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 o
pen 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.
Field theory tools are applied to analytically study fluctuation and correlation effects in spatially extended stochastic predator-prey systems. In the mean-field rate equation approximation, the classic Lotka-Volterra model is characterized by neutr
al cycles in phase space, describing undamped oscillations for both predator and prey populations. In contrast, Monte Carlo simulations for stochastic two-species predator-prey reaction systems on regular lattices display complex spatio-temporal structures associated with persistent erratic population oscillations. The Doi-Peliti path integral representation of the master equation for stochastic particle interaction models is utilized to arrive at a field theory action for spatial Lotka-Volterra models in the continuum limit. In the species coexistence phase, a perturbation expansion with respect to the nonlinear predation rate is employed to demonstrate that spatial degrees of freedom and stochastic noise induce instabilities toward structure formation, and to compute the fluctuation corrections for the oscillation frequency and diffusion coefficient. The drastic downward renormalization of the frequency and the enhanced diffusivity are in excellent qualitative agreement with Monte Carlo simulation data.
We investigate the competing effects and relative importance of intrinsic demographic and environmental variability on the evolutionary dynamics of a stochastic two-species Lotka-Volterra model by means of Monte Carlo simulations on a two-dimensional
lattice. Individuals are assigned inheritable predation efficiencies; quenched randomness in the spatially varying reaction rates serves as environmental noise. We find that environmental variability enhances the population densities of both predators and prey while demographic variability leads to essentially neutral optimization.
We employ Monte Carlo simulations to study the non-equilibrium relaxation of driven Ising lattice gases in two dimensions. Whereas the temporal scaling of the density auto-correlation function in the non-equilibrium steady state does not allow a prec
ise measurement of the critical exponents, these can be accurately determined from the aging scaling of the two-time auto-correlations and the order parameter evolution following a quench to the critical point. We obtain excellent agreement with renormalization group predictions based on the standard Langevin representation of driven Ising lattice gases.
These notes provide a concise introduction to important applications of the renormalization group (RG) in statistical physics. After reviewing the scaling approach and Ginzburg-Landau theory for critical phenomena, Wilsons momentum shell RG method is
presented, and the critical exponents for the scalar Phi^4 model are determined to first order in an eps expansion about d_c = 4. Subsequently, the technically more versatile field-theoretic formulation of the perturbational RG for static critical phenomena is described. It is explained how the emergence of scale invariance connects UV divergences to IR singularities, and the RG equation is employed to compute the critical exponents for the O(n)-symmetric Landau-Ginzburg-Wilson theory. The second part is devoted to field theory representations of non-linear stochastic dynamical systems, and the application of RG tools to critical dynamics. Dynamic critical phenomena in systems near equilibrium are efficiently captured through Langevin equations, and their mapping onto the Janssen-De Dominicis response functional, exemplified by the purely relaxational models with non-conserved (model A) / conserved order parameter (model B). The Langevin description and scaling exponents for isotropic ferromagnets (model J) and for driven diffusive non-equilibrium systems are also discussed. Finally, an outlook is presented to scale-invariant phenomena and non-equilibrium phase transitions in interacting particle systems. It is shown how the stochastic master equation associated with chemical reactions or population dynamics models can be mapped onto imaginary-time, non-Hermitian `quantum mechanics. In the continuum limit, this Doi-Peliti Hamiltonian is represented through a coherent-state path integral, which allows an RG analysis of diffusion-limited annihilation processes and phase transitions from active to inactive, absorbing states.
It is well-established that including spatial structure and stochastic noise in models for predator-prey interactions invalidates the classical deterministic Lotka-Volterra picture of neutral population cycles. In contrast, stochastic models yield lo
ng-lived, but ultimately decaying erratic population oscillations, which can be understood through a resonant amplification mechanism for density fluctuations. In Monte Carlo simulations of spatial stochastic predator-prey systems, one observes striking complex spatio-temporal structures. These spreading activity fronts induce persistent correlations between predators and prey. In the presence of local particle density restrictions (finite prey carrying capacity), there exists an extinction threshold for the predator population. The accompanying continuous non-equilibrium phase transition is governed by the directed-percolation universality class. We employ field-theoretic methods based on the Doi-Peliti representation of the master equation for stochastic particle interaction models to (i) map the ensuing action in the vicinity of the absorbing state phase transition to Reggeon field theory, and (ii) to quantitatively address fluctuation-induced renormalizations of the population oscillation frequency, damping, and diffusion coefficients in the species coexistence phase.
We numerically investigate the long-time behavior of the density-density auto-correlation function in driven lattice gases with particle exclusion and periodic boundary conditions in one, two, and three dimensions using precise Monte Carlo simulation
s. In the one-dimensional asymmetric exclusion process on a ring with half the lattice sites occupied, we find that correlations induce extremely slow relaxation to the asymptotic power law decay. We compare the crossover functions obtained from our simulations with various analytic results in the literature, and analyze the characteristic oscillations that occur in finite systems away from half-filling. As expected, in three dimensions correlations are weak and consequently the mean-field description is adequate. We also investigate the relaxation towards the nonequilibrium steady state in the two-time density-density auto-correlations, starting from strongly correlated initial conditions. We obtain simple aging scaling behavior in one, two, and three dimensions, with the expected power laws.
