Core Viewpoint - The article discusses a significant breakthrough in addressing Hilbert's sixth problem, which aims to establish a rigorous mathematical foundation for physics, particularly the transition from microscopic particle dynamics to macroscopic fluid behavior [2][13][35]. Summary by Sections Historical Context - Hilbert's sixth problem, proposed in 1900, questions whether physics can be constructed on a strict mathematical basis similar to Euclidean geometry [1][3]. - The challenge involves linking reversible microscopic laws of motion (Newtonian mechanics) with irreversible macroscopic behaviors (described by the Boltzmann equation) [8][9]. Breakthrough Achieved - Mathematicians Deng Yu, Ma Xiao, and Zaher Hani have made a significant advancement by deriving macroscopic gas behavior from microscopic particle models, bridging the gap between Newtonian mechanics and the Boltzmann equation [10][11][13]. - They provided a rigorous proof of the complete transition from Newtonian mechanics to the Boltzmann equation, addressing the "arrow of time" paradox left by Boltzmann [13][35]. Methodology - The solution involves two main steps: first, deriving the Boltzmann equation from Newton's laws through a "dynamical limit," and second, deriving fluid dynamics equations from the Boltzmann equation through a "fluid dynamical limit" [15][23]. - The team initially focused on wave systems before transitioning to particle systems, recognizing the complexity of particle collisions compared to wave interference [18][21]. Detailed Steps - In the first step, they demonstrated that as the number of hard sphere particles approaches infinity and their diameter approaches zero, the single-particle density can be described by the Boltzmann equation [17]. - In the second step, they showed that as the collision rate in the Boltzmann equation approaches infinity, its solution converges to the local Maxwell distribution, corresponding to macroscopic fluid parameters [24][30]. Implications - This work not only marks a major advancement in solving Hilbert's sixth problem but also provides a rigorous mathematical solution to the long-standing paradox of time irreversibility in physics [35][37]. - The findings establish a complete logical chain from Newtonian mechanics to statistical mechanics and fluid mechanics, enhancing the understanding of physical laws across different scales [31][34].

Core Viewpoint - The article discusses a significant breakthrough in addressing Hilbert's sixth problem, which aims to establish a rigorous mathematical foundation for physics, particularly the transition from microscopic particle dynamics to macroscopic fluid behavior [2][13][35]. Group 1: Historical Context and Problem Definition - Hilbert's sixth problem, proposed in 1900, questions whether a strict mathematical foundation for physics can be constructed similar to Euclidean geometry [3][5]. - The challenge involves linking reversible microscopic laws, governed by Newtonian mechanics, to irreversible macroscopic behaviors described by the Boltzmann equation [8][9]. Group 2: Breakthrough Achievements - Mathematicians Deng Yu, Ma Xiao, and Zaher Hani have made a significant advancement by deriving macroscopic gas behavior from microscopic particle models, bridging the gap between Newtonian mechanics and the Boltzmann equation [10][11][13]. - They provided a rigorous proof of the complete transition from Newtonian mechanics to the Boltzmann equation, addressing a long-standing logical gap in statistical mechanics [13][35]. Group 3: Methodology and Steps - The solution involves two main steps: first, deriving the Boltzmann equation from Newton's laws through a "dynamical limit," and second, deriving fluid equations from the Boltzmann equation via a "fluid dynamical limit" [15][23]. - The team initially focused on wave systems before transitioning to particle systems, developing new methods to track particle collisions and their effects on trajectories [18][21]. Group 4: Mathematical Techniques and Results - They utilized the Boltzmann-Grad limit to show that the single-particle density of a system of hard spheres can be described by the Boltzmann equation as the number of particles approaches infinity and their diameter approaches zero [17]. - The researchers introduced the Knudsen number to assess the thinness of gases and determine the applicable equations for different conditions [26]. Group 5: Implications and Future Directions - The work not only marks a major breakthrough in Hilbert's sixth problem but also provides a rigorous mathematical solution to the ancient paradox of time irreversibility in gas dynamics [35][37]. - The findings establish a complete logical chain from Newtonian mechanics to statistical mechanics and fluid dynamics, although limitations exist for complex turbulent phenomena [31][32].

Core Viewpoint - The article discusses the resolution of Hilbert's sixth problem, a significant mathematical challenge posed by David Hilbert in 1900, which has been solved by a team of Chinese researchers after 125 years [1][11]. Group 1: Authors and Their Backgrounds - The research was conducted by three authors: Deng Yu, a professor at the University of Chicago; Zaher Hani, an assistant professor at the University of Michigan; and Ma Xiao, also an assistant professor at the University of Michigan [2][27]. - Deng Yu graduated from Peking University and MIT, and completed his PhD at Princeton [28]. - Zaher Hani completed his undergraduate studies at the American University of Beirut and his master's and PhD at UCLA, studying under renowned mathematician Terence Tao [31][34]. - Ma Xiao graduated from the Young Scholars Program at the University of Science and Technology of China and completed his PhD at Princeton in 2023 [35]. Group 2: Significance of the Research - The resolution of Hilbert's sixth problem is not only a theoretical milestone but also provides new mathematical tools for the study of fluid mechanics [5][6]. - The achievement has been celebrated within the mathematical community, with some referring to it as a "miracle year for Chinese mathematics" [6][8]. Group 3: Methodology and Findings - The authors approached the problem by deriving fluid dynamics equations from microscopic Newtonian mechanics using Boltzmann kinetic theory [4][25]. - They introduced a cumulative quantity analysis method to track the complete history of particle collisions, leading to the proof of the long-term validity of the Boltzmann equation [18][20]. - The research culminated in the derivation of the Euler equations for compressible fluids and the Navier-Stokes-Fourier equations under incompressible conditions [21][25].