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Testing the validity of the local and global GKLS master equations on an exactly solvable model
J. Onam González, Luis A. Correa, Giorgio Nocerino, José P. Palao, Daniel Alonso, Gerardo Adesso
TL;DR
The paper asks when local and global master equations reliably describe heat transport in weakly interacting multipartite systems. It derives both approaches for an exactly solvable two-node quantum wire and benchmarks them against the exact steady state. The local equation is accurate near resonant weak coupling, where the global equation's secular approximation fails, while the global equation becomes correct as the inter-node coupling grows.
Problem
Local master equations simplify multipartite modelling by ignoring coherent couplings but can be thermodynamically inconsistent, whereas global equations require a potentially fragile secular approximation.
Method
The paper derives local and global second-order master equations for a two-node harmonic wire and compares their steady states, currents, and correlations with an exact solution.
Results
For nearly resonant weak coupling, the LME accurately reproduces stationary properties while the GME becomes qualitatively wrong; for larger coupling, the LME breaks down and the GME becomes correct.
Takeaways & Limitations
The local and global approaches are complementary, with their reliability determined by the relation between inter-node coupling, detuning, and dissipation.
Takeaways & Limitations
The local approach can produce qualitatively wrong conclusions in quantum thermodynamic cycles, including failures to capture heat leaks and internal dissipation.
Abstract
from arXiv · showhide
When deriving a master equation for a multipartite weakly-interacting open quantum systems, dissipation is often addressed \textit{locally} on each component, i.e. ignoring the coherent couplings, which are later added `by hand'. Although simple, the resulting local master equation (LME) is known to be thermodynamically inconsistent. Otherwise, one may always obtain a consistent \textit{global} master equation (GME) by working on the energy basis of the full interacting Hamiltonian. Here, we consider a two-node `quantum wire' connected to two heat baths. The stationary solution of the LME and GME are obtained and benchmarked against the exact result. Importantly, in our model, the validity of the GME is constrained by the underlying secular approximation. Whenever this breaks down (for resonant weakly-coupled nodes), we observe that the LME, in spite of being thermodynamically flawed: (a) predicts the correct steady state, (b) yields the exact asymptotic heat currents, and (c) reliably reflects the correlations between the nodes. In contrast, the GME fails at all three tasks. Nonetheless, as the inter-node coupling grows, the LME breaks down whilst the GME becomes correct. Hence, the global and local approach may be viewed as \textit{complementary} tools, best suited to different parameter regimes.
I. INTRODUCTION
GKLS master equations provide thermodynamic consistency under their standard assumptions, but local dissipation in multipartite systems can violate those assumptions. This paper tests local and global approaches against an exactly solvable two-node quantum wire, showing that their accuracy depends on parameter regime.
- GKLS dynamics ensures complete positivity and, under mild assumptions, relaxation toward the thermal state.
- Stationary heat currents generated by GKLS equations satisfy the Clausius inequality, ensuring thermodynamic consistency.
- The local master equation adds dissipators for individual components while ignoring their coherent interactions.
- Because local dissipators match the non-interacting Hamiltonian rather than the full Hamiltonian, local modelling can produce thermodynamic inconsistencies.Reported examples include heat flowing against the temperature gradient and nonzero currents when all reservoirs share one temperature.
- The global construction is more difficult for large systems because dense spectra can invalidate the assumption that dissipation is the slowest timescale.
- For a weakly coupled two-node wire, the local approach accurately approximates steady states, stationary currents, and asymptotic correlations precisely where the global approach can fail qualitatively.
A. The model, the Markovian master equation and its steady state
The model is a two-node harmonic wire, with each oscillator weakly coupled to a thermal bosonic bath. A second-order Markovian treatment yields Gaussian steady states characterized by a covariance matrix.
- The wire consists of mechanically coupled harmonic oscillators with frequencies ωc and ωh, coupling strength k, and bath temperatures Tc < Th.
- The system-bath interaction is treated perturbatively because the dissipation strength λ is weak.The bath coupling operators scale as O(λ), and the master-equation treatment is retained to second order in the coupling.
- The Born-Markov approximation replaces the finite memory integral with an infinite one when bath correlations decay sufficiently rapidly.The stated condition is λ^2 ≪ min{T,Λ}.
- The secular approximation averages away terms oscillating rapidly compared with the dissipation timescale, producing the canonical GKLS equation.
- Because the Hamiltonian is quadratic, the steady state is Gaussian and fully specified by its first and second moments.The stationary first moments vanish, leaving a 4 × 4 covariance matrix as the relevant state description.
B. The global master equation
The global master equation is derived in the normal-mode basis, where its validity depends on separating decay channels sufficiently for the secular approximation. Near resonance and weak inter-node coupling, that separation fails, limiting the GME.
- The system Hamiltonian is transformed into normal modes, whose frequencies determine the global dissipative channels.
- The interaction-picture dynamics contains 16 terms associated with five decay channels and several frequency differences.
- The secular approximation is justified for sufficiently detuned nodes when the detuning greatly exceeds the dissipation scale.
- For resonant nodes, validity requires λ^2 ≪ k/ωc, so the GME is expected to fail for nearly resonant weakly coupled nodes.
- The global steady state and heat currents follow from closed equations for normal-mode occupations and covariance elements.
- The GME predicts heat flow from the hotter bath to the colder bath when Th > Tc.
C. The local master equation
The local master equation constructs separate dissipators for each node while retaining the full system Hamiltonian, offering a simpler alternative when full energy-basis decompositions are difficult. Its steady-state solution can nevertheless violate thermodynamic consistency.
- Construction: The local approach uses dissipators derived for uncoupled nodes and combines them with the interacting system Hamiltonian.This approximates the dynamics by adding independent local dissipators to coherent evolution under H_S.
- Motivation: This strategy is convenient when finding all energy eigenstates of the full interacting Hamiltonian is difficult.The global decomposition of each bath-coupled operator requires those eigenstates.
- Dynamics: In the two-node model, the local approach requires all 10 independent covariances to obtain a closed set of equations of motion.Despite its simpler dissipator construction, its resulting dynamics are more complicated than the global approach in this example.
- Construction: The local master equation decomposes each node coordinate into positive- and negative-frequency components built from its annihilation operator.This decomposition defines the local jump operators used in the dissipative terms.
- Steady state: The stationary solution is cumbersome, although its steady-state heat currents can be written compactly.The covariance equations provide the route to the stationary solution and currents.
- Thermodynamic consistency: The local equation can yield thermodynamically inconsistent steady states, including heat flowing against the temperature gradient or nonzero currents at equal temperatures.These examples correspond respectively to ˙Q_c < 0 for T_h > T_c and ˙Q_α ≠ 0 for T_h = T_c.
D. Comment on the general validity of the local approach for modelling heat transport under weak internal coupling
The local approach is reliable in the weak-internal-coupling regime because its secular approximation remains valid there, whereas the global approach can fail near resonance when its secular approximation breaks down. The two methods therefore apply in complementary regimes.
- Perturbative interpretation: The local master equation is formally the lowest-order expansion of the global dissipator in the internal coupling.Its thermodynamic inconsistencies are therefore expected within an error scale O(λ^2k).
- Secular approximation: The global master equation is thermodynamically consistent because it has GKLS form, but its secular approximation can fail near resonance.The preceding Redfield equation can break positivity, while secularization supplies the consistency guarantee for the GME.
- Validity regime: For k/ω_c ≲ λ^2, the local master equation should correctly describe stationary properties because its non-secular terms oscillate rapidly when ω_α ≫ λ^2.This condition is independent of the detuning between the nodes.
- Validity regime: Energy transport through an arbitrarily long harmonic chain is correctly captured by the local approach whenever inter-node couplings are weak.The same weak-internal-coupling rationale extends to heat fluxes in spin chains.
- Alternative approaches: A partial Redfield equation can retain the problematic decay channel, but scaling it to many nodes quickly becomes impractical.This offers an alternative for the two-node problem without providing a scalable general solution.
III. EXACT NON-EQUILIBRIUM STEADY STATE
The exact nonequilibrium steady state is obtained from quantum Langevin equations for the coupled oscillators and their bath-induced noise and dissipation. Under the stated assumptions, the resulting covariances and heat currents coincide with those from the Markovian Redfield equation.
- Exact solution: The exact steady state is calculated by solving the quantum Langevin equations for the linear two-node system.The procedure yields stationary covariances from the Fourier-transformed oscillator equations and bath correlations.
- Assumptions: The calculation assumes a factorized initial system-bath state and takes the initial time to the remote past.These assumptions are used to obtain stationary covariances and the exact steady state.
- Exact solution: For the selected spectral density, the response function and Kramers-Kronig relation determine the covariance matrix and exact steady-state heat currents.The spectral density is specified as ˆχ_h(ω) = ˆχ_c(ω) = λ^2Λ^2/(Λ − iω).
- Benchmark: The exact steady-state covariances and heat currents coincide with those from the Markovian Redfield equation whenever the Born-Markov approximation holds.This identifies the Redfield solution with the exact benchmark within that approximation regime.
A. Steady state and stationary heat currents
The steady-state and heat-current comparisons show complementary validity regimes: the GME is reliable under large detuning, while near resonance the LME remains accurate as the secular approximation breaks down.
- Under large detuning, both local and global approaches agree with the exact steady state except when k becomes comparable to node frequencies, where the LME breaks down.
- For quasi-resonant nodes, the GME disagrees with the exact steady state when k/ωc approaches or falls below λ^2, whereas the LME remains valid for weak inter-node coupling.The GME error results from eliminating the non-secular decay channel at frequency Ω+ − Ω−.
- Under large detuning, both approaches reproduce the nearly vanishing exact heat currents except where strong coupling invalidates the LME, which also violates the second law.The LME predicts heat transport against the temperature gradient, with violations bounded by O(λ^2k).
- In the quasi-resonant regime, the GME largely overestimates steady-state heat-current magnitudes, while the LME quantitatively follows the exact result throughout its validity range.
- The secular approximation can make the GME qualitatively wrong even as the thermodynamically inconsistent LME remains accurate within its applicability range.
B. Steady-state correlations
When the secular approximation fails, the LME captures inter-node correlations more faithfully than the GME, while strong-coupling entanglement exposes the LME’s separate breakdown.
- Total correlations: The GME can both underestimate and overestimate inter-node correlations near resonance, whereas the LME faithfully captures them within its validity range.
- Total correlations: The GME’s correlation errors arise from neglecting stationary covariances ⟨xc ph⟩ and ⟨pcxh⟩ associated with the excluded non-secular channel.
- Correlation decomposition: The GME may overestimate or underestimate total, classical, and quantum correlations, while the LME can overestimate quantum correlations beyond its validity range and underestimate all correlations at sufficiently large coupling.
- Entanglement: Non-vanishing steady-state entanglement requires large ωα/Tα ratios and very large k, preventing observation in the problematic weak-coupling region.
- Entanglement: At strong coupling, the GME gives the correct entanglement scaling, whereas the LME wrongly predicts saturation of the stationary logarithmic negativity.
V. CONCLUSIONS
The exactly solvable two-node wire benchmarks global and local master equations against exact steady states, currents, and correlations, revealing complementary regimes of validity and important thermodynamic cautions.
- The study derives the global GKLS equation, constructs the local alternative, and benchmarks both against exact quantum-Langevin steady states for a two-node wire.
- The local approach is valid only for weak internal coupling and can violate the second law, although such violations can be bounded by suitably defined error bars.
- Near resonance with very weak coupling, secular breakdown makes GME heat currents and correlation predictions qualitatively wrong, while the LME accurately describes stationary wire properties.
- Despite outperforming the GME in some regimes, the LME can miss heat leaks and internal dissipation in quantum thermodynamic cycles, producing qualitatively wrong classifications.
- The paper notes related work comparing local and global approaches in a quantum heat-engine model.
Appendix: The partial Markovian Redfield master equation
The partial Markovian Redfield treatment reduces the steady-state calculation to a linear system for the relevant observables and then reconstructs covariances and heat currents.
- Including the non-secular Ω+ − Ω− channel is proposed as a way to compensate for deficiencies of the GME.
- Only four observables remain relevant because the stationary averages of the first six dynamical variables vanish.
- The remaining observables obey linear equations of motion of the form d⃗y/dt = B⃗y + b.
- The steady-state covariance matrix is specified in the normal-mode basis and rotated back to the original quadratures with the defined rotation matrix.
- Stationary heat currents are obtained from the stationary solution of the partial Redfield equations.