The penetrable-sphere fluid in the high-temperature, high-density limit

We consider a fluid of $d$-dimensional spherical particles interacting via a pair potential $\phi(r)$ which takes a finite value $\epsilon$ if the two spheres are overlapped ($r<\sigma$) and 0 otherwise. This penetrable-sphere model has been proposed to describe the effective interaction of micelles in a solvent. We derive the structural and thermodynamic functions in the limit where the reduced temperature $k_BT/\epsilon$ and density $\rho\sigma^d$ tend to infinity, their ratio being kept finite. The fluid exhibits a spinodal instability at a certain maximum scaled density where the correlation length diverges and a crystalline phase appears, even in the one-dimensional model. By using a simple free-volume theory for the solid phase of the model, the fluid-solid phase transition is located.Comment: 6 pages, 2 tables, 3 figures; v2: title has changed; new figure; v3: Correction of misprints (see http://dx.doi.org/10.1016/j.physleta.2012.05.043

On a correlation among azimuthal velocities and the flyby anomaly sign

Data of six flybys, those of Galileo I, Galileo II, NEAR, Cassini, Rosetta and Messenger were reported by Anderson et al \citep{Anderson}. Four of them: Galileo I, NEAR, Rosetta and Messenger gain Newtonian energy during the flyby transfer, while Galileo II and Cassini lose energy. This is, in both cases, a surprising anomaly since Newtonian forces derive from a potential and they are, therefore, conservative. We show here that the gravitational field of a rotating planet as derived from a new model introduces a non conservative force that gives a partial, but in our opinion satisfactory, explanation of these anomalies and suggests a correlation between the sign of the anomaly and the sign of the azimuthal velocity at perigee.Comment: 12 pages, 3 figure

I. Territory covered by N random walkers on deterministic fractals. The Sierpinski gasket

We address the problem of evaluating the number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all initially placed on one site of a deterministic fractal lattice. For a wide class of fractals, of which the Sierpinski gasket is a typical example, we propose that, after the short-time compact regime and for large N, $S_N(t) \approx \hat{S}_N(t) (1-\Delta)$, where $\hat{S}_N(t)$ is the number of sites inside a hypersphere of radius $R [\ln (N)/c]^{1/ u}$, R is the root-mean-square displacement of a single random walker, and u and c determine how fast $1-\Gamma_t({\bf r})$ (the probability that site ${\bf r}$ has been visited by a single random walker by time t) decays for large values of r/R: $1-\Gamma_t({\bf r})\sim \exp[-c(r/R)^u]$. For the deterministic fractals considered in this paper, $u =d_w/(d_w-1)$, $d_w$ being the random walk dimension. The corrective term $\Delta$ is expressed as a series in $\ln^{-n}(N) \ln^m \ln (N)$ (with $n\geq 1$ and $0\leq m\leq n$), which is given explicitly up to n=2. Numerical simulations on the Sierpinski gasket show reasonable agreement with the analytical expressions. The corrective term $\Delta$ contributes substantially to the final value of $S_N(t)$ even for relatively large values of N.Comment: 10 total pages (RevTex), 7 figures include

II. Territory covered by N random walkers on stochastic fractals. The percolation aggregate

The average number $S_N(t)$ of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in a disordered fractal is considered. This quantity $S_N(t)$ is the result of a double average: an average over random walks on a given lattice followed by an average over different realizations of the lattice. We show for two-dimensional percolation clusters at criticality (and conjecture for other stochastic fractals) that the distribution of the survival probability over these realizations is very broad in Euclidean space but very narrow in the chemical or topological space. This allows us to adapt the formalism developed for Euclidean and deterministic fractal lattices to the chemical language, and an asymptotic series for $S_N(t)$ analogous to that found for the non-disordered media is proposed here. The main term is equal to the number of sites (volume) inside a ``hypersphere'' in the chemical space of radius $L [\ln (N)/c]^{1/v}$ where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast $1-\Gamma_t(\ell)$ (the probability that a given site at chemical distance $\ell$ from the origin is visited by a single random walker by time t) decays for large values of $\ell/L$: $1-\Gamma_t(\ell)\sim \exp[-c(\ell/L)^v]$. The parameters appearing in the first two asymptotic terms of $S_N(t)$ are estimated by numerical simulation for the two-dimensional percolation cluster at criticality. The corresponding theoretical predictions are compared with simulation data, and the agreement is found to be very good.Comment: 11 total pages (RevTex), 8 figures include

Multiparticle random walks

An overview is presented of recent work on some statistical problems on multiparticle random walks. We consider a Euclidean, deterministic fractal or disordered lattice and N >> 1 independent random walkers initially (t=0) placed onto the same site of the substrate. Three classes of problems are considered: (i) the evaluation of the average number of distinct sites visited (territory explored) up to time t by the N random walkers, (ii) the statistical description of the first passage time t_{j,N} to a given distance of the first j random walkers (order statistics of exit times), and (iii) the statistical description of the time \mathbf{t}_{j,N} elapsed until the first j random walkers are trapped when a Euclidean lattice is randomly occupied by a concentration c of traps (order statistics of the trapping problem). Although these problems are very different in nature, their solutions share the same form of a series in ln^{-n}(N) \ln^m \ln (N) (with n>=1 and 0>1. These corrective terms contribute substantially to the statistical quantities even for relatively large values of N.Comment: 14 pages, 13 figures, RevTex

Moduli of regularity and rates of convergence for Fej\'er monotone sequences

In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the finite termination, for classes of Fej\'er monotone sequences which appear in fixed point theory, monotone operator theory, and convex optimization. This concept allows for a unified approach to several notions such as weak sharp minima, error bounds, metric subregularity, H\"older regularity, etc., as well as to obtain rates of convergence for Picard iterates, the Mann algorithm, the proximal point algorithm and the cyclic algorithm. As a byproduct we obtain a quantitative version of the well-known fact that for a convex lower semi-continuous function the set of minimizers coincides with the set of zeros of its subdifferential and the set of fixed points of its resolvent

Geodesic rays, the "Lion-Man" game, and the fixed point property

This paper focuses on the relation among the existence of different types of curves (such as directional ones, quasi-geodesic or geodesic rays), the (approximate) fixed point property for nonexpansive mappings, and a discrete lion and man game. Our main result holds in the setting of CAT(0) spaces that are additionally Gromov hyperbolic

"Lion-Man" and the Fixed Point Property

This paper focuses on the relation between the fixed point property for continuous mappings and a discrete lion and man game played in a strongly convex domain. Our main result states that in locally compact geodesic spaces, the compactness of the domain is equivalent to its fixed point property, as well as to the success of the lion. The common link among these properties involves the existence of different types of rays, which we also discuss.Comment: final version available in Geometriae Dedicata, https://doi.org/10.1007/s10711-018-0403-

Quantitative asymptotic regularity results for the composition of two mappings

In this paper, we use techniques which originate from proof mining to give rates of asymptotic regularity and metastability for a sequence associated to the composition of two firmly nonexpansive mappings

On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow

Uniform shear flow is a paradigmatic example of a nonequilibrium fluid state exhibiting non-Newtonian behavior. It is characterized by uniform density and temperature and a linear velocity profile $U_x(y)=a y$, where $a$ is the constant shear rate. In the case of a rarefied gas, all the relevant physical information is represented by the one-particle velocity distribution function $f({\bf r},{\bf v})=f({\bf V})$, with ${\bf V}\equiv {\bf v}-{\bf U}({\bf r})$, which satisfies the standard nonlinear integro-differential Boltzmann equation. We have studied this state for a two-dimensional gas of Maxwell molecules with grazing collisions in which the nonlinear Boltzmann collision operator reduces to a Fokker-Planck operator. We have found analytically that for shear rates larger than a certain threshold value the velocity distribution function exhibits an algebraic high-velocity tail of the form $f({\bf V};a)\sim |{\bf V}|^{-4-\sigma(a)}\Phi(\phi; a)$, where $\phi\equiv \tan V_y/V_x$ and the angular distribution function $\Phi(\phi; a)$ is the solution of a modified Mathieu equation. The enforcement of the periodicity condition $\Phi(\phi; a)=\Phi(\phi+\pi; a)$ allows one to obtain the exponent $\sigma(a)$ as a function of the shear rate. As a consequence of this power-law decay, all the velocity moments of a degree equal to or larger than $2+\sigma(a)$ are divergent. In the high-velocity domain the velocity distribution is highly anisotropic, with the angular distribution sharply concentrated around a preferred orientation angle which rotates counterclock-wise as the shear rate increases.Comment: 15 pages, 5 figures; change in title plus other minor changes; to be published in J. Stat. Phy