Carlo Vanderzande

Exact Current Statistics of the Asymmetric Simple Exclusion Process with Open Boundaries

Non-equilibrium systems are often characterized by the transport of some quantity at a macroscopic scale, such as, for instance, a current of particles through a wire. The Asymmetric Simple Exclusion Process (ASEP) is a paradigm for non-equilibrium transport that is amenable to exact analytical solution. In the present work, we determine the full statistics of the current in the finite size open ASEP for all values of the parameters. Our exact analytical results are checked against numerical calculations using DMRG techniques.

Strong disorder fixed point in absorbing-state phase transitions.

The effect of quenched disorder on nonequilibrium phase transitions in the directed percolation universality class is studied by a strong disorder renormalization group approach and by density matrix renormalization group calculations. We show that for sufficiently strong disorder the critical behavior is controlled by a strong disorder fixed point and in one dimension the critical exponents are conjectured to be exact: beta=(3-sqrt[5])/2 and nu( perpendicular )=2. For disorder strengths outside the attractive region of this fixed point, disorder dependent critical exponents are detected. Existing numerical results in two dimensions can be interpreted within a similar scenario.

Dynamics of a polymer in an active and viscoelastic bath

We study the dynamics of an ideal polymer chain in a viscoelastic medium and in the presence of active forces. The motion of the center of mass and of individual monomers is calculated. On time scales that are comparable to the persistence time of the active forces, monomers can move superdiffusively, while on larger time scales subdiffusive behavior occurs. The difference between this subdiffusion and that in the absence of active forces is quantified. We show that the polymer swells in response to active processes and determine how this swelling depends on the viscoelastic properties of the environment. Our results are compared to recent experiments on the motion of chromosomal loci in bacteria.

Absorbing state phase transitions with quenched disorder

Quenched disorder--in the sense of the Harris criterion--is generally a relevant perturbation at an absorbing state phase transition point. Here using a strong disorder renormalization group framework and effective numerical methods we study the properties of random fixed points for systems in the directed percolation universality class. For strong enough disorder the critical behavior is found to be controlled by a strong disorder fixed point, which is isomorph with the fixed point of random quantum Ising systems. In this fixed point dynamical correlations are logarithmically slow and the static critical exponents are conjecturedly exact for one-dimensional systems. The renormalization group scenario is confronted with numerical results on the random contact process in one and two dimensions and satisfactory agreement is found. For weaker disorder the numerical results indicate static critical exponents which vary with the strength of disorder, whereas the dynamical correlations are compatible with two possible scenarios. Either they follow a power-law decay with a varying dynamical exponent, like in random quantum systems, or the dynamical correlations are logarithmically slow even for a weak disorder. For models in the parity conserving universality class there is no strong disorder fixed point according to our renormalization group analysis.

Sandpiles on a Sierpinski gasket

We perform extensive simulations of the sandpile model on a Sierpinski gasket. Critical exponents for waves and avalanches are determined. We extend the existing theory of waves to the present case. This leads to an exact value for the exponent τw which describes the distribution of wave sizes: τw=ln(9/5)/ln3. Numerically, it is found that the number of waves in an avalanche is proportional to the number of distinct sites toppled in the avalanche. This leads to a conjecture for the exponent τ that determines the distribution of avalanche sizes: τ=1+τw=ln(27/5)/ln3. Our predictions are in good agreement with the numerical results.

Finite size scaling of current fluctuations in the totally asymmetric exclusion process

We study the fluctuations of the current J(t) of the totally asymmetric exclusion process with open boundaries. Using a density matrix renormalization group approach, we calculate the cumulant generating function of the current. This function can be interpreted as a free energy for an ensemble in which histories are weighted by exp ( − sJ(t)). We show that in this ensemble the model has a first-order spacetime phase transition at s = 0. We numerically determine the finite size scaling of the cumulant generating function near this phase transition, both in the non-equilibrium steady state and for large times.

Bulk, surface and hull fractal dimension of critical Ising clusters in d=2

The authors present accurate numerical calculations of the fractal dimension d and surface dimension ds of the critical Ising cluster, in d=2. The results clearly support the values d=187/96, ds=5/6 which are consistent with Ising clusters being described by tricritical q=1 Potts model exponents. From this, the hull dimension dH of critical Ising clusters is found to be dH=11/8, consistent with numerical work of other authors.

Density-matrix renormalization-group study of current and activity fluctuations near nonequilibrium phase transitions

Cumulants of a fluctuating current can be obtained from a free-energy-like generating function, which for Markov processes equals the largest eigenvalue of a generalized generator. We determine this eigenvalue with the density-matrix renormalization group for stochastic systems. We calculate the variance of the current in the different phases, and at the phase transitions, of the totally asymmetric exclusion process. Our results can be described in the terms of a scaling ansatz that involves the dynamical exponent z . We also calculate the generating function of the dynamical activity (total number of configuration changes) near the absorbing-state transition of the contact process. Its scaling properties can be expressed in terms of known critical exponents.

The size of knots in polymers

Circular DNA in viruses and bacteria is often knotted. While mathematically problematic, the determination of the knot size is crucial for the study of the physical and biological behaviour of long macromolecules. Here, we review work on the size distribution of these knots under equilibrium conditions. We discuss knot localization in good and poor solvents, or in polymers that are adsorbed on a surface. We also discuss recent evidence that knot size is a crucial quantity in relaxation processes of knotted polymers.

Real-space renormalization for reaction-diffusion systems

The stationary state of stochastic processes such as reaction-diffusion systems can be related to the ground state of a suitably defined quantum Hamiltonian. Using this analogy, we investigate the applicability of a real-space renormalization group approach, originally developed for quantum spin systems, to interacting particle systems. We apply the technique to an exactly solvable reaction-diffusion system and to the contact process (both in d = 1). In the former case, several exact results are recovered. For the contact process, surprisingly good estimates of critical parameters are obtained from a small-cell renormalization.