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Cavagna A, Cristín J, Giardina I, Veca M. From noise on the sites to noise on the links: Discretizing the conserved Kardar-Parisi-Zhang equation in real space. Phys Rev E 2024; 109:064136. [PMID: 39020940 DOI: 10.1103/physreve.109.064136] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2024] [Accepted: 05/29/2024] [Indexed: 07/20/2024]
Abstract
Numerical analysis of conserved field dynamics has been generally performed with pseudospectral methods. Finite differences integration, the common procedure for nonconserved field dynamics, indeed struggles to implement a conservative noise in the discrete spatial domain. In this work we present a method to generate a conservative noise in the finite differences framework, which works for any discrete topology and boundary conditions. We apply it to numerically solve the conserved Kardar-Parisi-Zhang (cKPZ) equation, widely used to describe surface roughening when the number of particles is conserved. Our numerical simulations recover the correct scaling exponents α, β, and z in d=1 and in d=2. To illustrate the potentiality of the method, we further consider the cKPZ equation on different kinds of nonstandard lattices and on the random Euclidean graph. This is a unique numerical study of conserved field dynamics on an irregular topology, paving the way for a broad spectrum of possible applications.
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Affiliation(s)
- Andrea Cavagna
- Istituto Sistemi Complessi (ISC-CNR), Via dei Taurini 19, 00185, Rome, Italy
- Dipartimento di Fisica, Sapienza Università di Roma, P. le Aldo Moro 2, 00185, Rome, Italy
- INFN, Unità di Roma 1, 00185 Rome, Italy
| | | | - Irene Giardina
- Istituto Sistemi Complessi (ISC-CNR), Via dei Taurini 19, 00185, Rome, Italy
- Dipartimento di Fisica, Sapienza Università di Roma, P. le Aldo Moro 2, 00185, Rome, Italy
- INFN, Unità di Roma 1, 00185 Rome, Italy
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Priyanka, Täuber UC, Pleimling M. Feedback control of surface roughness in a one-dimensional Kardar-Parisi-Zhang growth process. Phys Rev E 2020; 101:022101. [PMID: 32168635 DOI: 10.1103/physreve.101.022101] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2019] [Accepted: 01/13/2020] [Indexed: 11/07/2022]
Abstract
Control of generically scale-invariant systems, i.e., targeting specific cooperative features in nonlinear stochastic interacting systems with many degrees of freedom subject to strong fluctuations and correlations that are characterized by power laws, remains an important open problem. We study the control of surface roughness during a growth process described by the Kardar-Parisi-Zhang (KPZ) equation in (1+1) dimensions. We achieve the saturation of the mean surface roughness to a prescribed value using nonlinear feedback control. Numerical integration is performed by means of the pseudospectral method, and the results are used to investigate the coupling effects of controlled (linear) and uncontrolled (nonlinear) KPZ dynamics during the control process. While the intermediate time kinetics is governed by KPZ scaling, at later times a linear regime prevails, namely the relaxation toward the desired surface roughness. The temporal crossover region between these two distinct regimes displays intriguing scaling behavior that is characterized by nontrivial exponents and involves the number of controlled Fourier modes. Due to the control, the height probability distribution becomes negatively skewed, which affects the value of the saturation width.
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Affiliation(s)
- Priyanka
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | - Uwe C Täuber
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA
| | - Michel Pleimling
- Department of Physics and Center for Soft Matter and Biological Physics, Virginia Tech, Blacksburg, Virginia 24061-0435, USA and Academy of Integrated Science, Virginia Tech, Blacksburg, Virginia 24061-0563, USA
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Rodríguez MA, Wio HS. Stochastic entropies and fluctuation theorems for a discrete one-dimensional Kardar-Parisi-Zhang system. Phys Rev E 2019; 100:032111. [PMID: 31639943 DOI: 10.1103/physreve.100.032111] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2019] [Indexed: 11/07/2022]
Abstract
The statistical properties of stochastic entropies in a discrete stationary one-dimensional Kardar-Parisi-Zhang system are numerically studied. As the usual time-independent solution of the associated Fokker-Planck equation is not strictly stationary, it is necessary to transform the current variables to other variables with zero spatial mean. We resorted to discrete representations in order to prove the statistical properties of entropies, and we performed a direct test of the fluctuation theorem. We discuss the connection between fluctuation theorems and the deviation from Gaussian distribution for the entropy.
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Affiliation(s)
- Miguel A Rodríguez
- Instituto de Física de Cantabria (IFCA), CSIC-UNICAN, E-39005 Santander, Spain
| | - Horacio S Wio
- Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC), CSIC-UIB, E-07122 Palma de Mallorca, Spain
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Wio HS, Deza RR, Escudero C, Revelli JA. Invited review: KPZ. Recent developments via a variational formulation. PAPERS IN PHYSICS 2014. [DOI: 10.4279/pip.050010] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022] Open
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Aarão Reis FDA. Normal dynamic scaling in the class of the nonlinear molecular-beam-epitaxy equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022128. [PMID: 24032796 DOI: 10.1103/physreve.88.022128] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2013] [Revised: 07/31/2013] [Indexed: 06/02/2023]
Abstract
The scaling of local height fluctuations is studied numerically in lattice growth models of the class of the nonlinear stochastic equation of Villain-Lai-Das Sarma (VLDS) in substrate dimensions d=1 and 2. In d=1, the average local slopes of the conserved restricted solid-on-solid (CRSOS) models converge to a finite value in the long-time limit, with power-law corrections in time whose exponents are close to 0.1. Other VLDS models in d=1, such as that of Das Sarma and Tamborenea, show a divergence of local slopes up to 10(6) monolayers, typical of anomalous roughening, but a comparison of roughness distributions shows that they scale as the linear fourth-order growth equation in those time scales. Normal scaling is also obtained in a modified VLDS equation with instability suppression, in contrast to recent numerical works. In d=2, a CRSOS model and a model with lateral aggregation of diffusing particles show normal scaling of the local slopes, also with small correction exponents. These results consistently show that the VLDS class has normal dynamic scaling in d=1 and 2, in agreement with the theoretical predictions of Phys. Rev. Lett. 94, 166103 (2005), and they show that the apparently anomalous features observed in previous works are effects of large scaling correction terms or crossover effects.
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Affiliation(s)
- F D A Aarão Reis
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói RJ, Brazil
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Agoritsas E, Lecomte V, Giamarchi T. Static fluctuations of a thick one-dimensional interface in the 1+1 directed polymer formulation: numerical study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062405. [PMID: 23848695 DOI: 10.1103/physreve.87.062405] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2013] [Revised: 04/17/2013] [Indexed: 06/02/2023]
Abstract
We study numerically the geometrical and free-energy fluctuations of a static one-dimensional (1D) interface with a short-range elasticity, submitted to a quenched random-bond Gaussian disorder of finite correlation length ξ>0 and at finite temperature T. Using the exact mapping from the static 1D interface to the 1+1 directed polymer (DP) growing in a continuous space, we focus our analysis on the disorder free energy of the DP end point, a quantity which is strictly zero in the absence of disorder and whose sample-to-sample fluctuations at a fixed growing time t inherit the statistical translation invariance of the microscopic disorder explored by the DP. Constructing a new numerical scheme for the integration of the Kardar-Parisi-Zhang evolution equation obeyed by the free energy, we address numerically the time and temperature dependence of the disorder free-energy fluctuations at fixed finite ξ. We examine, on one hand, the amplitude D[over ̃](t) and effective correlation length ξ[over ̃](t) of the free-energy fluctuations and, on the other hand, the imprint of the specific microscopic disorder correlator on the large-time shape of the free-energy two-point correlator. We observe numerically the crossover to a low-temperature regime below a finite characteristic temperature T(c)(ξ), as previously predicted by Gaussian variational method computations and scaling arguments and extensively investigated analytically in [Phys. Rev. E 87, 042406 (2013)]. Finally, we address numerically the time and temperature dependence of the roughness B(t), which quantifies the DP end point transverse fluctuations, and we show how the amplitude D[over ̃](∞)(T,ξ) controls the different regimes experienced by B(t)-in agreement with the analytical predictions of a DP toy model approach.
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Affiliation(s)
- Elisabeth Agoritsas
- DPMC-MaNEP, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland.
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Wio HS, Escudero C, Revelli JA, Deza RR, de la Lama MS. Recent developments on the Kardar-Parisi-Zhang surface-growth equation. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2011; 369:396-411. [PMID: 21149379 DOI: 10.1098/rsta.2010.0259] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
The stochastic nonlinear partial differential equation known as the Kardar-Parisi-Zhang (KPZ) equation is a highly successful phenomenological mesoscopic model of surface and interface growth processes. Its suitability for analytical work, its explicit symmetries and its prediction of an exact dynamic scaling relation for a one-dimensional substratum led people to adopt it as a 'standard' model in the field during the last quarter of a century. At the same time, several conjectures deserving closer scrutiny were established as dogmas throughout the community. Among these, we find the beliefs that 'genuine' non-equilibrium processes are non-variational in essence, and that the exactness of the dynamic scaling relation owes its existence to a Galilean symmetry. Additionally, the equivalence among planar and radial interface profiles has been generally assumed in the literature throughout the years. Here--among other topics--we introduce a variational formulation of the KPZ equation, remark on the importance of consistency in discretization and challenge the mainstream view on the necessity for scaling of both Galilean symmetry and the one-dimensional fluctuation-dissipation theorem. We also derive the KPZ equation on a growing domain as a first approximation to radial growth, and outline the differences with respect to the classical case that arises in this new situation.
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Affiliation(s)
- Horacio S Wio
- Instituto de Física de Cantabria (UC and CSIC), Avda. de los Castros, s/n, 39005 Santander, Spain.
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Nicoli M, Vivo E, Cuerno R. Kardar-Parisi-Zhang asymptotics for the two-dimensional noisy Kuramoto-Sivashinsky equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:045202. [PMID: 21230337 DOI: 10.1103/physreve.82.045202] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2010] [Indexed: 05/30/2023]
Abstract
We study numerically the Kuramoto-Sivashinsky equation forced by external white noise in two space dimensions, that is a generic model for, e.g., surface kinetic roughening in the presence of morphological instabilities. Large scale simulations using a pseudospectral numerical scheme allow us to retrieve Kardar-Parisi-Zhang (KPZ) scaling as the asymptotic state of the system, as in the one-dimensional (1D) case. However, this is only the case for sufficiently large values of the coupling and/or system size, so that previous conclusions on non-KPZ asymptotics are demonstrated as finite size effects. Crossover effects are comparatively stronger for the two-dimensional case than for the 1D system.
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Affiliation(s)
- Matteo Nicoli
- Laboratoire de Physique de la Matière Condensée, École Polytechnique-CNRS, 91128 Palaiseau, France
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Wio HS, Revelli JA, Deza RR, Escudero C, de La Lama MS. Discretization-related issues in the Kardar-Parisi-Zhang equation: consistency, Galilean-invariance violation, and fluctuation-dissipation relation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:066706. [PMID: 20866543 DOI: 10.1103/physreve.81.066706] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/2010] [Indexed: 05/29/2023]
Abstract
In order to perform numerical simulations of the Kardar-Parisi-Zhang (KPZ) equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a Hopf-Cole transformation applied to a diffusion equation (with multiplicative noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on space and the Hopf-Cole transformation is local both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between real-space and pseudospectral discrete representations. In addition we discuss the relevance of the Galilean-invariance violation in these consistent discretization schemes and the alleged conflict of standard discretization with the fluctuation-dissipation theorem, peculiar of one dimension.
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Affiliation(s)
- H S Wio
- Instituto de Física de Cantabria (UC and CSIC), E-39005 Santander, Spain
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Nicoli M, Cuerno R, Castro M. Unstable nonlocal interface dynamics. PHYSICAL REVIEW LETTERS 2009; 102:256102. [PMID: 19659099 DOI: 10.1103/physrevlett.102.256102] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2008] [Indexed: 05/28/2023]
Abstract
Nonlocal effects occur in many nonequilibrium interfaces, due to diverse physical mechanisms like diffusive, ballistic, or anomalous transport, with examples from flame fronts to thin films. While dimensional analysis describes stable nonlocal interfaces, we show the morphologically unstable condition to be nontrivial. This is the case for a family of stochastic equations of experimental relevance, paradigmatically including the Michelson-Sivashinsky system. For a whole parameter range, the asymptotic dynamics is scale invariant with dimension-independent exponents reflecting a hidden Galilean symmetry. The usual Kardar-Parisi-Zhang nonlinearity, albeit irrelevant in that parameter range, plays a key role in this behavior.
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Affiliation(s)
- Matteo Nicoli
- Departamento de Matemáticas and Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Carlos III de Madrid, Avenida de la Universidad 30, E-28911 Leganés, Spain
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Nicoli M, Castro M, Cuerno R. Unified moving-boundary model with fluctuations for unstable diffusive growth. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:021601. [PMID: 18850840 DOI: 10.1103/physreve.78.021601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/12/2008] [Revised: 04/25/2008] [Indexed: 05/26/2023]
Abstract
We study a moving-boundary model of nonconserved interface growth that implements the interplay between diffusive matter transport and aggregation kinetics at the interface. Conspicuous examples are found in thin-film production by chemical vapor deposition and electrochemical deposition. The model also incorporates noise terms that account for fluctuations in the diffusive and attachment processes. A small-slope approximation allows us to derive effective interface evolution equations (IEEs) in which parameters are related to those of the full moving-boundary problem. In particular, the form of the linear dispersion relation of the IEE changes drastically for slow or for instantaneous attachment kinetics. In the former case the IEE takes the form of the well-known (noisy) Kuramoto-Sivashinsky equation, showing a morphological instability at short times that evolves into kinetic roughening of the Kardar-Parisi-Zhang (KPZ) class. In the instantaneous kinetics limit, the IEE combines the Mullins-Sekerka linear dispersion relation with a KPZ nonlinearity, and we provide a numerical study of the ensuing dynamics. In all cases, the long preasymptotic transients can account for the experimental difficulties in observing KPZ scaling. We also compare our results with relevant data from experiments and discrete models.
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Affiliation(s)
- Matteo Nicoli
- Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain
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Miranda VG, Aarão Reis FDA. Numerical study of the Kardar-Parisi-Zhang equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:031134. [PMID: 18517356 DOI: 10.1103/physreve.77.031134] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2007] [Revised: 01/10/2008] [Indexed: 05/26/2023]
Abstract
We integrate numerically the Kardar-Parisi-Zhang (KPZ) equation in 1+1 and 2+1 dimensions using a Euler discretization scheme and the replacement of (nablah)(2) by exponentially decreasing functions of that quantity to suppress instabilities. When applied to the equation in 1+1 dimensions, the method of instability control provides values of scaling amplitudes consistent with exactly known results, in contrast to the deviations generated by the original scheme. In 2+1 dimensions, we spanned a range of the model parameters where transients with Edwards-Wilkinson or random growth are not observed, in box sizes 8< or =L< or =128 . We obtain a roughness exponent of 0.37< or =alpha< or =0.40 and steady state height distributions with skewness S=0.25+/-0.01 and kurtosis Q=0.15+/-0.1 . These estimates are obtained after extrapolations to the large L limit, which is necessary due to significant finite-size effects in the estimates of effective exponents and height distributions. On the other hand, the steady state roughness distributions show weak scaling corrections and evidence of stretched exponential tails. These results confirm previous estimates from lattice models, showing their reliability as representatives of the KPZ class.
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Affiliation(s)
- Vladimir G Miranda
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói RJ, Brazil.
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