Reaction-Diffusion Processes, Critical Dynamics and Quantum Chains

The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schr\"odinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and $q$-deformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail.Comment: Latex 52 pages (2 figures appended at the end), UGVA-DPT 1992/12-79

On the universality of a class of annihilation-coagulation models

A class of $d$-dimensional reaction-diffusion models interpolating continuously between the diffusion-coagulation and the diffusion-annihilation models is introduced. Exact relations among the observables of different models are established. For the one-dimensional case, it is shown how correlations in the initial state can lead to non-universal amplitudes for time-dependent particles density.Comment: 18 pages with no figures. Latex file using REVTE

Temperature in nonequilibrium systems with conserved energy

We study a class of nonequilibrium lattice models which describe local redistributions of a globally conserved energy. A particular subclass can be solved analytically, allowing to define a temperature T_{th} along the same lines as in the equilibrium microcanonical ensemble. The fluctuation-dissipation relation is explicitely found to be linear, but its slope differs from the inverse temperature T_{th}^{-1}. A numerical renormalization group procedure suggests that, at a coarse-grained level, all models behave similarly, leading to a two-parameter description of their macroscopic properties.Comment: 4 pages, 1 figure, final versio

Can the post-Newtonian gravitational waveform of an inspiraling binary be improved by solving the energy balance equation numerically?

The detection of gravitational waves from inspiraling compact binaries using matched filtering depends crucially on the availability of accurate template waveforms. We determine whether the accuracy of the templates' phasing can be improved by solving the post-Newtonian energy balance equation numerically, rather than (as is normally done) analytically within the post-Newtonian perturbative expansion. By specializing to the limit of a small mass ratio, we find evidence that there is no gain in accuracy.Comment: 13 pages, RevTeX, 5 figures included via eps

Dynamical real-space renormalization group calculations with a new clustering scheme on random networks

We have defined a new type of clustering scheme preserving the connectivity of the nodes in network ignored by the conventional Migdal-Kadanoff bond moving process. Our new clustering scheme performs much better for correlation length and dynamical critical exponents in high dimensions, where the conventional Migdal-Kadanoff bond moving scheme breaks down. In two and three dimensions we find the dynamical critical exponents for the kinetic Ising Model to be z=2.13 and z=2.09, respectively at pure Ising fixed point. These values are in very good agreement with recent Monte Carlo results. We investigate the phase diagram and the critical behaviour for randomly bond diluted lattices in d=2 and 3, in the light of this new transformation. We also provide exact correlation exponent and dynamical critical exponent values on hierarchical lattices with power-law degree distributions, both in the pure and random cases.Comment: 8 figure

Dynamical phase transition in one-dimensional kinetic Ising model with nonuniform coupling constants

An extension of the Kinetic Ising model with nonuniform coupling constants on a one-dimensional lattice with boundaries is investigated, and the relaxation of such a system towards its equilibrium is studied. Using a transfer matrix method, it is shown that there are cases where the system exhibits a dynamical phase transition. There may be two phases, the fast phase and the slow phase. For some region of the parameter space, the relaxation time is independent of the reaction rates at the boundaries. Changing continuously the reaction rates at the boundaries, however, there is a point where the relaxation times begins changing, as a continuous (nonconstant) function of the reaction rates at the boundaries, so that at this point there is a jump in the derivative of the relaxation time with respect to the reaction rates at the boundaries.Comment: 17 page

A field theoretic approach to master equations and a variational method beyond the Poisson ansatz

We develop a variational scheme in a field theoretic approach to a stochastic process. While various stochastic processes can be expressed using master equations, in general it is difficult to solve the master equations exactly, and it is also hard to solve the master equations numerically because of the curse of dimensionality. The field theoretic approach has been used in order to study such complicated master equations, and the variational scheme achieves tremendous reduction in the dimensionality of master equations. For the variational method, only the Poisson ansatz has been used, in which one restricts the variational function to a Poisson distribution. Hence, one has dealt with only restricted fluctuation effects. We develop the variational method further, which enables us to treat an arbitrary variational function. It is shown that the variational scheme developed gives a quantitatively good approximation for master equations which describe a stochastic gene regulatory network.Comment: 13 pages, 2 figure

Hydrodynamic equations for self-propelled particles: microscopic derivation and stability analysis

Considering a gas of self-propelled particles with binary interactions, we derive the hydrodynamic equations governing the density and velocity fields from the microscopic dynamics, in the framework of the associated Boltzmann equation. Explicit expressions for the transport coefficients are given, as a function of the microscopic parameters of the model. We show that the homogeneous state with zero hydrodynamic velocity is unstable above a critical density (which depends on the microscopic parameters), signaling the onset of a collective motion. Comparison with numerical simulations on a standard model of self-propelled particles shows that the phase diagram we obtain is robust, in the sense that it depends only slightly on the precise definition of the model. While the homogeneous flow is found to be stable far from the transition line, it becomes unstable with respect to finite-wavelength perturbations close to the transition, implying a non trivial spatio-temporal structure for the resulting flow. We find solitary wave solutions of the hydrodynamic equations, quite similar to the stripes reported in direct numerical simulations of self-propelled particles.Comment: 33 pages, 11 figures, submitted to J. Phys.