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Thermodynamics of Quantum Coupled Transport

Shuvadip Ghosh, Arnab Ghosh

Abstract

This review presents a thermodynamic perspective on quantum coupled transport processes in nanoscale systems. Our analysis is formulated within the framework of entropy production rate, the central quantity governing non-equilibrium processes and expressed through conjugate force-flux pairs. Although thermodynamic laws are universal across classical and quantum domains, the discussion is developed within a microscopic open quantum system framework, focusing on quantum dots (QDs) coupled to electronic reservoirs. We first examine elementary single transport processes and highlight their strong thermodynamic constraints in the near-equilibrium regime. This motivates the study of coupled transport, where multiple force-flux pairs coexist and interact, leading to richer thermodynamic behaviour. Using entropy production as the guiding principle, we analyse coupled energy and particle transport in a minimal two-terminal single-QD setup and show how conventional thermoelectric phenomena, including Seebeck and Peltier effects as well as thermoelectric heat engines and refrigerators, naturally emerge as thermodynamic cross-effects. We then extend the framework to a three-terminal coupled quantum dot (CQD) geometry, which provides a versatile platform for studying coupled transport and reduces, under suitable constraints, to the well-known Sánchez-Büttiker configuration. Beyond standard cross-effects, we discuss the phenomenon of inverse currents in coupled transport (ICC), where a current flows against mutually parallel thermodynamic forces without violating the second law. We show that ICC requires breaking the symmetry between energy and particle transport and identify the conditions for its realization in coupled quantum-dot systems with attractive interdot interactions.

Thermodynamics of Quantum Coupled Transport

Abstract

This review presents a thermodynamic perspective on quantum coupled transport processes in nanoscale systems. Our analysis is formulated within the framework of entropy production rate, the central quantity governing non-equilibrium processes and expressed through conjugate force-flux pairs. Although thermodynamic laws are universal across classical and quantum domains, the discussion is developed within a microscopic open quantum system framework, focusing on quantum dots (QDs) coupled to electronic reservoirs. We first examine elementary single transport processes and highlight their strong thermodynamic constraints in the near-equilibrium regime. This motivates the study of coupled transport, where multiple force-flux pairs coexist and interact, leading to richer thermodynamic behaviour. Using entropy production as the guiding principle, we analyse coupled energy and particle transport in a minimal two-terminal single-QD setup and show how conventional thermoelectric phenomena, including Seebeck and Peltier effects as well as thermoelectric heat engines and refrigerators, naturally emerge as thermodynamic cross-effects. We then extend the framework to a three-terminal coupled quantum dot (CQD) geometry, which provides a versatile platform for studying coupled transport and reduces, under suitable constraints, to the well-known Sánchez-Büttiker configuration. Beyond standard cross-effects, we discuss the phenomenon of inverse currents in coupled transport (ICC), where a current flows against mutually parallel thermodynamic forces without violating the second law. We show that ICC requires breaking the symmetry between energy and particle transport and identify the conditions for its realization in coupled quantum-dot systems with attractive interdot interactions.
Paper Structure (26 sections, 73 equations, 6 figures)

This paper contains 26 sections, 73 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Thermodynamics of transport process
  3. Single Transport Process
  4. Coupled Transport Process
  5. The model for quantum coupled transport
  6. System
  7. Environment
  8. System-Bath Interaction
  9. Dynamics and steady-state current of a single QD setup
  10. Deriving entropy production rate for general QD-fermionic reservoir model
  11. Identifying the thermodynamic force-flux pairs for two-terminal single QD setup
  12. Thermodynamics of coupled transport in two terminal single QD setup
  13. Both $\mathcal{F}_{\rm E}^l$ and $\mathcal{F}_{\rm N}^l$ are zero
  14. Seebeck effect in presence of only energy force
  15. Peltier effect in presence of only particle force
  16. ...and 11 more sections

Figures (6)

  • Figure 1: A small quantum system interacts with multiple reservoirs
  • Figure 2: (a) Two-terminal single QD model for coupled transport (b) Two transitions are induced by the reservoirs.
  • Figure 3: Pictorial representation of various thermodynamic phenomena governed by the entropy production rate $\dot{\Sigma} =J_{\rm{E}}^{l}\mathcal{F}_{\rm{E}}^{l}+J_{\rm{N}}^{l}\mathcal{F}_{\rm{N}}^{l}$ i.e., normal cross-effect (red shaded), ICC (blue shaded), pseudo-ICC, Seebeck and Peltier effects that can be obtained.
  • Figure 4: (a) Schematic of the three-terminal CQD model. (b) Energy level diagram of CQD where transitions are mediated by the baths within the eigen-states of the composite system.
  • Figure 5: (a) Schematic of the reduced CQD-three terminal setup as a model for a quantum thermoelectric device, (b) Classical thermoelectric device, from which the model is inspired.
  • ...and 1 more figures