On extreme points of the diffusion polytope

Abstract

We consider a class of diffusion problems defined on simple graphs in which the populations at any two vertices may be averaged if they are connected by an edge. The diffusion polytope is the convex hull of the set of population vectors attainable using finite sequences of these operations. A number of physical problems have linear programming solutions taking the diffusion polytope as the feasible region, e.g. the free energy that can be removed from plasma using waves, so there is a need to describe and enumerate its extreme points. We review known results for the case of the complete graph K n , and study a variety of problems for the path graph P n and the cyclic graph C n . We describe the different kinds of extreme points that arise, and identify the diffusion polytope in a number of simple cases. In the case of increasing initial populations on P n the diffusion polytope is topologically an n-dimensional hypercube.

Continuous thermomajorization and a complete set of laws for Markovian thermal processes

“…[34]. In fact, the notion of continuous majorization has a decades-long history and appears in a variety of research fields from thermodynamics and order theory [35,36], through plasma physics [37,38], to social sciences [39]. Moreover, this notion was employed and studied in more detail in Ref.…”

Available Energy from Diffusive and Reversible Phase Space Rearrangements

“…Physically, previous work presents the discrete versions of these problems in two ways. 7,12,13 First, it can model an intrinsically discrete physical system, like transitions between atomic energy levels stimulated by lasers. Second, it can model a system with continuous phase space (e.g., a plasma) being mixed with some finite granularity.…”

“…10,11 The maximal extractable energy under diffusive phase space rearrangements in plasma was addressed by Hay, Schiff, and Fisch. 12, 13 Helander's recent calculation of the plasma free energy obeying phase space conservation, with the motion of individual particles constrained by adiabatic invariants, now points to a natural generalization: the free energy under diffusion in phase space, but with the motion of individual particles similarly constrained by adiabatic invariants. This paper will discuss that generalization.…”

Recovering Gardner Restacking with Purely Diffusive Operations

“…However, when the plasma distribution function is viewed with any finite granularity, many pro- * ekolmes@princeton.edu cesses can appear to diffuse particles between volumes of phase space rather than exchanging the contents of individual volumes [2,3]. As a result, it is often useful to consider an alternative to Gardner's problem, where the maximum accessible energy is determined by what can be extracted by diffusion between phase space volumes (including elements which are not adjacent) rather than Gardner restacking [4][5][6][7]. This is a qualitatively different process from the pairwise exchange of phase space densities that underlies Gardner restacking; for one thing, every diffusive step creates entropy, whereas restacking is reversible.…”

“…Fisch and Rax showed that f satisfies an H theorem, and that the system will reach a steady state, but they left the matter of the releasable free energy as an open problem, noting that it is "quite formidable" given the necessity to search over all possible kernels K(v, v , t). Indeed, in the years since, substantial progress has been made on the discrete diffusive exchange problem [5][6][7], as well as on continuous Gardner restacking [8][9][10], but the minimum energy state under continuous diffusive operations remains unsolved. This paper will show that, in fact, in the continuous limit, the free energy available under the diffusive constraint is equivalent to the Gardner free energy under the restacking constraint.…”