Renormalization of stochastic lattice models: basic formulation

Fixed-dimension renormalization group analysis of conserved surface roughening

Conserved surface roughening represents a special case of interface dynamics where the total height of the interface is conserved. Recently, it was suggested [Caballero et al., Phys. Rev. Lett. 121, 020601 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.020601] that the original continuum model known as the Conserved Kardar-Parisi-Zhang (CKPZ) equation is incomplete, as additional nonlinearity is not forbidden by any symmetry in d>1. In this work, we perform detailed field-theoretic renormalization group (RG) analysis of a general stochastic model describing conserved surface roughening. Systematic power counting reveals additional marginal interaction at the upper critical dimension, which appears also in the context of molecular beam epitaxy. Depending on the origin of the surface particle's mobility, the resulting model shows two different scaling regimes. If the particles move mainly due to the gravity, the leading dispersion law is ω∼k^{2}, and the mean-field approximation describing a flat interface is exact in any spatial dimension. On the other hand, if the particles move mainly due to the surface curvature, the interface becomes rough with the mean-field dispersion law ω∼k^{4}, and the corrections to scaling exponents must be taken into account. We show that the latter model consist of two subclasses of models that are decoupled in all orders of perturbation theory. Moreover, our RG analysis of the general model reveals that the universal scaling is described by a rougher interface than the CKPZ universality class. The universal exponents are derived within the one-loop approximation in both fixed d and ɛ-expansion schemes, and their relation is discussed. We point out all important details behind these two schemes, which are often overlooked in the literature, and their misinterpretation might lead to inconsistent results.

Stochastic self-assembly of reaction-diffusion patterns in synaptic membranes

Synaptic receptor and scaffold molecules self-assemble into membrane protein domains, which play an important role in signal transmission across chemical synapses. Experiment and theory have shown that the formation of receptor-scaffold domains of the characteristic size observed in nerve cells can be understood from the receptor and scaffold reaction and diffusion processes suggested by experiments. We employ here kinetic Monte Carlo (KMC) simulations to explore the self-assembly of synaptic receptor-scaffold domains in a stochastic lattice model of receptor and scaffold reaction-diffusion dynamics. For reaction and diffusion rates within the ranges of values suggested by experiments we find, in agreement with previous mean-field calculations, self-assembly of receptor-scaffold domains of a size similar to that observed in experiments. Comparisons between the results of our KMC simulations and mean-field solutions suggest that the intrinsic noise associated with receptor and scaffold reaction and diffusion processes accelerates the self-assembly of receptor-scaffold domains, and confers increased robustness to domain formation. In agreement with experimental observations, our KMC simulations yield a prevalence of scaffolds over receptors in receptor-scaffold domains. Our KMC simulations show that receptor and scaffold reaction-diffusion dynamics can inherently give rise to plasticity in the overall properties of receptor-scaffold domains, which may be utilized by nerve cells to regulate the receptor number at chemical synapses.

Local roughness exponent in the nonlinear molecular-beam-epitaxy universality class in one dimension

We report local roughness exponents, α_{loc}, for three interface growth models in one dimension which are believed to belong to the nonlinear molecular-beam-epitaxy (nMBE) universality class represented by the Villain-Lais-Das Sarma (VLDS) stochastic equation. We applied an optimum detrended fluctuation analysis (ODFA) [Luis et al., Phys. Rev. E 95, 042801 (2017)2470-004510.1103/PhysRevE.95.042801] and compared the outcomes with standard detrending methods. We observe in all investigated models that ODFA outperforms the standard methods providing exponents in the narrow interval α_{loc}^{}∈[0.96,0.98] quantitatively consistent with two-loop renormalization group predictions for the VLDS equation. In particular, these exponent values are calculated for the Clarke-Vvdensky and Das Sarma-Tamborenea models characterized by very strong corrections to the scaling, for which large deviations of these values had been reported. Our results strongly support the absence of anomalous scaling in the nMBE universality class and the existence of corrections in the form α_{loc}^{}=1-ε of the one-loop renormalization group analysis of the VLDS equation.

Distribution of randomly diffusing particles in inhomogeneous media

Diffusion can be conceptualized, at microscopic scales, as the random hopping of particles between neighboring lattice sites. In the case of diffusion in inhomogeneous media, distinct spatial domains in the system may yield distinct particle hopping rates. Starting from the master equations (MEs) governing diffusion in inhomogeneous media we derive here, for arbitrary spatial dimensions, the deterministic lattice equations (DLEs) specifying the average particle number at each lattice site for randomly diffusing particles in inhomogeneous media. We consider the case of free (Fickian) diffusion with no steric constraints on the maximum particle number per lattice site as well as the case of diffusion under steric constraints imposing a maximum particle concentration. We find, for both transient and asymptotic regimes, excellent agreement between the DLEs and kinetic Monte Carlo simulations of the MEs. The DLEs provide a computationally efficient method for predicting the (average) distribution of randomly diffusing particles in inhomogeneous media, with the number of DLEs associated with a given system being independent of the number of particles in the system. From the DLEs we obtain general analytic expressions for the steady-state particle distributions for free diffusion and, in special cases, diffusion under steric constraints in inhomogeneous media. We find that, in the steady state of the system, the average fraction of particles in a given domain is independent of most system properties, such as the arrangement and shape of domains, and only depends on the number of lattice sites in each domain, the particle hopping rates, the number of distinct particle species in the system, and the total number of particles of each particle species in the system. Our results provide general insights into the role of spatially inhomogeneous particle hopping rates in setting the particle distributions in inhomogeneous media.

Stochastic lattice model of synaptic membrane protein domains

Neurotransmitter receptor molecules, concentrated in synaptic membrane domains along with scaffolds and other kinds of proteins, are crucial for signal transmission across chemical synapses. In common with other membrane protein domains, synaptic domains are characterized by low protein copy numbers and protein crowding, with rapid stochastic turnover of individual molecules. We study here in detail a stochastic lattice model of the receptor-scaffold reaction-diffusion dynamics at synaptic domains that was found previously to capture, at the mean-field level, the self-assembly, stability, and characteristic size of synaptic domains observed in experiments. We show that our stochastic lattice model yields quantitative agreement with mean-field models of nonlinear diffusion in crowded membranes. Through a combination of analytic and numerical solutions of the master equation governing the reaction dynamics at synaptic domains, together with kinetic Monte Carlo simulations, we find substantial discrepancies between mean-field and stochastic models for the reaction dynamics at synaptic domains. Based on the reaction and diffusion properties of synaptic receptors and scaffolds suggested by previous experiments and mean-field calculations, we show that the stochastic reaction-diffusion dynamics of synaptic receptors and scaffolds provide a simple physical mechanism for collective fluctuations in synaptic domains, the molecular turnover observed at synaptic domains, key features of the observed single-molecule trajectories, and spatial heterogeneity in the effective rates at which receptors and scaffolds are recycled at the cell membrane. Our work sheds light on the physical mechanisms and principles linking the collective properties of membrane protein domains to the stochastic dynamics that rule their molecular components.

Self-assembly and plasticity of synaptic domains through a reaction-diffusion mechanism

Signal transmission across chemical synapses relies crucially on neurotransmitter receptor molecules, concentrated in postsynaptic membrane domains along with scaffold and other postsynaptic molecules. The strength of the transmitted signal depends on the number of receptor molecules in postsynaptic domains, and activity-induced variation in the receptor number is one of the mechanisms of postsynaptic plasticity. Recent experiments have demonstrated that the reaction and diffusion properties of receptors and scaffolds at the membrane, alone, yield spontaneous formation of receptor-scaffold domains of the stable characteristic size observed in neurons. On the basis of these experiments we develop a model describing synaptic receptor domains in terms of the underlying reaction-diffusion processes. Our model predicts that the spontaneous formation of receptor-scaffold domains of the stable characteristic size observed in experiments depends on a few key reactions between receptors and scaffolds. Furthermore, our model suggests novel mechanisms for the alignment of pre- and postsynaptic domains and for short-term postsynaptic plasticity in receptor number. We predict that synaptic receptor domains localize in membrane regions with an increased receptor diffusion coefficient or a decreased scaffold diffusion coefficient. Similarly, we find that activity-dependent increases or decreases in receptor or scaffold diffusion yield a transient increase in the number of receptor molecules concentrated in postsynaptic domains. Thus, the proposed reaction-diffusion model puts forth a coherent set of biophysical mechanisms for the formation, stability, and plasticity of molecular domains on the postsynaptic membrane.

Langevin equations for competitive growth models

Langevin equations for several competitive growth models in one dimension are derived. For models with crossover from random deposition (RD) to some correlated deposition (CD) dynamics, with small probability p of CD, the surface tension ν and the nonlinear coefficient λ of the associated equations have linear dependence on p due solely to this random choice. However, they also depend on the regularized step functions present in the analytical representations of the CD, whose expansion coefficients scale with p according to the divergence of local height differences when p→0. The superposition of those scaling factors gives ν~p(2) for random deposition with surface relaxation (RDSR) as the CD, and ν~p, λ~p(3/2) for ballistic deposition (BD) as the CD, in agreement with simulation and other scaling approaches. For bidisperse ballistic deposition (BBD), the same scaling of RD-BD model is found. The Langevin equation for the model with competing RDSR and BD, with probability p for the latter, is also constructed. It shows linear p dependence of λ, while the quadratic dependence observed in previous simulations is explained by an additional crossover before the asymptotic regime. The results highlight the relevance of scaling of the coefficients of step function expansions in systems with steep surfaces, which is responsible for noninteger exponents in some p-dependent stochastic equations, and the importance of the physical correspondence of aggregation rules and equation coefficients.

Statistics of interfacial fluctuations of radially growing clusters

The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scale-dependent fractality, and suggests the advent of a self-similar fractal dimension. The center-of-mass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the long-range surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilution-free universality class.

Transient regimes and crossover for epitaxial surfaces

We apply a formalism for deriving stochastic continuum equations associated with lattice models to obtain equations governing the transient regimes of epitaxial growth for various experimental scenarios and growth conditions. The first step of our methodology is the systematic transformation of the lattice model into a regularized stochastic equation of motion that provides initial conditions for differential renormalization-group (RG) equations for the coefficients in the regularized equation. The solutions of the RG equations then yield trajectories that describe the original model from the transient regimes, which are of primary experimental interest, to the eventual crossover to the asymptotically stable fixed point. We first consider regimes defined by the relative magnitude of deposition noise and diffusion noise. If the diffusion noise dominates, then the early stages of growth are described by the Mullins-Herring (MH) equation with conservative noise. This is the classic regime of molecular-beam epitaxy. If the diffusion and deposition noise are of comparable magnitude, the transient equation is the MH equation with nonconservative noise. This behavior has been observed in a recent report on the growth of aluminum on silicone oil surfaces [Z.-N. Fang, Thin Solid Films 517, 3408 (2009)]. Finally, the regime where deposition noise dominates over diffusion noise has been observed in computer simulations, but does not appear to have any direct experimental relevance. For initial conditions that consist of a flat surface, the Villain-Lai-Das Sarma (VLDS) equation with nonconservative noise is not appropriate for any transient regime. If, however, the initial surface is corrugated, the relative magnitudes of terms can be altered to the point where the VLDS equation with conservative noise does indeed describe transient growth. This is consistent with the experimental analysis of growth on patterned surfaces [H.-C. Kan, Phys. Rev. Lett. 92, 146101 (2004); T. Tadayyon-Eslami, Phys. Rev. Lett. 97, 126101 (2006)].

Investigation of film surface roughness and porosity dependence on lattice size in a porous thin film deposition process

The dependence of film surface roughness and porosity on lattice size in a porous thin film deposition process is studied via kinetic Monte Carlo simulations on a triangular lattice. For sufficiently large lattice size the steady-state value of the expected film porosity has a weak dependence on the lattice size and the steady-state value of the expected surface roughness square varies linearly with lattice size. An analysis of the film morphology based on a stochastic partial differential equation description of the film surface morphology supports and explains the findings of the numerical simulations.

Geometric principles of surface growth

I introduce a new equation describing epitaxial growth processes. This equation is derived from a simple variational geometric principle and it has a straightforward interpretation in terms of continuum and microscopic physics. It is also able to reproduce the critical behavior already observed, mound formation and mass conservation, but however does not fit a divergence form as the most commonly spread models of conserved surface growth. This formulation allows me to connect the results of the dynamic renormalization group analysis with intuitive geometric principles, whose generic character may well allow them to describe surface growth and other phenomena in different areas of physics.

Renormalization of stochastic lattice models: epitaxial surfaces

We present the application of a method [C. A. Haselwandter and D. D. Vvedensky, Phys. Rev. E 76, 041115 (2007)] for deriving stochastic partial differential equations from atomistic processes to the morphological evolution of epitaxial surfaces driven by the deposition of new material. Although formally identical to the one-dimensional (1D) systems considered previously, our methodology presents substantial additional technical issues when applied to two-dimensional (2D) surfaces. Once these are addressed, subsequent coarse-graining is accomplished as before by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic models. Our applications are to the Edwards-Wilkinson (EW) model [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London, Ser. A 381, 17 (1982)], the Wolf-Villain (WV) model [D. E. Wolf and J. Villain, Europhys. Lett. 13, 389 (1990)], and a model with concurrent random deposition and surface diffusion. With our rules for the EW model no appreciable crossover is obtained for either 1D or 2D substrates. For the 1D WV model, discussed previously, our analysis reproduces the crossover sequence known from kinetic Monte Carlo (KMC) simulations, but for the 2D WV model, we find a transition from smooth to unstable growth under repeated coarse-graining. Concurrent surface diffusion does not change this behavior, but can lead to extended transient regimes with kinetic roughening. This provides an explanation of recent experiments on Ge(001) with the intriguing conclusion that the same relaxation mechanism responsible for ordered structures during the early stages of growth also produces an instability at longer times that leads to epitaxial breakdown. The RG trajectories calculated for concurrent random deposition and surface diffusion reproduce the crossover sequences observed with KMC simulations for all values of the model parameters, and asymptotically always approach the fixed point corresponding to the equation proposed by Villain [J. Phys. I 1, 19 (1991)] and by Lai and Das Sarma [Phys. Rev. Lett. 66, 2899 (1991)]. We conclude with a discussion of the application of our methodology to other growth settings.