probe nonadiabatic FTs typically require two-point measurement schemes, environmental engineering to ensure well-defined dual operators, and pathwise reconstruction of trajectory probabilities to validate fluctuation symmetries (Manzano et al., 2017, Horowitz et al., 2013).

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References:

(Broeck et al., 2010, García-García, 2012, Horowitz et al., 2013, Horowitz et al., 2014, Sohn, 2014, Ford et al., 2015, Takahashi, 2017, Manzano et al., 2017, Weiß et al., 2020, Matsuo et al., 2022, Kolchinsky et al., 11 Dec 2024, Zhelnin et al., 19 Jun 2025).