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Pingwen Zhang

Publications and source records attributed to Pingwen Zhang.

67 records · Page 4Linked to original sources
arXiv2013

Is there a third order phase transition for supercritical fluids?

We prove that according to Molecular Dynamics (MD) simulations of liquid mixtures of Lennard-Jones (L-J) particles, there is no third order phase transition in the supercritical regime beyond Andrew's critical point. This result is in open contrast with recent theoretical studies and experiments which instead suggest not only its existence but also its universality regarding the chemical nature of the fluid. We argue that our results are solid enough to go beyond the limitations of MD and the generic character of L-J models, thus suggesting a rather smooth liquid-vapor thermodynamic behavior of fluids in supercritical regime.

physics.chem-ph↗
arXiv2013

Numerical Methods for Quasicrystals

Quasicrystals are one kind of space-filling structures. The traditional crystalline approximant method utilizes periodic structures to approximate quasicrystals. The errors of this approach come from two parts: the numerical discretization, and the approximate error of Simultaneous Diophantine Approximation which also determines the size of the domain necessary for accurate solution. As the approximate error decreases, the computational complexity grows rapidly, and moreover, the approximate error always exits unless the computational region is the full space. In this work we focus on the development of numerical method to compute quasicrystals with high accuracy. With the help of higher-dimensional reciprocal space, a new projection method is developed to compute quasicrystals. The approach enables us to calculate quasicrystals rather than crystalline approximants. Compared with the crystalline approximant method, the projection method overcomes the restrictions of the Simultaneous Diophantine Approximation, and can also use periodic boundary conditions conveniently. Meanwhile, the proposed method efficiently reduces the computational complexity through implementing in a unit cell and using pseudospectral method. For illustrative purpose we work with the Lifshitz-Petrich model, though our present algorithm will apply to more general systems including quasicrystals. We find that the projection method can maintain the rotational symmetry accurately. More significantly, the algorithm can calculate the free energy density to high precision.

physics.comp-ph↗
arXiv2013

Analytic Structure of the SCFT Energy Functional of Multicomponent Block Copolymers

This paper concerns the analytic structure of the self-consistent field theory (SCFT) energy functional of multicomponent block copolymer systems which contain more than two chemically distinct blocks. The SCFT has enjoyed considered success and wide usage in investigation of the complex phase behavior of block copolymers. It is well-known that the physical solutions of the SCFT equations are saddle points, however, the analytic structure of the SCFT energy functional has received little attention over the years. A recent work by Fredrickson and collaborators [see the monograph by Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, (2006), pp. 203-209] has analysed the mathematical structure of the field energy functional for polymeric systems, and clarified the index-1 saddle point nature of the problem produced by the incompressibility constraint. In this paper, our goals are to draw further attention to multicomponent block copolymers utilizing the Hubbard-Stratonovich transformation used by Fredrickson and co-workers. We first show that the saddle point character of the SCFT energy functional of multicomponent block copolymer systems may be high index, not only produced by the incompressibility constraint, but also by the Flory-Huggins interaction parameters. Our analysis will be beneficial to many theoretical studies, such as the nucleation theory of ordered phases, the mesoscopic dynamics. As an application, we then utilize the discovery to develop the gradient-based iterative schemes to solve the SCFT equations, and illustrate its performance through several numerical experiments taking ABC star triblock copolymers as an example.

physics.comp-ph↗
arXiv2013

From microscopic theory to macroscopic theory: a systematic study on static modeling for liquid crystals

In this paper, we propose a systematic way of liquid crystal modeling to build connection between microscopic theory and macroscopic theory. A new Q-tensor theory based on Onsager's molecular theory which leads to liquid crystals with certain shape has been proposed. Making uniaxial assumption, we can recover the Oseen-Frank theory from the derived $Q$-tensor theory, and the Oseen-Frank model coefficients can be examined. In addition, the smectic-A phase can also be characterized by the derived macroscopic model.

math.AP↗
arXiv2013

From microscopic theory to macroscopic theory: symmetries and order parameters of rigid molecules

We use density functional theory to describe the phase behaviors of rigid molecules. The construction of kernel function G(x, P, x, P) is discussed. Excluded-volume potential is calculated for two types of molecules with C_{2v} symmetry. Molecular symmetries lead to the symmetries of G and density function f(P), enabling a reduction of configuration space. By approximating G with a polynomial, the system can be fully characterized by some moments corresponding to the form of G. The symmetries of G determine the form of the polynomial, while the coefficients are determined by temperature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Order parameters for bent-core molecules are predicted.

math-ph↗
arXiv2013

Unstable Galaxy Models

The dynamics of collisionless galaxy can be described by the Vlasov-Poisson system. By the Jean's theorem, all the spherically symmetric steady galaxy models are given by a distribution of Φ(E,L), where E is the particle energy and L the angular momentum. In a celebrated Doremus-Feix-Baumann Theorem, the galaxy model Φ(E,L) is stable if the distribution Φ is monotonically decreasing with respect to the particle energy E. On the other hand, the stability of Φ(E,L) remains largely open otherwise. Based on a recent abstract instability criterion of Guo-Lin, we constuct examples of unstable galaxy models of f(E,L) and f(E) in which f fails to be monotone in E.

astro-ph.GA↗
arXiv2012

Well-posedness of the Ericksen-Leslie system

In this paper, we prove the local well-posedness of the Ericksen-Leslie system, and the global well-posednss for small initial data under the physical constrain condition on the Leslie coefficients, which ensures that the energy of the system is dissipated. Instead of the Ginzburg-Landau approximation, we construct an approximate system with the dissipated energy based on a new formulation of the system.

math.AP↗
arXiv2012

Optimizing working parameters of the twin-range cutoff method in terms of accuracy and efficiency

We construct a priori error estimation for the force error of the twin-range cutoff method, which is widely used to treat the short-range non-bonded interactions in molecular simulations. Based on the error and cost estimation, we develop a work flow that can automatically determine the nearly most efficient twin-range cutoff parameters (i.e. the cutoff radii and the neighbor list updating frequency) prior to a simulation for a predetermined accuracy. Both the error estimate and the parameter tuning method are demonstrated to be effective by testing simulations of the standard Lennard-Jones 6-12 fluid in gas, liquid as well as supercritical state. We recommend the tuned twin-range cutoff method that can save precious user time and computational resources.

physics.comp-ph↗
arXiv2012

The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation

We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen-Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional.

math.AP↗
arXiv2012

Well-posedness of Hydrodynamics on the Moving Elastic Surface

The dynamics of a membrane is a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system.

math.AP↗
arXiv2011

Dynamic Transitions and Pattern Formations for Cahn-Hilliard Model with Long-Range Repulsive Interactions

The main objective of this article is to study the order-disorder phase transition and pattern formation for systems with long-range repulsive interactions. The main focus is on the Cahn-Hilliard model with a nonlocal term in the corresponding energy functional, representing the long-range repulsive interaction. First, we show that as soon as the linear problem loses stability, the system always undergoes a dynamic transition to one of the three types, forming different patterns/structures. The types of transition are then dictated by a nondimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. The derived explicit form of this parameter offers precise information for the phase diagrams. Second, we obtain a novel and explicit pattern selection mechanism associated with the competition between the long-range repulsive interaction and the short-range attractive interactions. In particular, the hexagonal pattern is unique to the long-range interaction, and is associated with a novel two-dimensional reduced transition equations on the center manifold generated by the unstable modes, consisting of (degenerate) quadratic terms and non-degenerate cubic terms. Finally, explicit information on the metastability and basin of attraction of different disordered/ordered states and patterns are derived as well.

cond-mat.stat-mech↗