Dynamic mean-field models from a nonequilibrium thermodynamics perspective

Complicated dynamic models are often approximated by introducing mean-field approximations and closures. The focus here is on examining such mean-field models using nonequilibrium thermodynamics. Two illustrative examples are studied in terms of the double-generator general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework. First, it is shown that a model for the coil-stretch transition of long chains in strong elongation flows as proposed by de Gennes is thermodynamically admissible. In the second example, we study a Gaussian approximation, which is used to simplify the effect of hydrodynamic interactions in polymer solutions. This approximation, which is known to be in conflict with the fluctuation-dissipation theorem, is identified as defective directly when formulated in the thermodynamic formalism

An optimization principle for deriving nonequilibrium statistical models of Hamiltonian dynamics

A general method for deriving closed reduced models of Hamiltonian dynamical systems is developed using techniques from optimization and statistical estimation. As in standard projection operator methods, a set of resolved variables is selected to capture the slow, macroscopic behavior of the system, and the family of quasi-equilibrium probability densities on phase space corresponding to these resolved variables is employed as a statistical model. The macroscopic dynamics of the mean resolved variables is determined by optimizing over paths of these probability densities. Specifically, a cost function is introduced that quantifies the lack-of-fit of such paths to the underlying microscopic dynamics; it is an ensemble-averaged, squared-norm of the residual that results from submitting a path of trial densities to the Liouville equation. The evolution of the macrostate is estimated by minimizing the time integral of the cost function. The value function for this optimization satisfies the associated Hamilton-Jacobi equation, and it determines the optimal relation between the statistical parameters and the irreversible fluxes of the resolved variables, thereby closing the reduced dynamics. The resulting equations for the macroscopic variables have the generic form of governing equations for nonequilibrium thermodynamics, and they furnish a rational extension of the classical equations of linear irreversible thermodynamics beyond the near-equilibrium regime. In particular, the value function is a thermodynamic potential that extends the classical dissipation function and supplies the nonlinear relation between thermodynamics forces and fluxes

Note on Global Regularity for 2D Oldroyd-B Fluids with Diffusive Stress

We prove global regularity of solutions of Oldroyd-B equations in 2 spatial dimensions with spatial diffusion of the polymeric stresses

Modification of linear response theory for mean-field approximations

In the framework of statistical descriptions of many particle systems, the influence of mean-field approximations on the linear response theory is studied. A procedure, analogous to one where no mean-field approximation is involved, is used in order to determine the first order response of the distribution function to the perturbation. Subsequently, the effect of the mean-field approximations is examined when formulating Green-Kubo relations for transport coefficients on the deterministic Liouvillean level and the stochastic Fokker-Planck level. On the deterministic level, the Vlasov equation is employed to prove the Green-Kubo formula for the electric conductivity tensor in its well known form. One finds that the interpretation of the Green-Kubo formula is changed when applying Vlasovs mean-field approximation. On the stochastic level, the Gaussian approximation of the bead-spring model for dilute polymer solutions, including hydrodynamic interaction, is considered in homogeneous shear flow. The commonly known Green-Kubo formula for the viscosity is found to be invalid in the Gaussian approximation, and the appropriate modification to the formula is given

Fluctuation-dissipation theorem, kinetic stochastic integral, and efficient simulations

Diffusive systems respecting the fluctuation-dissipation theorem with multiplicative noise have been studied on the level of stochastic differential equations. We propose an efficient simulation scheme motivated by the direct definition of the "kinetic stochastic integral", which differs from the better known Ito and the Stratonovich integrals. This simulation scheme is based on introducing the identity matrix, expressed in terms of the diffusion tensor and its inverse, in front of the noise term, and evaluating these factors at different times

Dynamics of multiphase systems with complex microstructure. I. Development of the governing equations through nonequilibrium thermodynamics

In this paper we present a general model for the dynamic behavior of multiphase systems in which the bulk phases and interfaces have a complex microstructure (for example, immiscible polymer blends with added compatibilizers, or polymer stabilized emulsions with thickening agents dispersed in the continuous phase). The model is developed in the context of the GENERIC framework (general equation for the nonequilibrium reversible irreversible coupling). We incorporate scalar and tensorial structural variables in the set of independent bulk and surface excess variables, and these structural variables allow us to link the highly nonlinear rheological response typically observed in complex multiphase systems, directly to the time evolution of the microstructure of the bulk phases and phase interfaces. We present a general form of the Poisson and dissipative brackets for the chosen set of bulk and surface excess variables, and show that to satisfy the entropy degeneracy property, we need to add several contributions to the moving interface normal transfer term, involving the tensorial bulk and interfacial structural variables. We present the full set of balance equations, constitutive equations, and boundary conditions for the calculation of the time evolution of the bulk and interfacial variables, and this general set of equations can be used to develop specific models for a wide range of complex multiphase systems

Symbolic test of the Jacobi identity for given generalized ’Poisson’ bracket

We have developed and provide an algorithm which allows to test the Jacobi identity for a given generalized ‘Poisson’ bracket. Novel frameworks for nonequilibrium thermodynamics have been established, which require that the reversible part of motion of thermodynamically admissible models is described by Poisson brackets satisfying the Jacobi identity in order to ensure the full time-structure invariance of equations of motion for arbitrary function(al)s on state space. For a nonassociative algebra obeyed by objects such as the Lie bracket, the elements of Lie groups fulfill this identity. But the manual evaluation of Jacobi identities relevant for applications and even for basic examples is often very time consuming. The efficient algorithm presented here can be obtained as a package to be used within the framework of the symbolic programming language MathematicaTM. The tool handles Poisson brackets acting either on functions or on functionals, depending on whether the system is described in terms of discrete or of continuous variables

Energy elastic effects and the concept of temperature in flowing polymeric liquids

The incorporation of energy elastic effects in the modeling of flowing polymeric liquids is discussed. Since conformational energetic effects are determined by structural features much smaller than the end-to-end vector of the polymer chains, commonly employed single conformation tensor models are insufficient to describe energy elastic effects. The need for a local structural variable is substantiated by studying a microscopic toy model with energetic effects in the setting of a generalized canonical ensemble. In order to examine the dynamics of flowing polymeric liquids with energy elastic effects, a thermodynamically admissible set of evolution equations is presented that accounts for the evolution of the microstructure in terms of a slow tensor, as well as a fast, local scalar variable. It is demonstrated that the temperature used in the definition of the heat flux is directly related to the Lagrange multiplier of the microscopic energy in the generalized canonical partition function. The temperature equation is discussed with respect to, first, the dependence of the heat capacity on the polymer conformation and, second, the possibility to measure experimentally the effects of the conformational energy