Dynamics of end-to-end loop formation: A flexible chain in the presence of hydrodynamic interaction

Based on the Wilemski–Fixman approach [G. Wilemski, M. Fixman, J. Chem. Phys. 60 (1974) 866], we show that, for a flexible chain in θ$\theta$ solvent, hydrodynamic interaction treated with a pre-averaging approximation makes ring closing faster if the chain is not very short. We also show that the ring closing time for a long chain with hydrodynamic interaction in θ$\theta$ solvent scales with the chain length (N$N$) as N^1.5$N^{1.5}$, in agreement with the previous renormalization group calculation based prediction by Freidman and O’Shaughnessy [B. Friedman, B. O’Shaughnessy, Phys. Rev. A 40 (1989) 5950].     

The effect of initial spatial correlations on late time kinetics of bimolecular irreversible reactions

We study anomalous kinetics associated with incomplete mixing for a bimolecular irreversible kinetic reaction where the underlying transport of reactants is governed by a fractional dispersion equation. As has been previously shown, we demonstrate that at late times incomplete mixing effects dominate and the decay of reactants follows a fundamentally different scaling comparing to the idealized well mixed case. We do so in a fully analytical manner using moment equations. In particular the novel aspect of this work is that we focus on the role that the initial correlation structure of the distribution of reactants plays on the late time scalings. We focus on short range and long (power law) range correlations and demonstrate how long range correlations can give rise to different late time scalings than one would expect purely from the underlying transport model. For the short range correlations the late time scalings deviate from the well mixed t⁻¹$t^{-1}$ and scale like t^−1/2α$t^{-1/2\alpha }$, where 1<α≤2$1<\alpha \le 2$ is the fractional dispersion exponent, in agreement with previous studies. For the long range correlation case it scales like t^−β/2α$t^{-\beta /2\alpha }$, where 0<β<1$0<\beta <1$ is the power law correlation exponent.     

Comment on noncommutative deformation of instantons

It was argued in [Y. Maeda, A. Sako, Noncommutative deformation of instantons, J. Geom. Phys. 58 (2008) 1784] that the noncommutative deformation of instantons on a 4-torus T⁴$T^4$ should alter the instanton numbers for arbitrary noncommutativity parameter θ$\theta$. We show that this is not the case for the U(N²)$U\left(N^2\right)$ theory discussed there. And we discuss the instanton numbers in general gauge theories on the noncommutative T⁴$T^4$.     

Three ways to solve the Poisson equation on a sphere with Gaussian forcing

Motivated by the needs of vortex methods, we describe three different exact or approximate solutions to the Poisson equation on the surface of a sphere when the forcing is a Gaussian of the three-dimensional distance, ∇²ψ=exp(-2?²(1-cos(θ))-C^Gauss(?)${\nabla }^2\psi =\exp (-2{?}^2(1-\cos (\theta ))-C^{\mathit{Gauss}}(?)$. (More precisely, the forcing is a Gaussian minus the “Gauss constraint constant”, C^Gauss$C^{\mathit{Gauss}}$; this subtraction is necessary because ψ$\psi$ is bounded, for any type of forcing, only if the integral of the forcing over the sphere is zero [Y. Kimura, H. Okamoto, Vortex on a sphere, J. Phys. Soc. Jpn. 56 (1987) 4203–4206; D.G. Dritschel, Contour dynamics/surgery on the sphere, J. Comput. Phys. 79 (1988) 477–483]. The Legendre polynomial series is simple and yields the exact value of the Gauss constraint constant, but converges slowly for large ?$?$. The analytic solution involves nothing more exotic than the exponential integral, but all four terms are singular at one or the other pole, cancelling in pairs so that ψ$\psi$ is everywhere nice. The method of matched asymptotic expansions yields simpler, uniformly valid approximations as series of inverse even powers of ?$?$ that converge very rapidly for the large values of ?  (?>40)$(?>40)$ appropriate for geophysical vortex computations. The series converges to a nonzero O(exp(-4?²))$O(\exp (-4{?}^2))$ error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

Temporal breakdown and Borel resummation in the complex Langevin method

We reexamine the Parisi–Klauder conjecture for complex e^iθ/2?⁴$e^{i\theta /2}{?}^4$ measures with a Wick rotation angle 0≤θ/2≤π/2$0\le \theta /2\le \pi /2$ interpolating between Euclidean signature and Lorentzian signature. Our main result is that the asymptotics for short stochastic times t$t$ encapsulates information also about the equilibrium aspects. The moments evaluated with the complex measure and with the real measure defined by the stochastic Langevin equation have the same t→0$t\to 0$ asymptotic expansion which is shown to be Borel summable. The Borel transform correctly reproduces the time dependent moments of the complex measure for all t$t$, including their t→∞$t\to \infty$ equilibrium values. On the other hand the results of a direct numerical simulation of the Langevin moments are found to disagree from the ‘correct’ result for t$t$ larger than a finite t_c$t_c$. The breakdown time t_c$t_c$ increases powerlike for decreasing strength of the noise’s imaginary part but cannot be excluded to be finite for purely real noise. To ascertain the discrepancy we also compute the real equilibrium distribution for complex noise explicitly and verify that its moments differ from those obtained with the complex measure.     

Qubit disentanglement in local squeezed reservoirs

We consider the influence of the local squeezed vacuum fields on two initially entangled two-qubit system. By considering the upper bound of entanglement under time evolution, we find that the decay of the quantum entanglement shows different behavior for different time scales (t?max{(2βA)⁻¹,(2βB)⁻¹}$t?\max \{{(2{\beta }_A)}^{\mathrm{-1}},{(2{\beta }_B)}^{\mathrm{-1}}\}$ and t?min{(2βA)⁻¹,(2βB)⁻¹}$t?\min \{{(2{\beta }_A)}^{\mathrm{-1}},{(2{\beta }_B)}^{\mathrm{-1}}\}$). The relative phase of the squeezing environment can also affect the entanglement dynamics profoundly.     

Post-breakthrough scaling in reservoir field simulation

We study the oil displacement and production behavior in an isothermal thin layered reservoir model subjected to water flooding. We use the CMG’s (Computer Modelling Group  ) numerical simulators to solve mass balance equations. The influences of the viscosity ratio (m≡_μoil/_μwater$m\equiv {\mu }_{oil}/{\mu }_{water}$) and the inter-well (injector-producer) distance r$r$ on the oil production rate C(t)$C\left(t\right)$ and the breakthrough time _tbr$t_{br}$ are investigated. Two types of reservoir configuration are used, namely one with random porosities and another with a percolation cluster structure. We observe that the breakthrough time follows a power-law of m$m$ and r$r$, _tbr∝r^αm^β$t_{br}\propto r^{\alpha }{m}^{\beta }$, with α=1.8$\alpha =1.8$ and β=−0.25$\beta =-0.25$ for the random porosity type, and α=1.0$\alpha =1.0$ and β=−0.2$\beta =-0.2$ for the percolation cluster type. Moreover, our results indicate that the oil production rate is a power law of time. In the percolation cluster type of reservoir, we observe that P(t)∝t^γ$P\left(t\right)\propto t^{\gamma }$, with γ=−1.81$\gamma =-1.81$, where P(t)$P\left(t\right)$ is the time derivative of C(t)$C\left(t\right)$. The curves related to different values of m$m$ and r$r$ may be collapsed suggesting a universal behavior for the oil production rate.     

Universal behavior in a model of city traffic with unequal green/red times

The effect of green/red asymmetry is studied for the single-car traffic model proposed in [B.A. Toledo, V. Muñoz, J. Rogan, C. Tenreiro, J.A. Valdivia, Modeling traffic through a sequence of traffic lights, Phys. Rev. E 70 (1) (2004) 016107], on two different signal synchronization strategies, namely, all signals in phase, and a green wave. The asymmetry is characterized by the parameter g=_tgr/T$g=t_{\text{gr}}/T$, where _tgr$t_{\text{gr}}$ is the green time and T$T$ the signal period. Although the car dynamics turns simpler or more complex, as compared with the equivalent situation for the symmetric case g=0.5$g=0.5$, critical behavior around resonance is shown to be preserved. However, unlike the case g=0.5$g=0.5$, critical exponents at both sides of the resonance are not equal and depend on g$g$. Analytical expressions for them are found, and shown to be both consistent with simulation results and independent of the distribution of distances between signals for the green wave case. Also, it is found that the green wave strategy is more robust to changes in g$g$, with respect to the synchronized lights strategy, in the sense that larger departures from g=0.5$g=0.5$ are needed to have noticeable effects on the car dynamics.     

Surface and bulk properties of ballistic deposition models with bond breaking

We introduce a new class of growth models, with a surface restructuring mechanism in which impinging particles may dislodge suspended particles, previously aggregated on the same column in the deposit. The flux of these particles is controlled through a probability p$p$. These systems present a crossover, for small values of p$p$, from random to correlated (KPZ) growth of surface roughness, which is studied through scaling arguments and Monte Carlo simulations on one- and two-dimensional substrates. We show that the crossover characteristic time _t×$t_{\times }$ scales with p$p$ according to _t×∼p^−y$t_{\times }\sim p^{-y}$ with y=(n+1)$y=(n+1)$ and that the interface width at saturation _Wsat$W_{sat}$ scales as _Wsat∼p^−δ$W_{sat}\sim p^{-\delta }$ with δ=(n+1)/2$\delta =(n+1)/2$, where n$n$ is either the maximal number of broken bonds or of dislodged suspended particles. This result shows that the sets of exponents y=1$y=1$ and δ=1/2$\delta =1/2$ or y=2$y=2$ and δ=1$\delta =1$ found in all previous works focusing on systems with this same type of crossover are not universal. Using scaling arguments, we show that the bulk porosity P$P$ of the deposits scales as P∼p^y−δ$P\sim p^{y-\delta }$ for small values of p$p$. This general scaling relation is confirmed by our numerical simulations and explains previous results present in literature.     

Topological recursion for chord diagrams,RNA complexes,and cells in moduli spaces

We introduce and study the Hermitian matrix model with potential _Vs,t(x)=x²/2−stx/(1−tx)$V_{s,t}(x)=x^2/2-stx/(1-tx)$, which enumerates the number of linear chord diagrams with no isolated vertices of fixed genus with specified numbers of backbones generated by s and chords generated by t. For the one-cut solution, the partition function, correlators and free energies are convergent for small t and all s   as a perturbation of the Gaussian potential, which arises for st=0$st=0$. This perturbation is computed using the formalism of the topological recursion. The corresponding enumeration of chord diagrams gives at once the number of RNA complexes of a given topology as well as the number of cells in Riemann?s moduli spaces for bordered surfaces. The free energies are computed here in principle for all genera and explicitly in genus less than four.     

Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy

This paper studies the properties of the derivatives of differential entropy H(Xt) in Costa’s entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for m≥1, (−1)m+1(dm/dtm)H(Xt)≥0, while McKean conjectured a stronger statement, whereby (−1)m+1(dm/dtm)H(Xt)≥(−1)m+1(dm/dtm)H(XGt). Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: C1(m,n):(−1)m+1(dm/dtm)H(Xt)≥0, where n denotes that Xt is a random vector taking values in Rn, and similarly, C2(m,n):(−1)m+1(dm/dtm)H(Xt)≥(−1)m+1(dm/dtm)H(XGt)≥0. In this paper, we prove some new multivariate cases: C1(3,i),i=2,3,4. Motivated by our results, we further propose a weaker version of McKean’s conjecture C3(m,n):(−1)m+1(dm/dtm)H(Xt)≥(−1)m+11n(dm/dtm)H(XGt), which is implied by C2(m,n) and implies C1(m,n). We prove some multivariate cases of this conjecture under the log-concave condition: C3(3,i),i=2,3,4 and C3(4,2). A systematic procedure to prove Cl(m,n) is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure.     

Long-range memory elementary 1D cellular automata: Dynamics and nonextensivity

We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state _σi(t)∈{0,1}${\sigma }_i(t)\in \{0,1\}$ of a cell i   does not only depend on the states in its local neighborhood at time t-1$t-1$, but also on the memory of its own past states _σi(t-2),_σi(t-3),…,_σi(t-τ),…${\sigma }_i(t-2),{\sigma }_i(t-3),\dots ,{\sigma }_i(t-\tau ),\dots$ . We assume that the weight of this memory decays proportionally to τ^-α${\tau }^{-\alpha }$, with α?0$\alpha ?0$ (the limit α→∞$\alpha \to \infty$ corresponds to the usual CA). Since the memory function is summable for α>1$\alpha >1$ and nonsummable for 0?α?1$0?\alpha ?1$, we expect pronounced changes of the dynamical behavior near α=1$\alpha =1$. This is precisely what our simulations exhibit, particularly for the time evolution of the Hamming distance H   of initially close trajectories. We typically expect the asymptotic behavior H(t)∝t^1/(1-q)$H(t)\propto t^{1/(1-q)}$, where q   is the entropic index associated with nonextensive statistical mechanics. In all cases, the function q(α)$q(\alpha )$ exhibits a sensible change at α?1$\alpha ?1$. We focus on the class II rules 61, 99 and 111. For rule 61, q=0$q=0$ for 0?α?_αc?1.3$0?\alpha ?{\alpha }_c?1.3$, and q<0$q<0$ for α>_αc$\alpha >{\alpha }_c$, whereas the opposite behavior is found for rule 111. For rule 99, the effect of the long-range memory on the spread of damage is quite dramatic. These facts point at a rich dynamics intimately linked to the interplay of local lookup rules and the range of the memory. Finite size scaling studies varying system size N   indicate that the range of the power-law regime for H(t)$H(t)$ typically diverges ∝N^z$\propto N^z$ with 0?z?1$0?z?1$.     

Half-monopole in the Weinberg–Salam model

We present new axially symmetric half-monopole configuration of the SU(2)×U(1) Weinberg–Salam model of electromagnetic and weak interactions. The half-monopole configuration possesses net magnetic charge 2π/e$2\pi /e$ which is half the magnetic charge of a Cho–Maison monopole. The electromagnetic gauge potential is singular along the negative z$z$-axis. However the total energy is finite and increases only logarithmically with increasing Higgs field self-coupling constant λ^1/2${\lambda }^{1/2}$ at sin²θ_W=0.2312${sin}^2{\theta }_W=0.2312$. In the U(1) magnetic field, the half-monopole is just a one dimensional finite length line magnetic charge extending from the origin r=0$r=0$ and lying along the negative z$z$-axis. In the SU(2) ’t Hooft magnetic field, it is a point magnetic charge located at r=0$r=0$. The half-monopole possesses magnetic dipole moment that decreases exponentially fast with increasing Higgs field self-coupling constant λ^1/2${\lambda }^{1/2}$ at sin²θ_W=0.2312${sin}^2{\theta }_W=0.2312$.     

On hydrogen transport in the first wall material of fusion devices in the presence of a broadband distribution of traps over the trapping energy

The dynamics of hydrogen dissolved in a sample with continuous distribution of traps over trapping energy φ(ε)∝exp(−αε)$\phi (\epsilon )\propto \exp (-\alpha \epsilon )$ (ε=E/T$\epsilon =E/T$ is the ratio of trapping energy E to the sample's temperature T  ) is considered. Assuming that the hydrogen density is smaller than the trap density and the most of hydrogen is trapped, we found that the dynamics of hydrogen transport can be described by either sub-diffusion or non-linear diffusion equations. Analysis of the outgassing of the sample homogeneously loaded with hydrogen gives, in the most important cases, both power-law, _ΓH∝t^−p${\Gamma }_{\mathrm{H}}\propto t^{-p}$ (p≥1/2$p\ge 1/2$) and exponential, ln(_ΓH)∝−t^α$\ln ({\Gamma }_{\mathrm{H}})\propto -t^{\alpha }$, time dependencies of the outgassing flux, _ΓH(t)${\Gamma }_{\mathrm{H}}(t)$.