Volume 64, Issue 5
May 2005, pages 634a-827
pp 634a-634b May 2005
pp 635-643 May 2005
The struggle of Boltzmann with the proper description of the behavior of classical macroscopic bodies in equilibrium in terms of the properties of the particles out of which they consist will be sketched. He used both a dynamical and a statistical method. However, Einstein strongly disagreed with Boltzmann’s statistical method, arguing that a statistical description of a system should be based on the dynamics of the system. This opened the way, especially for complex systems, for other than Boltzmann statistics. The first non-Boltzmann statistics, not based on dynamics though, was proposed by Tsallis. A generalization of Tsallis’ statistics as a special case of a new class of superstatistics, based on Einstein’s criticism of Boltzmann, is discussed. It seems that perhaps a combination of dynamics and statistics is necessary to describe systems with complicated dynamics.
pp 645-660 May 2005
One challenge of biology, medicine, and economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture-crystalline or otherwise. Nonetheless, as if by magic, out of nothing but randomness one finds remarkably fine-tuned processes in time and remarkably fine-tuned structures in space. To understand this ‘miracle’, one might consider placing aside the human tendency to see the universe as a machine. Instead, one might address the challenge of uncovering how, through randomness (albeit, as we shall see, strongly correlated randomness), one can arrive at many spatial and temporal patterns in biology, medicine, and economics. Inspired by principles developed by statistical physics over the past 50 years-scale invariance and universality-we review some recent applications of correlated randomness to fields that might startle Boltzmann if he were alive today.
pp 661-677 May 2005
We present a time-dependent Ginzburg-Landau model of nonlinear elasticity in solid materials. We assume that the elastic energy density is a periodic function of the shear and tetragonal strains owing to the underlying lattice structure. With this new ingredient, solving the equations yields formation of dislocation dipoles or slips. In plastic flow high-density dislocations emerge at large strains to accumulate and grow into shear bands where the strains are localized. In addition to the elastic displacement, we also introduce the local free volumem. For very smallm the defect structures are metastable and long-lived where the dislocations are pinned by the Peierls potential barrier. However, if the shear modulus decreases with increasingm, accumulation ofm around dislocation cores eventually breaks the Peierls potential leading to slow relaxations in the stress and the free energy (aging). As another application of our scheme, we also study dislocation formation in two-phase alloys (coherency loss) under shear strains, where dislocations glide preferentially in the softer regions and are trapped at the interfaces.
pp 679-694 May 2005
Theoretical approaches to the development of an understanding of the behaviour of simple supercooled liquids near the structural glass transition are reviewed and our work on this problem, based on the density functional theory of freezing and replicated liquid state theory, are summarized in this context. A few directions for further work on this problem are suggested.
pp 695-707 May 2005
For systems in contact with two reservoirs at different densities or with two thermostats at different temperatures, the large deviation function of the density gives a possible way of extending the notion of free energy to non-equilibrium systems. This large deviation function of the density can be calculated explicitly for exclusion models in one dimension with open boundary conditions. For these models, one can also obtain the distribution of the current of particles flowing through the system and the results lead to a simple conjecture for the large deviation function of the current of more general diffusive systems.
pp 709-725 May 2005
Fluids adsorbed at micro-patterned and geometrically structured substrates can exhibit novel phase transitions and interfacial fluctuation effects distinct from those characteristic of wetting at planar, homogeneous walls. We review recent theoretical progress in this area paying particular attention to filling transitions pertinent to fluid adsorption near wedges, which have highlighted a deep connection between geometrical and contact angles. We show that filling transitions are not only characterized by large scale interfacial fluctuations leading to universal critical singularities but also reveal hidden symmetries with short-ranged critical wetting transitions and properties of dimensional reduction. We propose a non-local interfacial model which fulfills all these properties and throws light on long-standing problems regarding the order of the 3D short-range critical wetting transition.
pp 727-742 May 2005
This paper reviews the derivation of equations for slow dynamical processes in a variety of systems, including rotating rigid rotors, crystalline solids, isotropic and nematic elastomers, gels in an isotropic fluid background, and nematic liquid crystals. It presents a recent derivation of the Leslie-Ericksen equations for the dynamics of nematic liquid crystals that clarifies the nature of the nonhydrodynamic modes in these equations. As a final example of the phenomenological approach to slow dynamical processes, it discusses the dynamics of a driven nonequilibrium system: a two-dimensional gas of chiral ‘rattlebacks’ on a vibrating substrate.
pp 743-755 May 2005
We present a review of critical Casimir forces in connection with successive experiments on wetting near the critical point of helium mixtures.
pp 757-773 May 2005
It has been observed long ago that many systems from statistical physics behave randomly on macroscopic level at their critical temperature. In two dimensions, these phenomena have been classified by theoretical physicists thanks to conformal field theory, that led to the derivation of the exact value of various critical exponents that describe their behavior near the critical temperature. In the last couple of years, combining ideas of complex analysis and probability theory, mathematicians have constructed and studied a family of random fractals (called ‘Schramm-Loewner evolutions’ or SLE) that describe the only possible conformally invariant limits of the interfaces for these models. This gives a concrete construction of these random systems, puts various predictions on a rigorous footing, and leads to further understanding of their behavior. The goal of this paper is to survey some of these recent mathematical developments, and to describe a couple of basic underlying ideas. We will also briefly describe some very recent and ongoing developments relating SLE, Brownian loop soups and conformal field theory.
pp 775-783 May 2005
We present new results for the virial coefficientsBk for κ<- 10 for hard spheres in dimensionsD = 2,..., 8.
pp 785-801 May 2005
The structure of equilibrium density profiles in an electrolyte in the vicinity of an interface with an insulating or conductive medium is of crucial importance in chemical physics and colloidal science. The Coulomb interaction is responsible for screening effects, and in dilute solutions the latter effects give rise to universal leading corrections to nonideality, which distinguish electrolyte from nonelectrolyte solutions. An example is provided by the excess surface tension for an air-water interface, which is determined by the excess particle density, and which was first calculated by Onsager and Samaras. Because of the discrepancy between the dielectric constants on both sides of the interface, every charge in the electrolyte interacts with an electrostatic image, and the Boltzmann factor associated with the corresponding self-energy has an essential singularity over the length scalel from the wall. Besides Coulomb interactions, short-range repulsions must be taken into account in order to prevent the collapse between charges with opposite signs or between each charge and its image when the solvent dielectric constant is lower than that of the continuous medium on the other side of the interface. For a dilute and weaklycoupled electrolyte,l is negligible with respect to the bulk Debye screening length ξD. In the framework of the grand-canonical ensemble, systematic partial resummations in Mayer diagrammatics allow one to exhibit that, in this regime, the exact density profiles at leading order are the same as if they were calculated in a partially-linearized mean-field theory, where the screened pair interaction obeys an inhomogeneous Debye equation. In the latter equation the effective screening length depends on the distancex from the interface: it varies very fast over the lengthl and tends to its bulk value over a few ξDs. The equation can be solved iteratively at any distancex, and the exact density profiles are calculated analytically up to first order in the coupling parameter l/ξD. They show the interplay between three effects: (1) the geometric repulsion from the interface associated with the deformation of screening clouds, (2) the polarization effects described by the images on the other side of the interface, (3) the interaction between each charge and the potential drop created by the electric layer which appears as soon as the fluid has not a charge-symmetric composition. Moreover, the expressions allow us to go beyond Onsager-Samaras theory: the surface tension is calculated for charge-asymmetric electrolytes and for any value of the ratio between the dielectric constants on both sides of the interface. Similar diagrammatic techniques also allow one to investigate the charge renormalization in the dipolar effective pair interaction along the interface with an insulating medium.
pp 803-816 May 2005
Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard φ4 model. Analyzing these models systematically via modern field-theoretic renormalization group methods has been a long-standing challenge ever since their introduction in the middle of 1970s. We survey the recent progress made in this direction, discussing results obtained via dimensionality expansions, how they compare with Monte Carlo results, and open problems. These advances opened the way towards systematic studies of boundary critical behavior atm-axial Lifshitz points. The possible boundary critical behavior depends on whether the surface plane is perpendicular to one of them modulation axes or parallel to all of them. We show that the semi-infinite field theories representing the corresponding surface universality classes in these two cases of perpendicular and parallel surface orientation differ crucially in their Hamiltonian’s boundary terms and the implied boundary conditions, and explain recent results along with our current understanding of this matter.
pp 817-827 May 2005
In this paper, we discuss why functional renormalization is an essential tool to treat strongly disordered systems. More specifically, we treat elastic manifolds in a disordered environment. These are governed by a disorder distribution, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss how a renormalizable field theory can be constructed even beyond 2-loop order. We then consider an elastic manifold embedded inN dimensions, and give the exact solution forN →ɛ This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. Finally, the effective action at order 1/N is reported.
Volume 93 | Issue 6
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