• Volume 64, Issue 6

June 2005,   pages  828-1202

• Foreword

We investigate the solvability of a variety of well-known problems in lattice statistical mechanics. We provide a new numerical procedure which enables one to conjecture whether the solution falls into a class of functions called differentiably finite functions. Almost all solved problems fall into this class. The fact that one can conjecture whether a given problem is or is not 𝐷-finite then informs one as to whether the solution is likely to be tractable or not. We also show how, for certain problems, it is possible to prove that the solutions are not 𝐷-finite, based on the work of Rechnitzer $[1–3]$.

• The analytic structure of lattice models — why can’t we solve most models?

We investigate the solvability of a variety of well-known problems in lattice statistical mechanics. We provide a new numerical procedure which enables one to conjecture whether the solution falls into a class of functions calleddifferentiably finite functions. Almost all solved problems fall into this class. The fact that one can conjecture whether a given problem is or is not D-finite then informs one as to whether the solution is likely to be tractable or not. We also show how, for certain problems, it is possible to prove that the solutions are notD-finite, based on the work of Rechnitzer [1–3].

• Kardar—Parisi—Zhang equation in one dimension and line ensembles

For suitably discretized versions of the Kardar—Parisi—Zhang equation in one space dimension, exact scaling functions are available, amongst them the stationary two-point function. We explain one central piece from the technology through which such results are obtained, namely the method of line ensembles with purely entropic repulsion.

• Factorised steady states and condensation transitions in nonequilibrium systems

Systems driven out of equilibrium can often exhibit behaviour not seen in systems in thermal equilibrium —for example phase transitions in one-dimensional systems. In this talk I will review a simple model of a nonequilibrium system known as the ‘zero-range process’ and its recent developments. The nonequilibrium stationary state of this model factorises and this property allows a detailed analysis of several ‘condensation’ transitions wherein a finite fraction of the constituent particles condenses onto a single lattice site. I will then consider a more general class of mass transport models, encompassing continuous mass variables and discrete time updating, and present a necessary and sufficient condition for the steady state to factorise. The property of factorisation again allows an analysis of the condensation transitions which may occur.

• Nonequilibrium relaxation method — an alternative simulation strategy

One well-established simulation strategy to study the thermal phases and transitions of a given microscopic model system is the so-called equilibrium method, in which one first realizes the equilibrium ensemble of a finite system and then extrapolates the results to infinite system. This equilibrium method traces over the standard theory of the thermal statistical mechanics, and over the idea of the thermodynamic limit. Recently, an alternative simulation strategy has been developed, which analyzes the nonequilibrium relaxation (NER) process. It is called theNER method. NER method has some advantages over the equilibrium method. The NER method provides a simpler analyzing procedure. This implies less systematic error which is inevitable in the simulation and provides efficient resource usage. The NER method easily treats not only the thermodynamic limit but also other limits, for example, non-Gibbsian nonequilibrium steady states. So the NER method is also relevant for new fields of the statistical physics. Application of the NER method have been expanding to various problems: from basic first- and second-order transitions to advanced and exotic phases like chiral, KT spin-glass and quantum phases. These studies have provided, not only better estimations of transition point and exponents, but also qualitative developments. For example, the universality class of a random system, the nature of the two-dimensional melting and the scaling behavior of spin-glass aging phenomena have been clarified.

• Wetting and phase separation at surfaces

We study the problem ofsurfacedirected spinodal decomposition, viz., the dynamical interplay of wetting and phase separation at surfaces. In particular, we focus on the kinetics of wetting-layer growth in a semi-infinite geometry for arbitrary surface potentials and mixture compositions. We also present representative results for phase separation in confined geometries, e.g., cylindrical pores, thin films, etc.

• Structure and cluster formation in granular media

The two most important phenomena at the basis of granular media are excluded volume and dissipation. The former is captured by the hard sphere model and is responsible for, e.g., crystallization, the latter leads to interesting structures like clusters in non-equilibrium dynamical, freely cooling states. The freely cooling system is examined concerning the energy decay and the cluster evolution in time. Corrections for crystallization and multi-particle contacts are provided, which become more and more important with increasing density.

• Control and characterization of spatio-temporal disorder in parametrically excited surface waves

The nonlinear interactions of parametrically excited surface waves have been shown to yield a rich family of nonlinear states. When the system is driven by two commensurate frequencies, a variety of interesting superlattice type states are generated via a number of different 3-wave resonant interactions. These states occur either as symmetry-breaking bifurcations of hexagonal patterns composed of a single unstable mode or via nonlinear interactions between the two different unstable modes generated by the two forcing frequencies. Near the system’s bicritical point, a well-defined region of phase space exists in which a highly disordered state, both in space and time, is observed. We first show that this state results from the competition between two distinct nonlinear super-lattice states, each with different characteristic temporal and spatial symmetries. After characterizing the type of spatio-temporal disorder that is embodied in this disordered state, we will demonstrate that it can be controlled. Control to either of its neighboring nonlinear states is achieved by the application of a small-amplitude excitation at a third frequency, where the spatial symmetry of the selected pattern is determined by the temporal symmetry of the third frequency used. This technique can also excite rapid switching between different nonlinear states.

• Non-stationary probabilities for the asymmetric exclusion process on a ring

A solution of the master equation for a system of interacting particles for finite time and particle density is presented. By using a new form of the Bethe ansatz, the totally asymmetric exclusion process on a ring is solved for arbitrary initial conditions and time intervals.

• Pattern formations in chaotic spatio-temporal systems

Pattern formations in chaotic spatio-temporal systems modelled by coupled chaotic oscillators are investigated. We focus on various symmetry breakings and different kinds of chaos synchronization-desynchronization transitions, which lead to certain types of spontaneous spatial orderings and the emergence of some typical ordered patterns, such as rotating wave patterns with splay phase ordering (orientational symmetry breaking) and partially synchronous standing wave patterns with in-phase ordering (translational symmetry breaking). General pictures of the global behaviors of pattern formations and transitions in coupled chaotic oscillators are provided.

• Does the flatness of the velocity derivative blow up at a finite Reynolds number?

A tentative suggestion is made that the flatness of the velocity derivative could reach an infinite value at finite (though very large) Reynolds number, with possible implications for the singularities of the Navier—Stokes equations. A direct test of this suggestion requires measurements at Reynolds numbers presently outside the experimental capacity, so an alternative suggestion that can be tested at accessible Reynolds numbers is also made.

• Intermittency at critical transitions and aging dynamics at the onset of chaos

We recall that at both the intermittency transitions and the Feigenbaum attractor, in unimodal maps of non-linearity of order ζ &gt; 1, the dynamics rigorously obeys the Tsallis statistics. We account for theq-indices and the generalized Lyapunov coefficients λq that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the onset of chaos with the appearance of a special value for the entropic indexq. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.

• Where do ions solvate?

We study a simple model of ionic solvation inside a water cluster. The cluster is modeled as a spherical dielectric continuum. It is found that unpolarizable ions always prefer the bulk solvation. On the other hand, for polarizable ions, there exists a critical value of polarization above which surface solvation becomes energetically favorablefor large enough water clusters.

• Jamming patterns in a two-dimensional hopper

We report experimental studies of jamming phenomenon of monodisperse metal disks falling through a two-dimensional hopper when the hopper opening is larger than three times the size of the disks. For each jamming event, the configuration of the arch formed at the hopper opening is studied. The cumulative distribution functionsfd(X) for hoppers of opening sized are measured. (HereX is the horizontal component of the arch vector, which is defined as the displacement vector from the center of the first disk to the center of the last disk in the arch.) We found that the distribution offd(X) can be collasped into a master curveG(X) = fd(X)μ(d) that decays exponentially forX &gt; 4. The scaling factorμ(d) is a decreasing function ofd and is approximately proportional to the jamming probability.

• The depletion potential in one, two and three dimensions

We study the behavior of the depletion potential in binary mixtures of hard particles in one, two, and three dimensions within the framework of a general theory for depletion potential using density functional theory. By doing so we extend earlier studies of the depletion potential in three dimensions to the cases ofd = 1 and 2 about which little is known, despite their importance for experiments. We also verify scaling relations between depletion potentials in sphere-sphere and wall-sphere geometries ind = 3 and in disk-disk and wall-disk geometries ind = 2, which originate from geometrical considerations.

• Polymer mixtures in confined geometries: Model systems to explore phase transitions

While binary (A,B) symmetric polymer mixtures ind = 3 dimensions have an unmixing critical point that belongs to the 3d Ising universality class and crosses over to mean field behavior for very long chains, the critical behavior of mixtures confined into thin film geometry falls in the 2d Ising class irrespective of chain length. The critical temperature always scales linearly with chain length, except for strictly two-dimensional chains confined to a plane, for whichTcN5/8 (this unusual exponent describes the fractal contact line between segregated chains in dense melts in two spatial dimensions,d = 2). When the walls of the thin film are not neutral, but preferentially attract one species, complex phase diagrams occur due to the interplay between capillary condensation and wetting phenomena. For ‘competing walls’ (one wall prefers A, the other prefers B) particularly interesting interface localization-delocalization transitions occur, while analogous phenomena in wedges are related to the ‘filling transition’.

• Colloidal interactions in two-dimensional nematic emulsions

We review theoretical and experimental work on colloidal interactions in two-dimensional (2D) nematic emulsions. We pay particular attention to the effects of (i) the nematic elastic constants, (ii) the size of the colloids, and (iii) the boundary conditions at the particles and the container. We consider the interactions between colloids and fluid (deformable) interfaces and the shape of fluid colloids in smectic-C films.

• Local simulation algorithms for Coulombic interactions

We consider a problem in dynamically constrained Monte Carlo dynamics and show that this leads to the generation of long ranged effective interactions. This allows us to construct a local algorithm for the simulation of charged systems without ever having to evaluate pair potentials or solve the Poisson equation. We discuss a simple implementation of a charged lattice gas as well as more elaborate off-lattice versions of the algorithm. There are analogies between our formulation of electrostatics and the bosonic Hubbard model in the phase approximation. Cluster methods developed for this model further improve the efficiency of the electrostatics algorithm.

• Knots in polymers

Knots and topological entanglements play an important role in the statistical mechanics of polymers. While topological entanglement is a global property, it is possible to study the size of a knotted region both numerically and analytically. It can be shown that long-range repulsive interactions, as well as entropy favor small knots in dilute systems. However, in dense systems and at the Θ-point in two dimensions the uncontracted knot configuration is the most likely.

• Droplet dynamics on patterned substrates

We present a lattice Boltzmann algorithm which can be used to explore the spreading of droplets on chemically and topologically patterned substrates. As an example we use the method to show that the final configuration of a drop on a substrate comprising hydrophobic and hydrophilic stripes can depend sensitively on the dynamical pathway by which the state is reached. We also consider a substrate covered with micron-scale posts and investigate how this can lead to superhydrophobic behaviour. Finally we model how a Namibian desert beetle collects water from the wind.

• Fluctuation-induced forces in and out of equilibrium

In a fluctuating medium of quantum, thermal, or non-thermal origin, an interaction is induced between external objects that modify the fluctuations. These interactions can appear in a vast variety of systems, leading to a plethora of interesting phenomena. Notable examples of these include: (1) like-charge attraction in the presence of multivalent counterions, (2) Ludwig-Soret effect in charged colloids, (3) mass renormalization of moving defects in a phononic background and moving metallic objects in EM quantum vacuum, and (4) dissipation due to motion-induced radiation. The fluctuationinduced forces are statistical in nature, and this could make their measurement very difficult, because the actual value of the force might deviate most of the time from the predicted average value.

• Keldysh proximity action for disordered superconductors

We review a novel approach to the superconductive proximity effect in disordered normal-superconducting (N-S) structures. The method is based on the multicharge Keldysh action and is suitable for the treatment of interaction and fluctuation effects. As an application of the formalism, we study the subgap conductance and noise in two-dimensional N-S systems in the presence of the electron-electron interaction in the Cooper channel. It is shown that singular nature of the interaction correction at large scales leads to a nonmonotonuos temperature, voltage and magnetic field dependence of the Andreev conductance.

• Effect of interactions, disorder and magnetic field in the Hubbard model in two dimensions

The effects of both interactions and Zeeman magnetic field in disordered electronic systems are explored in the Hubbard model on a square lattice. We investigate the thermodynamic (density, magnetization, density of states) and transport (conductivity) properties using determinantal quantum Monte Carlo and inhomogeneous Hartree Fock techniques. We find that at half filling there is a novel metallic phase at intermediate disorder that is sandwiched between a Mott insulator and an Anderson insulator. The metallic phase is highly inhomogeneous and coexists with antiferromagnetic long-range order. At quarter filling also the combined effects of disorder and interactions produce a conducting state which can be destroyed by applying a Zeeman field, resulting in a magnetic field-driven transition. We discuss the implication of our results for experiments.

• A new theory of doped manganites exhibiting colossal magnetoresistance

Rare earth manganites doped with alkaline earths, namely Re1-xAxMnO3, exhibit colossal magnetoresistance, metal insulator transitions, competing magnetic, orbital and charge ordering, and many other interesting but poorly understood phenomena. In this article I outline our recent theory based on the idea that in the presence of strong Jahn-Teller, Coulomb and Hund’s couplings present in these materials, the low-energy electronic states dynamically reorganize themselves into two sets: one set (ℓ) which are polaronic, i.e., localized and accompanied by large local lattice distortion, and another (b) which are non-polaronic and band-like. The coexistence of the radically different ℓ andb states, and the sensitive dependence of their relative energies and occupation upon dopingx, temperatureT, magnetic fieldH, etc., underlies the unique effects seen in manganites. I present results from strong correlation calculations using dynamical mean-field theory and simulations on a new 2-fluid model which accord with a variety of observations.

• A sigma-model approach to glassy dynamics

In this contribution we review recent progress in understanding fluctuations in the aging process of macroscopic systems, and we propose further tests of these ideas. We discuss how the emergence of a symmetry in aging systems, global timereparametrization invariance, could be responsible for the observed ‘universal’ behavior of local and mesoscopic non-equilibrium fluctuations. We discuss (i) the two-time scaling and functional form of the distribution of local correlations and responses; (ii) the scaling of multi-time correlations and susceptibilities; (iii) how the above can be derived from a random surface effective action; (iv) the behavior of a diverging two-time dependent correlation length; (v) how these ideas apply to off-lattice particle systems.

• Some recent developments in spin glasses

I give some experimental and theoretical background to spin glasses, and then discuss the nature of the phase transition in spin glasses withvector spins. Results of Monte Carlo simulations of the Heisenberg spin glass model in three dimensions are presented. A finite-size scaling analysis of the correlation length of the spins and chiralities shows that there is a single, finite-temperature transition at which both spins and chiralities order.

• Models of plastic depinning of driven disordered systems

Two classes of models of driven disordered systems that exhibit historydependent dynamics are discussed. The first class incorporates local inertia in the dynamics via nonmonotonic stress transfer between adjacent degrees of freedom. The second class allows for proliferation of topological defects due to the interplay of strong disorder and drive. In mean field theory both models exhibit a tricritical point as a function of disorder strength. At weak disorder depinning is continuous and the sliding state is unique. At strong disorder depinning is discontinuous and hysteretic.

• Aging, rejuvenation and memory phenomena in spin glasses

In this paper, we review several important features of the out-of-equilibrium dynamics of spin glasses. Starting with the simplest experiments, we discuss the scaling laws used to describe the isothermal aging observed in spin glasses after a quench down to the low-temperature phase. We report in particular new results on the sub-aging behaviour of spin glasses. We then discuss the rejuvenation and memory effects observed when a spin glass is submitted to temperature variations during aging, from the point of view of both energy landscape pictures and real-space pictures. We highlight the fact that both approaches point out the necessity of hierarchical processes involved in aging. Finally, we report an investigation of the effect of small temperature variations on aging in spin glass samples with various anisotropies which indicates that this hierarchy depends on the spin anisotropy.

• Measuring information networks

Traffic and communication between different parts of a complex system are fundamental elements in maintaining its overall cooperativity. Because a complex system consists of many different parts, it matters where signals are transmitted. Thus signaling and traffic are in principle specific, with each message going from a unique sender to a specific recipient. In the current paper we review some measures of network topology that are related to its ability to direct specific communication.

• Fusion of biological membranes

The process of membrane fusion has been examined by Monte Carlo simulation, and is found to be very different than the conventional picture. The differences in mechanism lead to several predictions, in particular that fusion is accompanied by transient leakage. This prediction has recently been verified. Self-consistent field theory is applied to examine the free energy barriers in the different scenarios.

• Single-molecule experiments in biophysics: Exploring the thermal behavior of nonequilibrium small systems

Biomolecules carry out very specialized tasks inside the cell where energies involved are few tens ofκBT, small enough for thermal fluctuations to be relevant in many biomolecular processes. In this paper I discuss a few concepts and present some experimental results that show how the study of fluctuation theorems applied to biomolecules contributes to our understanding of the nonequilibrium thermal behavior of small systems.

• Scale-free random graphs and Potts model

We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertexi has a prescribed weight Pi ∝ i (0 &lt; μ&lt; 1) and an edge can connect verticesi andj with ratePiPj. Corresponding equilibrium ensemble is identified and the problem is solved by theq → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ &gt; 3 and 2 &lt; λ &lt; 3, whereλ = 1 +μ-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finiteN shows double peaks.

• Statistical physics, optimization and source coding

The combinatorial problem of satisfying a given set of constraints that depend on N discrete variables is a fundamental one in optimization and coding theory. Even for instances of randomly generated problems, the question “does there exist an assignment to the variables that satisfies all constraints?” may become extraordinarily difficult to solve in some range of parameters where a glass phase sets in. We shall provide a brief review of the recent advances in the statistical mechanics approach to these satisfiability problems and show how the analytic results have helped to design a new class of message-passing algorithms — the survey propagation (SP) algorithms — that can efficiently solve some combinatorial problems considered intractable. As an application, we discuss how the packing properties of clusters of solutions in randomly generated satisfiability problems can be exploited in the design of simple lossy data compression algorithms.

• Understanding search trees via statistical physics

We study the randomm-ary search tree model (wherem stands for the number of branches of the search tree), an important problem for data storage in computer science, using a variety of statistical physics techniques that allow us to obtain exact asymptotic results. In particular, we show that the probability distributions of extreme observables associated with a random search tree such as the height and the balanced height of a tree have a travelling front structure. In addition, the variance of the number of nodes needed to store a data string of a given sizeN is shown to undergo a striking phase transition at a critical value of the branching ratiomc = 26. We identified the mechanism of this phase transition and showed that it is generic and occurs in various other problems as well. New results are obtained when each element of the data string is a D-dimensional vector. We show that this problem also has a phase transition at a critical dimension,Dc = π/ sin-1 (l/√8) = 8.69363 …

• Subject Index of Volume 64

• Author Index of Volume 64

• # Pramana – Journal of Physics

Current Issue
Volume 93 | Issue 6
December 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019