Traffic by multiple species of molecular motors

We study the traffic of two types of molecular motors using the two-species symmetric simple exclusion process (ASEP) with periodic boundary conditions and with attachment and detachment of particles. We determine characteristic properties such as motor densities and currents by simulations and analytical calculations. For motors with different unbinding probabilities, mean field theory gives the correct bound density and total current of the motors, as shown by numerical simulations. For motors differing in their stepping probabilities, the particle-hole symmetry of the current-density relationship is broken and mean field theory fails drastically. The total motor current exhibits exponential finite-size scaling, which we use to extrapolate the total current to the thermodynamic limit. Finally, we also study the motion of a single motor in the background of many non-moving motors.Comment: 23 pages, 6 figures, late

Entanglement in Valence-Bond-Solid States

This article reviews the quantum entanglement in Valence-Bond-Solid (VBS) states defined on a lattice or a graph. The subject is presented in a self-contained and pedagogical way. The VBS state was first introduced in the celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki (abbreviation AKLT is widely used). It became essential in condensed matter physics and quantum information (measurement-based quantum computation). Many publications have been devoted to the subject. Recently entanglement was studied in the VBS state. In this review we start with the definition of a general AKLT spin chain and the construction of VBS ground state. In order to study entanglement, a block subsystem is introduced and described by the density matrix. Density matrices of 1-dimensional models are diagonalized and the entanglement entropies (the von Neumann entropy and Renyi entropy) are calculated. In the large block limit, the entropies also approach finite limits. Study of the spectrum of the density matrix led to the discovery that the density matrix is proportional to a projector.Comment: Published version, 80 pages, 8 figures; references update

A transition from river networks to scale-free networks

A spatial network is constructed on a two dimensional space where the nodes are geometrical points located at randomly distributed positions which are labeled sequentially in increasing order of one of their co-ordinates. Starting with $N$ such points the network is grown by including them one by one according to the serial number into the growing network. The $t$-th point is attached to the $i$-th node of the network using the probability: $\pi_i(t) \sim k_i(t)\ell_{ti}^{\alpha}$ where $k_i(t)$ is the degree of the $i$-th node and $\ell_{ti}$ is the Euclidean distance between the points $t$ and $i$. Here $\alpha$ is a continuously tunable parameter and while for $\alpha=0$ one gets the simple Barab\'asi-Albert network, the case for $\alpha \to -\infty$ corresponds to the spatially continuous version of the well known Scheidegger's river network problem. The modulating parameter $\alpha$ is tuned to study the transition between the two different critical behaviors at a specific value $\alpha_c$ which we numerically estimate to be -2.Comment: 5 pages, 5 figur

Theory of resistor networks: The two-point resistance

The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for two-point resistances are deduced for regular lattices in one, two, and three dimensions under various boundary conditions including that of a Moebius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyze large-size expansions of two-and-higher dimensional lattices.Comment: 30 pages, 5 figures now included; typos in Example 1 correcte

Correlation Functions of Dense Polymers and c=-2 Conformal Field Theory

The model of dense lattice polymers is studied as an example of non-unitary Conformal Field Theory (CFT) with $c=-2$. ``Antisymmetric'' correlation functions of the model are proved to be given by the generalized Kirchhoff theorem. Continuous limit of the model is described by the free complex Grassmann field with null vacuum vector. The fundamental property of the Grassmann field and its twist field (both having non-positive conformal weights) is that they themselves suppress zero mode so that their correlation functions become non-trivial. The correlation functions of the fields with positive conformal weights are non-zero only in the presence of the Dirichlet operator that suppresses zero mode and imposes proper boundary conditions.Comment: 5 pages, REVTeX, remark is adde

Inhomogeneous Superconductivity in Comb-Shaped Josephson Junction Networks

We show that some of the Josephson couplings of junctions arranged to form an inhomogeneous network undergo a non-perturbative renormalization provided that the network's connectivity is pertinently chosen. As a result, the zero-voltage Josephson critical currents $I_c$ turn out to be enhanced along directions selected by the network's topology. This renormalization effect is possible only on graphs whose adjacency matrix admits an hidden spectrum (i.e. a set of localized states disappearing in the thermodynamic limit). We provide a theoretical and experimental study of this effect by comparing the superconducting behavior of a comb-shaped Josephson junction network and a linear chain made with the same junctions: we show that the Josephson critical currents of the junctions located on the comb's backbone are bigger than the ones of the junctions located on the chain. Our theoretical analysis, based on a discrete version of the Bogoliubov-de Gennes equation, leads to results which are in good quantitative agreement with experimental results.Comment: 4 pages, 2 figures, revte

Parameterized Edge Hamiltonicity

We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set. We also consider the problem parameterized by treewidth or clique-width. Surprisingly, we show that the problem is FPT for both of these standard parameters, in contrast to its vertex version, which is W-hard for clique-width. Our technique, which may be of independent interest, relies on a structural characterization of clique-width in terms of treewidth and complete bipartite subgraphs due to Gurski and Wanke

Inductive Construction of 2-Connected Graphs for Calculating the Virial Coefficients

In this paper we give a method for constructing systematically all simple 2-connected graphs with n vertices from the set of simple 2-connected graphs with n-1 vertices, by means of two operations: subdivision of an edge and addition of a vertex. The motivation of our study comes from the theory of non-ideal gases and, more specifically, from the virial equation of state. It is a known result of Statistical Mechanics that the coefficients in the virial equation of state are sums over labelled 2-connected graphs. These graphs correspond to clusters of particles. Thus, theoretically, the virial coefficients of any order can be calculated by means of 2-connected graphs used in the virial coefficient of the previous order. Our main result gives a method for constructing inductively all simple 2-connected graphs, by induction on the number of vertices. Moreover, the two operations we are using maintain the correspondence between graphs and clusters of particles.Comment: 23 pages, 5 figures, 3 table

A rigorous implementation of the Jeans--Landau--Teller approximation

Rigorous bounds on the rate of energy exchanges between vibrational and translational degrees of freedom are established in simple classical models of diatomic molecules. The results are in agreement with an elementary approximation introduced by Landau and Teller. The method is perturbative theory ``beyond all orders'', with diagrammatic techniques (tree expansions) to organize and manipulate terms, and look for compensations, like in recent studies on KAM theorem homoclinic splitting.Comment: 23 pages, postscrip