RNA secondary structure formation: a solvable model of heteropolymer folding

The statistical mechanics of heteropolymer structure formation is studied in the context of RNA secondary structures. A designed RNA sequence biased energetically towards a particular native structure (a hairpin) is used to study the transition between the native and molten phase of the RNA as a function of temperature. The transition is driven by a competition between the energy gained from the polymer's overlap with the native structure and the entropic gain of forming random contacts. A simplified Go-like model is proposed and solved exactly. The predicted critical behavior is verified via exact numerical enumeration of a large ensemble of similarly designed sequences.Comment: 4 pages including 2 figure

Scaling of Selfavoiding Tethered Membranes: 2-Loop Renormalization Group Results

The scaling properties of selfavoiding polymerized membranes are studied using renormalization group methods. The scaling exponent \nu is calculated for the first time at two loop order. \nu is found to agree with the Gaussian variational estimate for large space dimension d and to be close to the Flory estimate for d=3.Comment: 4 pages, RevTeX + 20 .eps file

Disorder-Induced Depinning Transition

The competition in the pinning of a directed polymer by a columnar pin and a background of random point impurities is investigated systematically using the renormalization group method. With the aid of the mapping to the noisy-Burgers' equation and the use of the mode-coupling method, the directed polymer is shown to be marginally localized to an arbitrary weak columnar pin in 1+1 dimensions. This weak localization effect is attributed to the existence of large scale, nearly degenerate optimal paths of the randomly pinned directed polymer. The critical behavior of the depinning transition above 1+1 dimensions is obtained via an $\epsilon$-expansion.Comment: 47 pages in revtex; postscript files of 6 figures include

Recombination Models

We review the current status of recombination and coalescence models that have been successfully applied to describe hadronization in heavy ion collisions at RHIC energies. Basic concepts as well as actual implementations of the idea are discussed. We try to evaluate where we stand in our understanding at the moment and what remains to be done in the future.Comment: Plenary Talk at Quark Matter 2004, submitted to J. Phys. G, 8 pages, 3 figure

Statistical mechanics of secondary structures formed by random RNA sequences

The formation of secondary structures by a random RNA sequence is studied as a model system for the sequence-structure problem omnipresent in biopolymers. Several toy energy models are introduced to allow detailed analytical and numerical studies. First, a two-replica calculation is performed. By mapping the two-replica problem to the denaturation of a single homogeneous RNA in 6-dimensional embedding space, we show that sequence disorder is perturbatively irrelevant, i.e., an RNA molecule with weak sequence disorder is in a molten phase where many secondary structures with comparable total energy coexist. A numerical study of various models at high temperature reproduces behaviors characteristic of the molten phase. On the other hand, a scaling argument based on the extremal statistics of rare regions can be constructed to show that the low temperature phase is unstable to sequence disorder. We performed a detailed numerical study of the low temperature phase using the droplet theory as a guide, and characterized the statistics of large-scale, low-energy excitations of the secondary structures from the ground state structure. We find the excitation energy to grow very slowly (i.e., logarithmically) with the length scale of the excitation, suggesting the existence of a marginal glass phase. The transition between the low temperature glass phase and the high temperature molten phase is also characterized numerically. It is revealed by a change in the coefficient of the logarithmic excitation energy, from being disorder dominated to entropy dominated.Comment: 24 pages, 16 figure

Two-Loop Renormalization Group Analysis of the Burgers-Kardar-Parisi-Zhang Equation

A systematic analysis of the Burgers--Kardar--Parisi--Zhang equation in $d+1$ dimensions by dynamic renormalization group theory is described. The fixed points and exponents are calculated to two--loop order. We use the dimensional regularization scheme, carefully keeping the full $d$ dependence originating from the angular parts of the loop integrals. For dimensions less than $d_c=2$ we find a strong--coupling fixed point, which diverges at $d=2$, indicating that there is non--perturbative strong--coupling behavior for all $d \geq 2$. At $d=1$ our method yields the identical fixed point as in the one--loop approximation, and the two--loop contributions to the scaling functions are non--singular. For $d>2$ dimensions, there is no finite strong--coupling fixed point. In the framework of a $2+\epsilon$ expansion, we find the dynamic exponent corresponding to the unstable fixed point, which describes the non--equilibrium roughening transition, to be $z = 2 + {\cal O} (\epsilon^3)$, in agreement with a recent scaling argument by Doty and Kosterlitz. Similarly, our result for the correlation length exponent at the transition is $1/\nu = \epsilon + {\cal O} (\epsilon^3)$. For the smooth phase, some aspects of the crossover from Gaussian to critical behavior are discussed.Comment: 24 pages, written in LaTeX, 8 figures appended as postscript, EF/UCT--94/3, to be published in Phys. Rev. E

On Growth, Disorder, and Field Theory

This article reviews recent developments in statistical field theory far from equilibrium. It focuses on the Kardar-Parisi-Zhang equation of stochastic surface growth and its mathematical relatives, namely the stochastic Burgers equation in fluid mechanics and directed polymers in a medium with quenched disorder. At strong stochastic driving -- or at strong disorder, respectively -- these systems develop nonperturbative scale-invariance. Presumably exact values of the scaling exponents follow from a self-consistent asymptotic theory. This theory is based on the concept of an operator product expansion formed by the local scaling fields. The key difference to standard Lagrangian field theory is the appearance of a dangerous irrelevant coupling constant generating dynamical anomalies in the continuum limit.Comment: review article, 50 pages (latex), 10 figures (eps), minor modification of original versio

Universality Classes in Isotropic, Abelian and non-Abelian, Sandpile Models

Universality in isotropic, abelian and non-abelian, sandpile models is examined using extensive numerical simulations. To characterize the critical behavior we employ an extended set of critical exponents, geometric features of the avalanches, as well as scaling functions describing the time evolution of average quantities such as the area and size during the avalanche. Comparing between the abelian Bak-Tang-Wiesenfeld model [P. Bak, C. Tang and K. Wiensenfeld, Phys. Rev. Lett. 59, 381 (1987)], and the non-abelian models introduced by Manna [S. S. Manna, J. Phys. A. 24, L363 (1991)] and Zhang [Y. C. Zhang, Phys. Rev. Lett. 63, 470 (1989)] we find strong indications that each one of these models belongs to a distinct universality class.Comment: 18 pages of text, RevTeX, additional 8 figures in 12 PS file

Phase Diagram for Splay Glass Superconductivity

Localization of flux lines to splayed columnar pins is studied. A sine-Gordon type renormalization group study reveals the existence of a Splay glass phase and yields an analytic form for the transition temperature into the glass phase. As an independent test, the $I-V$ characteristics are determined via a Molecular Dynamics code. The glass transition temperature supports the RG results convincingly. The full phase diagram of the model is constructed.Comment: 14 pages, uuencoded compressed tar file with 3 postscript figure

Avalanche Merging and Continuous Flow in a Sandpile Model

A dynamical transition separating intermittent and continuous flow is observed in a sandpile model, with scaling functions relating the transport behaviors between both regimes. The width of the active zone diverges with system size in the avalanche regime but becomes very narrow for continuous flow. The change of the mean slope, Delta z, on increasing the driving rate, r, obeys Delta z ~ r^{1/theta}. It has nontrivial scaling behavior in the continuous flow phase with an exponent theta given, paradoxically, only in terms of exponents characterizing the avalanches theta = (1+z-D)/(3-D).Comment: Explanations added; relation to other model