Field-theoretic methods

Many complex systems are characterized by intriguing spatio-temporal structures. Their mathematical description relies on the analysis of appropriate correlation functions. Functional integral techniques provide a unifying formalism that facilitates the computation of such correlation functions and moments, and furthermore allows a systematic development of perturbation expansions and other useful approximative schemes. It is explained how nonlinear stochastic processes may be mapped onto exponential probability distributions, whose weights are determined by continuum field theory actions. Such mappings are madeexplicit for (1) stochastic interacting particle systems whose kinetics is defined through a microscopic master equation; and (2) nonlinear Langevin stochastic differential equations which provide a mesoscopic description wherein a separation of time scales between the relevant degrees of freedom and background statistical noise is assumed. Several well-studied examples are introduced to illustrate the general methodology.Comment: Article for the Encyclopedia of Complexity and System Science, B. Meyers (Ed.), Springer-Verlag Berlin, 200

Scale invariance and dynamic phase transitions in diffusion-limited reactions

Many systems that can be described in terms of diffusion-limited `chemical' reactions display non-equilibrium continuous transitions separating active from inactive, absorbing states, where stochastic fluctuations cease entirely. Their critical properties can be analyzed via a path-integral representation of the corresponding classical master equation, and the dynamical renormalization group. An overview over the ensuing universality classes in single-species processes is given, and generalizations to reactions with multiple particle species are discussed as well. The generic case is represented by the processes A A + A, and A -> 0, which map onto Reggeon field theory with the critical exponents of directed percolation (DP). For branching and annihilating random walks (BARW) A -> (m+1) A and A + A -> 0, the mean-field rate equation predicts an active state only. Yet BARW with odd m display a DP transition for d <= 2. For even offspring number m, the particle number parity is conserved locally. Below d_c' = 4/3, this leads to the emergence of an inactive phase that is characterized by the power laws of the pair annihilation process. The critical exponents at the transition are those of the `parity-conserving' (PC) universality class. For local processes without memory, competing pair or triplet annihilation and fission reactions k A -> (k - l) A, k A -> (k+m)A with k=2,3 appear to yield the only other universality classes not described by mean-field theory. In these reactions, site occupation number restrictions play a crucial role.Comment: 16 pages, talk given at 2003 German Physical Society Spring Meeting; four figures and style files include

Introduction to Library Trends 6 (2) 1957: Research in Librarianship

published or submitted for publicatio

Conservation Comes of Age

published or submitted for publicatio

Nonequilibrium transport through quantum-wire junctions and boundary defects for free massless bosonic fields

We consider a model of quantum-wire junctions where the latter are described by conformal-invariant boundary conditions of the simplest type in the multicomponent compactified massless scalar free field theory representing the bosonized Luttinger liquids in the bulk of wires. The boundary conditions result in the scattering of charges across the junction with nontrivial reflection and transmission amplitudes. The equilibrium state of such a system, corresponding to inverse temperature $\beta$ and electric potential $V$, is explicitly constructed both for finite and for semi-infinite wires. In the latter case, a stationary nonequilibrium state describing the wires kept at different temperatures and potentials may be also constructed. The main result of the present paper is the calculation of the full counting statistics (FCS) of the charge and energy transfers through the junction in a nonequilibrium situation. Explicit expressions are worked out for the generating function of FCS and its large-deviations asymptotics. For the purely transmitting case they coincide with those obtained in the litterature, but numerous cases of junctions with transmission and reflection are also covered. The large deviations rate function of FCS for charge and energy transfers is shown to satisfy the fluctuation relations and the expressions for FCS obtained here are compared with the Levitov-Lesovic formulae.Comment: 50 pages, 24 figure

Topological edge states in two-gap unitary systems: A transfer matrix approach

We construct and investigate a family of two-band unitary systems living on a cylinder geometry and presenting localized edge states. Using the transfer matrix formalism, we solve and investigate in details such states in the thermodynamic limit. Analitycity considerations then suggest the construction of a family of Riemman surfaces associated to the band structure of the system. In this picture, the corresponding edge states naturally wind around non contractile loops, defining by the way a topological invariant associated to each gap of the system.Comment: Accepted version for publication in New Journal of Physic