# Energetic Variational Approaches: General Diffusion and Stochastic Process

*Chun LIU*

*Professor of MathematicsDepartment of MathematicsPennsylvania State University*

In this talk, I will explore the variational structures in some specific types of non-ideal diffusion and stochastic processes. In particular, we will focus on those with nonlocal interactions and couplings.

# New PNP Type Systems for Ionic Liquids

*Tai-Chia LIN*

*ProfessorDepartment of MathematicsNational Taiwan University*

To describe ionic liquids with finite size effects involving different ion radii and valences, we derive new PNP (Poisson-Nernst-Planck) type systems and develop mathematical theorems for these systems. Symmetry and non-symmetry breaking conditions are represented by their coupling coefficients. When non-symmetry breaking condition holds true, we prove the existence theorems of solutions of new PNP type systems. On the other hand, when symmetry breaking condition holds true, two steady state solutions can be found and the excess currents (due to steric effects) associated with these two steady state solutions are derived and expressed as two distinct formulas. Our results indicate that new PNP type systems may become a useful model to study ionic liquids and related topics of liquid crystals.

# Biologically Active Nematics

*Michael Shelley*

*Lilian and George Lyttle Professor of Applied MathematicsProfessor of Mathematics and Neural Science Co-Director, Applied Mathematics Laboratory Courant Institute of Mathematical SciencesNew York University*

Active fluids are complex fluids with active microstructure that create non-thermodynamic stresses even in the absence of external forcing.

A typical example of such a out-of-equilibrium system is a bacterial bath where stresses created by bacterial swimming can create large-scale chaotic mixing flows. Other examples arise in cellular biophysics where the interactions of biopolymers with motor-proteins can yield "active nematic" states of matter. I will discuss the mathematical modeling and simulation of these systems by building Doi-Onsager descriptions based upon a microscopic conception of the microstructural dynamics.

# On Ground States of Spin-1 Bose-Einstein Condensates w/o External Magnetic Field

*I-Liang Chern*

*Professor Department of Mathematics National Taiwan University*

In this talk, I will first give a brief introduction to the spinor Bose-Einstein condensates (BECs). Then I will present two recent results, one is numerical, the other is analytical for spinor BECs w/o uniform external magnetic field. In the numerical study of spinor BECs, a pseudo-arclength continuation method (PACM) was proposed for investigating the ground state patterns and phase diagrams of the spin-1 Bose-Einstein condensates under the influence of a homogeneous magnetic field. Two types of phase transitions are found. The first type is a transition from a two-component (2C) state to a three-component (3C) state. The second type is a symmetry breaking in 3C state. After that, a phase separation of the spin component occurs. In the semi-classical regime, these two phase transition curves are gradually merged.

In the analytical study, the ground states of spin-1 BEC are characterized. First, we present the case when there is no external magnetic field. For ferromagnetic systems, we show the validity of the so-called single-mode approximation (SMA). For antiferromagnetic systems, there are two subcases. When the total magnetization M≠0, the corresponding ground states have vanishing zeroth (*mF* = 0) components (so call 2C state), thus are reduced to two-component systems. When M=0, the ground states are also reduced to the SMA, and there are one-parameter families of such ground states. Next, we study the case when an external magnetic field is applied. It is shown analytically that, for antiferromagnetic systems, there is a phase transition from 2C state to 3C state as the external magnetic field increases. The key idea in the proof is a redistribution of masses among different components, which reduces kinetic energy in all situations, and makes our proofs simple and unified. The numerical part is a joint work with Jen-Hao Chen and Weichung Wang, whereas the analytical part is jointly with Liren Lin.