1
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Takada S, Yoshinaga N, Doi N, Fujiwara K. Mode selection mechanism in traveling and standing waves revealed by Min wave reconstituted in artificial cells. SCIENCE ADVANCES 2022; 8:eabm8460. [PMID: 35675408 PMCID: PMC9177070 DOI: 10.1126/sciadv.abm8460] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
Reaction-diffusion coupling (RDc) generates spatiotemporal patterns, including two dynamic wave modes: traveling and standing waves. Although mode selection plays a substantial role in the spatiotemporal organization of living cell molecules, the mechanism for selecting each wave mode remains elusive. Here, we investigated a wave mode selection mechanism using Min waves reconstituted in artificial cells, emerged by the RDc of MinD and MinE. Our experiments and theoretical analysis revealed that the balance of membrane binding and dissociation from the membrane of MinD determines the mode selection of the Min wave. We successfully demonstrated that the transition of the wave modes can be regulated by controlling this balance and found hysteresis characteristics in the wave mode transition. These findings highlight a previously unidentified role of the balance between activators and inhibitors as a determinant of the mode selection of waves by RDc and depict an unexplored mechanism in intracellular spatiotemporal pattern formations.
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Affiliation(s)
- Sakura Takada
- Department of Biosciences and Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Natsuhiko Yoshinaga
- Mathematical Science Group, WPI Advanced Institute for Materials Research (WPI-AIMR), Tohoku University, Katahira 2-1-1, Aoba-Ku, Sendai 980-8577, Japan
- MathAM-OIL, AIST, Sendai 980-8577, Japan
- Corresponding author. (N.Y.); (K.F.)
| | - Nobuhide Doi
- Department of Biosciences and Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
| | - Kei Fujiwara
- Department of Biosciences and Informatics, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
- Corresponding author. (N.Y.); (K.F.)
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2
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Abstract
Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter1-6, non-equilibrium systems7-9, networks of neurons10,11, social groups with conformist and contrarian members12, directional interface growth phenomena13-15 and metamaterials16-20. Although wave propagation in non-reciprocal media has recently been closely studied1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points21. We describe the emergence of these phases using insights from bifurcation theory22,23 and non-Hermitian quantum mechanics24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.
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Rameshwar Y, Anuradha V, Srinivas G, Pérez LM, Laroze D, Pleiner H. Nonlinear convection of binary liquids in a porous medium. CHAOS (WOODBURY, N.Y.) 2018; 28:075512. [PMID: 30070520 DOI: 10.1063/1.5027468] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2018] [Accepted: 05/09/2018] [Indexed: 06/08/2023]
Abstract
Thermal convection of binary mixtures in a porous medium is studied with stress-free boundary conditions. The linear stability analysis is studied by using the normal mode method. The effects of the material parameters have been studied at the onset of convection. Using a multiple scale analysis near the onset of the stationary convection, a cubic-quintic amplitude equation is derived. The influence of the Lewis number and the separation ratio on the supercritical-subcritical transition is discussed. Stationary front solutions and localized states are analyzed at the Maxwell point. Near the threshold of the oscillatory convection, a set of two coupled complex cubic-quintic Ginzburg-Landau type amplitude equations is derived, and implicit analytical expressions for the coefficients are given.
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Affiliation(s)
- Y Rameshwar
- Department of Mathematics, University College of Engineering, Osmania University, Hyderabad 500007, India
| | - V Anuradha
- Department of Mathematics, Government Degree College for Women, Begumpet, Hyderabad 500007, India
| | - G Srinivas
- Department of Mathematics, University College of Engineering, Osmania University, Hyderabad 500007, India
| | - L M Pérez
- School of Physical Sciences and Nanotechnology, Yachay Tech University, Urcuquí 00119, Ecuador
| | - D Laroze
- School of Physical Sciences and Nanotechnology, Yachay Tech University, Urcuquí 00119, Ecuador
| | - H Pleiner
- Max Planck Institute for Polymer Research, Mainz D 55021, Germany
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4
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Contreras DAC, Clerc MG. Internal noise and system size effects induce nondiffusive kink dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032922. [PMID: 25871187 DOI: 10.1103/physreve.91.032922] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2015] [Indexed: 06/04/2023]
Abstract
We investigate the effects of inherent fluctuations and system size in the dynamics of domain between uniform symmetric states. In the case of monotonous kinks, this dynamics is characterized by exhibiting nonsymmetric random walks, being attracted to the system borders. For nonmonotonous interface, the dynamics is replaced by a hopping dynamic. Based on bistable universal models, we characterize the origin of these unexpected dynamics through use of the stochastic kinematic laws for the interface position and the survival probability. Numerical simulations show a quite good agreement with the theoretical predictions.
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Affiliation(s)
- Diego A C Contreras
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - Marcel G Clerc
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
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5
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Andrade-Silva I, Clerc MG, Odent V. Zig-zag wall lattice in a nematic liquid crystal with an in-plane switching configuration. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022504. [PMID: 25215746 DOI: 10.1103/physreve.90.022504] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2014] [Indexed: 06/03/2023]
Abstract
Liquid crystals displays with tailoring electrodes exhibit complex spatiotemporal dynamics when a large voltage is applied. We report experimental observations of the appearance of a programmable zig-zag lattice using an in-plane-switching cell filled with a nematic liquid crystal. Applying a small voltage to a wide range of frequencies, the system exhibits an Ising wall lattice. Increasing the voltage, this lattice presents a spatial instability generating an undulating wall lattice, and to higher voltages it becomes zig-zag type. Experimentally, we characterize the bifurcations and phase diagram of the wall lattice. Theoretically, we develop, from first principles, a descriptive model. This model has a good qualitative agreement with experimental observations.
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Affiliation(s)
- I Andrade-Silva
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - M G Clerc
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
| | - V Odent
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
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da Silva LA, Colombo EH, Anteneodo C. Effect of environment fluctuations on pattern formation of single species. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:012813. [PMID: 25122348 DOI: 10.1103/physreve.90.012813] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2014] [Indexed: 05/26/2023]
Abstract
System-environment interactions are intrinsically nonlinear and dependent on the interplay between many degrees of freedom. The complexity may be even more pronounced when one aims to describe biologically motivated systems. In that case, it is useful to resort to simplified models relying on effective stochastic equations. A natural consideration is to assume that there is a noisy contribution from the environment, such that the parameters that characterize it are not constant but instead fluctuate around their characteristic values. From this perspective, we propose a stochastic generalization of the nonlocal Fisher-KPP equation where, as a first step, environmental fluctuations are Gaussian white noises, both in space and time. We apply analytical and numerical techniques to study how noise affects stability and pattern formation in this context. Particularly, we investigate noise-induced coherence by means of the complementary information provided by the dispersion relation and the structure function.
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Affiliation(s)
- L A da Silva
- Center of Mathematics, Computation and Cognition, UFABC, São Paulo, Brazil
| | - E H Colombo
- Department of Physics, PUC-Rio, Rio de Janeiro, Brazil
| | - C Anteneodo
- Department of Physics, PUC-Rio, Rio de Janeiro, Brazil and Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil
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7
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Hilt M, Glässl M, Zimmermann W. Effects of a temperature-dependent viscosity on thermal convection in binary mixtures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052312. [PMID: 25353805 DOI: 10.1103/physreve.89.052312] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/02/2013] [Indexed: 06/04/2023]
Abstract
We investigate the effect of a temperature-dependent viscosity on the onset of thermal convection in a horizontal layer of a binary fluid mixture that is heated from below. For an exponential temperature dependence of the viscosity, we find, in binary mixtures as a function of a positive separation ratio ψ and beyond a certain viscosity contrast, a discontinuous transition between two stationary convection modes having different wavelengths. In the range of negative values of the separation ratio ψ, a (continuous or discontinuous) transition from an oscillatory to a stationary onset of convection occurs beyond a certain viscosity contrast, and for large values of the viscosity ratio, the oscillatory onset of convection is suppressed.
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Affiliation(s)
- Markus Hilt
- Theoretische Physik I, Universität Bayreuth, 95440 Bayreuth, Germany
| | - Martin Glässl
- Theoretische Physik I, Universität Bayreuth, 95440 Bayreuth, Germany
| | - Walter Zimmermann
- Theoretische Physik I, Universität Bayreuth, 95440 Bayreuth, Germany
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8
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Macías JE, Clerc MG, Falcón C, García-Ñustes MA. Spatially modulated kinks in shallow granular layers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:020201. [PMID: 24032762 DOI: 10.1103/physreve.88.020201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2013] [Revised: 06/30/2013] [Indexed: 06/02/2023]
Abstract
We report on the experimental observation of spatially modulated kinks in a shallow one-dimensional fluidized granular layer subjected to a periodic air flow. We show the appearance of these solutions as the layer undergoes a parametric instability. Due to the inherent fluctuations of the granular layer, the kink profile exhibits an effective wavelength, a precursor, which modulates spatially the homogeneous states and drastically modifies the kink dynamics. We characterize the average and fluctuating properties of this solution. Finally, we show that the temporal evolution of these kinks is dominated by a hopping dynamics, related directly to the underlying spatial structure.
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Affiliation(s)
- J E Macías
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
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Kengne E, Lakhssassi A, Liu WM, Vaillancourt R. Phase engineering, modulational instability, and solitons of Gross-Pitaevskii-type equations in 1+1 dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022914. [PMID: 23496598 DOI: 10.1103/physreve.87.022914] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2012] [Revised: 12/17/2012] [Indexed: 06/01/2023]
Abstract
Motivated by recent proposals of "collisionally inhomogeneous" Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we introduce a phase imprint into the macroscopic order parameter governing the dynamics of BECs with spatiotemporal varying scattering length described by a cubic Gross-Pitaevskii (GP) equation and then suitably engineer the imprinted phase to generate the modified GP equation, also called the cubic derivative nonlinear Schrödinger (NLS) equation. This equation describes the dynamics of condensates with two-body (attractive and repulsive) interactions in a time-varying quadratic external potential. We then carry out a theoretical analysis which invokes a lens-type transformation that converts the cubic derivative NLS equation into a modified NLS equation with only explicit temporal dependence. Our analysis suggests a particular interest in a specific time-varying potential with the strength of the magnetic trap ~1/(t+t(*))(2). For a time-varying quadratic external potential of this kind, an explicit expression for the growth rate of a purely growing modulational instability is presented and analyzed. We point out the effect of the imprint parameter and the parameter t(*) on the instability growth rate, as well as on the solitary waves of the BECs.
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Affiliation(s)
- E Kengne
- National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People's Republic of China
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10
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Glässl M, Hilt M, Zimmermann W. Convection in nanofluids with a particle-concentration-dependent thermal conductivity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:046315. [PMID: 21599303 DOI: 10.1103/physreve.83.046315] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/16/2010] [Revised: 03/09/2011] [Indexed: 05/30/2023]
Abstract
Thermal convection in nanofluids is investigated by means of a continuum model for binary-fluid mixtures, with a thermal conductivity depending on the local concentration of colloidal particles. The applied temperature difference between the upper and the lower boundary leads via the Soret effect to a variation of the colloid concentration and, therefore, to a spatially varying heat conductivity. An increasing difference between the heat conductivity of the mixture near the colder and the warmer boundary results in a shift of the onset of convection to higher values of the Rayleigh number for positive values of the separation ratio ψ>0 and to smaller values in the range ψ<0. Beyond some critical difference of the thermal conductivity between the two boundaries, we find an oscillatory onset of convection not only for ψ<0, but also within a finite range of ψ>0. This range can be extended by increasing the difference in the thermal conductivity, and it is bounded by two codimension-2 bifurcations.
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Affiliation(s)
- Martin Glässl
- Theoretische Physik I, Universität Bayreuth, DE-95440 Bayreuth, Germany
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11
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Glässl M, Hilt M, Zimmermann W. Convection in colloidal suspensions with particle-concentration-dependent viscosity. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2010; 32:265-272. [PMID: 20676720 DOI: 10.1140/epje/i2010-10652-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2010] [Revised: 06/14/2010] [Accepted: 06/17/2010] [Indexed: 05/29/2023]
Abstract
The onset of thermal convection in a horizontal layer of a colloidal suspension is investigated in terms of a continuum model for binary-fluid mixtures where the viscosity depends on the local concentration of colloidal particles. With an increasing difference between the viscosity at the warmer and the colder boundary the threshold of convection is reduced in the range of positive values of the separation ratio psi with the onset of stationary convection as well as in the range of negative values of psi with an oscillatory Hopf bifurcation. Additionally the convection rolls are shifted downwards with respect to the center of the horizontal layer for stationary convection psi>0 and upwards for the Hopf bifurcation (psi<0.
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Affiliation(s)
- M Glässl
- Theoretische Physik I, Universität Bayreuth, 95440, Bayreuth, Germany
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12
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Ortega I, Clerc MG, Falcón C, Mujica N. Subharmonic wave transition in a quasi-one-dimensional noisy fluidized shallow granular bed. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:046208. [PMID: 20481808 DOI: 10.1103/physreve.81.046208] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2009] [Indexed: 05/29/2023]
Abstract
We present an experimental and theoretical study of the pattern formation process of standing subharmonic waves in a fluidized quasi-one-dimensional shallow granular bed. The fluidization process is driven by means of a time-periodic air flow, analogous to a tapping type of forcing. Measurements of the amplitude of the critical mode close to the transition are in quite good agreement with those inferred from a universal stochastic amplitude equation. This allows us to determine both the bifurcation point of the deterministic system and the corresponding noise intensity. We also show that the probability density distribution is well described by a generalized Rayleigh distribution, which is the stationary solution of the corresponding Fokker-Planck equation of the universal stochastic amplitude equation that describes our system.
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Affiliation(s)
- Ignacio Ortega
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
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Huke B, Pleiner H, Lücke M. Thermal convection in mixtures with an inversion of the separation ratio. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:046315. [PMID: 18999534 DOI: 10.1103/physreve.78.046315] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2007] [Revised: 09/05/2008] [Indexed: 05/27/2023]
Abstract
We have numerically investigated Rayleigh-Bénard convection in binary mixtures assuming a thermal diffusion ratio that depends on the local temperature and changes its sign within the cell. The stationary instability has been found to precede the oscillatory one for a wide range of negative mean psi values. The bifurcation diagram for stationary rolls turns out to be qualitatively different from that for constant Soret effect. Nevertheless, it can be mapped onto this special case by using a scaling argument, taking into account the fact that for small convection amplitudes the rolls are restricted to parts of the cell where the sign of the Soret coefficient favors instability.
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Affiliation(s)
- B Huke
- Institut für Theoretische Physik, Universität des Saarlandes, D-66041 Saarbrücken, Germany
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14
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Agez G, Clerc MG, Louvergneaux E. Universal shape law of stochastic supercritical bifurcations: theory and experiments. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:026218. [PMID: 18352114 DOI: 10.1103/physreve.77.026218] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2006] [Revised: 01/11/2008] [Indexed: 05/26/2023]
Abstract
A universal analytical expression for the supercritical bifurcation shape of transverse one-dimensional (1D) systems in the presence of additive noise is given. The stochastic Langevin equation of such systems is solved by using a Fokker-Planck equation, leading to the expression for the most probable amplitude of the critical mode. From this universal expression, the shape of the bifurcation, its location, and its evolution with the noise level are completely defined. Experimental results obtained for a 1D transverse Kerr-type slice subjected to optical feedback are in excellent agreement.
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Affiliation(s)
- Gonzague Agez
- Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago, Chile
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15
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Hu HX, Li QS, Li S. Traveling and standing patterns induced by delay feedback in uniform oscillatory reaction–diffusion system. Chem Phys Lett 2007. [DOI: 10.1016/j.cplett.2007.09.031] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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16
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Alonso A, Batiste O, Meseguer A, Mercader I. Complex dynamical states in binary mixture convection with weak negative Soret coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:026310. [PMID: 17358425 DOI: 10.1103/physreve.75.026310] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2006] [Indexed: 05/14/2023]
Abstract
Binary convection in large-aspect-ratio annular containers heated from below is studied numerically for a water-ethanol mixture. High-resolution numerical tools based on spectral methods are used to solve the hydrodynamic equations in the two-dimensional approximation. The weakly nonlinear states arising very close to the onset of convection, the strongly nonlinear bursts of amplitude that precede the small-amplitude states, and the dispersive chaotic states encountered further above onset in experiments for mixtures with a weak negative Soret coupling are analyzed in detail in extended domains of aspect ratio 80. Steady localized states surrounded either by quiescent fluid or by small-amplitude waves are also obtained, and the role they play in the dynamics is elucidated.
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Affiliation(s)
- Arantxa Alonso
- Departament de Física Aplicada, Universitat Politècnica de Catalunya, Mòdul B4, 08034 Barcelona, Spain.
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17
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Hilt M, Zimmermann W. Hexagonal, square, and stripe patterns of the ion channel density in biomembranes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:016202. [PMID: 17358230 DOI: 10.1103/physreve.75.016202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/10/2006] [Indexed: 05/14/2023]
Abstract
Transmembrane ion flow through channel proteins undergoing density fluctuations may cause lateral gradients of the electrical potential across the membrane giving rise to electrophoresis of charged channels. A model for the dynamics of the channel density and the voltage drop across the membrane (cable equation) coupled to a binding-release reaction with the cell skeleton [P. Fromherz and W. Zimmerman, Phys. Rev. E 51, R1659 (1995)] is analyzed in one and two spatial dimensions. Due to the binding release reaction spatially periodic modulations of the channel density with a finite wave number are favored at the onset of pattern formation, whereby the wave number decreases with the kinetic rate of the binding-release reaction. In a two-dimensional extended membrane hexagonal modulations of the ion channel density are preferred in a large range of parameters. The stability diagrams of the periodic patterns near threshold are calculated and in addition the equations of motion in the limit of a slow binding-release kinetics are derived.
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Affiliation(s)
- Markus Hilt
- Theoretische Physik, Universität Bayreuth, D-95440 Bayreuth, Germany
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18
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Peter R, Zimmermann W. Traveling ion channel density waves affected by a conservation law. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:016206. [PMID: 16907176 DOI: 10.1103/physreve.74.016206] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2006] [Indexed: 05/11/2023]
Abstract
A model of mobile, charged ion channels embedded in a biomembrane is investigated. The ion channels fluctuate between an opened and a closed state according to a simple two-state reaction scheme whereas the total number of ion channels is a conserved quantity. Local transport mechanisms suggest that the ion channel densities are governed by electrodiffusionlike equations that have to be supplemented by a cable-type equation describing the dynamics of the transmembrane voltage. It is shown that the homogeneous distribution of ion channels may become unstable to either a stationary or an oscillatory instability. The nonlinear behavior immediately above threshold of an oscillatory bifurcation occurring at finite wave number is analyzed in terms of amplitude equations. Due to the conservation law imposed on ion channels, large-scale modes couple to the finite-wave-number instability and have thus to be included in the asymptotic analysis near the onset of pattern formation. A modified Ginzburg-Landau equation extended by long-wavelength stationary excitations is established, and it is highlighted how the global conservation law affects the stability of traveling ion channel density waves.
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Affiliation(s)
- Ronny Peter
- Theoretische Physik, Universität Bayreuth, D-95440 Bayreuth, Germany
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19
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Hammele M, Zimmermann W. Harmonic versus subharmonic patterns in a spatially forced oscillating chemical reaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:066211. [PMID: 16906950 DOI: 10.1103/physreve.73.066211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/15/2006] [Indexed: 05/11/2023]
Abstract
The effects of a spatially periodic forcing on an oscillating chemical reaction as described by the Lengyel-Epstein model are investigated. We find a surprising competition between two oscillating patterns, where one is harmonic and the other subharmonic with respect to the spatially periodic forcing. The occurrence of a subharmonic pattern is remarkable as well as its preference up to rather large values of the modulation amplitude. For small modulation amplitudes we derive from the model system a generic equation for the envelope of the oscillating reaction that includes an additional forcing contribution, compared to the amplitude equations known from previous studies in other systems. The analysis of this amplitude equation allows the derivation of analytical expressions even for the forcing corrections to the threshold and to the oscillation frequency, which are in a wide range of parameters in good agreement with the numerical analysis of the complete reaction equations. In the nonlinear regime beyond threshold, the subharmonic solutions exist in a finite range of the control parameter that has been determined by solving the reaction equations numerically for various sets of parameters.
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Affiliation(s)
- Martin Hammele
- Theoretische Physik, Universität Bayreuth, 95440 Bayreuth, Germany
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20
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Jung D, Lücke M. Traveling wave fronts and localized traveling wave convection in binary fluid mixtures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:026307. [PMID: 16196711 DOI: 10.1103/physreve.72.026307] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2005] [Indexed: 05/04/2023]
Abstract
Nonlinear fronts between spatially extended traveling wave (TW) convection and quiescent fluid and spatially localized traveling waves (LTWs) are investigated in quantitative detail in the bistable regime of binary fluid mixtures heated from below. A finite-difference method is used to solve the full hydrodynamic field equations in a vertical cross section of the layer perpendicular to the convection roll axes. Results are presented for ethanol-water parameters with several strongly negative separation ratios where TW solutions bifurcate subcritically. Fronts and LTWs are compared with each other and similarities and differences are elucidated. Phase propagation out of the quiescent fluid into the convective structure entails a unique selection of the latter while fronts and interfaces where the phase moves into the quiescent state behave differently. Interpretations of various experimental observations are suggested.
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Affiliation(s)
- D Jung
- Institut für Theoretische Physik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
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Jung D, Matura P, Lücke M. Oscillatory convection in binary mixtures: thermodiffusion, solutal buoyancy, and advection. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2004; 15:293-304. [PMID: 15592769 DOI: 10.1140/epje/i2004-10069-1] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2004] [Accepted: 10/21/2004] [Indexed: 05/24/2023]
Abstract
The role of thermodiffusive generation of concentration fluctuations via the Soret effect, their contribution to the buoyancy forces that drive convection, the advective mixing effect of the latter, and the diffusive homogenisation are compared and elucidated for oscillatory convection. Numerically obtained solutions of the field equations in the form of spatially extended relaxed traveling waves, of standing waves, and of the transient growth of standing waves and their transition to traveling waves are discussed as well as spatially localized convective states of traveling waves that are surrounded by the quiescent fluid.
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Affiliation(s)
- D Jung
- Institut für Theoretische Physik, Universität des Saarlandes, D-66041, Saarbrücken, Germany
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Matura P, Jung D, Lücke M. Standing-wave oscillations in binary mixture convection: from the onset via symmetry breaking to period doubling into chaos. PHYSICAL REVIEW LETTERS 2004; 92:254501. [PMID: 15245013 DOI: 10.1103/physrevlett.92.254501] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2003] [Indexed: 05/24/2023]
Abstract
Oscillatory solution branches of the hydrodynamic field equations describing convection in the form of a standing wave (SW) in binary fluid mixtures heated from below are determined completely for several negative Soret coefficients psi. Galerkin as well as finite-difference simulations were used. They were augmented by simple control methods to obtain also unstable SW states. For sufficiently negative psi, unstable SWs bifurcate subcritically out of the quiescent conductive state. They become stable via a saddle-node bifurcation when lateral phase pinning is exerted. Eventually their invariance under timeshift by half a period combined with reflection at midheight of the fluid layer gets broken. Thereafter, they terminate by undergoing a period-doubling cascade into chaos.
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Affiliation(s)
- P Matura
- Institut für Theoretische Physik, Universität des Saarlandes, D-66041 Saarbrücken, Germany
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Hammele M, Zimmermann W. Modeling oscillatory microtubule polymerization. PHYSICAL REVIEW E 2003; 67:021903. [PMID: 12636711 DOI: 10.1103/physreve.67.021903] [Citation(s) in RCA: 19] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2002] [Indexed: 11/07/2022]
Abstract
Polymerization of microtubules is ubiquitous in biological cells and under certain conditions it becomes oscillatory in time. Here, simple reaction models are analyzed that capture such oscillations as well as the length distribution of microtubules. We assume reaction conditions that are stationary over many oscillation periods, and it is a Hopf bifurcation that leads to a persistent oscillatory microtubule polymerization in these models. Analytical expressions are derived for the threshold of the bifurcation and the oscillation frequency in terms of reaction rates, and typical trends of their parameter dependence are presented. Both, a catastrophe rate that depends on the density of guanosine triphosphate liganded tubulin dimers and a delay reaction, such as the depolymerization of shrinking microtubules or the decay of oligomers, support oscillations. For a tubulin dimer concentration below the threshold, oscillatory microtubule polymerization occurs transiently on the route to a stationary state, as shown by numerical solutions of the model equations. Close to threshold, a so-called amplitude equation is derived and it is shown that the bifurcation to microtubule oscillations is supercritical.
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Affiliation(s)
- Martin Hammele
- Theoretical Physics, University of Saarland, D-66041 Saarbrücken, Germany
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Nicola EM, Or-Guil M, Wolf W, Bär M. Drifting pattern domains in a reaction-diffusion system with nonlocal coupling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:055101. [PMID: 12059625 DOI: 10.1103/physreve.65.055101] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2000] [Revised: 01/18/2002] [Indexed: 05/23/2023]
Abstract
Drifting pattern domains (DPDs), i.e., moving localized patches of traveling waves embedded in a stationary (Turing) pattern background and vice versa, are observed in simulations of a reaction-diffusion model with nonlocal coupling. Within this model, a region of bistability between Turing patterns and traveling waves arises from a codimension-2 Turing-wave bifurcation (TWB). DPDs are found within that region in a substantial distance from the TWB. We investigated the dynamics of single interfaces between Turing and wave patterns. It is found that DPDs exist due to a locking of the interface velocities, which is imposed by the absence of space-time defects near these interfaces.
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Affiliation(s)
- Ernesto M Nicola
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany
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Fütterer C, Lücke M. Growth of binary fluid convection: role of the concentration field. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:036315. [PMID: 11909252 DOI: 10.1103/physreve.65.036315] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2001] [Indexed: 05/23/2023]
Abstract
The growth of convection in binary fluid mixtures out of different perturbations of the quiescent conductive state is investigated using finite-difference numerical simulations for realistic ethanol-water parameters with strong negative Soret coupling between temperature and concentration fluctuations. Several different analysis tools are used to elucidate the complex spatiotemporal behavior associated with the dramatic concentration redistribution during the transients. It shows first the competition between counterpropagating waves that initially superimpose to form standing wave perturbations. Having reached a critical amplitude an advective breaking of the concentration wave triggers a very fast flow-induced transition from standing to traveling wave convection with large phase velocity and large concentration field amplitudes. Strongly nonlinear advective mixing and weak long-time diffusive homogenization then slow down the waves.
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Affiliation(s)
- C Fütterer
- Institut für Theoretische Physik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
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Jung D, Lücke M, Szprynger A. Influence of inlet and bulk noise on Rayleigh-Bénard convection with lateral flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:056301. [PMID: 11415000 DOI: 10.1103/physreve.63.056301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2000] [Indexed: 05/23/2023]
Abstract
Spatiotemporal properties of convective fluctuations and of their correlations are investigated theoretically in the vicinity of the threshold for onset of convection in the presence of a lateral through-flow using the full linearized equations of fluctuating hydrodynamics. The effect of external forcing by inlet boundary conditions on the downstream evolution of convective fields is separated from the effect of internal bulk thermal forcing with the use of spatial Laplace transformations. They show how the spatial variation of fluctuations and of their correlations are governed by the six spatial characteristic exponents of the field equations.
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Affiliation(s)
- D Jung
- Institut für Theoretische Physik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
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Büchel P, Lücke M. Localized perturbations in binary fluid convection with and without throughflow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:016307. [PMID: 11304353 DOI: 10.1103/physreve.63.016307] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2000] [Indexed: 05/23/2023]
Abstract
Dynamics and structure of spatially localized convective perturbations in binary fluid layers heated from below and the effect of a plane horizontal Poiseuille throughflow on them are investigated. Fronts and pulse-like wave packets formed out of the three relevant perturbations-two oscillatory ones and a stationary one-are analyzed after evaluating the appropriate saddle points of the three respective dispersion relations of the linear field equations over the complex wave number plane. Front and pulse properties are elucidated in quantitative detail as a function of small throughflow Reynolds numbers for different Soret coupling strengths psi including the pure fluid limit psi=0 in comparison with the appropriate Ginzburg-Landau amplitude equation approximations. Furthermore, small amplitude pulses and fronts obtained from solving the full nonlinear field equations numerically are presented to check and compare with the linear results.
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Affiliation(s)
- P Büchel
- Institut für Theoretische Physik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
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Buchel P, Lucke M. Influence of through flow on binary fluid convection. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:3793-3810. [PMID: 11088158 DOI: 10.1103/physreve.61.3793] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/1999] [Indexed: 05/23/2023]
Abstract
The influence of an externally imposed lateral Poiseuille through flow on linear, nonlinear, and transient behavior of transverse convective rolls in a horizontal layer of binary fluids heated from below is investigated. The convective roll solutions are determined numerically for realistic boundary conditions with a many-mode Galerkin expansion as well as with a finite-difference method. Bifurcation diagrams of various quantities like Nusselt number, frequency, and mixing behavior are determined as functions of heating rate and wave number for several through flow rates and Soret coupling strengths for ethanol-water parameters. The growth dynamics of small convective perturbations into different, strongly nonlinear convective states and the transition between the latter is studied also.
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Affiliation(s)
- P Buchel
- Institut fur Theoretische Physik, Universitat des Saarlandes, Postfach 151150,D-66041 Saarbrucken, Germany
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Jung C, Lücke M, Büchel P. Influence of through-flow on linear pattern formation properties in binary mixture convection. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:1510-1529. [PMID: 9965223 DOI: 10.1103/physreve.54.1510] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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31
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Colinet P, Géoris P, Legros JC, Lebon G. Spatially quasiperiodic convection and temporal chaos in two-layer thermocapillary instabilities. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:514-524. [PMID: 9965095 DOI: 10.1103/physreve.54.514] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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32
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Schmitz R, Zimmermann W. Spatially periodic modulated Rayleigh-Bénard convection. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:5993-6011. [PMID: 9964956 DOI: 10.1103/physreve.53.5993] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Lerman K, Ahlers G, Cannell DS. Different convection dynamics in mixtures with the same separation ratio. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:R2041-R2044. [PMID: 9964602 DOI: 10.1103/physreve.53.r2041] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Huntley DA, Davis SH. Effect of latent heat on oscillatory and cellular mode coupling in rapid directional solidification. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:3132-3144. [PMID: 9983822 DOI: 10.1103/physrevb.53.3132] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Treiber M. Analytic expressions for the stochastic amplitude equation for Taylor-Couette flow. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:577-585. [PMID: 9964288 DOI: 10.1103/physreve.53.577] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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36
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Dominguez-Lerma MA, Ahlers G, Cannell DS. Rayleigh-Bénard convection in binary mixtures with separation ratios near zero. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:6159-6174. [PMID: 9964133 DOI: 10.1103/physreve.52.6159] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Riecke H. Attractive interaction between pulses in a model for convection in binary mixtures. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:5685-5687. [PMID: 9964072 DOI: 10.1103/physreve.52.5685] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Wu M, Ahlers G, Cannell DS. Thermally Induced Fluctuations below the Onset of Rayleigh-Bénard Convection. PHYSICAL REVIEW LETTERS 1995; 75:1743-1746. [PMID: 10060380 DOI: 10.1103/physrevlett.75.1743] [Citation(s) in RCA: 23] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Barten W, Lücke M, Kamps M, Schmitz R. Convection in binary fluid mixtures. II. Localized traveling waves. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:5662-5680. [PMID: 9963299 DOI: 10.1103/physreve.51.5662] [Citation(s) in RCA: 50] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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40
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Barten W, Lücke M, Kamps M, Schmitz R. Convection in binary fluid mixtures. I. Extended traveling-wave and stationary states. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:5636-5661. [PMID: 9963298 DOI: 10.1103/physreve.51.5636] [Citation(s) in RCA: 76] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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41
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Fromherz P, Zimmermann W. Stable spatially periodic patterns of ion channels in biomembranes. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:R1659-R1662. [PMID: 9962950 DOI: 10.1103/physreve.51.r1659] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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42
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Quentin G, Rehberg I. Direct measurement of hydrodynamic fluctuations in a binary mixture. PHYSICAL REVIEW LETTERS 1995; 74:1578-1581. [PMID: 10059064 DOI: 10.1103/physrevlett.74.1578] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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43
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Kaplan E, Kuznetsov E, Steinberg V. Burst and collapse in traveling-wave convection of a binary fluid. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:3712-3722. [PMID: 9962425 DOI: 10.1103/physreve.50.3712] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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44
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Fang HP, Liu ZH. Predicting time-dependent motion in the double-diffusive convection system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 50:2790-2794. [PMID: 9962317 DOI: 10.1103/physreve.50.2790] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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45
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Karcher C, Müller U. Bénard convection in a binary mixture with a nonlinear density-temperature relation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:4031-4043. [PMID: 9961691 DOI: 10.1103/physreve.49.4031] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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46
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Treiber M, Kramer L. Stochastic envelope equations for nonequilibrium transitions and application to thermal fluctuations in electroconvection in nematic liquid crystals. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:3184-3198. [PMID: 9961587 DOI: 10.1103/physreve.49.3184] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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47
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Tsameret A, Goldner G, Steinberg V. Experimental evaluation of the intrinsic noise in the Couette-Taylor system with an axial flow. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:1309-1319. [PMID: 9961341 DOI: 10.1103/physreve.49.1309] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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