David Mukamel

An Exact Solution of a One-Dimensional Asymmetric Exclusion Model with Open Boundaries

A simple asymmetric exclusion model with open boundaries is solved exactly in one dimension. The exact solution is obtained by deriving a recursion relation for the steady state: if the steady state is known for all system sizes less thanN, then our equation (8) gives the steady state for sizeN. Using this recursion, we obtain closed expressions (48) for the average occupations of all sites. The results are compared to the predictions of a mean field theory. In particular, for infinitely large systems, the effect of the boundary decays as the distance to the power −1/2 instead of the inverse of the distance, as predicted by the mean field theory.

Why is the DNA denaturation transition first order

We study a model for the denaturation transition of DNA in which the molecules are considered as being composed of a sequence of alternating bound segments and denaturated loops. We take into account the excluded-volume interactions between denaturated loops and the rest of the chain by exploiting recent results on scaling properties of polymer networks of arbitrary topology. The phase transition is found to be first order in d = 2 dimensions and above, in agreement with experiments and at variance with previous theoretical results, in which only excluded-volume interactions within denaturated loops were taken into account. Our results agree with recent numerical simulations.

Criterion for phase separation in one-dimensional driven systems.

A general criterion for the existence of phase separation in driven density-conserving one-dimensional systems is proposed. It is suggested that phase separation is related to the size dependence of the steady-state currents of domains in the system. A quantitative criterion for the existence of phase separation is conjectured using a correspondence made between driven diffusive models and zero-range processes. The criterion is verified in all cases where analytical results are available, and predictions for other models are provided.

Asymmetric exclusion model with two species: Spontaneous symmetry breaking

A simple two-species asymmetric exclusion model is introduced. It consists of two types of oppositely charged particles driven by an electric field and hopping on an open chain. The phase diagram of the model is calculated in the meanfield approximation and by Monte Carlo simulations. Exact solutions are given for special values of the parameters defining its dynamics. The model is found to exhibit two phases in which spontaneous symmetry breaking takes place, where the two currents of the two species are not equal.

Phase Separation in One-Dimensional Driven Diffusive Systems

Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel(February 7, 2008)A driven diffusive model of three types of particles that exhibits phase separation on a ring isintroduced. The dynamics is local and comprises nearest neighbor exchanges that conserve each ofthe three species. For the case in which the three densities are equal, it is shown that the modelobeys detailed balance. The Hamiltonian governing the steady state distribution in this case is givenand is found to have long range asymmetric interactions. The partition sum and bounds on somecorrelation functions are calculated analytically demonstrating phase separation.PACS numbers: 02.50.Ey; 05.20.-y; 64.75.+g

Thermodynamics and dynamics of systems with long-range interactions

We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as 1/rd+σ at large distances r in d dimensions. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed “strong” LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions affect the thermodynamic behavior of systems near phase transitions, where long-range correlations are naturally present. Long-range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems with local dynamics are considered within the framework outlined above.

Duality relations and equivalences for models with O(N) and cubic symmetry

Formal expression for high-temperature series are derived for models with O(N) and cubic symmetry, with a special form of nearest neighbor interactions on the honeycomb lattice. By deriving low-temperature series for a class of generalized solid-on-solid and cubic models, a duality relation is established. Equivalences between cubic and SOS type models are also found. In the large-N limit, the series reduce to those of the hard hexagon model.

Breaking of Ergodicity and Long Relaxation Times in Systems with Long-Range Interactions

The thermodynamic and dynamical properties of an Ising model with both short-range and long-range, mean-field-like, interactions are studied within the microcanonical ensemble. It is found that the relaxation time of thermodynamically unstable states diverges logarithmically with system size. This is in contrast with the case of short-range interactions where this time is finite. Moreover, at sufficiently low energies, gaps in the magnetization interval may develop to which no microscopic configuration corresponds. As a result, in local microcanonical dynamics the system cannot move across the gap, leading to breaking of ergodicity even in finite systems. These are general features of systems with long-range interactions and are expected to be valid even when the interaction is slowly decaying with distance.

Phase Separation and Coarsening in One-Dimensional Driven Diffusive Systems: Local Dynamics Leading to Long-Range Hamiltonians

A driven system of three species of particles diffusing on a ring is studied in detail. The dynamics is local and conserves the three densities. A simple argument suggesting that the model should phase separate and break the translational symmetry is given. We show that for the special case where the three densities are equal the model obeys detailed balance, and the steady-state distribution is governed by a Hamiltonian with asymmetric long-range interactions. This provides an explicit demonstration of a simple mechanism for breaking of ergodicity in one dimension. The steady state of finite-size systems is studied using a generalized matrix product ansatz. The coarsening process leading to phase separation is studied numerically and in a mean-field model. The system exhibits slow dynamics due to trapping in metastable states whose number is exponentially large in the system size. The typical domain size is shown to grow logarithmically in time. Generalizations to a larger number of species are discussed.

Melting and unzipping of DNA

Abstract:Existing experimental studies of the thermal denaturation of DNA yield sharp steps in the melting curve suggesting that the melting transition is first order. This transition has been theoretically studied since the early sixties, mostly within an approach in which the microscopic configurations of a DNA molecule consist of an alternating sequence of non-interacting bound segments and denaturated loops. Studies of these models neglect the repulsive, self-avoiding, interaction between different loops and segments and have invariably yielded continuous denaturation transitions. In the present study we take into account in an approximate way the excluded-volume interaction between denaturated loops and the rest of the chain. This is done by exploiting recent results on scaling properties of polymer networks of arbitrary topology. We also ignore the heterogeneity of the polymer. We obtain a first-order melting transition in d = 2 dimensions and above, consistent with the experimental results. We also consider within our approach the unzipping transition, which takes place when the two DNA strands are pulled apart by an external force acting on one end. We find that the under equilibrium condition the unzipping transition is also first order. Although the denaturation and unzipping transitions are thermodynamically first order, they do exhibit critical fluctuations in some of their properties. For instance, the loop size distribution decays algebraically at the transition and the length of the denaturated end segment diverges as the transition is approached. We evaluate these critical properties within our approach.