In classical thermodynamics the Euler relation is an expression for the internal energy as a sum of the products of canonical pairs of extensive and intensive variables. For quantum systems the situation is more intricate, since one has to account for the effects of the measurement back action. To this end, we derive a quantum analog of the Euler relation, which is governed by the information retrieved by local quantum measurements. The validity of the relation is demonstrated for the collective dissipation model, where we find that thermodynamic behavior is exhibited in the weak-coupling regime.
In classical thermodynamics the Euler relation is an expression for the internal energy as a sum of the products of canonical pairs of extensive and intensive variables. For quantum systems the situation is more intricate, since one has to account for the effects of the measurement back action. To this end, we derive a quantum analog of the Euler relation, which is governed by the information retrieved by local quantum measurements. The validity of the relation is demonstrated for the collective dissipation model, where we find that thermodynamic behavior is exhibited in the weak-coupling regime.
Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir. For quantum thermodynamics at strong coupling, one needs to reexamine the meaning of the thermodynamic functions, the viability of the thermodynamic relations and the validity of the thermodynamic laws anew. After a brief motivation, this paper starts with a short overview of the quantum formulation based on Gelin & Thoss and Seifert. We then provide a quantum formulation of Jarzynski's two representations. We show how to construct the operator thermodynamic potentials, the expectation values of which provide the familiar thermodynamic variables. Constructing the operator thermodynamic functions and verifying or modifying their relations is a necessary first step in the establishment of a viable thermodynamics theory for quantum systems. We mention noteworthy subtleties for quantum thermodynamics at strong coupling, such as in issues related to energy and entropy, and possible ambiguities of their operator forms. We end by indicating some fruitful pathways for further developments.
Abstract Quantum thermodynamics describes dynamic processes by means of the operators of entropy production P and time t . P and t do not commute. It exists the non-vanishing t–P commutator [ t , P ]=ik. Here the Boltzmann constant k has the physical meaning of a quantum of entropy. The t–P commutator immediately leads us to the t–P uncertainty relation ΔtΔP≥k/2. Hence the observables t and P are not sharply defined simultaneously. Similar uncertainty relations can also be expected for other pairs of conjugate variables with products of the physical meaning of an entropy. The free energy F and the reciprocal temperature (1/T) are the respective conjugate variables of an isolated system of many particles, which leads us to the F–(1/T) uncertainty relation ǀΔFǁΔ(1/T)ǀ≥k/2. It can be traced back to the t–P uncertainty relation mentioned above. In this way the Helmholtz free energy F and the temperatur T are introduced into quantum thermodynamics. The uncertainties ǀΔFǀ→0 and ǀΔTǀ→0 are negligible at low temperatures T→0, and quantum thermodynamics turns into the time-independent classical thermodynamics. Against this the uncertainties ǀΔFǀ→∞ and ǀΔTǀ→∞ grow unlimited at high temperatures T→∞, and classical thermodynamics loses its sense. In the limit of one particle the uncertainties cannot be neglected even at low temperatures. However a detailed discussion shows that the free energy f of a single particle vanishes within the whole range of temperatures T. This defines the particle entropy σ=ε/T=ak. The dimensionless entropy number a connects the particle energy ε=akT with the temperature T. The entropy number a of a single (s) independent particle can be calculated with the extended, temperature-dependent Schrödinger equation A s φ=aφ. Here A s=−(Λ 2/4π)∇2 means the dimensionless entropy operator describing the entropy number a and thus the particle entropy σ=ak. Λ is the thermal de Broglie wave length. Finally we calculate by means of quantized particle entropies σ the internal energy E, the Helmholtz free energy F, the entropy S, the chemical potential μ, and the equation of state of an ideal gas of N monatomic free particles in full agreement with classical thermodynamics. We also calculate the partition function q=V/Λ 3 of a single free particle within the volume V. Here Λ 3 is a small volume element taking into account the wave–particle dualism of a single free particle of mass m at temperature T. Extension to a system of N free particles leads us to a simple geometrical model and to the conclusion that an ideal gas of independent particles becomes instable below a critical temperature T C. T C corresponds to the critical temperature T BE of Bose–Einstein condensation.
The quest to develop a general framework for thermodynamics, suitable for the regime of strong coupling and correlations between subsystems of an autonomous quantum "universe," has entailed diverging definitions for basic quantities, including internal energy. While most approaches focus solely on the system of interest, we propose that a universal notion of internal energy should also account for the environment in order to keep consistency with the closed-system energy of the universe. We introduce an abstract framework to describe all effective Hamiltonian-based approaches and address a rigorous definition of energy additivity in this context, in both a weak and a strong form, discussing the underlying subtleties. As an illustration, we study a particular two-qubit universe model, obtaining the exact master equations for both parties and calculating their effective Hamiltonians and internal energies as given by the recently devised minimal dissipation approach. In this case, we show that internal energies are neither additive nor conservative, which leads to unphysical features.
It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system's state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature's signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.
Abstract It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system’s state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature’s signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.
We extend classical coarse-grained entropy, commonly used in many branches of physics, to the quantum realm. We find two coarse grainings, one using measurements of local particle numbers and then total energy, and the second using local energy measurements, which lead to an entropy that is defined outside of equilibrium, is in accord with the thermodynamic entropy for equilibrium systems, and reaches the thermodynamic entropy in the long-time limit, even in genuinely isolated quantum systems. This answers the long-standing conceptual problem, as to which entropy is relevant for the formulation of the second thermodynamic law in closed quantum systems. This entropy could be in principle measured, especially now that experiments on such systems are becoming feasible.