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Monte Carlo analysis of non-equilibrium steady states and relaxation kinetics in driven lattice gases

George L. Daquila

Published Aug 10, 2011• 3 citations
OpenAlex

Abstract

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 simulations of larger system sizes than previous studies. 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, correlations are weak in three dimensions and consequently the mean-field description is adequate. We also investigate the relaxation towards the non-equilibrium 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. We numerically investigate the behavior of driven lattice gases with nearest neighbor interactions at half-filling with periodic boundary conditions below and at the critical temperature using Monte Carlo simulations of very large lattices in two dimensions. This work is one of few that explores the relaxation to a non-equilibrium steady state. We obtain data collapse for the finite-size scaling form of density-density auto-correlation function at the critical point. We achieve data collapse using finite-size scaling of the time-dependent order parameter during the transient regime starting from strongly correlated initial conditions. We present simple aging scaling of the density-density auto-correlation function at the critical point starting from strongly correlated initial conditions using Monte Carlo simulations of two different lattice anisotropies. We thus unambiguously confirm the critical exponents determined by renormalization group methods using measurement of dynamic quantities in the transient regime. Measuring these dynamic quantities in the transient regime provides more conclusive measurements of the critical exponents than previous studies measuring static quantities in the stationary state. We provide qualitative arguments that the lattice anisotropy determines the steady-state for sub-critical quenches.

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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 simulations of larger system sizes than previous studies. 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, correlations are weak in three dimensions and consequently the mean-field description is adequate. We also investigate the relaxation towards the non-equilibrium 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. We numerically investigate the behavior of driven lattice gases with nearest neighbor interactions at half-filling with periodic boundary conditions below and at the critical temperature using Monte Carlo simulations of very large lattices in two dimensions. This work is one of few that explores the relaxation to a non-equilibrium steady state. We obtain data collapse for the finite-size scaling form of density-density auto-correlation function at the critical point. We achieve data collapse using finite-size scaling of the time-dependent order parameter during the transient regime starting from strongly correlated initial conditions. We present simple aging scaling of the density-density auto-correlation function at the critical point starting from strongly correlated initial conditions using Monte Carlo simulations of two different lattice anisotropies. We thus unambiguously confirm the critical exponents determined by renormalization group methods using measurement of dynamic quantities in the transient regime. Measuring these dynamic quantities in the transient regime provides more conclusive measurements of the critical exponents than previous studies measuring static quantities in the stationary state. We provide qualitative arguments that the lattice anisotropy determines the steady-state for sub-critical quenches.
We investigate the relaxation dynamics of two distinct non-equilibrium processes: relaxation of three-dimensional antiferromagnetic lattice spin models with Heisenberg interaction following a critical quench, and a one-dimensional exclusion process inspired by the gear-like motion of molecular motors. In a system of three-dimensional Heisenberg antiferromagnets the non-conserved staggered magnetization components couple non-trivially to the conserved magnetization densities inducing fully reversible terms that enter the Langevin dynamic equation. We simulate the exact microscopic dynamics of such a system of antiferromagnets by employing a hybrid simulation algorithm that combines the reversible spin precession implemented by the fourth-order Runge-Kutta integration method with the standard relaxational dynamics at finite temperatures using Monte Carlo updates. We characterize the dynamic universality class of this system by probing the early temporal window where the system exhibits aging scaling properties. We also verify an earlier renormalization group prediction that the temporal decay exponent in the two-time spin autocorrelation function exhibits non-universality, specifically it depends on the width of the initial spin orientation distribution. We employ a similar numerical technique to study the critical dynamics of an anisotropic Heisenberg antiferromagnet in the presence of an external field. The phase diagram of this system exhibits two critical lines that meet at a bicritical point. We study the aging scaling dynamics for the model C critical line, probe the model F critical line by investigating the system size dependence of the characteristic spin-wave frequencies near criticality, and measure the dynamic critical exponents for the order parameter including its aging scaling at the bicritical point. We introduce a one-dimensional non-equilibrium lattice gas model representing the processive motion of dynein molecular motors over the microtubule. We study both dynamical and stationary state properties for the model consisting of hardcore particles hopping on the lattice with variable step sizes. We find that the stationary state gap-distribution exhibits striking peaks around gap sizes that are multiples of the maximum step size, for both open and periodic boundary conditions, and verify this using a mean-field calculation. For open boundary conditions, we observe intriguing damped oscillator-like distribution of particles over the lattice with a periodicity equal to the maximum step size. To characterize transient dynamics, we measure the mean square displacement that shows weak superdiffusive growth with exponent γ≈ 1.34 for periodic boundary and ballistic growth ( γ≈ 2) for open boundary conditions at early times. We also study the effect of Langmuir dynamics on the density profile.
We employ Monte Carlo simulations to study the nonequilibrium relaxation of driven Ising lattice gases in two dimensions. Whereas the temporal scaling of the density autocorrelation function in the nonequilibrium steady state does not allow a precise measurement of the critical exponents, these can be accurately determined from the aging scaling of the two-time autocorrelations 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.
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
This thesis describes the development of analytical and computational techniques for systems far from equilibrium and their application to three model systems. Each of the model systems reaches a non-equilibrium steady state and exhibits one or more phase transitions. We first introduce a new position-space renormalization-group approach and illustrate its application using the one-dimensional fully asymmetric exclusion process. We have constructed a recursion relation for the relevant dynamic parameters for this model and have reproduced all of the important critical features of the model, including the exact positions of the critical point and the first and second order phase boundaries. The method yields an approximate value for the critical exponent v which is very close to the known value. The second major part of this thesis combines information theoretic techniques for calculating the entropy and a Monte Carlo renormalization-group approach. We have used this method to study and compare infinitely driven lattice gases. This approach enables us to calculate the critical exponents associated with the correlation length v and the order parameter /3. These results are compared to the values predicted from different field theoretic treatments of the models. In the final set of calculations, we build position-space renormalization-group recursion relations from the master equations of small clusters. By obtaining the probability distributions for these clusters numerically, we develop a mapping connecting the parameters specifying the dynamics on different length scales. The resulting flow topology in some ways mimics equilibrium features, with sinks for each phase and fixed points associated with each phase boundary. In addition, though, there are added complexities in the flows, suggesting multiple regions within the ordered phase for some values of parameters, and the presence of an extra "source" fixed point within the ordered phase. Thus, this study illustrates the successful applicability of position-space renormalization- group and information theoretic approaches to driven lattice gases in one and two dimensions. These methods provide new insights into the critical properties and ordering in these systems, and set the stage for further development of these approaches and their application to additional, more realistic models.
We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behaviour of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1/rd + σ, with σ = 0.75. The critical temperature, as well as the critical exponents, are evaluated from the power-law behaviour of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different truncated interaction ranges. All the estimated static critical exponents (γ/ν, β/ν, and 1/ν) are in good agreement with renormalization group (RG) results and previously reported numerical data obtained under equilibrium conditions. On the other hand, the dynamic exponent of the initial increase of the magnetization (θ) was close to RG predictions. However, the dynamic exponent z of the time correlation length is slightly different to the RG results probably due to the fact that it may depend on the specific dynamics used or because the two-loop expansion used in the RG analysis may be insufficient.