Quantum Thermodynamics

TL;DR

The notes bridge quantum theory and thermodynamics by formulating open quantum systems with Markovian master equations, enabling analysis of heat, work, and temperature at the nanoscale. They cover foundational quantum mechanics, equilibrium ensembles, and the emergence of thermodynamic laws within a quantum framework, including entropy production and information-theoretic aspects. Central tools include Nakajima–Zwanzig, Born–Markov, and GKLS master equations, with detailed examples such as quantum dots, double-dot engines, entanglement generators, and absorption refrigerators. The work culminates with an exploration of fluctuations via fluctuation theorems and stochastic thermodynamics, highlighting how thermodynamic laws extend to nano-scale, non-equilibrium, and fluctuating regimes. Overall, it provides a rigorous framework for analyzing quantum thermal machines and the role of fluctuations in quantum thermodynamics.

Abstract

The theory of quantum thermodynamics investigates how the concepts of heat, work, and temperature can be carried over to the quantum realm, where fluctuations and randomness are fundamentally unavoidable. These lecture notes provide an introduction to the thermodynamics of small quantum systems. It is illustrated how the laws of thermodynamics emerge from quantum theory and how open quantum systems can be modeled by Markovian master equations. Quantum systems that are designed to perform a certain task, such as cooling or generating entanglement are considered. Finally, the effect of fluctuations on the thermodynamic description is discussed.

Figures (18)

• Figure 1: A gas in a container of volume $V$, at temperature $T$, with pressure $p$. The internal energy of the gas is given by $U$ and its entropy by $S$. Using a piston, the volume of the gas can be changed and work $W$ can be performed on the system. Energy may also be exchanged with the environment in the form of heat $Q$. Note that the temperature $T_\mathrm{B}$ and pressure $p_\mathrm{B}$ of the environment may differ from the temperature and pressure of the system.
• Figure 2: Perpetuum mobile of the first kind. A lamp illuminates a solar panel which in turn powers the lamp. Without any hidden energy source, such a device cannot work because of unavoidable losses. At long times, we may set $dU=0$. Furthermore, we split the work ${d\mkern-7mu\mathchar'26\mkern-2mu} W= {d\mkern-7mu\mathchar'26\mkern-2mu} W_\mathrm{L}-{d\mkern-7mu\mathchar'26\mkern-2mu} W_\mathrm{S}$, where $W_\mathrm{L}$ denotes the work exerted on the lamp by the solar panel and $W_\mathrm{S}$ denotes the work exerted on the solar panel by the lamp. In the presence of losses within the solar panel (any other loss mechanism results in similar conclusions), it provides strictly less work to the lamp than it received, i.e., ${d\mkern-7mu\mathchar'26\mkern-2mu} W_\mathrm{L}<{d\mkern-7mu\mathchar'26\mkern-2mu} W_\mathrm{S}$. By Eq. \ref{['eq:firstlawcl']}, this would necessitate another source of energy, either in the form of work or in the form of heat with ${d\mkern-7mu\mathchar'26\mkern-2mu} Q<0$.
• Figure 3: Perpetuum mobile of the second kind. A boat moves across the sea, being powered by the heat absorbed from the surrounding water. To move in the water, the boat needs to perform work against the friction force, i.e., ${d\mkern-7mu\mathchar'26\mkern-2mu} W<0$. At long times, we may neglect any changes in the internal energy or the entropy of the system, i.e., $dS =dU=0$. The work required to move against the friction force must then be provided by heat coming from the environment, requiring ${d\mkern-7mu\mathchar'26\mkern-2mu} Q<0$. Since the environment only has a single temperature, this violates the second law of thermodynamics given in Eq. \ref{['eq:secondlaw']} which requires ${d\mkern-7mu\mathchar'26\mkern-2mu} Q/T\geq 0$.
• Figure 4: Working principle of a heat engine. Heat from a hot reservoir $Q_\mathrm{h}$ is partly converted into work $W$ and partly dissipated into the cold reservoir $Q_\mathrm{c}$. The dissipation ensures that the entropy production is positive, as required by the second law of thermodynamics, and limits the efficiency to be below the Carnot efficiency, see Eq. \ref{['eq:eff']}.
• Figure 5: System exchanging energy and particles with the environment.