Standard Scaling Arguments Research Articles

Free-energy cost of localizing a single monomer of a confined polymer

We describe a simple Monte Carlo simulation method to calculate the free-energy cost of localizing a single monomer of a polymer confined to a cavity. The localization position is chosen to be on the inside surface of the confining cavity. The method is applied to a freely jointed hard-sphere polymer chain confined to cavities of spherical and cubic geometries. In the latter case, we consider localization at a corner and at the center of a face of the confining cube. We consider cases of end-monomer localization both with and without tethering of the other end monomer to a point on the surface. We also examine localization of monomers at arbitrary positions along the contour of the polymer. We characterize the dependence of the free energy on the cavity size and shape, the localization position, and the polymer length. The quantitative trends can be understood using standard scaling arguments and use of a simple theoretical model. The results are relevant to those theories of polymer translocation that focus on the importance of the free-energy barrier as the translocation process requires an initial localization of a monomer to the position of a nanopore.

Properties of dislocation lines in crystals with strong atomic-scale disorder

We use discrete dislocation dynamics (DDD) to study the motion of a dislocation under strong stochastic forces that may cause bending and roughening of the dislocation line on scales that are comparable to the dislocation core radius. In such situations, which may be relevant in high entropy alloys (HEAs) exhibiting strong atomic scale disorder, standard scaling arguments based upon a line tension approximation may be no longer adequate and corrections to scaling need to be considered. We first study the wandering of the dislocation under thermal Langevin forces. This leads to a linear stochastic differential equation which can be exactly solved. From the Fourier modes of the thermalized dislocation line we can directly deduce the scale dependent effective line tension. We then use this information to investigate the wandering of a dislocation in a crystal with spatial, time-independent (‘quenched’) disorder. We establish the pinning length and show how this length can be used as a predictor of the flow stress. Implications for the determination of flow stresses in HEAs from molecular dynamics simulations are discussed.

Universality of crossover scaling for the adsorption transition of lattice polymers.

Recently, it has been proposed that the adsorption transition for a single polymer in dilute solution, modeled by lattice walks in three dimensions, is not universal with respect to intermonomer interactions. Moreover, it has been conjectured that key critical exponents ϕ, measuring the growth of the contacts with the surface at the adsorption point, and 1/δ, which measures the finite-size shift of the critical temperature, are not the same. However, applying standard scaling arguments the two key critical exponents should rather be identical, hence pointing to a potential breakdown of these standard scaling arguments. Both of these conjectures are in contrast to the well-studied situation in two dimensions, where there are exact results from conformal field theory: these exponents are both accepted to be 1/2 and universal. We use the flatPERM algorithm to simulate self-avoiding walks and trails on the hexagonal, square, and simple cubic lattices up to length 1024 to investigate these claims. Walks can be seen as a repulsive limit of intermonomer interaction for trails, allowing us to probe the universality of adsorption. For each lattice model we analyze several thermodynamic properties to produce different methods of estimating the critical temperature and the key exponents. We test our methodology on the two-dimensional cases, and the resulting spread in values for ϕ and 1/δ indicates that there is a systematic error which can far exceed the statistical error usually reported. We further suggest a methodology for consistent estimation of the key adsorption exponents which gives ϕ=1/δ=0.484(4) in three dimensions. Hence, we conclude that in three dimensions these critical exponents indeed differ from the mean-field value of 1/2, as had previously been calculated, but cannot find evidence that they differ from each other. Importantly, we also find no substantive evidence of any nonuniversality in the polymer adsorption transition.

Singular ferromagnetic susceptibility of the transverse-field Ising antiferromagnet on the triangular lattice

A transverse magnetic field $\Gamma$ is known to induce antiferromagnetic three-sublattice order of the Ising spins $\sigma^z$ in the triangular lattice Ising antiferromagnet at low enough temperature. This low-temperature order is known to melt on heating in a two-step manner, with a power-law ordered intermediate temperature phase characterized by power-law correlations at the three-sublattice wavevector ${\bf Q}$: $\langle \sigma^z(\vec{R}) \sigma^z(0)\rangle \sim \cos({\mathbf Q}\cdot \vec{R}) /|\vec{R}|^{\eta(T)}$ with the temperature-dependent power-law exponent $\eta(T) \in (1/9,1/4)$. Here, we use a newly developed quantum cluster algorithm to study the {\em ferromagnetic} easy-axis susceptibility $\chi_{u}(L)$ of an $L \times L$ sample in this power-law ordered phase. Our numerical results are consistent with a recent prediction of a singular $L$ dependence $\chi_{u}(L)\sim L^{2- 9 \eta}$ when $\eta(T)$ is in the range $(1/9,2/9)$. This finite-size result implies, via standard scaling arguments, that the ferromagnetic susceptibility $\chi_{u}(B)$ to a uniform field $B$ along the easy axis is singular at intermediate temperatures in the small $B$ limit, $\chi_{u}(B) \sim |B|^{-\frac{4 - 18 \eta}{4-9\eta}}$ for $\eta(T) \in (1/9, 2/9)$, although there is no ferromagnetic long-range order in the low temperature state.

Interfacial adsorption in two-dimensional pure and random-bond Potts models.

We use Monte Carlo simulations to study the finite-size scaling behavior of the interfacial adsorption of the two-dimensional square-lattice q-states Potts model. We consider the pure and random-bond versions of the Potts model for q=3,4,5,8, and 10, thus probing the interfacial properties at the originally continuous, weak, and strong first-order phase transitions. For the pure systems our results support the early scaling predictions for the size dependence of the interfacial adsorption at both first- and second-order phase transitions. For the disordered systems, the interfacial adsorption at the (disordered induced) continuous transitions is discussed, applying standard scaling arguments and invoking findings for bulk critical properties. The self-averaging properties of the interfacial adsorption are also analyzed by studying the infinite limit-size extrapolation of properly defined signal-to-noise ratios.

Fixed-point structure and effective fractional dimensionality for O(N) models with long-range interactions.

We study, by renormalization group methods, O(N) models with interactions decaying as power law with exponent d+σ. When only the long-range momentum term p(σ) is considered in the propagator, the critical exponents can be computed from those of the corresponding short-range O(N) models at an effective fractional dimension D(eff). Neglecting wave function renormalization effects the result for the effective dimension is D(eff)=2d/σ, which turns to be exact in the spherical model limit (N→∞). Introducing a running wave function renormalization term the effective dimension becomes instead D(eff)=(2-η(SR))d/σ. The latter result coincides with the one found using standard scaling arguments. Explicit results in two and three dimensions are given for the exponent ν. We propose an improved method to describe the full theory space of the models where both short- and long-range propagator terms are present and no a priori choice among the two in the renormalization group flow is done. The eigenvalue spectrum of the full theory for all possible fixed points is drawn and a full description of the fixed-point structure is given, including multicritical long-range universality classes. The effective dimension is shown to be only approximate, and the resulting error is estimated.

Existence of PrescribedL2-Norm Solutions for a Class of Schrödinger-Poisson Equation

By using the standard scaling arguments, we show that the infimum of the following minimization problem:Iρ2=inf{(1/2)∫ℝ3|∇u|2dx+(1/4)∬ℝ3(|u(x)|2|u(y)|2/|x-y|)dx dy− (1/p)∫ℝ3|u|pdx:u∈Bρ}can be achieved forp∈(2,3)andρ>0small, whereBρ:={u∈H1(ℝ3):∥u∥2=ρ}. Moreover, the properties ofIρ2/ρ2and the associated Lagrange multiplierλρare also given ifp∈(2,8/3].

Dynamical and stationary critical behavior of the Ising ferromagnet in a thermal gradient

In this paper we present and discuss results of Monte Carlo numerical simulations of the two-dimensional Ising ferromagnet in contact with a heat bath that intrinsically has a thermal gradient. The extremes of the magnet are at temperatures $T_1<T_c<T_2$, where $T_c$ is the Onsager critical temperature. In this way one can observe a phase transition between an ordered phase ($T<T_c$) and a disordered one ($T>T_c$) by means of a single simulation. By starting the simulations with fully disordered initial configurations with magnetization $m\equiv 0$ corresponding to $T=\infty$, which are then suddenly annealed to a preset thermal gradient, we study the short-time critical dynamic behavior of the system. Also, by setting a small initial magnetization $m=m_0$, we study the critical initial increase of the order parameter. Furthermore, by starting the simulations from fully ordered configurations, which correspond to the ground state at T=0 and are subsequently quenched to a preset gradient, we study the critical relaxation dynamics of the system. Additionally, we perform stationary measurements ($t\rightarrow\infty$) that are discussed in terms of the standard finite-size scaling theory. We conclude that our numerical simulation results of the Ising magnet in a thermal gradient, which are rationalized in terms of both dynamic and standard scaling arguments, are fully consistent with well established results obtained under equilibrium conditions.

Impurity spin texture at the critical point between Néel-ordered and valence-bond-solid states in two-dimensional SU(3) quantum antiferromagnets

We study the impurity physics at a continuous quantum phase transition from an SU(3) symmetric N\'eel-ordered state to a valence-bond-solid state that breaks lattice symmetries, using quantum Monte Carlo techniques. This continuous transition is expected to be an example of ``deconfined criticality'' in an SU(3) symmetric system. We find that the spin-texture induced by a missing-spin defect at the transition takes on a finite-size scaling form consistent with expectations from standard scaling arguments at a scale-invariant quantum critical point, albeit with significant subleading power-law finite size corrections that we analyze in detail. Together with recently found logarithmic violations of scaling at similar continuous transitions in the SU(2) case, our results provide indirect evidence of the existence of operators that become marginal as $N$ is reduced to $2$ in the field theoretical description of these deconfined critical points.

The equilibrium limit of a constitutive model for two-phase granular mixtures and its numerical approximation

In this paper we analyze the equilibrium limit of the constitutive model for two-phase granular mixtures introduced in Papalexandris (2004) [13], and develop an algorithm for its numerical approximation. At, equilibrium, the constitutive model reduces to a strongly coupled, overdetermined system of quasilinear elliptic partial differential equations with respect to the pressure and the volume fraction of the solid granular phase. First we carry a perturbation analysis based on standard hydrostatic-type scaling arguments which reduces the complexity of the coupling of the equations. The perturbed system is then supplemented by an appropriate compatibility condition which arises from the properties of the gradient operator. Further, based on the Helmholtz decomposition and Ladyzhenskaya's decomposition theorem, we develop a projection-type, Successive-Over-Relaxation numerical method. This method is general enough and can be applied to a variety of continuum models of complex mixtures and mixtures with micro-structure. We also prove that this method is both stable and consistent hence, under standard assumptions, convergent. The paper concludes with the presentation of representative numerical results.

Noninteracting Electrons and the Metal-Insulator Transition in Two Dimensions with Correlated Impurities

While standard scaling arguments show that a system of noninteracting electrons in two dimensions and in the presence of uncorrelated disorder is insulating, in this paper we discuss the case where interimpurity correlations are included. We find that for pointlike impurities and an infinite interimpurity correlation length, a mobility edge exists in 2D even if the individual impurity potentials are random. In the uncorrelated system we recover the scaling results, while in the intermediate regime for length scales comparable to the correlation length, the system behaves like a metal but with increasing fluctuations, before strong localization eventually takes over for length scales much larger than the correlation length. In the intermediate regime, the relevant length scale is given by the interimpurity correlation length, with important consequences for high mobility systems.

Epidemic spreading with immunization and mutations.

The spreading of infectious diseases with and without immunization of individuals can be modeled by stochastic processes that exhibit a transition between an active phase of epidemic spreading and an absorbing phase, where the disease dies out. In nature, however, the transmitted pathogen may also mutate, weakening the effect of immunization. In order to study the influence of mutations, we introduce a model that mimics epidemic spreading with immunization and mutations. The model exhibits a line of continuous phase transitions and includes the general epidemic process (GEP) and directed percolation (DP) as special cases. Restricting to perfect immunization in two spatial dimensions, we analyze the phase diagram and study the scaling behavior along the phase transition line as well as in the vicinity of the GEP point. We show that mutations lead generically to a crossover from the GEP to DP. Using standard scaling arguments, we also predict the form of the phase transition line close to the GEP point. The protection gained by immunization is vitally decreased by the occurrence of mutations.

Ill-posedness issues for a class of parabolic equations

We prove that the Cauchy problem for the one-dimensional parabolic equations , with initial data in Hs(R), cannot be solved by an iterative scheme based on the Duhamel formula for s &lt; −1 if (k, d) = (2, 0) and s &lt; sc(k, d) = ½ − (2 − d)/(k − 1) otherwise. This exactly completes the positive results on the Cauchy problem in Hs(R) for these equations and shows the particularity of the case (k, d) = (2, 0), for which we prove that the critical space Hsc(R) = H−3/2(R), by standard scaling arguments, cannot be reached. Our results also hold in the periodic setting.

The Hausdorff dimension of random walks and the correlation length critical exponent in Euclidean field theory

We study the random walk representation of the two-point function in statistical mechanics models near the critical point. Using standard scaling arguments, we show that the critical exponentv describing the vanishing of the physical mass at the critical point is equal tov Θ /dw, whered w is the Hausdorff dimension of the walk, andv Θ is the exponent describing the vanishing of the energy per unit length of the walk at the critical point. For the case ofO(N) models, we show thatv 0=ϕ, whereϕ is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk isϕ/v forO(N) models.

Extreme ultraviolet spectroscopy of Capella

The X-ray active binary system Capella was observed with a moderate-resolution extreme ultraviolet spectrograph from 200 to 330 A. Two low-level emission features were detected. One most likely is geocoronal He II 304 A emission, while the other probably originates from the corona of Capella. The weak stellar emission at 304 A is in direct conflict with predictions of the intrinsic stellar He II flux based on standard scaling arguments but is consistent with the only previous observation of Capella in the EUV. The most plausible explanation for the lack of stellar 304 A emission is a warm wind from the active G0 III star.