We recognize some universal features of macroscopic dynamics describing the approach of a well-established level of description (that is, successfully tested by experimental observations) to equilibrium. The universal features are collected in a general equation for the nonequilibrium reversible-irreversible coupling (abbreviated as GENERIC). In this paper we formulate a GENERIC, derive properties of its solutions, and discuss their physical interpretation. The relation of the GENERIC with thermodynamics is most clearly displayed in a formulation that uses contact structures. The GENERIC is also discussed in the presence of noise. In applications we either search for new governing equations expressing our insight into a particular complex fluid or take well-established governing equations and cast them into the form of a GENERIC. In the former case we obtain the governing equations as particular realizations of the GENERIC structure; in the latter case we justify the universality of the GENERIC and derive some properties of solutions. Both types of applications are discussed mainly in the following paper [Phys. Rev. E 56, 6633 (1997)].

Dynamics and thermodynamics of complex fluids. I. Development of a general formalism

http://web.mit.edu/doylegroup/pubs/BiophysJ-Savin05.pdf

Static and Dynamic Errors in Particle Tracking Microrheology

On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

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Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients

The pdf approach to turbulent polydispersed two-phase flows

Simple models for complex nonequilibrium fluids

1 Stochastic Processes, Polymer Dynamics, and Fluid Mechanics.- 1.1 Approach to Kinetic Theory Models.- 1.2 Flow Calculation and Material Functions.- 1.2.1 Shear Flows.- 1.2.2 General Extensional Flows.- 1.2.3 The CONNFFESSIT Idea.- References.- I Stochastic Processes.- 2 Basic Concepts from Stochastics.- 2.1 Events and Probabilities.- 2.1.1 Events and ?-Algebras.- 2.1.2 Probability Axioms.- 2.1.3 Gaussian Probability Measures.- 2.2 Random Variables.- 2.2.1 Definitions and Examples.- 2.2.2 Expectations and Moments.- 2.2.3 Joint Distributions and Independence.- 2.2.4 Conditional Expectations and Probabilities.- 2.2.5 Gaussian Random Variables.- 2.2.6 Convergence of Random Variables.- 2.3 Basic Theory of Stochastic Processes.- 2.3.1 Definitions and Distributions.- 2.3.2 Gaussian Processes.- 2.3.3 Markov Processes.- 2.3.4 Martingales.- References.- 3 Stochastic Calculus.- 3.1 Motivation.- 3.1.1 Naive Approach to Stochastic Differential Equations.- 3.1.2 Criticism of the Naive Approach.- 3.2 Stochastic Integration.- 3.2.1 Definition of the Ito Integral.- 3.2.2 Properties of the Ito Integral.- 3.2.3 Ito's Formula.- 3.3 Stochastic Differential Equations.- 3.3.1 Definitions and Basic Theorems.- 3.3.2 Linear Stochastic Differential Equations.- 3.3.3 Fokker-Planck Equations.- 3.3.4 Mean Field Interactions.- 3.3.5 Boundary Conditions.- 3.3.6 Stratonovich's Stochastic Calculus.- 3.4 Numerical Integration Schemes.- 3.4.1 Euler's Method.- 3.4.2 Mil'shtein's Method.- 3.4.3 Weak Approximation Schemes.- 3.4.4 More Sophisticated Methods.- References.- II Polymer Dynamics.- 4 Bead-Spring Models for Dilute Solutions.- 4.1 Rouse Model.- 4.1.1 Analytical Solution for the Equations of Motion.- 4.1.2 Stress Tensor.- 4.1.3 Material Functions in Shear and Extensional Flows.- 4.1.4 A Primer in Brownian Dynamics Simulations.- 4.1.5 Variance Reduced Simulations.- 4.2 Hydrodynamic Interaction.- 4.2.1 Description of Hydrodynamic Interaction.- 4.2.2 Zimm Model.- 4.2.3 Long Chain Limit and Universal Behavior.- 4.2.4 Gaussian Approximation.- 4.2.5 Simulation of Dumbbells.- 4.3 Nonlinear Forces.- 4.3.1 Excluded Volume.- 4.3.2 Finite Polymer Extensibility.- References.- 5 Models with Constraints.- 5.1 General Bead-Rod-Spring Models.- 5.1.1 Philosophy of Constraints.- 5.1.2 Formulation of Stochastic Differential Equations.- 5.1.3 Generalized Coordinates Versus Constraint Conditions.- 5.1.4 Numerical Integration Schemes.- 5.1.5 Stress Tensor.- 5.2 Rigid Rod Models.- 5.2.1 Dilute Solutions of Rod-like Molecules.- 5.2.2 Liquid Crystal Polymers.- References.- 6 Reptation Models for Concentrated Solutions and Melts.- 6.1 Doi-Edwards and Curtiss-Bird Models.- 6.1.1 Polymer Dynamics.- 6.1.2 Stress Tensor.- 6.1.3 Simulations in Steady Shear Flow.- 6.1.4 Efficiency of Simulations.- 6.2 Reptating-Rope Model.- 6.2.1 Basic Model Equations.- 6.2.2 Results for Steady Shear Flow.- 6.3 Modified Reptation Models.- 6.3.1 A Model Related to "Double Reptation".- 6.3.2 Doi-Edwards Model Without Independent Alignment.- References.- Landmark Papers and Books.- Solutions to Exercises.- References.- Author Index.

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H. Ch. Öttinger

Papers

Stochastic Processes in Polymeric Fluids: Tools and Examples for Developing Simulation Algorithms