Quantization of Energy

The Boltzmann distribution describes the probability that a system in thermal equilibrium at temperature T will be in a specific microstate with energy E. This probability is proportional to the Boltzmann factor, [latex]e^{-E / k_B T}[/latex]. It implies that states with lower energy are exponentially more likely to be occupied than states with higher energy, with temperature modulating this preference.

The Reynolds number ([latex]\text{Re}[/latex]) is a dimensionless quantity in fluid mechanics used to predict flow patterns by representing the ratio of inertial forces to viscous forces. Low Reynolds numbers characterize smooth, orderly laminar flow, while high Reynolds numbers indicate chaotic, eddy-filled turbulent flow. It is crucial for determining the dynamic behavior of a fluid and for scaling experiments.

The Mach number (M) is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound: [latex]M = v/a[/latex], where v is the flow velocity and a is the speed of sound. It is the primary indicator of compressibility effects. As Mach number approaches and exceeds 1, the density of the air changes significantly, altering aerodynamic forces.

A statistical ensemble is a conceptual tool consisting of a large number of virtual copies of a system, each representing a possible microstate. By averaging properties over all systems in the ensemble, one can calculate macroscopic observables. The main types are the microcanonical (isolated system with fixed N, V, E), canonical (closed system, fixed N, V, T), and grand canonical (open system, fixed µ, V, T).

The equivalence principle is a cornerstone of general relativity. It posits that the effects of a uniform gravitational field are indistinguishable from the effects of uniform acceleration. This means that an observer in a windowless, free-falling elevator experiences weightlessness, just like an observer in deep space, far from any gravitational source, forming the conceptual basis for gravity as a geometric phenomenon.

A key prediction of general relativity is that time passes at different rates in different gravitational potentials. A clock in a stronger gravitational field (closer to a massive object) will tick slower than a clock in a weaker field. This effect, known as gravitational time dilation, has been experimentally verified and is a crucial factor in modern technology like GPS.

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like honey, resist shear flow and strain linearly with time when a stress is applied. Elastic materials, like a rubber band, strain when stretched and quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both.

This foundational formula connects the macroscopic thermodynamic quantity of entropy (S) with the number of possible microscopic arrangements, or microstates (W), corresponding to the system's macroscopic state. The equation, [latex]S = k_B \ln W[/latex], reveals that entropy is a measure of statistical disorder or randomness. The constant [latex]k_B[/latex] is the Boltzmann constant, linking energy at the particle level with temperature.

Shear-thickening, or dilatancy, is a non-Newtonian behavior where viscosity increases with the rate of shear stress. A classic example is a suspension of cornstarch in water (oobleck), which feels liquid when stirred slowly but becomes almost solid when struck or stirred rapidly. This phenomenon is caused by the jamming of suspended particles under high shear.

The Reynolds-Averaged Navier-Stokes (RANS) equations are time-averaged equations of motion for turbulent fluid flow. This approach, called Reynolds decomposition, separates flow variables into a mean and a fluctuating component. The averaging process introduces an additional term, the Reynolds stress tensor, which represents the effect of turbulence and must be modeled to achieve closure, making simulations computationally tractable.

The Kutta-Joukowski theorem quantifies the lift force generated by an airfoil. It states that the lift per unit span ([latex]L'[/latex]) is directly proportional to the fluid density ([latex]\rho[/latex]), the free-stream velocity ([latex]V[/latex]), and the circulation ([latex]\Gamma[/latex]) around the body: [latex]L' = \rho V \Gamma[/latex]. This links the abstract concept of circulation to the physical force of lift.

The boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. Introduced by Ludwig Prandtl, this concept simplifies fluid dynamics problems by dividing the flow into two regions: the thin boundary layer where viscosity dominates and the outer region where inviscid flow theory can be applied.

The von Mises yield criterion predicts that yielding of a ductile material begins when the second deviatoric stress invariant, [latex]J_2[/latex], reaches a critical value. It is often expressed in terms of the von Mises stress, [latex]\sigma_v[/latex], a scalar value that must be less than the material's yield strength, [latex]\sigma_y[/latex]. Yielding occurs when [latex]\sigma_v = \sigma_y[/latex].

General relativity provided the first accurate explanation for the anomalous precession of Mercury's perihelion. Newtonian gravity could not fully account for the slow, gradual shift in the orientation of Mercury's elliptical orbit. Einstein's theory correctly predicted the missing 43 arcseconds per century, attributing it to the curvature of spacetime around the Sun, a major early triumph for the theory.