Rigorous Results for Hierarchical Models of Structural Glasses

We consider two non-mean-field models of structural glasses built on a hierarchical lattice. First, we consider a hierarchical version of the random energy model, and we prove the existence of the thermodynamic limit and self-averaging of the free energy. Furthermore, we prove that the infinite-volume entropy is positive in a high-temperature region bounded from below, thus providing an upper bound on the Kauzmann critical temperature. In addition, we show how to improve this bound by leveraging the hierarchical structure of the model. Finally, we introduce a hierarchical version of the \(p\) -spin model of a structural glass, and we prove the existence of the thermodynamic limit and self-averaging of the free energy.     

Thermodynamics for the zero-level set of the Brownian bridge

The random set of instants where the Brownian bridge vanishes is constructed in terms of a random branching process. The Hausdorff measure supported by this set is shown to be equivalent to the partition function of a special class of disordered systems. This similarity is used to show rigorously the existence of a phase transition for this particular class of disordered systems. Moreover, it is shown that at high temperature the specific free energy has the strong self-averaging property and that at low temperature it has no self-averaging property. The unicity at high-temperature and the existence of many limits at low temperature are established almost surely in the disorder.Work supported by the Swiss National Science Foundation     

A Self-Averaging “Order Parameter” for the Sherrington-Kirkpatrick Spin Glass Model

Following an idea of van Enter and Griffiths, we define a self-averaging parameter for the Sherrington-Kirkpatrick (SK) spin glass which is a self-averaging version of the order parameter introduced by Aizenman, Lebowitz and Ruelle. It is strictly positive at low temperature and zero at sufficiently high temperature. The proof is based on the recent construction of the thermodynamic limit of the free energy by Guerra and Toninelli. We also discuss how our definition compares with various existing definitions of order-parameter like quantities.     

On the Distribution of Overlaps in the Sherrington–Kirkpatrick Spin Glass Model

This paper describes some of the analytic tools developed recently by Ghirlanda and Guerra in the investigation of the distribution of overlaps in the Sherrington–Kirkpatrick spin glass model and of Parisi's ultrametricity. In particular, we introduce to this task a simplified (but also generalized) model on which the Gaussian analysis is made easier. Moments of the Hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps. Under the essential tool of self-averaging, we describe with full rigour, various overlap identities and replica independence that actually hold in a rather large generality. The results are presented in a language accessible to probabilists and analysts.     

The Mean Field Ising Model trough Interpolating Techniques

Aim of this paper is to illustrate how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfill our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and investigated with the interpolation techniques. We show the existence of the thermodynamic limit, bounds for the free energy density, the explicit expression for the free energy with its suitable expansion via the order parameter, the self-consistency relation, the phase transition, the critical behavior and the self-averaging properties. At the end a formulation of a Parisi-like theory is tried and discussed.     

Energy transformation in biological molecular motors

Biological molecular motors transform the metabolic free energy into the directed movement. The physical principles governing this transformation are very different from the principles underlying the manmade macroscopic motors. Theoretical analysis shows that the internal thermal diffusion in motor proteins is a key element of the process, and the chemical energy performs no mechanical work directly but instead it is used for rectifying the diffusion. A few specific motor systems are considered to illustrate the general principle. The principle of rectified thermal diffusion has recently received a great support from the single-molecule studies.     

Designing protein beta-sheet surfaces by Z-score optimization

Studies of lattice models of proteins have suggested that the appropriate energy expression for protein design may include nonthermodynamic terms to accommodate negative design concerns. One method, developed in lattice model studies, maximizes a quantity known as the " Z-score," which compares the lowest energy sequence whose ground state structure is the target structure to an ensemble of random sequences. Here we show that, in certain circumstances, the technique can be applied to real proteins. The resulting energy expression is used to design the beta-sheet surfaces of two real proteins. We find experimentally that the designed proteins are stable and well folded, and in one case is even more thermostable than the wild type.     

Gibbs states of the Hopfield model with extensively many patterns

We consider the Hopfield model withM(N)=N patterns, whereN is the number of neurons. We show that if is sufficiently small and the temperature sufficiently low, then there exist disjoint Gibbs states for each of the stored patterns, almost surely with respect to the distribution of the random patterns. This solves a provlem left open in previous work. The key new ingredient is a self-averaging result on the free energy functional. This result has considerable additional interest and some consequences are discussed. A similar result for the free energy of the Sherrington-Kirkpatrick model is also given.     

Folding/unfolding kinetics of lattice proteins studied using a simple statistical mechanical model for protein folding, I: Dependence on native structures and amino acid sequences

The folding/unfolding kinetics of a three-dimensional lattice protein was studied using a simple statistical mechanical model for protein folding that we developed earlier. We calculated a characteristic relaxation rate for the free energy profile starting from a completely unfolded structure (or native structure) that is assumed to be associated with a folding rate (or an unfolding rate). The chevron plot of these rates as a function of the inverse temperature was obtained for four lattice proteins, namely, proteins a1, a2, b1, and b2, in order to investigate the dependency of the folding and unfolding rates on their native structures and amino acid sequences. Proteins a1 and a2 fold to the same native conformation, but their amino acid sequences differ. The same is the case for proteins b1 and b2, but their native conformation is different from that of proteins a1 and a2. However, the chevron plots of proteins a1 and a2 are very similar to each other, and those of proteins b1 and b2 differ considerably. Since the contact orders of proteins b1 and b2 are identical, the differences in their kinetics should be attributed to the amino acid sequences and consequently to the interactions between the amino acid residues. A detailed analysis revealed that long-range interactions play an important role in causing the difference in the folding rates. The chevron plots for the four proteins exhibit a chevron rollover under both strongly folding and strongly unfolding conditions. The slower relaxation time on the broad and flat free energy surfaces of the unfolding conformations is considered to be the main origin of the chevron rollover, although the free energy surfaces have features that are rather complicated to be described in detail here. Finally, in order to concretely examine the relationship between changes in the free energy profiles and the chevron plots, we illustrate some examples of single amino acid substitutions that increase the folding rate.     

The dependence of surface tension of solid nanoscale films on their thickness

This paper presents a theoretical method to calculate the surface energy dependence on thickness of nanofilms based on the Landau theory. For the first time, it is shown that the surface energy of thin films having free surfaces is greater than the surface energy of macroscopic objects. For nano-objects having free surfaces, it is stated that their interior order parameter is always less than that of macroscopic solids of the same composition. It is obtained that the surface energy of thin films increases with decrease in their thickness passing its maximum meaning. A further decrease in the solid film thickness leads to a monotonic decrease in the surface energy.     

Nematics with quenched disorder: violation of self-averaging

We consider the isotropic-to-nematic transition in liquid crystals confined to aerogel hosts, and assume that the aerogel acts as a random field. We generally find that self-averaging is violated. For a bulk transition that is weakly first order, the violation of self-averaging is so severe that even the correlation length becomes non-self-averaging: no phase transition remains in this case. For a bulk transition that is more strongly first order, the violation of self-averaging is milder, and a phase transition is observed.     

The free energy of a class of Hopfield models

We show that in the limitp ,N 0,=p/N 0 the limit free energy of the Hopfield model equals in probability the Curie-Weiss free energy. We prove also that the free energy of the Hopfield model is self-averaging for any finite .     

Trees at an Interface

A lattice tree at an interface between two solvents of different quality is examined as a model of a branched polymer at an interface. Existence of the free energy is shown, and the existence of critical lines in its phase diagram is proven. In particular, there is a line of first order transitions separating a positive phase from a negative phase (the tree being predominantly on either side of the interface in these phases), and a curve of localization–delocalization transitions which separate the delocalized positive and negative phases from a phase where the tree is localized at the interface. This model is generalized to a branched copolymer which is examined in a certain averaged quenched ensemble. Existence of a thermodynamic limit is shown for this model, and it is also shown that the model is self-averaging. Lastly, a model of branched polymers interacting with one another through a membrane is considered. The existence of a limiting free energy is shown, and it is demonstrated that if the interaction is strong enough, then the two branched polymers will adsorb on one another.     

Multicanonical chain-growth algorithm

We present a temperature-independent Monte Carlo method for the determination of the density of states of lattice proteins that combines the fast ground-state search strategy of the new pruned-enriched Rosenbluth chain-growth method and multicanonical reweighting for sampling the complete energy space. Since the density of states contains all energetic information of a statistical system, we can directly calculate the mean energy, specific heat, Helmholtz free energy, and entropy for all temperatures. We apply this method to lattice proteins consisting of hydrophobic and polar monomers, and for the examples of sequences considered, we identify the transitions between native, globule, and random coil states. Since no special properties of heteropolymers are involved in this algorithm, the method applies to polymer models as well.     

Landau–Ginzburg theory for ‘star’-shaped droplets

ABSTRACT

Molecular simulations have shown that when a nano-drop comprising a single spherical central ion and a dielectric solvent is charged above a well-defined threshold, it acquires a stable star morphology. A linear continuum model of the ‘star’-shapes comprised electrostatic and surface energy is not sufficient to describe these shapes. We employ combined molecular dynamics, continuum electrostatics and macroscopic modelling in order to construct a unified free energy functional that describes the observed star-shaped droplets. We demonstrate that the Landau free energy coupled to the third-order Steinhardt invariant mimics the shapes of droplets detected in molecular simulations. Using the maximum likelihood technique we build a universal free energy functional that describes droplets for a range of Rayleigh fissility parameter. The analysis of the macroscopic free energy demonstrates the origin of the finite amplitude perturbations just above the Rayleigh limit. We argue that the presence of the finite amplitude perturbations precludes the use of the small parameter perturbation method for the analysis of the shapes above the Rayleigh limit of the corresponding spherical shape.     

Self-Averaging of Wigner Transforms in Random Media

We establish the self-averaging properties of the Wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance. The main ingredients in the proof are the error estimates for the semiclassical approximation of the Wigner transform by the solution of the Liouville equations, and the limit theorem for two-particle motion along the characteristics of the Liouville equations. The results are applied to a mathematical model of the time-reversal experiments for the acoustic waves, and self-averaging properties of the re-transmitted wave are proved.     

On the theory of the condensed states of heteropolymers

The collapse (globulization) of an ideal heteropolymer chain under the action of an external attractive field is considered. The problem of the collapse of different types of primary structures, including mobile, periodic, large-block, and statistical structures, is formulated. It is shown that for a random heteropolymer, the mathematical image of the globular state is the chain-length independence of the probability distribution of a random thermal distribution function of the end monomer coordinates. The free energy per monomer of a chain in a globular state and local densities of monomers of all types are shown to be a self-averaging quantities. An exactly solvable model is proposed for a globule formed by a statistical heteropolymer chain. In this model, different types of monomers are attracted to different centers by linear elastic forces with identical elastic constants. The modulus of elasticity is obtained for a heteropolymer globule with respect to the attraction of different types of monomers in different directions. It is shown that this modulus is higher for a short-periodic polymer than for a statistical one.     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

Fluctuations and spectral relaxation in dispersion transfer over traps

Based on the self-consistent cluster approximation of an effective medium for random walk over a lattice with stochastically positioned traps in the dispersion regime, the kinetics of the spectral relaxation of the partial populations was analyzed and the self-averaging of the diffusion coefficient was studied. It was demonstrated that the relaxation of partial populations at long time occurs anomalously slow, according to a power law. It was shown that, in the one-dimensional case, no self-averaging of the diffusion coefficient occurs even at long time (its fluctuation amplitude remain comparable with its value) and that in the two- and three-dimensional cases, self-averaging proceeds very slowly, in accordance with a logarithmic and power dependence, respectively.