First passages for a search by a swarm of independent random searchers

In this paper we study some aspects of search for an immobile target by a swarm of N non-communicating, randomly moving searchers (numbered by the index k, k = 1, 2,..., N), which all start their random motion simultaneously at the same point in space. For each realization of the search process, we record the unordered set of time moments \{\tau_k\}, where \tau_k is the time of the first passage of the k-th searcher to the location of the target. Clearly, \tau_k's are independent, identically distributed random variables with the same distribution function \Psi(\tau). We evaluate then the distribution P(\omega) of the random variable \omega \sim \tau_1/bar{\tau}, where bar{\tau} = N^{-1} \sum_{k=1}^N \tau_k is the ensemble-averaged realization-dependent first passage time. We show that P(\omega) exhibits quite a non-trivial and sometimes a counterintuitive behaviour. We demonstrate that in some well-studied cases e.g., Brownian motion in finite d-dimensional domains) the \textit{mean} first passage time is not a robust measure of the search efficiency, despite the fact that \Psi(\tau) has moments of arbitrary order. This implies, in particular, that even in this simplest case (not saying about complex systems and/or anomalous diffusion) first passage data extracted from a single particle tracking should be regarded with an appropriate caution because of the significant sample-to-sample fluctuations.Comment: 35 pages, 18 figures, to appear in JSTA

Negative response to an excessive bias by a mixed population of voters

We study an outcome of a vote in a population of voters exposed to an externally applied bias in favour of one of two potential candidates. The population consists of ordinary individuals, that are in majority and tend to align their opinion with the external bias, and some number of contrarians --- individuals who are always hostile to the bias but are not in a conflict with ordinary voters. The voters interact among themselves, all with all, trying to find an opinion reached by the community as a whole. We demonstrate that for a sufficiently weak external bias, the opinion of ordinary individuals is always decisive and the outcome of the vote is in favour of the preferential candidate. On the contrary, for an excessively strong bias, the contrarians dominate in the population's opinion, producing overall a negative response to the imposed bias. We also show that for sufficiently strong interactions within the community, either of two subgroups can abruptly change an opinion of the other group.Comment: 11 pages, 6 figure

Entanglement Across a Transition to Quantum Chaos

We study the relation between entanglement and quantum chaos in one- and two-dimensional spin-1/2 lattice models, which exhibit mixing of the noninteracting eigenfunctions and transition from integrability to quantum chaos. Contrary to what occurs in a quantum phase transition, the onset of quantum chaos is not a property of the ground state but take place for any typical many-spin quantum state. We study bipartite and pairwise entanglement measures, namely the reduced Von Neumann entropy and the concurrence, and discuss quantum entanglement sharing. Our results suggest that the behavior of the entanglement is related to the mixing of the eigenfunctions rather than to the transition to chaos.Comment: 14 pages, 14 figure

Fourier's Law in a Quantum Spin Chain and the Onset of Quantum Chaos

We study heat transport in a nonequilibrium steady state of a quantum interacting spin chain. We provide clear numerical evidence of the validity of Fourier law. The regime of normal conductivity is shown to set in at the transition to quantum chaos.Comment: 4 pages, 5 figures, RevTe

Memory Effects in Nonequilibrium Transport for Deterministic Hamiltonian Systems

We consider nonequilibrium transport in a simple chain of identical mechanical cells in which particles move around. In each cell, there is a rotating disc, with which these particles interact, and this is the only interaction in the model. It was shown in \cite{eckmann-young} that when the cells are weakly coupled, to a good approximation, the jump rates of particles and the energy-exchange rates from cell to cell follow linear profiles. Here, we refine that study by analyzing higher-order effects which are induced by the presence of external gradients for situations in which memory effects, typical of Hamiltonian dynamics, cannot be neglected. For the steady state we propose a set of balance equations for the particle number and energy in terms of the reflection probabilities of the cell and solve it phenomenologically. Using this approximate theory we explain how these asymmetries affect various aspects of heat and particle transport in systems of the general type described above and obtain in the infinite volume limit the deviation from the theory in \cite{eckmann-young} to first-order. We verify our assumptions with extensive numerical simulations.Comment: Several change

Symmetry breaking between statistically equivalent, independent channels in a few-channel chaotic scattering

We study the distribution function $P(\omega)$ of the random variable $\omega = \tau_1/(\tau_1 + ... + \tau_N)$, where $\tau_k$'s are the partial Wigner delay times for chaotic scattering in a disordered system with $N$ independent, statistically equivalent channels. In this case, $\tau_k$'s are i.i.d. random variables with a distribution $\Psi(\tau)$ characterized by a "fat" power-law intermediate tail $\sim 1/\tau^{1 + \mu}$, truncated by an exponential (or a log-normal) function of $\tau$. For $N = 2$ and N=3, we observe a surprisingly rich behavior of $P(\omega)$ revealing a breakdown of the symmetry between identical independent channels. For N=2, numerical simulations of the quasi one-dimensional Anderson model confirm our findings.Comment: 4 pages, 5 figure

High order non-unitary split-step decomposition of unitary operators

We propose a high order numerical decomposition of exponentials of hermitean operators in terms of a product of exponentials of simple terms, following an idea which has been pioneered by M. Suzuki, however implementing it for complex coefficients. We outline a convenient fourth order formula which can be written compactly for arbitrary number of noncommuting terms in the Hamiltonian and which is superiour to the optimal formula with real coefficients, both in complexity and accuracy. We show asymptotic stability of our method for sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models.Comment: 10 pages, 4 figures (5 eps files) Submitted to J. of Phys. A: Math. Ge

Reconstructing Fourier's law from disorder in quantum wires

The theory of open quantum systems is used to study the local temperature and heat currents in metallic nanowires connected to leads at different temperatures. We show that for ballistic wires the local temperature is almost uniform along the wire and Fourier's law is invalid. By gradually increasing disorder, a uniform temperature gradient ensues inside the wire and the thermal current linearly relates to this local temperature gradient, in agreement with Fourier's law. Finally, we demonstrate that while disorder is responsible for the onset of Fourier's law, the non-equilibrium energy distribution function is determined solely by the heat baths