In this work the influence of bulk reservoirs on lattice models for many-particle systems far from equilibrium is studied. A hydrodynamic equation is presented that allows to compute the steady state density profile for single species driven diffusive systems with open boundaries and a coupling of the bulk reservoir scaling with the inverse system size. Furthermore, a related model for a limit order market as well as the case of periodic single species driven models with a weak coupling to the reservoir are treated. New results on driven systems with two particle species are presented both for the cases with and without coupling to a bulk reservoir. For a two species model with periodic boundaries and conserving dynamics a novel type of phase transition is observed that is formally similar to Bose-Einstein condensation. In absence of a bulk reservoir, the existence of a spontaneously symmetry broken phase in a two species model with open boundaries and deterministic bulk behavior is demonstrated by regarding the dynamics of the system. For a related model with non-conserving bulk dynamics a rich phase diagram is found. In systems with an absorbing phase transition the action of the reservoir is that of an external field. For the universality class of directed percolation the scaling functions are determined in several dimensions.
In this work the influence of bulk reservoirs on lattice models for many-particle systems far from equilibrium is studied. A hydrodynamic equation is presented that allows to compute the steady state density profile for single species driven diffusive systems with open boundaries and a coupling of the bulk reservoir scaling with the inverse system size. Furthermore, a related model for a limit order market as well as the case of periodic single species driven models with a weak coupling to the reservoir are treated. New results on driven systems with two particle species are presented both for the cases with and without coupling to a bulk reservoir. For a two species model with periodic boundaries and conserving dynamics a novel type of phase transition is observed that is formally similar to Bose-Einstein condensation. In absence of a bulk reservoir, the existence of a spontaneously symmetry broken phase in a two species model with open boundaries and deterministic bulk behavior is demonstrated by regarding the dynamics of the system. For a related model with non-conserving bulk dynamics a rich phase diagram is found. In systems with an absorbing phase transition the action of the reservoir is that of an external field. For the universality class of directed percolation the scaling functions are determined in several dimensions.
Using field theoretic renormalization group methods we study the critical behavior of a driven diffusive system near a boundary perpendicular to the driving force. The boundary acts as a particle reservoir, which is necessary to maintain the critical particle density in the bulk. The scaling behavior of correlation and response functions is governed by a new exponent ${\ensuremath{\eta}}_{1},$ which is related to the anomalous scaling dimension of the chemical potential of the boundary. The new exponent and a universal amplitude ratio for the density profile are calculated at first order in $\ensuremath{\epsilon}=5\ensuremath{-}d.$ Some of our results are checked by computer simulations.
In this work non-equilibrium phenomena in statistical models with more than nearest neighbor interactions are examined. The investigated models belong to three different types of non-equilibrium, i.e. systems undergoing aging, absorbing phase transitions and systems being intrinsically out of equilibrium. The behavior of the mean spherical model after a temperature quench from the disordered phase into the ordered phase in an external magnetic field is calculated. For a sufficiently small external field, the kinetics of the magnetization reversal transition from the metastable to the ground state is comparable to the aging behavior in the case without external field. For an oscillating external magnetic field the absence of a dynamic phase transition is demonstrated. The bosonic pair contact process with diffusion shows unusual behavior of the second moment of the probability distribution for which a phase transition from a constant asymptotic value to exponential divergence is found. The control parameter of this transition is the ratio of reaction and diffusion rates which gives insight into the role of diffusion in this process. The divergence of the second moment can lead to a negative correlation time while the correlation length is positive. Coupling conservative diffusion system to reaction dynamics leads to interesting behavior if the rates are tuned adequately. The general conditions for ergodicity breaking in those systems are derived and an example is constructed. For this example hysteresis is demonstrated. The behavior, including the exact steady state phase diagram, of this non-equilibrium many particle system is well described by an effective equilibrium one-particle problem. The motion of domain walls in reaction diffusion systems is investigated for three site interactions. A variety of models is found and they show phenomena like double shocks or additional symmetries not possible for two-site interaction models. While the two-site interaction models leading to exact shock measures are connected by the transformability to free fermion systems, this is not valid for the three site interaction models.
In this thesis we examine universal scaling properties of strongly-correlated systems near and far from equilibrium. We discuss quantum phase transitions at vanishing temperature, multicritical and dynamic critical behavior near thermal equilibrium, and scaling properties of nonequilibrium steady states. We employ nonperturbative methods including the functional renormalization group as well as Monte Carlo simulations. A general outline of the functional renormalization group is given in the introductory chapters. \n \nIn the first part of this thesis, we investigate spinless fermions on the honeycomb lattice interacting via short-range repulsive interactions. Such a system can be seen as a simple model for suspended graphene. The short-range interactions control the ground state properties of the system that may lead to a chiral phase transition from the semimetal to the charge density wave (CDW)/Kekulé ordered state. We determine the universal scaling properties at the chiral transition, and establish the presence of large anomalous dimensions indicating the importance of strong fluctuations. \n \nThe competition of two nonvanishing order parameters and their corresponding multicritical behavior are investigated in the subsequent chapter. We characterize the bicritical and tetracritical behavior in the purely bosonic O(N_1) + O(N_2) symmetric model and comment on possible applications to condensed-matter and high-energy physics. \n \nIn the following chapter we discuss the long-time relaxational behavior at criticality of an order parameter with O(N) symmetry coupled to an additional conserved density. We find an anomalous diffusion phase with new dynamic scaling properties. Using the functional renormalization group we determine the complete dynamic critical behavior of the model in 2 < d < 4 dimensions and compare our results to experiments. \n \nFinally, we investigate the scaling properties of stationary states far from equilibrium. At the example of the one-dimensional Burgers’ equation we develop a novel approach to hydrodynamic turbulence using lattice Monte Carlo methods. We apply these techniques to determine the statistical properties of small-scale fluctuations in this model and identify the anomalous scaling behavior. \n
Many non-equilibrium systems display dynamic phase transitions from active to absorbing states, where fluctuations cease entirely. Based on a field theory representation of the master equation, the critical behavior can be analyzed by means of the renormalization group. The resulting universality classes for single-species systems are reviewed here. Generically, the critical exponents are those of directed percolation (Reggeon field theory), with critical dimension d_c = 4. Yet local particle number parity conservation in even-offspring branching and annihilating random walks implies an inactive phase (emerging below d_c' = 4/3) that is characterized by the power laws of the pair annihilation reaction, and leads to different critical exponents at the transition. For local processes without memory, the pair contact process with diffusion represents the only other non-trivial universality class. The consistent treatment of restricted site occupations and quenched random reaction rates are important open issues.
We review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of these models, we devote the main part of this review to analyzing the impact of walls on their critical behavior. We discuss the possible boundary universality classes for the DP and BARW models, which can be described by a general scaling theory which allows for two independent surface exponents in addition to the bulk critical exponents. Above the upper critical dimension d c , we review the use of mean field theories, whereas in the regime d<d c , where fluctuations are important, we examine the application of field theoretic methods. Of particular interest is the situation in d=1, which has been extensively investigated using numerical simulations and series expansions. Although DP and BARW fit into the same scaling theory, they can still show very different surface behavior: for DP some exponents are degenerate, a property not shared with the BARW model. Moreover, a "hidden" duality symmetry of BARW in d=1 is broken by the boundary and this relates exponents and boundary conditions in an intricate way.
Recently, considerable progress has been made in understanding finite-size scaling in equilibrium systems. Here, we study finite-size scaling in nonequilibrium systems at the instance of directed percolation (DP), which has become the paradigm of nonequilibrium phase transitions into absorbing states, above, at, and below the upper critical dimension. We investigate the finite-size scaling behavior of DP analytically and numerically by considering its steady state generated by a homogeneous constant external source on a d-dimensional hypercube of finite edge length L with periodic boundary conditions near the bulk critical point. In particular, we study the order parameter and its higher moments using renormalized field theory. We derive finite-size scaling forms of the moments in a one-loop calculation. Moreover, we introduce and calculate a ratio of the order parameter moments that plays a similar role in the analysis of finite size scaling in absorbing nonequilibrium processes as the famous Binder cumulant in equilibrium systems and that, in particular, provides a signature of the DP universality class. To complement our analytical work, we perform Monte Carlo simulations which confirm our analytical results.