The price of anarchy in basketball

Optimizing the performance of a basketball offense may be viewed as a network problem, wherein each play represents a "pathway" through which the ball and players may move from origin (the in-bounds pass) to goal (the basket). Effective field goal percentages from the resulting shot attempts can be used to characterize the efficiency of each pathway. Inspired by recent discussions of the "price of anarchy" in traffic networks, this paper makes a formal analogy between a basketball offense and a simplified traffic network. The analysis suggests that there may be a significant difference between taking the highest-percentage shot each time down the court and playing the most efficient possible game. There may also be an analogue of Braess's Paradox in basketball, such that removing a key player from a team can result in the improvement of the team's offensive efficiency.Comment: 9 pages, 6 figures; extra example and some discussion added; formatting errors fixed; typo in Sec. IIID fixe

Chemical potential and compressibility of quantum Hall bilayer excitons

This paper considers a system of two parallel quantum Hall layers with total filling factor $0$ or $1$. When the distance between the layers is small enough, electrons and holes in opposite layers form inter-layer excitons, which have a finite effective mass and interact via a dipole-dipole potential. Results are presented for the chemical potential $\mu$ of the resulting bosonic system as a function of the exciton concentration $n$ and the interlayer separation $d$. Both $\mu$ and the interlayer capacitance have an unusual nonmonotonic dependence on $d$, owing to the interplay between an increasing dipole moment and an increasing effective mass with increasing $d$. A phase transition between superfluid and Wigner crystal phases is shown to occur at $d \propto n^{-1/10}$. Results are derived first via simple intuitive arguments, and then verified with more careful analytic derivations and numeric calculations.Comment: 7 pages, 5 figures; improved discussion and references; published versio

Large, nonsaturating thermopower in a quantizing magnetic field

The thermoelectric effect is the generation of an electrical voltage from a temperature gradient in a solid material due to the diffusion of free charge carriers from hot to cold. Identifying materials with large thermoelectric response is crucial for the development of novel electric generators and coolers. In this paper we consider theoretically the thermopower of Dirac/Weyl semimetals subjected to a quantizing magnetic field. We contrast their thermoelectric properties with those of traditional heavily-doped semiconductors and we show that, under a sufficiently large magnetic field, the thermopower of Dirac/Weyl semimetals grows linearly with the field without saturation and can reach extremely high values. Our results suggest an immediate pathway for achieving record-high thermopower and thermoelectric figure of merit, and they compare well with a recent experiment on Pb$_{1-x}$Sn$_x$Se.Comment: 6+3 pages, 4 figures; update discussion of experiments and device performanc

Enhancement of hopping conductivity by spontaneous fractal ordering of low-energy sites

Variable-range hopping conductivity has long been understood in terms of a canonical prescription for relating the single-particle density of states to the temperature-dependent conductivity. Here we demonstrate that this prescription breaks down in situations where a large and long-ranged random potential develops. In particular, we examine a canonical model of a completely compensated semiconductor, and we show that at low temperatures hopping proceeds along self-organized, low-dimensional subspaces having fractal dimension $d = 2$. We derive and study numerically the spatial structure of these subspaces, as well as the conductivity and density of states that result from them. One of our prominent findings is that fractal ordering of low energy sites greatly enhances the hopping conductivity, and allows Efros-Shklovskii type conductivity to persist up to unexpectedly high temperatures.Comment: 9 pages, 6 figures; published version with added references and discussio

Thermoelectric Hall conductivity and figure of merit in Dirac/Weyl materials

We calculate the thermoelectric response coefficients of three-dimensional Dirac or Weyl semimetals as a function of magnetic field, temperature, and Fermi energy. We focus in particular on the thermoelectric Hall coefficient $\alpha_{xy}$ and the Seebeck coefficient $S_{xx}$, which are well-defined even in the dissipationless limit. We contrast the behaviors of $\alpha_{xy}$ and $S_{xx}$ with those of traditional Schr\"{o}dinger particle systems, such as doped semiconductors. Strikingly, we find that for Dirac materials $\alpha_{xy}$ acquires a constant, quantized value at sufficiently large magnetic field, which is independent of the magnetic field or the Fermi energy, and this leads to unprecedented growth in the thermopower and the thermoelectric figure of merit. We further show that even relatively small fields, such that $\omega_c \tau \sim 1$ (where $\omega_c$ is the cyclotron frequency and $\tau$ is the scattering time), are sufficient to produce a more than $100\%$ increase in the figure of merit.Comment: 10 pages, 5 figure