On the Local and Global Approaches to Quantum Transport and Violation of the Second-law of Thermodynamics (1402.3825v2)

Abstract: Clausius' statement of the second law of thermodynamics reads: Heat will flow spontaneously from a hot to cold reservoir. This statement should hold for transport of energy through a quantum network composed of small subsystems each coupled to a heat reservoir. When the coupling between nodes is small, it seems reasonable to construct a local master equation for each node in contact with the local reservoir. The energy transport through the network is evaluated by calculating the energy flux after the individual nodes are coupled. We show by analysing the most simple network composed of two quantum nodes coupled to a hot and cold reservoir, that the local description can result in heat flowing from cold to hot reservoirs, even in the limit of vanishing coupling between the nodes. A global derivation of the master equation which prediagonalizes the total network Hamiltonian, and within this framework derives the master equation, is always consistent with the second-law of thermodynamics.

• The paper reveals that local master equations, by neglecting inter-node interactions, can inaccurately predict global energy flows and lead to apparent second-law violations.
• The paper demonstrates that the global approach, which diagonalizes the network Hamiltonian, robustly ensures positive entropy production in quantum systems.
• The paper employs numerical and analytical models of a two-node quantum network to underscore the necessity for comprehensive, system-wide energy exchange frameworks.

Local and Global Approaches to Quantum Transport: An Examination of Thermodynamic Consistency

This paper by Amikam Levy and Ronnie Kosloff addresses the theoretical understanding of energy transport through quantum networks, specifically investigating the conditions under which the second law of thermodynamics may be violated. The research differentiates between local and global approaches to quantum transport, examining how each methodology handles energy exchange in quantum systems composed of nodes connected to reservoirs at different temperatures.

Quantum Transport Framework

Quantum networks are increasingly integral to technologies such as molecular electronics and quantum heat engines. In such networks, quantum nodes interact with their respective heat reservoirs and other network nodes, facilitating energy transport. The dynamics of these open quantum systems are described using quantum master equations, typically under the assumption of weak coupling between nodes.

Local vs. Global Descriptions

1. Local Approach: The paper critiques the use of local master equations, which treat each node's coupling to its reservoir independently of inter-node interactions. This simplified model, while capturing local observables like node populations, fails to accurately predict global energy flows, leading to potential violations of the second law. Specifically, for certain parameter ranges, it predicts scenarios where heat could erroneously flow from colder to hotter nodes.
2. Global Approach: Levy and Kosloff advocate for a global approach that diagonalizes the entire network Hamiltonian before deriving the master equation. This method inherently respects the thermodynamic laws by accounting for system-wide energy exchanges, even at weak inter-node coupling limits. This approach prevents the aforementioned thermodynamic inconsistencies inherent in the local model.

Numerical and Analytical Insights

The authors employ a minimal model of two quantum nodes connected to hot and cold reservoirs. They highlight instances where the local description incorrectly suggests negative entropy production, a clear violation of the second law. In contrast, the global approach successfully upholds positivity in entropy production across all explored conditions, reinforcing its theoretical stability.

Implications and Future Prospects

The findings underscore the inadequacy of local master equations for capturing complete thermodynamic consistency in quantum networks. This insight is crucial as research delves deeper into quantum thermodynamics and the development of nanoscale devices where quantum effects are predominant. The work suggests that as technology progresses, particularly at the quantum level, relying on more comprehensive, system-wide models is essential.

Conclusion

Levy and Kosloff present a compelling argument for considering global approaches in quantum transport analysis to ensure thermodynamic consistency, especially in weak coupling scenarios. Their exploration serves as an important reminder of the intricacies of quantum systems and the necessity for rigorous theoretical frameworks as we advance technologically. Future research may further extend these findings, exploring broader classes of quantum systems or incorporating additional environmental interactions to refine the global master equation approach.