Resolution of the ellipsoid paradox in thermodynamics
Geometric configurations for two variants of the ellipsoid paradox, shown in cross-section. Sources A and B are located at the foci of an ellipsoid section E 1. Attached to E 1 is a section of a sphere S with its center at B. In the two-ellipsoid geometry of (a) there is a second and larger ellipsoid section, E 2, with the same foci as E 1. The spherical section in (a) is positioned and constrained so that no rays coming directly from A can hit anywhere on S. The “Chinese furnace” geometry of (b) is formed entirely of the ellipsoid E 1 and sphere S. The actual cavities are surfaces of revolution obtained from these cross-sections by revolution on the line containing A and B.
A spread of 11 rays leaves body A from a point on its surface. These rays do not come to a focus at the center of body B, and several of the rays do not hit body B at all. The dots show where each ray hits after one reflection. Of the four that miss B after one reflection, three are eventually absorbed by A and one by B. Rays that reflect off E 2 must intersect body B (which in this example has the same radius as A) because the distance from the point of reflection to B is less than the distance from A to the point of reflection.
Ray diagrams for typical two-dimensional paths, shown in order of increasing numbers of reflections. These paths can be thought of as two-dimensional projections of paths in the cylindrical geometry, or as special cases of the three-dimensional geometry that happen to lie in a plane. The ray shown in (a) is typical of a single-reflection path from A to B. Ray (b) bounces 8 times before returning to A. Ray (c) from B to A has 12 reflections. Ray (d) from A to A retraces its path after a reflection at nearly a right angle. Ray (e) from A to B traverses the perimeter of the cavity in a precessing triangle. Ray (f) from B to A enters a five-sided nearly resonant situation with 140 reflections. Rays (g) from B to A and (h) from A to B form some of the striking patterns that often emerge with high numbers of reflections (here 194 and 352). It is possible to find paths that nearly fill the cavity and have many thousands of reflections.
(Color) Examples of the fate and number of reflections for rays traveling between the sources in the two-dimensional ellipsoid geometry with a 1 = 2.5, , ξ = 0.9, and RA = RB = 0.2. Parts (a) and (b) show the destinations of rays given off by source A. The colors in (a) show the number of reflections undergone by rays from A that end on A (with black for rays ending on B) and (b) carries the same information for rays from A to B (with black for rays ending on A). The angles identify which ray is being followed: θ describes the position on the source of the emitted ray and φ gives its angle with respect to the normal. The angles are defined so that θ = 0 is the position on one source closest to the other source, and both θ and φ increase in the counterclockwise direction. Rays with -- π ≤ θ ≤ 0 duplicate those with 0 ≤ θ ≤ π, and so are not shown. Most rays from A to B undergo one or two reflections—a few suffer many reflections. A sampling of the colors used to represent numbers of reflections is shown on a continuous logarithmic scale in (c). Parts (d) and (e) give the same information as (a) and (b), only for rays from B to A in (d) and B to B in (e).
(Color) The same information as in Fig. 4, except for the Chinese furnace geometry with a 1 = 2.5, , α = π/4, RA = 0.1, and RB = 0.2. Most of the rays from A to B and from B to A suffer only zero or one reflections while a few undergo many reflections. Many of the rays from A to A have complicated paths.
A bundle of one-reflection rays from S 1 on A to S 3 on B by way of S 2 on the reflecting envelope. The sizes of S 1, S 2, S 3 and, consequently, the sizes of the solid angles are exaggerated for clarity. The regions S 1, S 2, and S 3 are infinitesimal, so all rays from S 1 to S 2 are approximately parallel, as are all rays from S 2 to S 3. The normals and to S 1 and S 2 are shown along with some representative rays.
(a) An ellipse with semi-major axis a and semi-minor axis b. The foci and center are at A, B, and O, and c is half the distance between the foci. A generic point P is on the ellipse. The ellipse has eccentricity . (b) The construction of the ellipsoid paradox geometry using a 1, , and α as parameters. The continuations of ellipsoid cross-section E 1 and sphere cross-section S are shown as dashed lines.
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