Partial survival and inelastic collapse for a randomly accelerated particle

We present an exact derivation of the survival probability of a randomly accelerated particle subject to partial absorption at the origin. We determine the persistence exponent and the amplitude associated to the decay of the survival probability at large times. For the problem of inelastic reflection at the origin, with coefficient of restitution $r$, we give a new derivation of the condition for inelastic collapse, $r<r_c=e^{-\pi/\sqrt{3}}$, and determine the persistence exponent exactly.Comment: 6 page

Casimir interaction of rod-like particles in a two-dimensional critical system

We consider the fluctuation-induced interaction of two thin, rod-like particles or "needles" immersed in a two-dimensional critical fluid of Ising symmetry right at the critical point. Conformally mapping the plane containing the needles onto a simpler geometry in which the stress tensor is known, we analyze the force and torque between needles of arbitrary length, separation, and orientation. For infinite and semi-infinite needles we utilize the mapping of the plane bounded by the needles onto the half plane, and for two needles of finite length the mapping onto an annulus. For semi-infinite and infinite needles the force is expressed in terms of elementary functions, and we also obtain analytical results for the force and torque between needles of finite length with separation much greater than their length. Evaluating formulas in our approach numerically for several needle geometries and surface universality classes, we study the full crossover from small to large values of the separation to length ratio. In these two limits the numerical results agree with results for infinitely long needles and with predictions of the small-particle operator expansion, respectively.Comment: 68 pages, 9 figure

Radial Distribution Function for Semiflexible Polymers Confined in Microchannels

An analytic expression is derived for the distribution $G(\vec{R})$ of the end-to-end distance $\vec{R}$ of semiflexible polymers in external potentials to elucidate the effect of confinement on the mechanical and statistical properties of biomolecules. For parabolic confinement the result is exact whereas for realistic potentials a self-consistent ansatz is developed, so that $G(\vec{R})$ is given explicitly even for hard wall confinement. The theoretical result is in excellent quantitative agreement with fluorescence microscopy data for actin filaments confined in rectangularly shaped microchannels. This allows an unambiguous determination of persistence length $L_P$ and the dependence of statistical properties such as Odijk's deflection length $\lambda$ on the channel width $D$. It is shown that neglecting the effect of confinement leads to a significant overestimation of bending rigidities for filaments

A nonperturbative Real-Space Renormalization Group scheme

Based on the original idea of the density matrix renormalization group (DMRG), i.e. to include the missing boundary conditions between adjacent blocks of the blocked quantum system, we present a rigorous and nonperturbative mathematical formulation for the real-space renormalization group (RG) idea invented by L.P. Kadanoff and further developed by K.G. Wilson. This is achieved by using additional Hilbert spaces called auxiliary spaces in the construction of each single isolated block, which is then named a superblock according to the original nomenclature. On this superblock we define two maps called embedding and truncation for successively integrating out the small scale structure. Our method overcomes the known difficulties of the numerical DMRG, i.e. limitation to zero temperature and one space dimension.Comment: 13 pages, 5 figures, late

Surface Critical Behavior of Binary Alloys and Antiferromagnets: Dependence of the Universality Class on Surface Orientation

The surface critical behavior of semi-infinite (a) binary alloys with a continuous order-disorder transition and (b) Ising antiferromagnets in the presence of a magnetic field is considered. In contrast to ferromagnets, the surface universality class of these systems depends on the orientation of the surface with respect to the crystal axes. There is ordinary and extraordinary surface critical behavior for orientations that preserve and break the two-sublattice symmetry, respectively. This is confirmed by transfer-matrix calculations for the two-dimensional antiferromagnet and other evidence.Comment: Final version that appeared in PRL, some minor stylistic changes and one corrected formula; 4 pp., twocolumn, REVTeX, 3 eps fig

Spatial Constraint Corrections to the Elasticity of dsDNA Measured with Magnetic Tweezers

In this paper, we have studied, within a discrete WLC model, the spatial constraints in magnetic tweezers used in single molecule experiments. Two elements are involved: first, the fixed plastic slab on which is stuck the initial strand, second, the magnetic bead which pulls (or twists) the attached molecule free end. We have shown that the bead surface can be replaced by its tangent plane at the anchoring point, when it is close to the bead south pole relative to the force. We are led to a model with two parallel repulsive plates: the fixed anchoring plate and a fluctuating plate, simulating the bead, in thermal equilibrium with the system. The bead effect is a slight upper shift of the elongation, about four times smaller than the similar effect induced by the fixed plate. This rather unexpected result, has been qualitatively confirmed within the soluble Gaussian model. A study of the molecule elongation versus the countour length exhibits a significant non-extensive behaviour. The curve for short molecules (with less than 2 kbp) is well fitted by a straight line, with a slope given by the WLC model, but it does not go through the origin. The non-extensive offset gives a 15% upward shift to the elongation of a 2 kbp molecule stretched by a 0.3 pN force.Comment: 28 pages, 6 figures An explanatory figure has been added. The physical interpretation of the results has been made somewhat more transparen

Simulation of a semiflexible polymer in a narrow cylindrical pore

The probability that a randomly accelerated particle in two dimensions has not yet left a simply connected domain ${\cal A}$ after a time $t$ decays as $e^{-E_0t}$ for long times. The same quantity $E_0$ also determines the confinement free energy per unit length $\Delta f=k_BT\thinspace E_0$ of a semiflexible polymer in a narrow cylindrical pore with cross section ${\cal A}$. From simulations of a randomly accelerated particle we estimate the universal amplitude of $\Delta f$ for both circular and rectangular cross sections.Comment: 10 pages, 2 eps figure

Two-dimensional critical systems with mixed boundary conditions: Exact Ising results from conformal invariance and boundary-operator expansions

With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane $y>0$ with different boundary conditions $a$ and $b$ on the negative and positive $x$ axes. For $ab=-+$ and $f+$, they determined the one and two-point averages of the spin $\sigma$ and energy $\epsilon$. Here $+$, $-$, and $f$ stand for spin-up, spin-down, and free-spin boundaries, respectively. The case $+-+-+\dots$, where the boundary conditions switch between $+$ and $-$ at arbitrary points, $\zeta_1$, $\zeta_2$, $\dots$ on the $x$ axis was also analyzed. In this paper the alternating boundary conditions $+f+f+\dots$ and the case $-f+$ of three different boundary conditions are considered. Exact results for the one and two-point averages of $\sigma$, $\epsilon$, and the stress tensor $T$ are derived. Using the results for $\langle T\rangle$, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is analyzed for mixed boundary conditions. The paper also includes a comprehensive discussion of boundary-operator expansions in two-dimensional critical systems with mixed boundary conditions. Two types of expansions - away from switching points of the boundary condition and at switching points - are considered. The asymptotic behavior of two-point averages is expressed in terms of one-point averages with the help of the expansions. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge using the boundary-operator expansions. The predictions of the boundary-operator expansions are consistent with exact results for Ising systems.Comment: 50 pages, 1 figur

Casimir Forces between Spherical Particles in a Critical Fluid and Conformal Invariance

Mesoscopic particles immersed in a critical fluid experience long-range Casimir forces due to critical fluctuations. Using field theoretical methods, we investigate the Casimir interaction between two spherical particles and between a single particle and a planar boundary of the fluid. We exploit the conformal symmetry at the critical point to map both cases onto a highly symmetric geometry where the fluid is bounded by two concentric spheres with radii R_- and R_+. In this geometry the singular part of the free energy F only depends upon the ratio R_-/R_+, and the stress tensor, which we use to calculate F, has a particularly simple form. Different boundary conditions (surface universality classes) are considered, which either break or preserve the order-parameter symmetry. We also consider profiles of thermodynamic densities in the presence of two spheres. Explicit results are presented for an ordinary critical point to leading order in epsilon=4-d and, in the case of preserved symmetry, for the Gaussian model in arbitrary spatial dimension d. Fundamental short-distance properties, such as profile behavior near a surface or the behavior if a sphere has a `small' radius, are discussed and verified. The relevance for colloidal solutions is pointed out.Comment: 37 pages, 2 postscript figures, REVTEX 3.0, published in Phys. Rev. B 51, 13717 (1995