Logarithmic Potential Research Articles

Anomalous diffusion: fractional Brownian motion vs fractional Ito motion

Generalizing Brownian motion (BM), fractional Brownian motion (FBM) is a paradigmatic selfsimilar model for anomalous diffusion. Specifically, varying its Hurst exponent, FBM spans: sub-diffusion, regular diffusion, and super-diffusion. As BM, also FBM is a symmetric and Gaussian process, with a continuous trajectory, and with a stationary velocity. In contrast to BM, FBM is neither a Markov process nor a martingale, and its velocity is correlated. Based on a recent study of selfsimilar Ito diffusions, we explore an alternative selfsimilar model for anomalous diffusion: fractional Ito motion (FIM). The FIM model exhibits the same Hurst-exponent behavior as FBM, and it is also a symmetric process with a continuous trajectory. In sharp contrast to FBM, we show that FIM: is not a Gaussian process; is a Markov process; is a martingale; and its velocity is not stationary and is not correlated. On the one hand, FBM is hard to simulate, its analytic tractability is limited, and it generates only a Gaussian dissipation pattern. On the other hand, FIM is easy to simulate, it is analytically tractable, and it generates non-Gaussian dissipation patterns. Moreover, we show that FIM has an intimate linkage to diffusion in a logarithmic potential. With its compelling properties, FIM offers researchers and practitioners a highly workable analytic model for anomalous diffusion.

Method of variable separation for investigating exact solutions and dynamical properties of the time-fractional Fokker–Planck equation

In this paper, the traditional separation method of variables is improved by fixing a part of the variables called the separation method of semi-fixed variables. By using this improved method, the time-fractional Fokker–Planck equation with external force field is studied. Some new exact solutions and dynamical properties of the equation are investigated in various external potential functions such as linear potential, harmonic potential, logarithmic potential, exponential potential, and quartic potential. Some interesting dynamical behaviors and phenomena are discovered. The profiles of several representative exact solutions are illustrated by 3D-graphs and 2D-graphs

Trudinger–Moser‐type inequality with logarithmic convolution potentials

We establish Moser–Trudinger-type inequalities in the presence of a logarithmic convolution potential when the domain is a ball or the entire space R 2 $\mathbb {R}^2$ . Moreover, we characterize critical nonlinear growth rates for these inequalities to hold and for the existence of corresponding extremal functions. In addition, we show that extremal functions satisfy corresponding Euler–Lagrange equations, and we derive general symmetry and uniqueness results for solutions of these equations.

Extremal particles of two-dimensional Coulomb gases and random polynomials on a positive background

We study the outliers for two models which have an interesting connection. On the one hand, we study a specific class of planar Coulomb gases which are determinantal. It corresponds to the case where the confining potential is the logarithmic potential of a radial probability measure. On the other hand, we study the zeros of random polynomials that appear to be closely related to the first model. Their behavior far from the origin is shown to depend only on the decaying properties of the probability measure generating the potential. A similar feature is observed for their behavior near the origin. Furthermore, in some cases, the appearance of outliers is observed, and the zeros of random polynomials and the Coulomb gases are seen to exhibit exactly the same behavior, which is related to the unweighted Bergman kernel.

An Optimal Estimate for the Anisotropic Logarithmic Potential

This paper introduces the new annulus body to establish the optimal lower bound for the anisotropic logarithmic potential as the complement to the theory of its upper bound estimate which has already been investigated. The connections with convex geometry analysis and some metric properties are also established. For the application, a polynomial dual log-mixed volume difference law is deduced from the optimal estimate.

Global and exponential attractors for a Cahn–Hilliard equation with logarithmic potentials and mass source

We investigate the long-time behaviour of strong solutions to a generalised Cahn–Hilliard equation with singular logarithmic potentials and a solution-dependent mass source term. Under appropriate choices of the latter, such models have been used for image inpainting and various biological applications. The logarithmic potential is used to ensure the phase field variable stays in the physically relevant interval. Under rather general assumptions on the source term, we first demonstrate a dissipative estimate, leading to uniform-in-time bounds and regularity assertions for previously established weak solutions. Using these, for two spatial dimensions, we prove global strong well-posedness for the model, and demonstrate the existence of the global attractor and exponential attractors with finite fractal dimensions. Moreover, a backwards uniqueness property is shown for the dynamics restricted to the global attractor.

A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials

<p style='text-indent:20px;'>We introduce a regularization-free approach for the wellposedness of the classic Cahn-Hilliard equation with logarithmic potentials.</p>

Wave optics and weak gravitational lensing of lights from spherically symmetric static scalar vector tensor Brans Dicke wormholes

This paper has three parts. In first step we use modified scalar tensor vector Brans Dicke gravity \cite{GH0} to obtain metric solution of a spherically symmetric static black hole via perturbation method which asymptotically behaves as modified Schwarzschild de Sitter black hole in weak field limits. Corrections on the line element with respect to the point mass Schwarzschild de Sitter black hole are two parts: power law function for black hole mass instead of point mass and additional logarithmic metric potential which they are produced from effects of interacting timelike dynamical vector fields. Second part of the paper is dedicated to review Fresnel-Kirchhoff diffraction theory for massless scalar field instead of the vector electromagnetic waves. In third part of the work we consider the obtained black hole metric to be lens and we study weak gravitational lensing of moving light originated from point star. Production of stationary images is investigated via both interference of waves and geometric optics approaches. At last we compare results of these approaches and give some outlooks.

A Second-Order Strang Splitting Scheme with Exponential Integrating Factor for the Allen-Cahn Equation with Logarithmic Flory-Huggins Potential

A Second-Order Strang Splitting Scheme with Exponential Integrating Factor for the Allen-Cahn Equation with Logarithmic Flory-Huggins Potential

A generalization of the discrete Rodrigues formula for Meixner polynomials

A generalization of Meixner polynomials leading to a new construction of Apéry approximations is put forward. The limiting distribution of the zeros of scaled polynomials is described in terms of algebraic functions. The resulting distribution is shown to be a solution of some vector equilibrium problem in the theory of logarithmic potential. Bibliography: 21 titles.

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term

<p style='text-indent:20px;'>The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in <inline-formula><tex-math id="M1">\begin{document}$ {{\mathbb{R}}}^d $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ 1\leq d\leq 3 $\end{document}</tex-math></inline-formula>, with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case <inline-formula><tex-math id="M3">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula>. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain.</p>

An iteration solver for the Poisson–Nernst–Planck system and its convergence analysis

In this paper, we provide a theoretical analysis for an iteration solver to implement a finite difference numerical scheme for the Poisson-Nernst–Planck (PNP) system, based on the Energetic Variational Approach (EnVarA), in which a non-constant mobility H−1 gradient flow is formulated. In particular, the nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty. In the numerical design, the mobility function is explicitly updated, for the sake of unique solvability analysis. The logarithmic and the electric potential diffusion terms, which come from the gradient of convex energy functional parts, are implicitly computed. The positivity-preserving property for all the concentrations, an unconditional energy stability, and the optimal rate error estimate have been established in a recent work. A modified Newton iteration for the nonlinear and logarithmic part, combined with a linear iteration for the electric potential part, is proposed to implement the given numerical scheme, in which a non-constant linear elliptic equation needs to be solved at each iteration stage. A theoretical analysis is presented in this article, and a linear convergence is proved for such an iteration, with an asymptotic error constant in the same order of the time step size. A numerical test is also presented in this article, which demonstrates the linear convergence rate of the proposed iteration solver.

On the viscous Cahn–Hilliard–Oono system with chemotaxis and singular potential

We analyze a diffuse interface model that couples a viscous Cahn–Hilliard equation for the phase function φ with a diffusion‐reaction equation for the nutrient concentration σ. The system under consideration also takes into account some important mechanisms like chemotaxis, active transport, and nonlocal interaction of Oono's type. When the spatial dimension is three, we prove the existence and uniqueness of a global weak solution to the model with singular potentials including the physically relevant logarithmic potential. Then we obtain some regularity properties of the weak solution when t > 0. In particular, with the aid of the viscous term ϵ∂tφ, we prove the so‐called instantaneous separation property of the phase variable φ such that it stays away from the pure states ±1 as long as t > 0. Next, we study longtime behavior of the system, by proving the existence of the global attractor in a suitable phase space and characterizing the ω‐limit set. Moreover, we show the existence of an exponential attractor, which implies that the global attractor is of finite fractal dimension.

Nonlocal Cahn–Hilliard–Hele–Shaw Systems with Singular Potential and Degenerate Mobility

We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $\varphi$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Korteweg force $\mu\nabla \varphi$, where $\mu$ is the nonlocal chemical potential. In addition, the kinematic viscosity $\eta$ may depend on $\varphi$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we prove the existence of a strong solution. Further regularity results on the pressure and on $\mathbf{u}$ are also obtained. Weak-strong uniqueness is demonstrated in the two dimensional case. In the three-dimensional case, uniqueness of weak solutions holds if $\eta$ is constant. Otherwise, weak-strong uniqueness is shown by assuming that the pressure of the strong solution is $\alpha$-H\"{o}lder continuous in space for $\alpha\in (1/5,1)$.

Geometries on the cone of positive-definite matrices derived from the power potential and their relation to the power means

We study a Riemannian metric on the cone of symmetric positive-definite matrices obtained from the Hessian of the power potential function (1−det⁡(X)β)/β. We give explicit expressions for the geodesics and distance function, under suitable conditions. In the scalar case, the geodesic between two positive numbers coincides with a weighted power mean, while for matrices of size at least two it yields a notion of weighted power mean different from the ones given in the literature. As β tends to zero, the power potential converges to the logarithmic potential, that yields a well-known metric associated with the matrix geometric mean; we show that the geodesic and the distance associated with the power potential converge to the weighted matrix geometric mean and the distance associated with the logarithmic potential, respectively.

Passage of test particles through oscillating spherically symmetric dark matter configurations

Applying the perturbative approach to geodesic equations, we study motion of the test particles in time-dependent spherically symmetric spacetimes created by oscillating dark matter. Assuming the weakness of the gravitational field, we derive general formulas that describe infinite trajectories of the test particles and determine the total deflection angle in the leading order approximation. The obtained formulas are valid for both time-dependent and static matter configurations. Using these results, we calculate the deflection angle of a test particle passing through a spherically symmetric oscillating distribution of a self-gravitating scalar field with a logarithmic potential. It turned out that, in a wide range of amplitudes, oscillations in the deflection angle are sinusoidal and become small for ultrarelativistic particles.

Asymptotics of the minimum values of Riesz and logarithmic potentials generated by greedy energy sequences on the unit circle

In this work we investigate greedy energy sequences on the unit circle for the logarithmic and Riesz potentials. By definition, if (an)n=0∞ is a greedy s-energy sequence on the unit circle, the Riesz potential UN,s(x):=∑k=0N−1|ak−x|−s, s>0, generated by the first N points of the sequence attains its minimum value at the point aN, for every N≥1. In the case s=0 we minimize instead the logarithmic potential UN,0(x):=−∑k=0N−1log⁡|ak−x|. We analyze the asymptotic properties of these extremal values UN,s(aN), studying separately the cases s=0, 0<s<1, s=1, and s>1. We obtain second-order asymptotic formulas for UN,s(aN) in the cases s=0, 0<s<1, and s=1 (the corresponding first-order formulas are well known). A first-order result for s>1 is proved, and it is shown that the normalized sequence UN,s(aN)/Ns is bounded and divergent in this case. We also consider, briefly, greedy energy sequences in which the minimization condition is required starting from the point ap+1 (instead of the point a1 as previously stated), for some p≥1. For this more general class of greedy sequences, we prove a first-order asymptotic result for 0≤s<1.

Selfsimilar stochastic differential equations

Diffusion in a logarithmic potential (DLP) attracted significant interest in physics recently. The dynamics of DLP are governed by a Langevin stochastic differential equation (SDE) whose underpinning potential is logarithmic, and that is driven by Brownian motion. The SDE that governs DLP is a particular case of a selfsimilar SDE: one that is driven by a selfsimilar motion, and that produces a selfsimilar motion. This paper establishes the pivotal role of selfsimilar SDEs via two novel universality results. I) Selfsimilar SDEs emerge universally, on the macro level, when applying scaling limits to micro-level SDEs. II) Selfsimilar SDEs emerge universally when applying the Lamperti transformation to stationary SDEs. Using the universality results, this paper further establishes: a novel statistical-analysis approach to selfsimilar Ito diffusions; and the focal importance of DLP.

Theory space of one unitary matrix model and its critical behavior associated with Argyres–Douglas theory

The lowest critical point of one unitary matrix model with cosine plus logarithmic potential is known to correspond with the [Formula: see text] Argyres–Douglas (AD) theory and its double scaling limit derives the Painlevé II equation with parameter. Here, we consider the critical points associated with all cosine potentials and determine the scaling operators, their vacuum expectation values (vevs) and their scaling dimensions from perturbed string equations at planar level. These dimensions agree with those of [Formula: see text] AD theory.

Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions

We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.