Departmental Overviews

## Mysteries in Science: Physics and Information Science ― Can we derive the second law of thermodynamics from quantum mechanics?

#### Masahito Ueda

#### (Professor, Department of Physics)

Thermodynamics originated during the Industrial Revolution, and from that period to the present day, its experimental accuracy has improved to the point that it can exceed ten digits and be applied at all scales. In fact, thermodynamics is established at any arbitrarily chosen coarse-graining level, and even if quantum effects appear, the laws generally do not change. This is the universality of thermodynamics.

Entropy in thermodynamics characterizes the number of microscopic states that are indistinguishable from one another at a coarse-grained level. However, imagine a “demon” that could perform feedback control by measuring microscopic degrees of freedom; the entropy of a system would then be reduced as more microscopic states are identified, violating the second law of thermodynamics. This intelligent creature is known as “Maxwell’s demon”. If Maxwell’s demon exists, different levels of coarse-graining coexist within the thermodynamic system and, without proper adjustment, cause inconsistencies in the theoretical system. Thus, an interesting problem arises in terms of how to determine the thermodynamic cost of feedback control, specifically measuring the microscopic degrees of freedom and incorporating this information into the theoretical system of thermodynamics. This has led to the emergence of a new science that combines thermodynamics with information theory – nanoscience that can artificially control microscopic degrees of freedom.

The second law of thermodynamics distinguishes what macroscopic thermodynamic operations, such as a piston, can and cannot do. For instance, a thermodynamic operation cannot cause entropy to decrease. However, what plays an important role in allowing us to control degrees of freedom is not microscopic-macroscopic but information content that distinguishes experimentally accessible and controllable degrees of freedom from those that are not. Hence, we must expand the second law of thermodynamics in order for information content and thermodynamic entropy to play equal roles.

We can calculate physical quantities in thermodynamics from a microscopic point of view by using statistical mechanics. However, as thermodynamics and statistical mechanics generally have different levels of coarse graining, the entropy of statistical mechanics and that of thermodynamics typically vary, resulting in a constant difference. This discrepancy in entropy is the physical origin of the Gibbs paradox.

Recently, there has been increased research on ultracold atomic gases in order to study the properties of laser-cooled groups of atoms that are isolated in a vacuum chamber. As the law of quantum mechanics governs closed systems, the total (Von Neumann) entropy of the system remains constant. In actual experiments, however, such an isolated quantum system does show thermal equilibration. Therefore, the thermodynamic entropy of an isolated quantum system must increase. Furthermore, the "principle of equal a priori probabilities" in statistical mechanics may potentially be replaced by the "eigenstate thermalization hypothesis" in quantum mechanics in which all the energy eigenstates are at thermal equilibration at the thermodynamic limit. Von Neumann thought that if quantum mechanics is a microscopic fundamental theory, the process of thermal equilibrium could be obtained from quantum mechanics. I anticipate that in the near future, we will indeed derive the second law of thermodynamics from quantum mechanics.

― *This article is from the "Mysteries in Science" series in The Rigakubu News* ―

*Translated by Kristina Awatsu, Office of Communication*

― Office of Communication ―