Dimensional dependence of the Stokes-Einstein relation and its violation.

We generalize to higher spatial dimensions the Stokes-Einstein relation (SER) as well as the leading correction to diffusivity in finite systems with periodic boundary conditions, and validate these results with numerical simulations. We then investigate the evolution of the high-density SER violation with dimension in simple hard sphere glass formers. The analysis suggests that this SER violation disappears around dimension d(u) = 8, above which it is not observed. The critical exponent associated with the violation appears to evolve linearly in 8 - d, below d = 8, as predicted by Biroli and Bouchaud [J. Phys.: Condens. Matter 19, 205101 (2007)], but the linear coefficient is not consistent with the prediction. The SER violation with d establishes a new benchmark for theory, and its complete description remains an open problem.

[1]  F. Sciortino,et al.  Mixing effects for the structural relaxation in binary hard-sphere liquids. , 2003, Physical review letters.

[2]  J. Skinner,et al.  Hydrodynamic boundary conditions, the Stokes–Einstein law, and long-time tails in the Brownian limit , 2003 .

[3]  F. Stillinger,et al.  Relaxation processes in liquids: variations on a theme by Stokes and Einstein. , 2013, The Journal of chemical physics.

[4]  Pablo G. Debenedetti,et al.  Supercooled liquids and the glass transition , 2001, Nature.

[5]  Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions , 2006, cond-mat/0503525.

[6]  M. Nakahara.  Geometry, Topology and Physics , 2018 .

[7]  Gerhard Hummer,et al.  System-Size Dependence of Diffusion Coefficients and Viscosities from Molecular Dynamics Simulations with Periodic Boundary Conditions , 2004 .

[8]  J. Lutsko.  Transport properties of dense dissipative hard-sphere fluids for arbitrary energy loss models. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Patrick Charbonneau,et al.  Geometrical frustration and static correlations in hard-sphere glass formers. , 2012, The Journal of chemical physics.

[10]  Stephen Childress,et al.  An Introduction to Theoretical Fluid Mechanics , 2009 .

[11]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[12]  T. R. Kirkpatrick,et al.  Scaling concepts for the dynamics of viscous liquids near an ideal glassy state. , 1989, Physical review. A, General physics.

[13]  Jean-Philippe Bouchaud,et al.  Critical fluctuations and breakdown of the Stokes–Einstein relation in the mode-coupling theory of glasses , 2006, cond-mat/0609705.

[14]  Steven W. Smith,et al.  Molecular dynamics study of transport coefficients for hard‐chain fluids , 1995 .

[15]  H. Sillescu,et al.  Translational and rotational diffusion in supercooled orthoterphenyl close to the glass transition , 1992 .

[16]  A. Einstein.  Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen [AdP 17, 549 (1905)] , 2005, Annalen der Physik.

[17]  I. R. Mcdonald,et al.  Theory of simple liquids , 1998 .

[18]  T. R. Kirkpatrick,et al.  Dynamics of the structural glass transition and the p-spin-interaction spin-glass model. , 1987, Physical review letters.

[19]  The short-time behavior of the velocity autocorrelation function of smooth, hard hyperspheres in three, four and five dimensions , 1985 .

[20]  D. Frenkel,et al.  Thermodynamic properties of binary hard sphere mixtures , 1991 .

[21]  D. Heyes.  System size dependence of the transport coefficients and Stokes–Einstein relationship of hard sphere and Weeks–Chandler–Andersen fluids , 2007 .

[22]  F. Sciortino,et al.  α-relaxation processes in binary hard-sphere mixtures , 2004 .

[23]  R. Kapral,et al.  Molecular theory of translational diffusion: Microscopic generalization of the normal velocity boundary condition , 1979 .

[24]  J. G. Powles,et al.  Viscoelasticity of fluids with steeply repulsive potentials , 2003 .

[25]  Stillinger,et al.  Translation-rotation paradox for diffusion in fragile glass-forming liquids. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  M. Cicerone,et al.  Enhanced translation of probe molecules in supercooled o‐terphenyl: Signature of spatially heterogeneous dynamics? , 1996 .

[27]  Giorgio Parisi,et al.  Quantitative field theory of the glass transition , 2012, Proceedings of the National Academy of Sciences.

[28]  K. Schweizer,et al.  Derivation of a microscopic theory of barriers and activated hopping transport in glassy liquids and suspensions. , 2005, The Journal of chemical physics.

[29]  Srikanth Sastry,et al.  Breakdown of the Stokes-Einstein relation in two, three, and four dimensions. , 2012, The Journal of chemical physics.

[30]  Berend Smit,et al.  Accelerating Monte Carlo Sampling , 2002 .

[31]  Fast simulation of facilitated spin models , 2005, cond-mat/0510356.

[32]  Length scale for the onset of Fickian diffusion in supercooled liquids , 2004, cond-mat/0409428.

[33]  D. Lévesque,et al.  Molecular-dynamics investigation of tracer diffusion in a simple liquid: test of the Stokes-Einstein law. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  H. Hasimoto.  On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres , 1959, Journal of Fluid Mechanics.

[35]  David R Reichman,et al.  Spatial dimension and the dynamics of supercooled liquids , 2009, Proceedings of the National Academy of Sciences.

[36]  Dynamical exchanges in facilitated models of supercooled liquids. , 2005, The Journal of chemical physics.

[37]  Kunimasa Miyazaki,et al.  Cooperativity beyond caging: Generalized mode-coupling theory. , 2006, Physical review letters.

[38]  Giorgio Parisi,et al.  Glass transition and random close packing above three dimensions. , 2011, Physical review letters.

[39]  Marco Zoppi,et al.  Dynamics of the liquid state , 1994 .

[40]  F. Stillinger,et al.  Jammed hard-particle packings: From Kepler to Bernal and beyond , 2010, 1008.2982.

[41]  O. Blondel,et al.  Is there a fractional breakdown of the Stokes-Einstein relation in kinetically constrained models at low temperature? , 2013, 1307.1651.

[42]  Toy model for the mean-field theory of hard-sphere liquids , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[43]  J. P. Garrahan,et al.  Excitations Are Localized and Relaxation Is Hierarchical in Glass-Forming Liquids , 2011, 1107.3628.

[44]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[45]  Salvatore Torquato,et al.  Densest binary sphere packings. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  I. Procaccia,et al.  Finite-size scaling for the glass transition: the role of a static length scale. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  Peter G Wolynes,et al.  Facilitation, complexity growth, mode coupling, and activated dynamics in supercooled liquids , 2008, Proceedings of the National Academy of Sciences.

[48]  F. Ricci-Tersenghi,et al.  Field theory of fluctuations in glasses , 2011, The European physical journal. E, Soft matter.

[49]  B. Alder,et al.  Studies in Molecular Dynamics. VIII. The Transport Coefficients for a Hard-Sphere Fluid , 1970 .

[50]  P. Wolynes,et al.  Structural Glasses and Supercooled Liquids: Theory, Experiment, and Applications , 2012 .

[51]  P. Harrowell,et al.  Origin of the Difference in the Temperature Dependences of Diffusion and Structural Relaxation in a Supercooled Liquid , 1998 .

[52]  Thirumalai,et al.  Comparison between dynamical theories and metastable states in regular and glassy mean-field spin models with underlying first-order-like phase transitions. , 1988, Physical review. A, General physics.

[53]  Jack F Douglas,et al.  Nature of the breakdown in the Stokes-Einstein relationship in a hard sphere fluid. , 2006, The Journal of chemical physics.

[54]  Jean-Philippe Bouchaud,et al.  Inhomogeneous mode-coupling theory and growing dynamic length in supercooled liquids. , 2006, Physical review letters.

[55]  P. Charbonneau,et al.  Hard-sphere crystallization gets rarer with increasing dimension. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  R. Goodman,et al.  Representations and Invariants of the Classical Groups , 1998 .

[57]  M. Smoluchowski.  Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen , 1906 .

[58]  H L Frisch,et al.  High dimensionality as an organizing device for classical fluids. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[59]  P. Daivis,et al.  Comparison of constant pressure and constant volume nonequilibrium simulations of sheared model decane , 1994 .

[60]  Ludovic Berthier,et al.  Universal nature of particle displacements close to glass and jamming transitions. , 2007, Physical review letters.

[61]  P. Charbonneau,et al.  Numerical and theoretical study of a monodisperse hard-sphere glass former. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[62]  H. Brenner.  The translational and rotational motions of an n-dimensional hypersphere through a viscous fluid at small Reynolds numbers , 1981, Journal of Fluid Mechanics.

[63]  T. R. Kirkpatrick,et al.  Connections between some kinetic and equilibrium theories of the glass transition. , 1987, Physical review. A, General physics.

[64]  R. Goodman,et al.  Symmetry, Representations, and Invariants , 2009 .

[65]  H. Sillescu,et al.  Heterogeneity at the Glass Transition: Translational and Rotational Self-Diffusion , 1997 .

[66]  M. Cicerone,et al.  Photobleaching technique for measuring ultraslow reorientation near and below the glass transition : tetracene in o-terphenyl , 1993 .

[67]  Giorgio Parisi,et al.  Dimensional study of the caging order parameter at the glass transition , 2012, Proceedings of the National Academy of Sciences.

[68]  Claudio Procesi,et al.  Lie Groups: An Approach through Invariants and Representations , 2006 .

[69]  Srikanth Sastry,et al.  Growing length and time scales in glass-forming liquids , 2008, Proceedings of the National Academy of Sciences.

[70]  L. Berthier,et al.  Superdiffusive, heterogeneous, and collective particle motion near the fluid-solid transition in athermal disordered materials , 2010, 1001.0914.

[71]  Berend Smit,et al.  Understanding Molecular Simulation , 2001 .

[72]  Kurt Kremer,et al.  Molecular dynamics simulation of a polymer chain in solution , 1993 .

[73]  G. Tarjus,et al.  BREAKDOWN OF THE STOKES-EINSTEIN RELATION IN SUPERCOOLED LIQUIDS , 1995 .

[74]  G. Szamel,et al.  Analysis of a growing dynamic length scale in a glass-forming binary hard-sphere mixture. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[75]  R. Zwanzig,et al.  High‐Frequency Elastic Moduli of Simple Fluids , 1965 .

[76]  H. Flanders.  Differential Forms with Applications to the Physical Sciences , 1964 .

[77]  Daan Frenkel,et al.  Simulations: The dark side , 2012, The European Physical Journal Plus.

[78]  Monica L. Skoge,et al.  Packing hyperspheres in high-dimensional Euclidean spaces. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[79]  P. Charbonneau,et al.  Decorrelation of the static and dynamic length scales in hard-sphere glass formers. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.