XIII. Statistical Hydrodynamics. (L. Onsager, New Haven, Conn.) It is a well-known fact in hydrodynamics that when the Reynolds number exceeds a critical value, steady flow becomes unstable. Unsteady flow under these conditions requires statistical analysis, but early attempts faced significant challenges. In recent years, C. C. Lin resolved key questions about the stability of laminar flow, and Kolmogorov made progress toward a quantitative theory of turbulence. Kolmogorov's main result was rediscovered at least twice. These theories focus on the mechanism of turbulent dissipation. We will return to this topic, but first, we will discuss a new application of statistics to hydrodynamics. ## Ergodic Motion of Parallel Vortices The formation of large, isolated vortices is a common and spectacular phenomenon in unsteady flow. Its prevalence suggests a statistical explanation. We consider n parallel vortices with intensities $ k_1, \ldots, k_n $ in an incompressible, frictionless fluid. This is a two-dimensional system with a finite number of degrees of freedom, allowing the use of statistical mechanics. The equations of motion can be written as: $$ \left\{ \begin{array}{l} k_i \frac{dx_i}{dt} = \frac{\partial H}{\partial y_i}, \\ k_i \frac{dy_i}{dt} = -\frac{\partial H}{\partial x_i}, \end{array} \right. $$ where $ H $ is the energy integral. In an unbounded fluid, $ H $ has the form: $$ \left\{ \begin{array}{l} H = -\frac{1}{2\pi} \sum_{i>j} k_i k_j \log r_{ij}, \\ r_{ij}^2 = (x_i - x_j)^2 + (y_i - y_j)^2. \end{array} \right. $$ The equations of motion still apply when the fluid is bounded, with the Hamiltonian modified to account for image forces. The phase-space is finite, and the energy can range from $ +\infty $ to $ -\infty $. The phase-volume corresponding to energies less than a given value is a differentiable function of energy. The temperature $ \Theta = \Phi' / \Phi'' $ is positive when $ E < E_m $.XIII. Statistical Hydrodynamics. (L. Onsager, New Haven, Conn.) It is a well-known fact in hydrodynamics that when the Reynolds number exceeds a critical value, steady flow becomes unstable. Unsteady flow under these conditions requires statistical analysis, but early attempts faced significant challenges. In recent years, C. C. Lin resolved key questions about the stability of laminar flow, and Kolmogorov made progress toward a quantitative theory of turbulence. Kolmogorov's main result was rediscovered at least twice. These theories focus on the mechanism of turbulent dissipation. We will return to this topic, but first, we will discuss a new application of statistics to hydrodynamics. ## Ergodic Motion of Parallel Vortices The formation of large, isolated vortices is a common and spectacular phenomenon in unsteady flow. Its prevalence suggests a statistical explanation. We consider n parallel vortices with intensities $ k_1, \ldots, k_n $ in an incompressible, frictionless fluid. This is a two-dimensional system with a finite number of degrees of freedom, allowing the use of statistical mechanics. The equations of motion can be written as: $$ \left\{ \begin{array}{l} k_i \frac{dx_i}{dt} = \frac{\partial H}{\partial y_i}, \\ k_i \frac{dy_i}{dt} = -\frac{\partial H}{\partial x_i}, \end{array} \right. $$ where $ H $ is the energy integral. In an unbounded fluid, $ H $ has the form: $$ \left\{ \begin{array}{l} H = -\frac{1}{2\pi} \sum_{i>j} k_i k_j \log r_{ij}, \\ r_{ij}^2 = (x_i - x_j)^2 + (y_i - y_j)^2. \end{array} \right. $$ The equations of motion still apply when the fluid is bounded, with the Hamiltonian modified to account for image forces. The phase-space is finite, and the energy can range from $ +\infty $ to $ -\infty $. The phase-volume corresponding to energies less than a given value is a differentiable function of energy. The temperature $ \Theta = \Phi' / \Phi'' $ is positive when $ E < E_m $.