Unifying various thermodynamically compatible modeling approaches

Lecture "Introduction to Nonequilibrium Thermodynamics - Onsager's variational principle"

Hydrodynamic Boundary Conditions Derived from Onsager's Variational Principle

Onsager's variational principle is generalized to address the diffusive dynamics of an electrolyte solution composed of charge-regulated macro-ions and counterions. The free energy entering the Rayleighian corresponds to the Poisson-Boltzmann theory augmented by the charge-regulation mechanism. The dynamical equations obtained by minimizing the Rayleighian include the classical Poisson-Nernst-Planck equations, the Debye-Falkenhagen equation, and their modifications in the presence of charge regulation. By analyzing the steady state, we show that the charge regulation impacts the non-equilibrium macro-ion spatial distribution and their effective charge, deviating significantly from their equilibrium values. Our model, based on Onsager's variational principle, offers a unified approach to the diffusive dynamics of electrolytes containing components that undergo various charge association/dissociation processes.

A brief outline of classical and extended irreversible thermodynamics is presented. Classical irreversible thermodynamics (CIT) is known as an active and fast developing field with numerous applications in continuum mechanics, chemistry and statistical mechanics. Its foundations and principal results are shortly commented. Extended irreversible thermodynamics (EIT) has fuelled an increasing interest during the last decade. Its main objective is to extend the domain of validity of classical non-equilibrium thermodynamics. Here, a short analysis of Lebon-Jou-Casas' version of EIT is presented, it is compared with CIT and the Gyarmati wave approach.

We discuss the general energetic variational approaches for hydrodynamic systems of complex fluids. In these energetic variational approaches, the least action principle (LAP) with action functional gives the Hamiltonian parts (conservative force) of the hydrodynamic systems, and the maximum/minimum dissipation principle (MDP), i.e., Onsager's principle, gives the dissipative parts (dissipative force) of the systems. When we combine the two systems derived from the two different principles, we obtain a whole coupled nonlinear system of equations satisfying the dissipative energy law. We will discuss the important roles of MDP in designing numerical method for computations of hydrodynamic systems in complex fluids. We will reformulate the dissipation in energy equation in terms of a rate in time by using an appropriate evolution equations, then the MDP is employed in the reformulated dissipation to obtain the dissipative force for the hydrodynamic systems. The systems are consistent with the Hamiltonian parts which are derived from LAP. This procedure allows the usage of lower order element (a continuous $C^0$ finite element) in numerical method to solve the system rather than high order elements, and at the same time preserves the dissipative energy law. We also verify this method through some numerical experiments in simulating the free interface motion in the mixture of two different fluids.

We propose a systematic formulation of the migration behaviors of a vesicle in a Poiseuille flow based on Onsager's variational principle, which can be used to determine the most stable steady state. Our model is described by a combination of the phase field theory for the vesicle and the hydrodynamics for the flow field. The dynamics is governed by the bending elastic energy and the dissipation functional, the latter being composed of viscous dissipation of the flow field, dissipation of the bending energy of the vesicle, and the friction between the vesicle and the flow field. We performed a series of simulations on 2-dimensional systems by changing the bending elasticity of the membrane and observed 3 types of steady states, i.e., those with slipper shape, bullet shape, and snaking motion, and a quasi-steady state with zig-zag motion. We show that the transitions among these steady states can be quantitatively explained by evaluating the dissipation functional, which is determined by the competition between the friction on the vesicle surface and the viscous dissipation in the bulk flow.

Using Onsager's variational principle, we derive dynamical equations for a nonequilibrium active system with odd elasticity. The elimination of the extra variable that is coupled to the nonequilibrium driving force leads to the nonreciprocal set of equations for the material coordinates. The obtained nonreciprocal equations manifest the physical origin of the odd elastic constants that are proportional to the nonequilibrium force and the friction coefficients. Our approach offers a systematic and consistent way to derive nonreciprocal equations for active matter in which the time-reversal symmetry is broken.

In the theory of extended irreversible thermodynamics (EIT), the flux-dependent entropy function plays a key role; it is a fundamental distinction between EIT and the usual flux-independent entropy function adopted by classical irreversible thermodynamics (CIT). However, its existence, as a prerequisite for EIT, and its statistical origin have never been justified. In this work, by studying the macroscopic limit of an ϵ-dependent Langevin dynamics, which admits a large deviations (LD) principle, we show that the stationary LD rate functions of probability density p ϵ (x, t) and joint probability density actually turn out to be the desired flux-independent entropy function in CIT and flux-dependent entropy function in EIT respectively. The difference of the two entropy functions is determined by the time resolution for Brownian motions times a Lagrangian, the latter arises from the LD Hamilton–Jacobi equation and can be used for constructing conserved Lagrangian/Hamiltonian dynamics.

Application of Onsager's variational principle to the dynamics of a solid toroidal island on a substrate

The Cahn--Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn--Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints such as conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of asymptotic limit as $t\to+\infty$ and characterize the stability of local energy minimizers for the system.

Using the Onsager variational principle, we study the dynamic coupling between the stress and the composition in a polymer solution. In the original derivation of the two-fluid model of Doi and Onuki the polymer stress was introduced a priori; therefore, a constitutive equation is required to close the equations. Based on our previous study of viscoelastic fluids with homogeneous composition, we start with a dumbbell model for the polymer, and derive all dynamic equations using the Onsager variational principle.

By examining the deterministic limit of a general ε-dependent generator for Markovian dynamics, which includes the continuous Fokker-Planck equations and discrete chemical master equations as two special cases, the intrinsic connections among mesoscopic stochastic dynamics, deterministic ordinary differential equations or partial differential equations, large deviation rate functions, and macroscopic thermodynamic potentials are established. Our result not only solves the long-lasting question of the origin of the entropy function in classical irreversible thermodynamics, but also reveals an emergent feature that arises automatically during the deterministic limit, through its large deviation rate function, with both time-reversible dynamics equipped with a Hamiltonian function and time-irreversible dynamics equipped with an entropy function.

Mass and heat diffusion in ternary polymer solutions: A classical irreversible thermodynamics approach

Bendotaxis has recently been proposed as a mechanism for self-transport of droplets at small scales. When an active droplet undergoes self-transport via bendotaxis, interfacial, elastic, and active forces jointly determine the droplet motion in a deformable channel. Through simulations of thin-film dynamics and a model reduction based on Onsager's variational principle, we show that wettability and activity can jointly operate to enhance or weaken the self-transport effect of bendotaxis, depending on the sign of wettability (hydrophobic or hydrophilic) on the channel wall and the sign of activity (contractile or extensile) in the droplet.

Flow and director fields strongly couple with each other in liquid crystalline systems, and herein we discuss the coupling effect in cylindrical and spherical-cap droplets formed by nematic liquid crystal. Applying a temperature gradient to droplets dispersed in a liquid solvent, we observed a crosslike texture in the droplets moved toward the high-temperature side, indicating that the director field was deformed from equilibrium. Additionally, measurement of the flow field revealed that a convective flow was induced in the droplets under temperature gradient. These results suggested that the director deformation in the droplet was induced by convection. By designing a simplified model based on this, we theoretically analyzed the above phenomenon based on Onsager's variational principle. The results show that the phenomenon was well described by a balance of surface energy gradient with viscous and elastic forces.