David R. Nelson

Localization Transitions in Non-Hermitian Quantum Mechanics.

We study the localization transitions which arise in both one and two dimensions when quantum mechanical particles described by a random Schrodinger equation are subjected to a constant imaginary vector potential. A path-integral formulation relates the transition to flux lines depinned from columnar defects by a transverse magnetic field in superconductors. The theory predicts that the transverse Meissner effect is accompanied by stretched exponential relaxation of the field into the bulk and a diverging penetration depth at the transition.

Vortex pinning and non-Hermitian quantum mechanics

A delocalization phenomenon is studied in a class of non-Hermitian random quantum-mechanical problems. Delocalization arises in response to a sufficiently large constant imaginary vector potential. The transition is related to depinning of flux lines from extended defects in type-II superconductors subject to a tilted external magnetic field. The physical meaning of the complex eigenvalues and currents of the non-Hermitian system is elucidated in terms of properties of tilted vortex lines. The singular behavior of the penetration length describing stretched exponential screening of a perpendicular magnetic field (transverse Meissner effect), the surface transverse magnetization, and the trapping length is determined near the flux-line depinning point. {copyright} {ital 1997} {ital The American Physical Society}

NON-HERMITIAN LOCALIZATION AND POPULATION BIOLOGY

The time evolution of spatial fluctuations in inhomogeneous

Single molecule statistics and the polynucleotide unzipping transition.

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Pulling Pinned Polymers and Unzipping DNA

-dimensional biological systems is analyzed. A single species continuous growth model, in which the population disperses via diffusion and convection is considered. Time-independent environmental heterogeneities, such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate. Linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size. Using an analogy with a Schrodinger equation subject to a constant imaginary vector potential, we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states. In the limit of high convection velocity, the linearized growth problem in

Dynamics of Molecular Motors and Polymer Translocation with Sequence Heterogeneity

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Winding Numbers, Complex Currents, and Non-Hermitian Localization

dimensions exhibits singular scaling behavior described by a

Population dynamics and non-Hermitian localization

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Dynamics of molecular motors with finite processivity on heterogeneous tracks

-dimensional generalization of the noisy Burgers equation, with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane. The Burgers mapping leads to unusual transverse spreading of convecting delocalized populations.

Localization and population biology

We present an extensive theoretical investigation of the mechanical unzipping of double-stranded DNA under the influence of an applied force. In the limit of long polymers, there is a thermodynamic unzipping transition at a critical force value of order 10 pN, with different critical behavior for homopolymers and for random heteropolymers. We extend results on the disorder-averaged behavior of DNAs with random sequences [D. K. Lubensky and D. R. Nelson, Phys. Rev. Lett. 85, 1572 (2000)] to the more experimentally accessible problem of unzipping a single DNA molecule. As the applied force approaches the critical value, the double-stranded DNA unravels in a series of discrete, sequence-dependent steps that allow it to reach successively deeper energy minima. Plots of extension versus force thus take the striking form of a series of plateaus separated by sharp jumps. Similar qualitative features should reappear in micromanipulation experiments on proteins and on folded RNA molecules. Despite their unusual form, the extension versus force curves for single molecules still reveal remnants of the disorder-averaged critical behavior. Above the transition, the dynamics of the unzipping fork is related to that of a particle diffusing in a random force field; anomalous, disorder-dominated behavior is expected until the applied force exceeds the critical value for unzipping by roughly 5 pN.