Thermodynamics

Jen-Tsung Hsiang, B. L. Hu

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.

Manabendra Nath Bera, Andreas Winter, Maciej Lewenstein

Peter Hertel

Samuel Duffield, Maxwell Aifer, Gavin E. Crooks +2 more

Thermodynamic computing exploits fluctuations and dissipation in physical systems to efficiently solve various mathematical problems. It was recently shown that certain linear algebra problems can be solved thermodynamically, leading to a speedup scaling with the matrix dimension. Here, we provide a thermodynamic algorithm for exponentiating a real matrix. We describe a simple electrical circuit involving coupled oscillators, which can implement our algorithm. We also show that this algorithm provides an asymptotic speedup that is linear in the dimension. Finally, we introduce the concept of thermodynamic parallelism to explain this speedup, stating that thermodynamic noise provides a resource leading to effective parallelization of computations.

Ruo-Xun Zhai, Che Sun

We extend the Carathéodory principle of the second law to quantum thermodynamics, where energy levels depend on macroscopic variables such as volume and magnetic field. This extension introduces the concept of quantum thermodynamic integrability (QTI), providing an alternative foundation for statistical mechanics. QTI is characterized by the path independence of work and heat within the thermodynamic manifold, locally described by energy levels and specific thermodynamic parameters. Within this framework, temperature naturally emerges as an integrating factor, enabling the derivation of both canonical and noncanonical states from the entropy integrable equations based on QTI. Notably, noncanonical states, which become particularly significant outside the thermodynamic limit, reveal the existence of informational correlations in finite-size thermodynamic systems.