Pair-factorized steady states on arbitrary graphs

Stochastic mass transport models are usually described by specifying hopping rates of particles between sites of a given lattice, and the goal is to predict the existence and properties of the steady state. Here we ask the reverse question: given a stationary state that factorizes over links (pairs of sites) of an arbitrary connected graph, what are possible hopping rates that converge to this state? We define a class of hopping functions which lead to the same steady state and guarantee current conservation but may differ by the induced current strength. For the special case of anisotropic hopping in two dimensions we discuss some aspects of the phase structure. We also show how this case can be traced back to an effective zero-range process in one dimension which is solvable for a large class of hopping functions.Comment: IOP style, 9 pages, 1 figur

Mass condensation on networks

We construct classical stochastic mass transport processes for stationary states which are chosen to factorize over pairs of sites of an undirected, connected, but otherwise arbitrary graph. For the special topology of a ring we derive static properties such as the critical point of the transition between the liquid and the condensed phase, the shape of the condensate and its scaling with the system size. It turns out that the shape is not universal, but determined by the interplay of local and ultralocal interactions. In two dimensions the effect of anisotropic interactions of hopping rates can be treated analytically, since the partition function allows a dimensional reduction to an effective one-dimensional zero-range process. Here we predict the onset, shape and scaling of the condensate on a square lattice. We indicate further extensions in the outlook

Self-gravitating Brownian systems and bacterial populations with two or more types of particles

We study the thermodynamical properties of a self-gravitating gas with two or more types of particles. Using the method of linear series of equilibria, we determine the structure and stability of statistical equilibrium states in both microcanonical and canonical ensembles. We show how the critical temperature (Jeans instability) and the critical energy (Antonov instability) depend on the relative mass of the particles and on the dimension of space. We then study the dynamical evolution of a multi-components gas of self-gravitating Brownian particles in the canonical ensemble. Self-similar solutions describing the collapse below the critical temperature are obtained analytically. We find particle segregation, with the scaling profile of the slowest collapsing particles decaying with a non universal exponent that we compute perturbatively in different limits. These results are compared with numerical simulations of the two-species Smoluchowski-Poisson system. Our model of self-attracting Brownian particles also describes the chemotactic aggregation of a multi-species system of bacteria in biology

Tuning the shape of the condensate in spontaneous symmetry breaking

We investigate what determines the shape of a particle condensate in situations when it emerges as a result of spontaneous breaking of translational symmetry. We consider a model with particles hopping between sites of a one-dimensional grid and interacting if they are at the same or at neighboring nodes. We predict the envelope of the condensate and the scaling of its width with the system size for various interaction potentials and show how to tune the shape from a delta-peak to a rectangular or a parabolic-like form.Comment: 4 pages, 2 figures, major revision, the title has been change

Self-gravitating Brownian particles in two dimensions: the case of N=2 particles

We study the motion of N=2 overdamped Brownian particles in gravitational interaction in a space of dimension d=2. This is equivalent to the simplified motion of two biological entities interacting via chemotaxis when time delay and degradation of the chemical are ignored. This problem also bears some similarities with the stochastic motion of two point vortices in viscous hydrodynamics [Agullo & Verga, Phys. Rev. E, 63, 056304 (2001)]. We analytically obtain the density probability of finding the particles at a distance r from each other at time t. We also determine the probability that the particles have coalesced and formed a Dirac peak at time t (i.e. the probability that the reduced particle has reached r=0 at time t). Finally, we investigate the variance of the distribution and discuss the proper form of the virial theorem for this system. The reduced particle has a normal diffusion behaviour for small times with a gravity-modified diffusion coefficient =r_0^2+(4k_B/\xi\mu)(T-T_*)t, where k_BT_{*}=Gm_1m_2/2 is a critical temperature, and an anomalous diffusion for large times ~t^(1-T_*/T). As a by-product, our solution also describes the growth of the Dirac peak (condensate) that forms in the post-collapse regime of the Smoluchowski-Poisson system (or Keller-Segel model) for T<T_c=GMm/(4k_B). We find that the saturation of the mass of the condensate to the total mass is algebraic in an infinite domain and exponential in a bounded domain.Comment: Revised version (20/5/2010) accepted for publication in EPJ

Recurrent mutations of BRCA1, BRCA2 and PALB2 in the population of breast and ovarian cancer patients in Southern Poland

Background Mutations in the BRCA1, BRCA2 and PALB2 genes are well-established risk factors for the development of breast and/or ovarian cancer. The frequency and spectrum of mutations in these genes has not yet been examined in the population of Southern Poland. Methods We examined the entire coding sequences of the BRCA1 and BRCA2 genes and genotyped a recurrent mutation of the PALB2 gene (c.509_510delGA) in 121 women with familial and/or early-onset breast or ovarian cancer from Southern Poland. Results A BRCA1 mutation was identified in 11 of 121 patients (9.1 %) and a BRCA2 mutation was identified in 10 of 121 patients (8.3 %). Two founder mutations of BRCA1 accounted for 91 % of all BRCA1 mutation carriers (c.5266dupC was identified in six patients and c.181 T > G was identified in four patients). Three of the seven different BRCA2 mutations were detected in two patients each (c.9371A > T, c.9403delC and c.1310_1313delAAGA). Three mutations have not been previously reported in the Polish population (BRCA1 c.3531delT, BRCA2 c.1310_1313delAAGA and BRCA2 c.9027delT). The recurrent PALB2 mutation c.509_510delGA was identified in two patients (1.7 %). Conclusions The standard panel of BRCA1 founder mutations is sufficiently sensitive for the identification of BRCA1 mutation carriers in Southern Poland. The BRCA2 mutations c.9371A > T and c.9403delC as well as the PALB2 mutation c.509_510delGA should be included in the testing panel for this population

Mass condensation on networks

We construct classical stochastic mass transport processes for stationary states which are chosen to factorize over pairs of sites of an undirected, connected, but otherwise arbitrary graph. For the special topology of a ring we derive static properties such as the critical point of the transition between the liquid and the condensed phase, the shape of the condensate and its scaling with the system size. It turns out that the shape is not universal, but determined by the interplay of local and ultralocal interactions. In two dimensions the effect of anisotropic interactions of hopping rates can be treated analytically, since the partition function allows a dimensional reduction to an effective one-dimensional zero-range process. Here we predict the onset, shape and scaling of the condensate on a square lattice. We indicate further extensions in the outlook

Mass condensation on networks

We construct classical stochastic mass transport processes for stationary states which are chosen to factorize over pairs of sites of an undirected, connected, but otherwise arbitrary graph. For the special topology of a ring we derive static properties such as the critical point of the transition between the liquid and the condensed phase, the shape of the condensate and its scaling with the system size. It turns out that the shape is not universal, but determined by the interplay of local and ultralocal interactions. In two dimensions the effect of anisotropic interactions of hopping rates can be treated analytically, since the partition function allows a dimensional reduction to an effective one-dimensional zero-range process. Here we predict the onset, shape and scaling of the condensate on a square lattice. We indicate further extensions in the outlook