Maximum Distance Between the Leader and the Laggard for Three Brownian Walkers

We consider three independent Brownian walkers moving on a line. The process terminates when the left-most walker (the `Leader') meets either of the other two walkers. For arbitrary values of the diffusion constants D_1 (the Leader), D_2 and D_3 of the three walkers, we compute the probability distribution P(m|y_2,y_3) of the maximum distance m between the Leader and the current right-most particle (the `Laggard') during the process, where y_2 and y_3 are the initial distances between the leader and the other two walkers. The result has, for large m, the form P(m|y_2,y_3) \sim A(y_2,y_3) m^{-\delta}, where \delta = (2\pi-\theta)/(\pi-\theta) and \theta = cos^{-1}(D_1/\sqrt{(D_1+D_2)(D_1+D_3)}. The amplitude A(y_2,y_3) is also determined exactly

Maximum of N Independent Brownian Walkers till the First Exit From the Half Space

We consider the one-dimensional target search process that involves an immobile target located at the origin and $N$ searchers performing independent Brownian motions starting at the initial positions $\vec x = (x_1,x_2,..., x_N)$ all on the positive half space. The process stops when the target is first found by one of the searchers. We compute the probability distribution of the maximum distance $m$ visited by the searchers till the stopping time and show that it has a power law tail: $P_N(m|\vec x)\sim B_N (x_1x_2... x_N)/m^{N+1}$ for large $m$. Thus all moments of $m$ up to the order $(N-1)$ are finite, while the higher moments diverge. The prefactor $B_N$ increases with $N$ faster than exponentially. Our solution gives the exit probability of a set of $N$ particles from a box $[0,L]$ through the left boundary. Incidentally, it also provides an exact solution of the Laplace's equation in an $N$-dimensional hypercube with some prescribed boundary conditions. The analytical results are in excellent agreement with Monte Carlo simulations.Comment: 18 pages, 9 figure

Brane-world cosmology with black strings

We consider the simplest scenario when black strings / cigars penetrate the cosmological brane. As a result, the brane has a Swiss-cheese structure, with Schwarzschild black holes immersed in a Friedmann-Lema\^{\i}tre-Robertson-Walker brane. There is no dark radiation in the model, the cosmological regions of the brane are characterized by a cosmological constant $\Lambda$ and flat spatial sections. Regardless of the value of $\Lambda$, these brane-world universes forever expand and forever decelerate. The totality of source terms in the modified Einstein equation sum up to a dust, establishing a formal equivalence with the general relativistic Einstein-Straus model. However in this brane-world scenario with black strings the evolution of the cosmological fluid strongly depends on $\Lambda$. For $\Lambda \leq 0$ it has positive energy density $\rho$ and negative pressure $p$ and at late times it behaves as in the Einstein-Straus model. For (not too high) positive values of $\Lambda$ the cosmological evolution begins with positive $\rho$ and negative $p$, but this is followed by an epoch with both $\rho$ and $p$ positive. Eventually, $\rho$ becomes negative, while $p$ stays positive. A similar evolution is present for high positive values of $\Lambda$, however in this case the evolution ends in a pressure singularity, accompanied by a regular behaviour of the cosmic acceleration. This is a novel type of singularity appearing in brane-worlds.Comment: 6 pages, 4 figures; expanded version, references added, to appear in Physical Review

Morphological instability of a nonequilibrium icecolloid interface

We assess the morphological stability of a nonequilibrium icecolloidal suspension interface, and apply the theory to bentonite clay. An experimentally convenient scaling is employed which takes advantage of the vanishing segregation coefficient at low freezing velocities, and when anisotropic kinetic effects are included the interface is shown to be unstable to travelling waves. The potential for traveling wave modes reveals a possible mechanism for the polygonal and spiral ice lenses observed in frozen clays. A weakly nonlinear analysis yields a long-wave evolution equation for the interface shape containing a new parameter related to the highly nonlinear liquidus curve in colloidal systems. We discuss the implications of these results for the frost susceptibility of soils and the fabrication of microtailored porous materials

Spatial survival probability for one-dimensional fluctuating interfaces in the steady state

We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the `sampling interval' used in the measurement for both `steady-state' and `finite' initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A `deterministic approximation' is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.Comment: RevTeX4, 21 pages, 8 .eps figures, changes in sections IIIB and IIIC and in Figs 7 and 8, version to be published in Physical Review

Anderson-Mott Transition Driven by Spin Disorder: Spin Glass Transition and Magnetotransport in Amorphous GdSi

A zero temperature Anderson-Mott transition driven by spin disorder can be `tuned' by an applied magnetic field to achieve colossal magnetoconductance. Usually this is not possible since spin disorder by itself cannot localise a high density electron system. However, the presence of strong structural disorder can realise this situation, self consistently generating a disordered magnetic ground state. We explore such a model, constructed to understand amorphous GdSi, and highlight the emergence of a spin glass phase, Anderson-Mott signatures in transport and tunneling spectra, and unusual magneto-optical conductivity. We solve a disordered strong coupling fermion-spin-lattice problem essentially exactly on finite systems, and account for all the qualitative features observed in magnetism, transport, and the optical spectra in this system.Comment: Final version of cond-mat/0209579, to appear in Phys. Rev. Let

Residual resistivity ratio and its relation to the positive magnetoresistance behavior in natural multilayer LaMn2Ge2; relevance to artificial multilayer physics

Results of low temperature magnetoresistance ($\Delta\rho/\rho$) and isothermal magnetization (M) measurements on polycrystalline ferromagnetic (T_C close to 300 K) natural multilayers, LaMn_{2+x}Ge_{2-y}Si_y, are reported. It is found that the samples with large residual resistivity ratio, $\rho(300K)/\rho(4.2K)$, exhibit large positive magnetoresistance at high magnetic fields. The Kohler's rule is not obeyed in these alloys. In addition, at 4.5 K, there is a tendency towards linear variation of $\Delta\rho/\rho$ with magnetic field with increasing $\rho(300K)/\rho(4.2K$); however, the field dependence of $\Delta\rho/\rho$ does not track that of M, thereby suggesting that the magnetoresistance originates from non-magnetic layers. It is interesting that these experimental findings on bulk polycrystals are qualitatively similar to what is seen in artificially grown multilayer systems recently.Comment: 5 pages, 3 figures, separate figures. This work is a follow-up of our earlier paper in APL, Ref. : APL Vol 71, pp 2385 (1997