The free path in a high velocity random flight process associated to a Lorentz gas in an external field

We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight.Comment: 30 page

Radial Density in Apollonian Packings

Given an Apollonian Circle Packing $\mathcal{P}$ and a circle $C_0 = \partial B(z_0, r_0)$ in $\mathcal{P}$, color the set of disks in $\mathcal{P}$ tangent to $C_0$ red. What proportion of the concentric circle $C_{\epsilon} = \partial B(z_0, r_0 + \epsilon)$ is red, and what is the behavior of this quantity as $\epsilon \rightarrow 0$? Using equidistribution of closed horocycles on the modular surface $\mathbb{H}^2/SL(2, \mathbb{Z})$, we show that the answer is $\frac{3}{\pi} = 0.9549\dots$ We also describe an observation due to Alex Kontorovich connecting the rate of this convergence in the Farey-Ford packing to the Riemann Hypothesis. For the analogous problem for Soddy Sphere packings, we find that the limiting radial density is $\frac{\sqrt{3}}{2V_T}=0.853\dots$, where $V_T$ denotes the volume of an ideal hyperbolic tetrahedron with dihedral angles $\pi/3$.Comment: New section based on an observation due to Alex Kontorovich connecting the rate of this convergence in the Farey-Ford packing to the Riemann Hypothesi

Critical point of Nf=3N_f = 3 QCD from lattice simulations in the canonical ensemble

A canonical ensemble algorithm is employed to study the phase diagram of $N_f = 3$ QCD using lattice simulations. We lock in the desired quark number sector using an exact Fourier transform of the fermion determinant. We scan the phase space below $T_c$ and look for an S-shape structure in the chemical potential, which signals the coexistence phase of a first order phase transition in finite volume. Applying Maxwell construction, we determine the boundaries of the coexistence phase at three temperatures and extrapolate them to locate the critical point. Using an improved gauge action and improved Wilson fermions on lattices with a spatial extent of 1.8 \fm and quark masses close to that of the strange, we find the critical point at $T_E = 0.925(5) T_c$ and baryon chemical potential $\mu_B^E = 2.60(8) T_c$.Comment: 5 pages, 7 figures, references added, published versio

Analytical Results for the Classical and Quantum Tsallis Hadron Transverse Momentum Spectra: the Zeroth Order Approximation and beyond

We derive the analytical expressions for the first and second order terms in the hadronic transverse momentum spectra obtained from the Tsallis normalized (Tsallis-1) statistics. We revisit the zeroth order quantum Tsallis distributions and obtain the corresponding analytical closed form expressions. It is observed that unlike the classical case, the analytical closed forms of the zeroth order quantum spectra do not resemble the phenomenological distributions used in the literature after $q \to q^{-1}$ substitution, where $q$ is the Tsallis entropic parameter. However, the factorization approximation increases the extent of similarity.Comment: 12 pages, 6 figure

Not All Traces on the Circle Come from Functions of Least Gradient in the Disk

We provide an example of an L-1 function on the unit circle that cannot be the trace of a function of bounded variation of least gradient in the unit disk

Not All Traces On the Circle Come From Functions of Least Gradient in the Disk

We provide an example of an L¹ function on the circle, which cannot be the trace of a function of bounded variation of least gradient in the disk

Diffusion coefficient and shear viscosity of rigid water models

We report the diffusion coefficient and viscosity of popular rigid water models: Two non polarizable ones (SPC/E with 3 sites, and TIP4P/2005 with 4 sites) and a polarizable one (Dang-Chang, 4 sites). We exploit the dependence of the diffusion coefficient on the system size [Yeh and Hummer, J. Phys. Chem. B 108, 15873 (2004)] to obtain the size-independent value. This also provides an estimate of the viscosity of all water models, which we compare to the Green-Kubo result. In all cases, a good agreement is found. The TIP4P/2005 model is in better agreement with the experimental data for both diffusion and viscosity. The SPC/E and Dang-Chang water overestimate the diffusion coefficient and underestimate the viscosity.Comment: 10 pages, 2 figures. To be published in J. Phys.: Condens. Matte