In this paper we study a Landau-type operator with an external force. It is a linear model of the Landau equation near Maxwellian distributions. Making use of multiplier method, we get the global hypoelliptic estimate under suitable assumptions on the external potential.

We establish existence of partially regular weak solutions for the Landau-Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. The construction is based on Ginzburg-Landau approximation. The new key ingredient is a nonlocal representation formula for the penalty term that permits to take advantage of the special trilinear structure of the limiting nonl...

The renormalization group (RG) method is extended for global asymptotic analysis of discrete systems. We show that the RG equation in the discretized form leads to difference equations corresponding to the Stuart-Landau or Ginzburg-Landau equations. We propose a discretization scheme which leads to a faithful discretization of the reduced dynamics of the original differential equations.

we describe how to obtain electronic transport properties of disordered graphene, including the tight binding model and nearest neighbor hopping. we present a new method for computing, electronic transport wave function and greens function of the disordered graphene. in this method, based on the small rectangular approximation, break up the potential barriers in to small parts. then using the f...

Part 3: (H, PB) Interacting systems, phase transitions and critical phenomena (11 lectures) Interacting spin systems, Ising model. Interacting classical gas, cluster expansion, van der Waals equation of state, Virial Expansion, phase equilibrium, chemical equilibrium. Interacting quantum gases in atom traps. BCS theory of Superconductivity, Landau and Ginzburg Landau theory. Topological excitat...

We consider the Ginzburg-Landau equation for a complex scalar field in one dimension and show that small phase and amplitude perturbations of a stationary solution repair diffusively to converge to a stationary solution. Our methods explain the range of validity of the phase equation, and the coupling between the “fast” amplitude equation and the “slow” phase equation.

We discuss numerical schemes for various forms of the Landau–Lifshitz equation. A new simple projection method is introduced and is shown to be unconditionally stable. The advantages over other schemes are also demonstrated numerically.

We consider the stochastic Ginzburg-Landau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The low-lying frequencies are then only connected to this forcing through the non-linear (cubic) term of the Ginzburg-Landau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof comb...

Part I Near-equilibrium critical dynamics 1 Introduction 3 1 Equilibrium critical phenomena 5 1.1 Mean-field theory 6 1.1.1 Van-der-Waals equation of state 6 1.1.2 Mean-field theory for the ferromagnetic Ising model 9 1.1.3 Ornstein–Zernicke correlation function 14 1.2 Landau–Ginzburg theory 17 1.2.1 Mean-field theory revisited: Landau expansion 17 1.2.

We have studied patterns of electrically-driven convective vortices in thin freely-suspended lms of smectic liquid crystal at and above the onset of con-vection. We present measurements of the convective amplitude above onset, the spatial variation of the amplitude due to the presence of end walls, and the relaxation of the amplitude following a sudden change in the experimental control paramet...