Feb 7, 2025 10:50 PM
(This post was last modified: Feb 8, 2025 07:48 AM by C C.)
New proofs probe the limits of mathematical truth
https://www.quantamagazine.org/new-proof...-20250203/
EXCERPTS: In 1900, the eminent mathematician David Hilbert announced a list of 23 key problems to guide the next century of mathematical research. [...] A key part of this vision was that mathematics should be “complete.” That is, all its statements should be provably true or false.
In the 1930s, Kurt Gödel demonstrated that this is impossible: In any mathematical system, there are statements that can be neither proved nor disproved. A few years later, Alan Turing and others built on his work, showing that mathematics is riddled with “undecidable” statements — problems that cannot be solved by any computer algorithm.
These results demonstrated that there are fundamental limits to what proof and computation are capable of. Some mathematics can simply never be known. Hilbert’s dream was dead. But it lived on in fragments. Many of the questions from his turn-of-the-century list still evoked his vision, allowing the idea of a complete mathematics to survive in narrower contexts.... (MORE - details)
No quantum exorcism for Maxwell's demon (but it doesn't need one)
https://www.eurekalert.org/news-releases/1072526
INTRO: In a groundbreaking discovery, researchers from Nagoya University in Japan and the Slovak Academy of Sciences have unveiled new insights into the interplay between quantum theory and thermodynamics. The team demonstrated that while quantum theory does not inherently forbid violations of the second law of thermodynamics, quantum processes may be implemented without actually breaching the law.
This discovery, published in npj Quantum Information, highlights a harmonious coexistence between the two fields, despite their logical independence. Their findings open up new avenues for understanding the thermodynamic boundaries of quantum technologies, such as quantum computing and nanoscale engines.
This breakthrough contributes to the long-standing exploration of the second law of thermodynamics, a principle often regarded as one of the most profound and enigmatic in physics. The second law asserts that entropy—a measure of disorder in a system—never decreases spontaneously. It also states that a cyclically operating engine cannot produce mechanical work by extracting heat from a single thermal environment and underscores the concept of a unidirectional flow of time.
Despite its foundational role, the second law remains one of the most debated and misunderstood principles in science. Central to this debate is the paradox of “Maxwell's Demon,” a thought experiment proposed by physicist James Clerk Maxwell in 1867.
Maxwell envisioned a hypothetical being—the demon—capable of sorting fast and slow molecules within a gas at thermal equilibrium without expending energy. By separating these molecules into distinct regions, the demon could create a temperature difference. As the system returns to equilibrium, mechanical work is extracted, seemingly defying the second law of thermodynamics.
The paradox has intrigued physicists for over a century, raising questions about the law’s universality and whether it depends on the observer’s knowledge and capabilities. Solutions to the paradox have largely centered on treating the demon as a physical system subject to thermodynamic laws. A proposed solution is erasing the demon’s memory, which would require an expenditure of mechanical work, effectively offsetting the violation of the second law.
To explore this phenomenon further, the researchers developed a mathematical model for a “demonic engine,” a system powered by Maxwell’s demon. Their approach is rooted in the theory of quantum instruments, a framework introduced in the 1970s and 1980s to describe the most general forms of quantum measurement.
The model involves three steps: the demon measures a target system, then extracts work from it by coupling it to a thermal environment, and finally erases its memory by interacting with the same environment.
Using this framework, the team derived precise equations for the work expended by the demon and the work it extracts, expressed in terms of quantum information measures such as von Neumann entropy and Groenewold-Ozawa information gain. When comparing these equations, they got a surprising result.
“Our results showed that under certain conditions permitted by quantum theory, even after accounting for all costs, the work extracted can exceed the work expended, seemingly violating the second law of thermodynamics,” explained Shintaro Minagawa, a lead researcher on the project. “This revelation was as exciting as it was unexpected, challenging the assumption that quantum theory is inherently ‘demon-proof.’ There are hidden corners in the framework where Maxwell’s Demon could still work its magic.”
Despite these loopholes, the researchers emphasize that they don't pose a threat to the second law... (MORE - details)
https://www.quantamagazine.org/new-proof...-20250203/
EXCERPTS: In 1900, the eminent mathematician David Hilbert announced a list of 23 key problems to guide the next century of mathematical research. [...] A key part of this vision was that mathematics should be “complete.” That is, all its statements should be provably true or false.
In the 1930s, Kurt Gödel demonstrated that this is impossible: In any mathematical system, there are statements that can be neither proved nor disproved. A few years later, Alan Turing and others built on his work, showing that mathematics is riddled with “undecidable” statements — problems that cannot be solved by any computer algorithm.
These results demonstrated that there are fundamental limits to what proof and computation are capable of. Some mathematics can simply never be known. Hilbert’s dream was dead. But it lived on in fragments. Many of the questions from his turn-of-the-century list still evoked his vision, allowing the idea of a complete mathematics to survive in narrower contexts.... (MORE - details)
No quantum exorcism for Maxwell's demon (but it doesn't need one)
https://www.eurekalert.org/news-releases/1072526
INTRO: In a groundbreaking discovery, researchers from Nagoya University in Japan and the Slovak Academy of Sciences have unveiled new insights into the interplay between quantum theory and thermodynamics. The team demonstrated that while quantum theory does not inherently forbid violations of the second law of thermodynamics, quantum processes may be implemented without actually breaching the law.
This discovery, published in npj Quantum Information, highlights a harmonious coexistence between the two fields, despite their logical independence. Their findings open up new avenues for understanding the thermodynamic boundaries of quantum technologies, such as quantum computing and nanoscale engines.
This breakthrough contributes to the long-standing exploration of the second law of thermodynamics, a principle often regarded as one of the most profound and enigmatic in physics. The second law asserts that entropy—a measure of disorder in a system—never decreases spontaneously. It also states that a cyclically operating engine cannot produce mechanical work by extracting heat from a single thermal environment and underscores the concept of a unidirectional flow of time.
Despite its foundational role, the second law remains one of the most debated and misunderstood principles in science. Central to this debate is the paradox of “Maxwell's Demon,” a thought experiment proposed by physicist James Clerk Maxwell in 1867.
Maxwell envisioned a hypothetical being—the demon—capable of sorting fast and slow molecules within a gas at thermal equilibrium without expending energy. By separating these molecules into distinct regions, the demon could create a temperature difference. As the system returns to equilibrium, mechanical work is extracted, seemingly defying the second law of thermodynamics.
The paradox has intrigued physicists for over a century, raising questions about the law’s universality and whether it depends on the observer’s knowledge and capabilities. Solutions to the paradox have largely centered on treating the demon as a physical system subject to thermodynamic laws. A proposed solution is erasing the demon’s memory, which would require an expenditure of mechanical work, effectively offsetting the violation of the second law.
To explore this phenomenon further, the researchers developed a mathematical model for a “demonic engine,” a system powered by Maxwell’s demon. Their approach is rooted in the theory of quantum instruments, a framework introduced in the 1970s and 1980s to describe the most general forms of quantum measurement.
The model involves three steps: the demon measures a target system, then extracts work from it by coupling it to a thermal environment, and finally erases its memory by interacting with the same environment.
Using this framework, the team derived precise equations for the work expended by the demon and the work it extracts, expressed in terms of quantum information measures such as von Neumann entropy and Groenewold-Ozawa information gain. When comparing these equations, they got a surprising result.
“Our results showed that under certain conditions permitted by quantum theory, even after accounting for all costs, the work extracted can exceed the work expended, seemingly violating the second law of thermodynamics,” explained Shintaro Minagawa, a lead researcher on the project. “This revelation was as exciting as it was unexpected, challenging the assumption that quantum theory is inherently ‘demon-proof.’ There are hidden corners in the framework where Maxwell’s Demon could still work its magic.”
Despite these loopholes, the researchers emphasize that they don't pose a threat to the second law... (MORE - details)