Owen Maroney: Is there a unique thermal entropy?

The speaker begins by challenging the traditional view of entropy as a unique, objective feature of an individual system. They argue that the quest for a globally unique thermodynamic entropy is misguided and that a better understanding of phenomenological thermodynamics could have led to a more comfortable acceptance of multiple thermal entropy functions. The speaker plans to show that the reasons for having such families of thermal entropy functions in statistical mechanics are similar to those in phenomenological thermodynamics, emphasizing the importance of revisiting 19th-century thermal analysis.

This paragraph delves into the mathematical foundation of thermodynamics, discussing the conditions under which an entropy function can exist. It explains that for any two states in a state space, if a certain quantity is path-dependent and changes by a finite amount, then an entropy function exists. The speaker further explores the implications of reversibility in processes, stating that reversible cycles can lead to a unique entropy function, while irreversible cycles result in a family of such functions. The emphasis is on understanding the principles of thermodynamics without relying on quasi-static processes or thermal equilibrium.

The speaker introduces the concept of thermodynamic fluctuations, suggesting that the traditional statements of the second law of thermodynamics may not fully account for these fluctuations. They propose a generalization of the second law that includes the possibility of fluctuations, leading to a principal theory of thermal fluctuations. This approach allows for the derivation of a single unified form for fluctuation laws, which can be applied to various scenarios, including heat pumps and Carnot engines, and results in a relationship between the efficiency of energy transfer and the temperatures involved.

Building on the concept of fluctuations, the speaker discusses the relationship between entropy, temperature, and the likelihood of fluctuations. They propose that the probability of fluctuations depends on the ratio of temperatures, leading to a specific form of the entropy function. The speaker also addresses the possibility of large fluctuations and the role of Boltzmann's constant in this context, highlighting the puzzling nature of its physical significance and the challenges in determining the exact form of the entropy function in the presence of fluctuations.

The paragraph explores the implications of reversibility on the uniqueness of entropy functions. While reversibility in processes can lead to a unique fluctuation-free thermodynamic entropy function, the presence of fluctuations does not necessarily yield a globally unique entropy function. The speaker argues that additional conditions are required to achieve uniqueness, such as the assumption of maximal fluctuations between all pairs of states. However, this assumption is considered strong and potentially unjustified.

The speaker discusses the impact of fluctuations on the concept of entropy, suggesting that incorporating fluctuations into phenomenological thermodynamics can lead to a family of entropy functions rather than a single unique function. They propose a method to combine reliable paths with fluctuations to maintain the concept of entropy, even in the presence of irreversibility and fluctuations. The speaker emphasizes that this approach does not invalidate the understanding of entropy or the second law of thermodynamics, even for non-equilibrium systems.

Shifting the focus to quantum mechanics, the speaker explores the application of thermodynamic principles to quantum systems. They introduce the concept of adiabatic availability and discuss the implications of mechanical work in quantum systems. The speaker also addresses the challenges of applying Hamiltonian dynamics to time-variable systems and the need to consider the interaction between systems in the context of quantum thermodynamics.

This paragraph delves into the concept of adiabatic availability in quantum thermodynamics, contrasting it with the Gibbs fine-grained entropy. The speaker discusses the conditions under which energy can be extracted from a system while preserving certain properties, highlighting the difference between Hamiltonian dynamics and unitary evolution. They also touch upon the historical development of these concepts and their significance in understanding quantum thermodynamic processes.

The speaker proposes a foundation for statistical mechanics and thermal physics, starting with the identification of thermal states based on their passive properties. They argue that thermal states should not exchange heat on average when in contact and that these states are passive. The speaker also discusses the implications of considering multiple systems at the same temperature and the role of passivity in defining thermal states without relying on notions of equilibrium.

The speaker discusses the derivation of entropy functions in statistical mechanics, emphasizing that these functions can be defined without assuming thermal equilibrium or the thermodynamic limit. They highlight the importance of understanding the assumptions behind the derivation of these functions and the conditions under which they hold. The speaker also addresses the potential issues with certain assumptions, such as unitary evolution and the negligible variation interaction engine.

In this paragraph, the speaker reflects on the various assumptions made in the analysis of thermodynamic entropy, such as reversibility, constant interaction, and statistical independence. They question the validity of these assumptions in different contexts and suggest alternative approaches, such as passivism or considering different optimization criteria. The speaker also discusses the implications of these assumptions for the understanding of entropy and the development of statistical mechanics.

The speaker concludes by emphasizing that entropy is an expression of constraints on the conversion of heat into work, derived from cyclic conditions. They argue that these constraints yield families of functions of state, which can be considered objective physical properties. The speaker also discusses the role of reversibility, fluctuations, and statistical dependence in generating these families of entropy functions and the importance of being open to multiple entropy functions.

The speaker acknowledges the open questions and challenges in understanding thermodynamics and entropy, particularly in the context of quantum mechanics. They discuss the potential issues with achieving reversible processes and the implications of quantum effects on traditional thermodynamic assumptions. The speaker also touches upon the importance of considering different optimization criteria and the potential for alternative extensive entropies in various physical scenarios.

In the final paragraph, the speaker highlights the complexity of understanding entropy and the process of reaching equilibrium in different physical systems. They discuss the potential impact of long-range correlations, memories, and the role of practical choices in defining entropy functions. The speaker also emphasizes the importance of questioning assumptions and exploring alternative approaches in the study of thermodynamics and statistical mechanics.