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Dumonteil E. Rough neutron fields and nuclear reactor noise. Phys Rev E 2024; 110:014119. [PMID: 39160932 DOI: 10.1103/physreve.110.014119] [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: 06/13/2024] [Indexed: 08/21/2024]
Abstract
Nuclear reactor cores achieve sustained fission chain reactions through the so-called "critical state"-a subtle equilibrium between their material properties and their geometries. Observed at macroscopic scales during operations, the resulting stationary neutron field is tainted by a noise term that hinders various fluctuations occurring at smaller scales. These fluctuations are either of a stochastic nature (whenever the core is operated at low power) or related to various perturbations and vibrations within the core, even operated in its power regime. For reasons that are only partially understood using linear noise theory, incidental events have been reported, characterized by an increase of the power noise. Such events of power noise growth, sometimes up to seemingly unbounded levels, have already led in the past to voluntary scramming of reactors. In this paper, we will use a statistical field theory of critical processes to model the effects of neutron power noise. We will show that the evolution of the neutron field in a reactor is intimately connected to the dynamic of surface growths given by the Kardar-Parisi-Zhang equation. Recent numerical results emerging from renormalization-group approaches will be used to calculate a threshold in the amplitude of the reactor noise above which the core enters a new criticality state, and to estimate the critical exponents characterizing this phase transition to rough neutron fields. The theoretical model of nonlinear noise built in this paper from ab initio statistical mechanics principles will be correlated and compared to data of misunderstood reactor noise levels and reactor instabilities and will be shown to provide both qualitative and quantitative insights into this long-standing issue of reactor physics.
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Oliveira TJ. Kardar-Parisi-Zhang universality class in (d+1)-dimensions. Phys Rev E 2022; 106:L062103. [PMID: 36671175 DOI: 10.1103/physreve.106.l062103] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2022] [Accepted: 12/06/2022] [Indexed: 06/17/2023]
Abstract
The determination of the exact exponents of the KPZ class in any substrate dimension d is one of the most important open issues in Statistical Physics. Based on the behavior of the dimensional variation of some exact exponent differences for other growth equations, I find here that the KPZ growth exponents (related to the temporal scaling of the fluctuations) are given by β_{d}=7/8d+13. These exponents present an excellent agreement with the most accurate estimates for them in the literature. Moreover, they are confirmed here through extensive Monte Carlo simulations of discrete growth models and real-space renormalization group (RG) calculations for directed polymers in random media (DPRM), up to d=15. The left-tail exponents of the probability density functions for the DPRM energy provide another striking verification of the analytical result above.
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Affiliation(s)
- Tiago J Oliveira
- Departamento de Física, Universidade Federal de Viçosa, 36570-900 Viçosa, MG, Brazil
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Pál G, Domokos G, Kun F. Curvature flows, scaling laws and the geometry of attrition under impacts. Sci Rep 2021; 11:20661. [PMID: 34667174 PMCID: PMC8526698 DOI: 10.1038/s41598-021-00030-1] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2021] [Accepted: 10/05/2021] [Indexed: 11/30/2022] Open
Abstract
Impact induced attrition processes are, beyond being essential models of industrial ore processing, broadly regarded as the key to decipher the provenance of sedimentary particles. Here we establish the first link between microscopic, particle-based models and the mean field theory for these processes. Based on realistic computer simulations of particle-wall collision sequences we first identify the well-known damage and fragmentation energy phases, then we show that the former is split into the abrasion phase with infinite sample lifetime (analogous to Sternberg's Law) at finite asymptotic mass and the cleavage phase with finite sample lifetime, decreasing as a power law of the impact velocity (analogous to Basquin's Law). This splitting establishes the link between mean field models (curvature-driven partial differential equations) and particle-based models: only in the abrasion phase does shape evolution emerging in the latter reproduce with startling accuracy the spatio-temporal patterns (two geometric phases) predicted by the former.
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Affiliation(s)
- Gergő Pál
- grid.7122.60000 0001 1088 8582Department of Theoretical Physics, Doctoral School of Physics, Faculty of Science and Technology, University of Debrecen, P.O. Box 400, 4002 Debrecen, Hungary ,grid.418861.20000 0001 0674 7808Institute of Nuclear Research (Atomki), P.O. Box 51, 4001 Debrecen, Hungary
| | - Gábor Domokos
- grid.6759.d0000 0001 2180 0451Department of Mechanics, Materials and Structures, Budapest University of Technology and Economics, Műegyetem rkp. 3., K261, 1111 Budapest, Hungary ,grid.5018.c0000 0001 2149 4407MTA-BME Morphodynamics Reserarch Group, Műegyetem rkp. 3., K261, 1111 Budapest, Hungary
| | - Ferenc Kun
- grid.7122.60000 0001 1088 8582Department of Theoretical Physics, Doctoral School of Physics, Faculty of Science and Technology, University of Debrecen, P.O. Box 400, 4002 Debrecen, Hungary
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Nakayama Y, Nishida Y. Efimov effect at the Kardar-Parisi-Zhang roughening transition. Phys Rev E 2021; 103:012117. [PMID: 33601523 DOI: 10.1103/physreve.103.012117] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2020] [Accepted: 12/23/2020] [Indexed: 11/07/2022]
Abstract
Surface growth governed by the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than two undergoes a roughening transition from smooth to rough phases with increasing the nonlinearity. It is also known that the KPZ equation can be mapped onto quantum mechanics of attractive bosons with a contact interaction, where the roughening transition corresponds to a binding transition of two bosons with increasing the attraction. Such critical bosons in three dimensions actually exhibit the Efimov effect, where a three-boson coupling turns out to be relevant under the renormalization group so as to break the scale invariance down to a discrete one. On the basis of these facts linking the two distinct subjects in physics, we predict that the KPZ roughening transition in three dimensions shows either the discrete scale invariance or no intrinsic scale invariance.
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Affiliation(s)
- Yu Nakayama
- Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan
| | - Yusuke Nishida
- Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro, Tokyo 152-8551, Japan
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Alves SG, Ferreira SC. Scaling, cumulant ratios, and height distribution of ballistic deposition in 3+1 and 4+1 dimensions. Phys Rev E 2016; 93:052131. [PMID: 27300853 DOI: 10.1103/physreve.93.052131] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2016] [Indexed: 06/06/2023]
Abstract
We investigate the origin of the scaling corrections in ballistic deposition models in high dimensions using the method proposed by Alves et al. [Phys. Rev. E 90, 052405 (2014)PLEEE81539-375510.1103/PhysRevE.90.052405] in d=2+1 dimensions, where the intrinsic width associated with the fluctuations of the height increments during the deposition processes is explicitly taken into account. In the present work, we show that this concept holds for d=3+1 and 4+1 dimensions. We have found that growth and roughness exponents and dimensionless cumulant ratios are in agreement with other models, presenting small finite-time corrections to the scaling, that in principle belong to the Kardar-Parisi-Zhang (KPZ) universality class in both d=3+1 and 4+1. Our results constitute further evidence that the upper critical dimension of the KPZ class, if it exists, is larger than 4.
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Affiliation(s)
- Sidiney G Alves
- Departamento de Física e Matemática, Universidade Federal de São João Del Rei, 36420-000 Ouro Branco, MG, Brazil
| | - Silvio C Ferreira
- Departamento de Física, Universidade Federal de Viçosa, 36570-000 Viçosa, MG, Brazil
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Alves SG, Oliveira TJ, Ferreira SC. Universality of fluctuations in the Kardar-Parisi-Zhang class in high dimensions and its upper critical dimension. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:020103. [PMID: 25215669 DOI: 10.1103/physreve.90.020103] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2014] [Indexed: 06/03/2023]
Abstract
We show that the theoretical machinery developed for the Kardar-Parisi-Zhang (KPZ) class in low dimensions is obeyed by the restricted solid-on-solid model for substrates with dimensions up to d=6. Analyzing different restriction conditions, we show that the height distributions of the interface are universal for all investigated dimensions. It means that fluctuations are not negligible and, consequently, the system is still below the upper critical dimension at d=6. The extrapolation of the data to dimensions d≥7 predicts that the upper critical dimension of the KPZ class is infinite.
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Affiliation(s)
- S G Alves
- Departamento de Física, Universidade Federal de Viçosa, 36570-000, Viçosa, MG, Brazil
| | - T J Oliveira
- Departamento de Física, Universidade Federal de Viçosa, 36570-000, Viçosa, MG, Brazil
| | - S C Ferreira
- Departamento de Física, Universidade Federal de Viçosa, 36570-000, Viçosa, MG, Brazil
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Ódor G, Kelling J, Gemming S. Aging of the (2+1)-dimensional Kardar-Parisi-Zhang model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:032146. [PMID: 24730828 DOI: 10.1103/physreve.89.032146] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2013] [Indexed: 06/03/2023]
Abstract
Extended dynamical simulations have been performed on a (2+1)-dimensional driven dimer lattice-gas model to estimate aging properties. The autocorrelation and the autoresponse functions are determined and the corresponding scaling exponents are tabulated. Since this model can be mapped onto the (2+1)-dimensional Kardar-Parisi-Zhang surface growth model, our results contribute to the understanding of the universality class of that basic system.
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Affiliation(s)
- Géza Ódor
- MTA TTK MFA Research Institute for Natural Sciences, P. O. Box 49, H-1525 Budapest, Hungary
| | - Jeffrey Kelling
- Institute of Ion Beam Physics and Materials Research Helmholtz-Zentrum, Dresden-Rossendorf, P. O. Box 51 01 19, 01314 Dresden, Germany and Institute of Physics, TU Chemnitz 09107 Chemnitz, Germany
| | - Sibylle Gemming
- Institute of Ion Beam Physics and Materials Research Helmholtz-Zentrum, Dresden-Rossendorf, P. O. Box 51 01 19, 01314 Dresden, Germany and Institute of Physics, TU Chemnitz 09107 Chemnitz, Germany
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Kim JM, Kim SW. Restricted solid-on-solid model with a proper restriction parameter N in 4+1 dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:034102. [PMID: 24125386 DOI: 10.1103/physreve.88.034102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2013] [Indexed: 06/02/2023]
Abstract
A restricted solid-on-solid growth model is studied for various restriction parameters N in d=4+1 dimensions. The interface width W grows as t^{β} with β=0.158 ± 0.006 and W follows W∼L{α} at saturation with α=0.273 ± 0.009, where L is the system size. The dynamic exponent z≈1.73 is obtained from the relation z=α/β. The estimated exponents satisfy the scaling relation α+z=2 very well. Our results indicate that the upper critical dimension of the Kardar-Parisi-Zhang equation is larger than d=4+1 dimensions. With a proper choice of the restriction parameter N, we can reduce the discrete effect of the height to the width and obtain the values of the exponents accurately.
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Affiliation(s)
- Jin Min Kim
- Department of Physics and Institute for Integrative Basic Sciences, Soongsil University, Seoul 156-743, Korea
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Pagnani A, Parisi G. Multisurface coding simulations of the restricted solid-on-solid model in four dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:010102. [PMID: 23410269 DOI: 10.1103/physreve.87.010102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2012] [Indexed: 06/01/2023]
Abstract
We study the restricted solid-on-solid model for surface growth in spatial dimension d=4 by means of a multisurface coding technique that allows us to analyze samples of size up to 256(4) in the steady-state regime. For such large systems we are able to achieve a controlled asymptotic regime where the typical scale of the fluctuations are larger than the lattice spacing used in the simulations. A careful finite-size scaling analysis of the critical exponents clearly indicate that d=4 is not the upper critical dimension of the model.
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Affiliation(s)
- Andrea Pagnani
- Human Genetics Foundation (HuGeF), Via Nizza 52, I-10126 Turin, Italy
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Kelling J, Ódo G. Extremely large-scale simulation of a Kardar-Parisi-Zhang model using graphics cards. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:061150. [PMID: 22304083 DOI: 10.1103/physreve.84.061150] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2011] [Indexed: 05/31/2023]
Abstract
The octahedron model introduced recently has been implemented onto graphics cards, which permits extremely large-scale simulations via binary lattice gases and bit-coded algorithms. We confirm scaling behavior belonging to the two-dimensional Kardar-Parisi-Zhang universality class and find a surface growth exponent: β = 0.2415(15) on 2(17) × 2(17) systems, ruling out β = 1/4 suggested by field theory. The maximum speedup with respect to a single CPU is 240. The steady state has been analyzed by finite-size scaling and a growth exponent α = 0.393(4) is found. Correction-to-scaling-exponent are computed and the power-spectrum density of the steady state is determined. We calculate the universal scaling functions and cumulants and show that the limit distribution can be obtained by the sizes considered. We provide numerical fitting for the small and large tail behavior of the steady-state scaling function of the interface width.
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Affiliation(s)
- Jeffrey Kelling
- Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, PO Box 51 01 19, D-01314 Dresden, Germany
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Odor G, Liedke B, Heinig KH. Surface pattern formation and scaling described by conserved lattice gases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:051114. [PMID: 20866192 DOI: 10.1103/physreve.81.051114] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/15/2009] [Revised: 02/26/2010] [Indexed: 05/29/2023]
Abstract
We extend our 2+1 -dimensional discrete growth model [Odor, Phys. Rev. E 79, 021125 (2009)] with conserved, local exchange dynamics of octahedra, describing surface diffusion. A roughening process was realized by uphill diffusion and curvature dependence. By mapping the slopes onto particles, two-dimensional nonequilibrium binary lattice model emerges, in which the (smoothing or roughening) surface diffusion can be described by attracting or repelling motion of oriented dimers. The binary representation allows simulations on very large size and time scales. We provide numerical evidence for Mullins-Herring or molecular-beam epitaxy class scaling of the surface width. The competition of inverse Mullins-Herring diffusion with a smoothing deposition, which corresponds to a Kardar-Parisi-Zhang (KPZ) process, generates different patterns: dots or ripples. We analyze numerically the scaling and wavelength growth behavior in these models. In particular, we confirm by large size simulations that the KPZ type of scaling is stable against the addition of this surface diffusion, hence this is the asymptotic behavior of the Kuramoto-Sivashinsky equation as conjectured by field theory in two dimensions, but has been debated numerically. If very strong, normal surface diffusion is added to a KPZ process, we observe smooth surfaces with logarithmic growth, which can describe the mean-field behavior of the strong-coupling KPZ class. We show that ripple coarsening occurs if parallel surface currents are present, otherwise logarithmic behavior emerges.
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Affiliation(s)
- Géza Odor
- Research Institute for Technical Physics and Materials Science, P.O. Box 49, H-1525 Budapest, Hungary
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