A derivation (and quantification) of the third law of thermodynamics (1412.3828v4)

Abstract: The third law of thermodynamics has a controversial past and a number of formulations due to Planck, Einstein, and Nernst. It's most accepted version, the unattainability principle, states that "any thermodynamic process cannot reach the temperature of absolute zero by a finite number of steps and within a finite time." Although formulated in 1912, there has been no general proof of the principle, and the only evidence we have for it is that particular cooling methods become less efficient as the temperature decreases. Here we provide the first derivation of a general unattainability principle, which applies to arbitrary cooling processes, even those exploiting the laws of quantum mechanics or involving an infinite-dimensional reservoir. We quantify the resources needed to cool a system to any particular temperature, and translate these resources into a minimal time or number of steps by considering the notion of a Thermal Machine which obeys similar restrictions to universal computers. We generally find that the obtainable temperature can scale as an inverse power of the cooling time. Our argument relies on the heat capacity of the bath being positive, and we show that if this is not the case then perfect cooling in finite time is in principle possible. Our results also clarify the connection between two versions of the third law (the Unattainability Principle and the Heat Theorem), and place ultimate bounds on the speed at which information can be erased.

A Derivation and Quantification of the Third Law of Thermodynamics

The paper "A derivation (and quantification) of the third law of thermodynamics" explores the foundations and implications of the third law of thermodynamics, particularly focusing on the unattainability principle. This principle, originally proposed by Walther Nernst in 1912, posits that no thermodynamic process can reach absolute zero in a finite number of steps or finite time. Despite its long-standing acceptance, a general proof validating this principle has remained elusive, and the authors tackle this challenge using a rigorous approach that incorporates quantum mechanics and resource theory.

Key Contributions

The authors present the first derivation of a universal unattainability principle that can be applied to a variety of cooling processes, including those that leverage quantum mechanics and involve infinite-dimensional reservoirs. Notably, they quantify the resources needed to cool a system to a specified temperature. This quantification is presented in terms of either the time required or the number of steps, paralleling the functions of a Thermal Machine which can be likened to universal computers.

• General Applicability: The derivation covers cooling processes that do not ignore quantum effects and considers reservoirs with infinite-dimensional Hilbert spaces, extending the scope of the unattainability principle.
• Resource Quantification: By mapping cooling processes onto resource-intensive tasks, the authors derive bounds that show the obtainable temperature can scale inversely with the cooling time.
• Physical Assumptions: The argument is contingent on the heat capacity of the bath being positive. If this condition fails, then perfect cooling in finite time could theoretically be possible, which contradicts the third law.

The paper also establishes a link with the Heat Theorem, another formulation of the third law that describes how the entropy approaches zero as temperature approaches absolute zero. By clarifying these connections, the authors place ultimate bounds on the speed at which information can be erased, which has profound implications for quantum information theory.

Implications in AI and Quantum Computing

The practical implications of this paper stretch beyond traditional thermodynamic systems, touching on areas integral to quantum computing and the development of artificial intelligence systems. Extreme cooling has been pivotal in enabling quantum phenomena, facilitating processes like quantum computation and precision measurements.

• Quantum Systems Management: The ability to estimate and manage resources effectively for cooling processes is crucial for the viability and scalability of quantum systems, particularly those based on superconducting circuits where maintaining low temperatures is non-negotiable.
• Thermodynamic Transitions: Understanding these ultimate bounds could steer the future design of thermal machines within computational frameworks, advancing current models used within AI systems dealing with large-scale quantum data.

Speculations on Future Developments

The results and methodologies presented in the paper set the stage for future explorations that might extend the applicability of thermodynamic principles to varied non-classical systems. Considering the integration with quantum mechanics, there’s a stark potential for redefining system constraints within AI applications where thermodynamic effects are significant.

Furthermore, innovative designs in thermal management can be anticipated, holding the promise of lower boundaries on achievable temperatures that revolutionize operational limits for digital and quantum information processing. These developments could redefine computational complexity and efficiency, influencing AI advancements significantly.

Conclusion

This paper rigorously addresses a century-old principle within thermodynamics through contemporary quantum frameworks and resource theories. By quantifying the third law of thermodynamics and exploring its boundaries, it opens dialogues on the fundamental limits of cooling processes with real-world ramifications for quantum technologies and AI systems. Overall, this work symbolizes a stride towards consolidating classic thermodynamic principles with modern scientific realms, providing a cornerstone for future explorations in the field.