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].