arXiv subjects

Tian Ma

Publications and source records attributed to Tian Ma.

At least 37 records · Page 2Linked to original sources

arXiv2012

Relatively Coherent Sets as a Hierarchical Partition Method

Finite time coherent sets [8] have recently been defined by a measure based objective function describing the degree that sets hold together, along with a Frobenius-Perron transfer operator method to produce optimally coherent sets. Here we present an extension to generalize the concept to hierarchially defined relatively coherent sets based on adjusting the finite time coherent sets to use relative mesure restricted to sets which are developed iteratively and hierarchically in a tree of partitions. Several examples help clarify the meaning and expectation of the techniques, as they are the nonautonomous double gyre, the standard map, an idealized stratospheric flow, and empirical data from the Mexico Gulf during the 2010 oil spill. Also for sake of analysis of computational complexity, we include an appendic concerning the computational complexity of developing the Ulam-Galerkin matrix extimates of the Frobenius-Perron operator centrally used here.

math.DS↗

arXiv2012

Dynamic Transition and Pattern Formation for Chemotactic Systems

The main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.

nlin.AO↗

arXiv2011

Tropical Atmospheric Circulations with Humidity Effects

The main objective of this article is to study the effect of the moisture on the planetary scale atmospheric circulation over the tropics. The modeling we adopt is the Boussinesq equations coupled with a diffusive equation of humidity and the humidity dependent heat source is modeled by a linear approximation of the humidity. The rigorous mathematical analysis is carried out using the dynamic transition theory. In particular, we obtain the same types of transitions and hence the scenario of the El Niño mechanism as described in \cite{MW2,MW3}. The effect of the moisture only lowers slightly the magnitude of the critical thermal Rayleigh number.

physics.ao-ph↗

arXiv2010

Phase Transitions for the Brusselator Model

Dynamic phase transitions of the Brusselator model is carefully analyzed, leading to a rigorous characterization of the types and structure of the phase transitions of the model from basic homogeneous states. The study is based on the dynamic transition theory developed recently by the authors.

math-ph↗

arXiv2010

Phase Transitions for Belousov-Zhabotinsky Reactions

The main objective of this article is to study the dynamic phase transitions associated with the spatial-temporal oscillations of the BZ reactions, given by Field, Koros and Noyes, also referred as the Oregonator. Two criteria are derived to determine 1) existence of either multiple equilibria or spatiotemporal oscillations, and 2) the types of transitions. These criteria gives a complete characterization of the dynamic transitions of the BZ systems from the homogeneous states. The analysis is carried out using a dynamic transition theory developed recently by the authors, which has been successfully applied to a number of problems in science.

math-ph↗

arXiv2010

Third-Order Gas-Liquid Phase Transition and the Nature of Andrews Critical Point

The main objective of this article is to study the nature of the Andrews critical point in the gas-liquid transition in a physical-vapor transport (PVT) system. A dynamical model, consistent with the van der Waals equation near the Andrews critical point, is derived. With this model, we deduce two physical parameters, which interact exactly at the Andrews critical point, and which dictate the dynamic transition behavior near the Andrews critical point. In particular, it is shown that 1) the Andrews critical point is a switching point where the phase transition changes from the first order to the third order, 2) the gas-liquid co-existence curve can be extended beyond the Andrews critical point, and 3) the liquid-gas phase transition going beyond Andrews point is of the third order. This clearly explains why it is hard to observe the gas-liquid phase transition beyond the Andrews critical point. Furthermore, the analysis leads naturally the introduction of a general asymmetry principle of fluctuations and the preferred transition mechanism for a thermodynamic system.

cond-mat.soft↗

arXiv2010

Dynamic Transition and Pattern Formation in Taylor Problem

The main objective of this article is to study both dynamic and structural transitions of the Taylor-Couette flow, using the dynamic transition theory and geometric theory of incompressible flows developed recently by the authors. In particular we show that as the Taylor number crosses the critical number, the system undergoes either a continuous or a jump dynamic transition, dictated by the sign of a computable, nondimensional parameter $R$. In addition, we show that the new transition states have the Taylor vortex type of flow structure, which is structurally stable.

math-ph↗

arXiv2009

Dynamic Transition Theory for Thermohaline Circulation

The main objective of this and its accompanying articles is to derive a mathematical theory associated with the thermohaline circulations (THC). This article provides a general transition and stability theory for the Boussinesq system, governing the motion and states of the large-scale ocean circulation. First, it is shown that the first transition is either to multiple steady states or to oscillations (periodic solutions), determined by the sign of a nondimensional parameter $K$, depending on the geometry of the physical domain and the thermal and saline Rayleigh numbers. Second, for both the multiple equilibria and periodic solutions transitions, both Type-I (continuous) and Type-II (jump) transitions can occur, and precise criteria are derived in terms of two computable nondimensional parameters $b_1$ and $b_2$. Associated with Type-II transitions are the hysteresis phenomena, and the physical reality is represented by either metastable states or by a local attractor away from the basic solution, showing more complex dynamical behavior. Third, a convection scale law is introduced, leading to an introduction of proper friction terms in the model in order to derive the correct circulation length scale. In particular, the dynamic transitions of the model with the derived friction terms suggest that the THC favors the continuous transitions to stable multiple equilibria. Applications of the theoretical analysis and results to different flow regimes will be explored in the accompanying articles.

physics.ao-ph↗

arXiv2009

Tropical Atmospheric Circulations: Dynamic Stability and Transitions

In this article, we present a mathematical theory of the Walker circulation of the large-scale atmosphere over the tropics. This study leads to a new metastable state oscillation theory for the El Nino Southern Oscillation (ENSO), a typical inter-annual climate low frequency oscillation. The mathematical analysis is based on 1) the dynamic transition theory, 2) the geometric theory of incompressible flows, and 3) the scaling law for proper effect of the turbulent friction terms, developed recently by the authors.

math-ph↗

arXiv2008

El Nino Southern Oscillation as Sporadic Oscillations between Metastable States

The main objective of this article is to establish a new mechanism of the El Nino Southern Oscillation (ENSO), as a self-organizing and self-excitation system, with two highly coupled processes. The first is the oscillation between the two metastable warm (El Nino phase) and cold events (La Nina phase), and the second is the spatiotemporal oscillation of the sea surface temperature (SST) field. The interplay between these two processes gives rises the climate variability associated with the ENSO, leads to both the random and deterministic features of the ENSO, and defines a new natural feedback mechanism, which drives the sporadic oscillation of the ENSO. The new mechanism is rigorously derived using a dynamic transition theory developed recently by the authors, which has also been successfully applied to a wide range of problems in nonlinear sciences.

physics.ao-ph↗

arXiv2008

Phase Separation of Binary Systems

In this Letter, three physical predictions on the phase separation of binary systems are derived based on a dynamic transition theory developed recently by the authors. First, the order of phase transitions is precisely determined by the sign of a parameter Kd (or a nondimensional parameter K) such that if Kd>0, the transition is first-order with latent heat and if Kd <0, the transition is second-order. Second, a theoretical transition diagram is derived, leading in particular to a prediction that there is only second-order transition for molar fraction near 1/2. This is different from the prediction made by the classical transition diagram. Third, a critical length scale is derived such that no phase separation occurs at any temperature if the length of the container is smaller than the critical length scale.

cond-mat.stat-mech↗

arXiv2008

Cahn-Hilliard Equations and Phase Transition Dynamics for Binary Systems

The process of phase separation of binary systems is described by the Cahn-Hilliard equation. The main objective of this article is to give a classification on the dynamic phase transitions for binary systems using either the classical Cahn-Hilliard equation or the Cahn-Hilliard equation coupled with entropy, leading to some interesting physical predictions. The analysis is based on dynamic transition theory for nonlinear systems and new classification scheme for dynamic transitions, developed recently by the authors.

math-ph↗

arXiv2008

Phase Transition and Separation for Mixture of Liquid He-3 and He-4

This article introduces a dynamical Ginzburg-Landau phase transition/separation model for the mixture of liquid helium-3 and helium-4, using a unified dynamical Ginzburg-Landau model for equilibrium phase transitions. The analysis of this model leads to three critical length scales L1 < L2 < L3, detailed theoretical phase diagrams and transition properties with different length scales of the container.

cond-mat.soft↗

arXiv2008

Superfluidity of Helium-3

This article presents a phenomenological dynamic phase transition theory -- modeling and analysis -- for liquid helium-3. We derived two new models, for liquid helium-3 with or without applied field, by introducing three wave functions and using a unified dynamical Ginzburg-Landau model. The analysis of these new models leads to predictions of existence of 1) a unstable region, 2) a new phase C in a narrow region, and 3) switch points of transition types on the coexistence curves near two triple points. It is hoped that these predictions will be useful for designing better physical experiments and lead to better understanding of the physical mechanism of superfluidity.

cond-mat.other↗

arXiv2008

Dynamic Model and Phase Transitions for Liquid Helium

This article presents a phenomenological dynamic phase transition theory -- modeling and analysis -- for superfluids. As we know, although the time-dependent Ginzburg-Landau model has been successfully used in superconductivity, and the classical Ginzburg-Landau free energy is still poorly applicable to liquid helium in a quantitative sense. The study in this article is based on 1) a new dynamic classification scheme of phase transitions, 2) new time-dependent Ginzburg-Landau models for general equilibrium transitions, and 3) the general dynamic transition theory. The results in this article predict the existence of a unstable region H, where both solid and liquid He II states appear randomly depending on fluctuations and the existence of a switch point M on the lambda-curve, where the transitions changes types.

cond-mat.stat-mech↗

arXiv2007

Dynamic Phase Transitions in PVT Systems

The main objective of this article are two-fold. First, we introduce some general principles on phase transition dynamics, including a new dynamic transition classification scheme, and a Ginzburg-Landau theory for modeling equilibrium phase transitions. Second, apply the general principles and the recently developed dynamic transition theory to study dynamic phase transitions of PVT systems. In particular, we establish a new time-dependent Ginzburg-Landau model, whose dynamic transition analysis is carried out. It is worth pointing out that the new dynamic transition theory, along with the dynamic classification scheme and new time-dependent Ginzburg Landau models for equilibrium phase transitions can be used in other phase transition problems, including e.g. the ferromagnetism and superfluidity, which will be reported elsewhere. In addition, the analysis for the PVT system in this article leads to a few physical predications, which are otherwise unclear from the physical point of view.

math-ph↗

arXiv2007

Dynamic Phase Transitions for Ferromagnetic Systems

This article presents a phenomenological dynamic phase transition theory for ferromagnetism, leading to a precise description of the dynamic transitions, and to a physical predication on the spontaneous magnetization. The analysis also suggests asymmetry of fluctuations in both the ferromagnetism and the PVT systems.

cond-mat.mtrl-sci↗

arXiv2007

Dynamic Phase Transitions in Superconductivity

In this Letter, the dynamic phase transitions of the time-dependent Ginzburg-Landau equations are analyzed using a newly developed dynamic transition theory and a new classification scheme of dynamics phase transitions. First, we demonstrate that there are two type of dynamic transitions, jump and continuous, dictated by the sign of a nondimensional parameter R. This parameter is computable, and depends on the material property, the applied field, and the geometry of domain that the sample occupies. Second, using the parameter R, precise analytical formulas for critical domain size, and for critical magnetic fields are derived.

cond-mat.supr-con↗