Numerically "exact" simulations of entropy production in the fully quantum regime: Boltzmann entropy versus von Neumann entropy

Abstract

We present a scheme to evaluate thermodynamic variables for a system coupled to a heat bath under a time-dependent external force using the quasi-static Helmholtz energy from the numerically "exact" hierarchical equations of motion (HEOM). We computed the entropy produced by a spin system strongly coupled to a non-Markovian heat bath for various temperatures. We showed that when changes to the external perturbation occurred sufficiently slowly, the system always reached thermal equilibrium. Thus, we calculated the Boltzmann entropy and the von Neumann entropy for an isothermal process, as well as various thermodynamic variables, such as changes of internal energies, heat, and work, for a system in quasi-static equilibrium based on the HEOM. We found that, although the characteristic features of the system entropies in the Boltzmann and von Neumann cases as a function of the system--bath coupling strength are similar, those for the total entropy production are completely different. The total entropy production in the Boltzmann case is always positive, whereas that in the von Neumann case becomes negative if we chose a thermal equilibrium state of the total system (an unfactorized thermal equilibrium state) as the initial state. This is because the total entropy production in the von Neumann case does not properly take into account the contribution of the entropy from the system--bath interaction. Thus, the Boltzmann entropy must be used to investigate entropy production in the fully quantum regime. Finally, we examined the applicability of the Jarzynski equality.

25 pages, 4 figures

References in corpus (16)

• Quantum Thermodynamic Cycles and quantum heat engines
• The Physics of Maxwell's demon and information
• Perspective: Numerically "exact" approach to open quantum dynamics: The hierarchical equations of motion (HEOM)
• Fluctuation Theorem for Arbitrary Open Quantum Systems
• Fundamental Aspects of Quantum Brownian Motion
• Markovian master equations for quantum thermal machines: local vs global approach
• Long-Lived Electronic Coherence in Dissipative Exciton-Dynamics of Light-Harvesting Complexes
• Reduced hierarchical equations of motion in real and imaginary time: Correlated initial states and thermodynamic quantities
• Testing the validity of the local and global GKLS master equations on an exactly solvable model
• Performance of a quantum heat engine at strong reservoir coupling
• Lower bounds on dissipation upon coarse graining
• Real-Time and Imaginary-Time Quantum Hierarchal Fokker-Planck Equations
• Impact of environmentally induced fluctuations on quantum mechanically mixed electronic and vibrational pigment states in photosynthetic energy transfer and 2D electronic spectra
• Energy Dissipation and Fluctuation-Response in Driven Quantum Langevin Dynamics
• An exciton-coupled electron transfer process controlled by non-Markovian environments
• Open Quantum Dynamics Theory of Spin Relaxation: Application to $\mu$SR and Low-Field NMR Spectroscopies

Cited by in corpus (10)

• Fingerprint and universal Markovian closure of structured bosonic environments
• Numerically "exact" simulations of a quantum Carnot cycle: Analysis using thermodynamic work diagrams
• Thermodynamic quantum Fokker-Planck equations and their application to thermostatic Stirling engine
• Dynamics of a strongly coupled quantum heat engine -- computing bath observables from the hierarchy of pure states
• Open Quantum Dynamics Theory for Non-Equilibrium Work: Hierarchical Equations of Motion Approach
• The laws of thermodynamics for quantum dissipative systems: A quasi-equilibrium Helmholtz energy approach
• Hierarchical equations of motion for multiple baths (HEOM-MB) and their application to Carnot cycle
• Classical and quantum thermodynamics in a non-equilibrium regime: Application to Stirling engine
• Comparative Microscopic Study of Entropies and their Production
• Classical and quantum thermodynamics described as a system-bath model: The dimensionless minimum work principle