Additional energy-information relations in thermodynamics of small systems

Abstract

The Clausius inequality (CI) form of the second law of thermodynamics relates information changes (entropy) to changes in the first moment of the energy (heat and indirectly also work). Are there similar relations between other moments of the energy distribution, and other information measures, or is the Clausius inequality a one of a kind instance of the energy-information paradigm? If there are additional relations, can they be used to make predictions on measurable quantities? Changes in the energy distribution beyond the first moment (average heat or work) are especially important in small systems which are often very far from thermal equilibrium. The generalized Clausius inequalities (GCI's), here derived, provide positive answers to the two questions above and add another layer to the fundamental connection between energy and information. To illustrate the utility of the new GCI's, we find scenarios where the GCI's yield tighter constraints on performance (e.g. in thermal machines) compared to the second law. To obtain the GCI's we use the Bregman divergence -a mathematical tool found to be highly suitable for energy-information studies. The quantum version of the GCI's provides a thermodynamic meaning to various quantum coherence measures. It is intriguing to fully map the regime of validity of the GCI's and extend the present results to more general scenarios including continuous systems and particles exchange with the baths.

Optimal Work Extraction and Thermodynamics of Quantum Measurements and Correlations

“…[40] (see also Refs. [41,42] for similar analysis), where isothermal processes are constructed by means of alternating infinitesimal adiabatic and isochoric processes in an infinite sequence. Similarly, we assume, here, an infinite sequence of maps E 1 • E 2 • · · · • E N , with N → ∞, each of them describing an infinitessimal time step of the dynamics, with E n (ρ n−1 ) = ρ n , and the Hamiltonian changing as H n−1 → H n , where we set H N ≡ H.…”

Global Passivity in Microscopic Thermodynamics

“…With this combination of system information and observables, the CI neatly expresses the energyinformation relation that appears in fundamental processes such as Landauer's erasure, the Szilard engine, and reversible state preparation [25,26]. When extending the CI it is desirable to maintain this information-expectation value structure.…”

“…See Ref. [26] for an extension of the second law that preserves the information-observable structure, and Refs. [9,10] for an extension that does not.…”

Markovian heat sources with the smallest heat capacity

“…To define the heat capacity of a small bath we consider two additional macroscopic baths at temperatures T and d ¢ = + T T T. We first thermalize the small bath by connecting it to the large bath at temperature T. This is repeated for bath T′. The heat flow Q (which is equal to the change in the energy of the small bath) is recorded and the small bath heat capacity is given by (8). Note that this process is an isochore (constant volume) since we did not change the energy levels of the small bath (no work involved, only heat).…”

“…Apart from the practical importance of understanding and experimenting with thermodynamics at the smallest scales, the study of quantum thermodynamics has also provided exciting theoretical developments. As an example, it has been shown that there are additional second law-like constraints on small systems interacting with a thermal bath [6][7][8]. In addition, there are thermodynamic effects in heat machines that can be observed only in systems that are sufficiently 'quantum' [9][10][11][12].…”