32 research outputs found
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
Perturbative Field-Theoretical Renormalization Group Approach to Driven-Dissipative Bose-Einstein Criticality
The universal critical behavior of the driven-dissipative non-equilibrium
Bose-Einstein condensation transition is investigated employing the
field-theoretical renormalization group method. Such criticality may be
realized in broad ranges of driven open systems on the interface of quantum
optics and many-body physics, from exciton-polariton condensates to cold atomic
gases. The starting point is a noisy and dissipative Gross-Pitaevski equation
corresponding to a complex valued Landau-Ginzburg functional, which captures
the near critical non-equilibrium dynamics, and generalizes Model A for
classical relaxational dynamics with non-conserved order parameter. We confirm
and further develop the physical picture previously established by means of a
functional renormalization group study of this system. Complementing this
earlier numerical analysis, we analytically compute the static and dynamical
critical exponents at the condensation transition to lowest non-trivial order
in the dimensional epsilon expansion about the upper critical dimension d_c =
4, and establish the emergence of a novel universal scaling exponent associated
with the non-equilibrium drive. We also discuss the corresponding situation for
a conserved order parameter field, i.e., (sub-)diffusive Model B with complex
coefficients.Comment: 17 pages, 6 figures, to appear in Phys. Rev. X (2014
Non-equilibrium behavior at a liquid-gas critical point
Second-order phase transitions in a non-equilibrium liquid-gas model with
reversible mode couplings, i.e., model H for binary-fluid critical dynamics,
are studied using dynamic field theory and the renormalization group. The
system is driven out of equilibrium either by considering different values for
the noise strengths in the Langevin equations describing the evolution of the
dynamic variables (effectively placing these at different temperatures), or
more generally by allowing for anisotropic noise strengths, i.e., by
constraining the dynamics to be at different temperatures in d_par- and
d_perp-dimensional subspaces, respectively. In the first, case, we find one
infrared-stable and one unstable renormalization group fixed point. At the
stable fixed point, detailed balance is dynamically restored, with the two
noise strengths becoming asymptotically equal. The ensuing critical behavior is
that of the standard equilibrium model H. At the novel unstable fixed point,
the temperature ratio for the dynamic variables is renormalized to infinity,
resulting in an effective decoupling between the two modes. We compute the
critical exponents at this new fixed point to one-loop order. For model H with
spatially anisotropic noise, we observe a critical softening only in the
d_perp-dimensional sector in wave vector space with lower noise temperature.
The ensuing effective two-temperature model H does not have any stable fixed
point in any physical dimension, at least to one-loop order. We obtain formal
expressions for the novel critical exponents in a double expansion about the
upper critical dimension d_c = 4 - d_par and with respect to d_par, i.e., about
the equilibrium theory.Comment: 17 pages, revtex, one figure and EPJB style files include