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Affiliation(s)
- Jean-Louis Bretonnet
- Institut of Physics, Université de Lorraine, LCP-A2MC, EA 3469, 1 Bd. François Arago, Metz F-57078, France
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Bidhoodi N, Das SP. Slow dynamics of a tagged particle in a supercooled liquid. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062309. [PMID: 26764694 DOI: 10.1103/physreve.92.062309] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2015] [Indexed: 06/05/2023]
Abstract
The ergodicity-nonergodicity (ENE) transition of the self-consistent mode-coupling theory (MCT) is marked by the point at which the time correlation of collective density fluctuations is not zero in the long-time limit. The nonergodic state, reaching beyond the ENE transition of simple MCT, is characterized by a finite shear modulus. The MCT, formulated in the current set of papers, predicts that the single-particle density correlation, unlike the collective density correlation, decays to zero at long times on either side of the ENE transition. The self-diffusion coefficient remains finite. This differs from the existing MCT results in which both collective and single-particle correlations are simultaniously frozen at the ENE transition. We discuss in this paper mechanisms by which a sharp fall in self-diffusion coefficient may occur within the present model. This overdamping or the so-called adiabatic approximation for the supercooled state does not maintain microscopic momentum conservation. Within this approximation, the self-diffusion constant approaches zero at the ENE transition point. This approximate result, which is similar to the prediction of the existing MCT models, further illustrates the process of cage formation with increase of density. At a qualitative level, our analysis shows that the self-diffusion process depends on the structure as well as short-time transport properties of the supercooled liquid. We solve the integral equations for the nonergodicity parameters to analyze the full implications of the adiabatic approximation.
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Affiliation(s)
- Neeta Bidhoodi
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Shankar P Das
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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McCowan DD. Numerical study of long-time dynamics and ergodic-nonergodic transitions in dense simple fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022107. [PMID: 26382344 DOI: 10.1103/physreve.92.022107] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2014] [Indexed: 06/05/2023]
Abstract
Since the mid-1980s, mode-coupling theory (MCT) has been the de facto theoretic description of dense fluids and the transition from the fluid state to the glassy state. MCT, however, is limited by the approximations used in its construction and lacks an unambiguous mechanism to institute corrections. We use recent results from a new theoretical framework--developed from first principles via a self-consistent perturbation expansion in terms of an effective two-body potential--to numerically explore the kinetics of systems of classical particles, specifically hard spheres governed by Smoluchowski dynamics. We present here a full solution for such a system to the kinetic equation governing the density-density time correlation function and show that the function exhibits the characteristic two-step decay of supercooled fluids and an ergodic-nonergodic transition to a dynamically arrested state. Unlike many previous numerical studies--and in stark contrast to experiment--we have access to the full time and wave-number range of the correlation function with great precision and are able to track the solution unprecedentedly close to the transition, covering nearly 15 decades in scaled time. Using asymptotic approximation techniques analogous to those developed for MCT, we fit the solution to predicted forms and extract critical parameters. We find complete qualitative agreement with known glassy behavior (e.g. power-law divergence of the α-relaxation time scale in the ergodic phase and square-root growth of the glass form factors in the nonergodic phase), as well as some limited quantitative agreement [e.g. the transition at packing fraction η*=0.60149761(10)], consistent with previous static solutions under this theory and with comparable colloidal suspension experiments. However, most importantly, we establish that this new theory is able to reproduce the salient features seen in other theories, experiments, and simulations but has the advantages of being derived from first principles and possessing a clear mechanism for making systematic corrections.
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Affiliation(s)
- David D McCowan
- The James Franck Institute and the Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
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Viermann C, Fabis F, Kozlikin E, Lilow R, Bartelmann M. Nonequilibrium statistical field theory for classical particles: Basic kinetic theory. Phys Rev E 2015; 91:062120. [PMID: 26172674 DOI: 10.1103/physreve.91.062120] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2014] [Indexed: 11/07/2022]
Abstract
Recently Mazenko and Das and Mazenko [Phys. Rev. E 81, 061102 (2010); J. Stat. Phys. 149, 643 (2012); J. Stat. Phys. 152, 159 (2013); Phys. Rev. E 83, 041125 (2011)] introduced a nonequilibrium field-theoretical approach to describe the statistical properties of a classical particle ensemble starting from the microscopic equations of motion of each individual particle. We use this theory to investigate the transition from those microscopic degrees of freedom to the evolution equations of the macroscopic observables of the ensemble. For the free theory, we recover the continuity and Jeans equations of a collisionless gas. For a theory containing two-particle interactions in a canonical perturbation series, we find the macroscopic evolution equations to be described by the Born-Bogoliubov-Green-Kirkwood-Yvon hierarchy with a truncation criterion depending on the order in perturbation theory. This establishes a direct link between the classical and the field-theoretical approaches to kinetic theory that might serve as a starting point to investigate kinetic theory beyond the classical limits.
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Affiliation(s)
- Celia Viermann
- Heidelberg University, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Philosophenweg 12, 69120 Heidelberg, Germany
| | - Felix Fabis
- Heidelberg University, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Philosophenweg 12, 69120 Heidelberg, Germany
| | - Elena Kozlikin
- Heidelberg University, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Philosophenweg 12, 69120 Heidelberg, Germany
| | - Robert Lilow
- Heidelberg University, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Philosophenweg 12, 69120 Heidelberg, Germany
| | - Matthias Bartelmann
- Heidelberg University, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Philosophenweg 12, 69120 Heidelberg, Germany
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Bidhoodi N, Das SP. Renormalized dynamics of the Dean-Kawasaki model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012325. [PMID: 26274179 DOI: 10.1103/physreve.92.012325] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2015] [Indexed: 06/04/2023]
Abstract
We study the model of a supercooled liquid for which the equation of motion for the coarse-grained density ρ(x,t) is the nonlinear diffusion equation originally proposed by Dean and Kawasaki, respectively, for Brownian and Newtonian dynamics of fluid particles. Using a Martin-Siggia-Rose (MSR) field theory we study the renormalization of the dynamics in a self-consistent form in terms of the so-called self-energy matrix Σ. The appropriate model for the renormalized dynamics involves an extended set of field variables {ρ,θ}, linked through a nonlinear constraint. The latter incorporates, in a nonperturbative manner, the effects of an infinite number of density nonlinearities in the dynamics. We show that the contributing element of Σ which renormalizes the bare diffusion constant D(0) to D(R) is same as that proposed by Kawasaki and Miyazima [Z. Phys. B Condens. Matter 103, 423 (1997)]. D(R) sharply decreases with increasing density. We consider the likelihood of a ergodic-nonergodic (ENE) transition in the model beyond a critical point. The transition is characterized by the long-time limit of the density correlation freezing at a nonzero value. From our analysis we identify an element of Σ which arises from the above-mentioned nonlinear constraint and is key to the viability of the ENE transition. If this self-energy would be zero, then the model supports a sharp ENE transition with D(R)=0 as predicted by Kawasaki and Miyazima. With the full model having nonzero value for this self-energy, the density autocorrelation function decays to zero in the long-time limit. Hence the ENE transition is not supported in the model.
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Affiliation(s)
- Neeta Bidhoodi
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Shankar P Das
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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Mazenko GF. Universal long-time dynamics in dense simple fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:022110. [PMID: 25353425 DOI: 10.1103/physreve.89.022110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/24/2013] [Indexed: 06/04/2023]
Abstract
Dense simple fluids appear to have a long-time, very slowly evolving state which--for high-enough density--approaches a glassy, nonergodic phase. In this high-density regime, the equilibrating system decays via a three-step process as identified in mode-coupling theory (MCT). MCT, however, is an inherently phenomenological theory without prospects for improvement, and so a systematic theory has been recently developed which naturally allows one to calculate cumulants between the fundamental particle density and response fields self-consistently in a perturbation expansion in a pseudopotential. This paper details the extension of this theory and specifically addresses the universal nature of this erogodic-nonergodic transition. We find that the nonergodic state can be characterized by the associated static equilibrium state, and--nontrivially--that these states are identical regardless of whether a system is driven by Smoluchowski or Newtonian dynamics. In addition, though the specific response fields differ between Smoluchowski and Newtonian dynamics, the two share identical linear fluctuation-dissipation relations connecting the ρ-B cumulants. While we show this universality of nonergodic states within perturbation theory, we expect it to be true more generally. At leading order in the perturbation expansion, one obtains a kinetic kernel quadratic in the density analogous to the "one-loop" theory of MCT. At this one-loop level one finds vertex corrections which depend on the three-point equilibrium cumulants. Here we assume these vertex-corrections can be ignored and instead focus on computing the contributions of higher-order loops. We show that it is possible to sum up all of the loop contributions and find that these higher-order loops do not change the nonergodic state parameters substantially.
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Affiliation(s)
- Gene F Mazenko
- The James Franck Institute and the Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
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Kim B, Kawasaki K, Jacquin H, van Wijland F. Equilibrium dynamics of the Dean-Kawasaki equation: mode-coupling theory and its extension. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012150. [PMID: 24580215 DOI: 10.1103/physreve.89.012150] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/04/2013] [Indexed: 06/03/2023]
Abstract
We extend a previously proposed field-theoretic self-consistent perturbation approach for the equilibrium dynamics of the Dean-Kawasaki equation presented in [Kim and Kawasaki, J. Stat. Mech. (2008) P02004]. By taking terms missing in the latter analysis into account we arrive at a set of three new equations for correlation functions of the system. These correlations involve the density and its logarithm as local observables. Our new one-loop equations, which must carefully deal with the noninteracting Brownian gas theory, are more general than the historic mode-coupling one in that a further approximation corresponding to Gaussian density fluctuations leads back to the original mode-coupling equation for the density correlations alone. However, without performing any further approximation step, our set of three equations does not feature any ergodic-nonergodic transition, as opposed to the historical mode-coupling approach.
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Affiliation(s)
- Bongsoo Kim
- Department of Physics and Institute for Soft and Bio Matter Science, Changwon National University, Changwon 641-773, Korea
| | - Kyozi Kawasaki
- Department of Physics, Faculty of Science, Kyushu University, Fukuoka 812-8581, Japan
| | - Hugo Jacquin
- Laboratoire Matière et Systèmes Complexes, UMR7057 CNRS/Paris 7, Université Denis Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France and Laboratoire de Physique, UMR5672 CNRS/ENS Lyon, École Normale Supérieure de Lyon, 46 allé d'Italie, 69364 Lyon Cedex 7, France
| | - Frédéric van Wijland
- Laboratoire Matière et Systèmes Complexes, UMR7057 CNRS/Paris 7, Université Denis Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France
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Jacquin H, Zamponi F. Systematic expansion in the order parameter for replica theory of the dynamical glass transition. J Chem Phys 2013; 138:12A542. [DOI: 10.1063/1.4792641] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Mazenko GF, McCowan DD, Spyridis P. Kinetic equations governing Smoluchowski dynamics in equilibrium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051105. [PMID: 23004701 DOI: 10.1103/physreve.85.051105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2011] [Revised: 04/03/2012] [Indexed: 06/01/2023]
Abstract
We continue our study of the statistical properties of particles in equilibrium obeying Smoluchowski dynamics. We show that the system is governed by a kinetic equation of the memory function form and that the memory function is given by one of the self-energies available via perturbation theory as introduced in previous work. We determine the memory function explicitly to second order in an expansion in a pseudopotential. The method we use allows for a straightforward computation of corrections via a formal expansion and we therefore view it as an improvement over the conventional mode-coupling theory (MCT) formalism where it is not clear how to make systematic corrections. In addition, the formalism we have introduced is flexible enough to allow for a wide array of different approximation schemes, including density expansions. The convergence criteria for our formal series are not worked out here, but the second-order equation that we derive is promising in the sense that it leads to analytic and numerical results consistent with expectations from computer simulations of the hard-sphere system in addition to replicating the desired features from conventional MCT (e.g., a two-step decay). These particular solutions will be discussed in forthcoming work.
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Affiliation(s)
- Gene F Mazenko
- The James Franck Institute and the Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
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