Two dimensional Hotelling model : analytical results and numerical simulations

We present an analytical solution to a two dimensional Hotelling model with quadratic transportation costs for two stores in a square city. We assume that the consumers choice as to which store to patronize is tempered by a logit function. As in the one-dimensional case, stores are led to aggregate spatially as the disorder introduced by the logit increases. This solution is confirmed by numerical simulations

Statistical properties of volume in the Bitcoin/USD market

It is widely known that the volumes of limit orders and market orders display non-trivial statistical properties. For example, the sizes of volumes can take on many different values across several orders of magnitude, with a marked preference for whole numbers like 5, 10, 100. These studies have been centered on the volume of incoming orders or of realized transactions. In the present work we analyze the statistical properties of volumes stored at the best ask and best bid. We measured both dynamical properties as well as properties that do not directly depend on time. Of the properties found some are also present in the time series of price returns while others turn out to be particularly idiosyncratic and distinguish the stochastic nature of volume time series from that of price returns time series

On the energy density in quantum mechanics

There are several definitions of energy density in quantum mechanics. These yield expressions that differ locally, but all satisfy a continuity equation and integrate to the value of the expected energy of the system under consideration. Thus, the question of whether there are physical grounds to choose one definition over another arises naturally. In this work, we propose a way to probe a system by varying the size of a well containing a quantum particle. We show that the mean work done by moving the wall is closely related to one of the definitions for energy density. Specifically, the appropriate energy density, evaluated at the wall corresponds to the force exerted by the particle locally, against which the work is done. We show that this identification extends to two and three dimensional systems

Kinetics of shape equilibration for two-dimensional islands

We study the relaxation to equilibrium of two dimensional islands containing up to 20000 atoms by Kinetic Monte Carlo simulations. We find that the commonly assumed relaxation mechanism - curvature-driven relaxation via atom diffusion - cannot explain the results obtained at low temperatures, where the island edges consist in large facets. Specifically, our simulations show that the exponent characterizing the dependence of the equilibration time on the island size is different at high and low temperatures, in contradiction with the above cited assumptions. Instead, we propose that - at low temperatures - the relaxation is limited by the nucleation of new atomic rows on the large facets : this allows us to explain both the activation energy and the island size dependence of the equilibration time.Comment: 9 pages, revte