Onsager's variational principle (OVP) was originally proposed by Lars Onsager in 1931 [L. Onsager, $Phys. Rev.$, 1931, $37$, 405]. This fundamental principle provides a very powerful tool for formulating thermodynamically consistent models. It can also be employed to find approximate solutions, especially in the study of soft matter dynamics. In this work, OVP is extended and applied to the dynamic modeling of active soft matter such as suspensions of bacteria and aggregates of animal cells. We first extend the general formulation of OVP to active matter dynamics where active forces are included as external non-conservative forces. We then use OVP to analyze the directional motion of individual active units: a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer. Next, we use OVP to formulate a diffuse-interface model for an active polar droplet on a solid substrate. In addition to the generalized hydrodynamic equations for active polar fluids in the bulk region, we have also derived thermodynamically consistent boundary conditions. Finally, we consider the dynamics of a thin active polar droplet under the lubrication approximation. We use OVP to derive a generalized thin film equation and then employ OVP as an approximation tool to find the spreading laws for the thin active polar droplet. By incorporating the activity of biological systems into OVP, we develop a general approach to construct thermodynamically consistent models for better understanding the emergent behaviors of individual animal cells and cell aggregates or tissues.
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 symm…
Using Onsager variational principle, we study the dynamic coupling between the stress and the composition in polymer solution. In the original derivation of the two-fluid model [Doi and Onuki, J. Phys. II France {\bf 2}, 1631 (1992)], the polymer stress was introduced \emph{a priopri}, therefore a constitutive equation is required to close the equations. Based on our previous study of viscoelastic fluids with homogeneous composition [Phys. Rev. Fluids {\bf 3}, 084004 (2018)], we start with a dumbbell model for the polymer, and derive all dynamic equations using the Onsager variational principl…
Covariant Onsager and Onsager-Machlup principles for active and inertial dynamics
Nihon University（日本大学）；University of Chinese Academy of Sciences（中国科学院大学）；Kyoto University（京都大学）；Technion - Israel Institute of Technology（以色列理工学院）；Tel Aviv University（特拉维夫大学）
The variational principle of the Onsager-Machlup integral is used to describe the stochastic dynamics of a micromachine, such as an enzyme, characterized by odd elasticity. The obtained most probable path is found to become non-reciprocal in the presence of odd elasticity and is further related to the entropy production.
In this paper, we consider the capillarity-driven evolution of a solid toroidal island on a flat rigid substrate, where mass transport is controlled by surface diffusion. This problem is representative of the geometrical complexity associated with the solid-state dewetting of thin films on substrates. We apply Onsager's variational principle to develop a general approach for describing surface diffusion-controlled problems. Based on this approach, we derive a simple, reduced-order model and obtain an analytical expression for the rate of island shrinking and validate this prediction by numeric…
In 1931, Onsager proposed a variational principle which has become the base of many kinetic equations for non-equilibrium systems. We have been showing that this principle is useful in obtaining approximate solutions for the kinetic equations, but our previous method has a weakness that it can be justified, strictly speaking, only for small incremental time. Here we propose an improved method which does not have this drawback. The new method utilizes the integral proposed by Onsager and Machlup in 1953, and can tell us which of the approximate solutions is the best solution without knowing the…
Generalized Onsager theory for strongly anisometric patchy colloids
用Onsager-van der Waals密度泛函理论研究了软“补丁”相互作用对强伸长胶体棒和扁平圆盘取向无序-有序转变的影响。该理论提供了一个通用的框架，研究液晶相行为的高度不等轴的圆柱形胶体，它携带一个独特的几何图案的排斥或吸引力的软相互作用定位在颗粒表面上。在本文中，我们应用我们的理论的情况下，带电杆和磁盘的局部静电相互作用可以描述的屏蔽库仑势。我们考虑无限薄的棒状圆柱体与均匀线电荷和无限薄dispersive圆柱体与几个明显不同的表面电荷模式。不考虑骨架形状，带电胶体的各向同性-均质相图的特征在于均质有序的一般不稳定性、两相密度区域的急剧变窄以及在减少静电屏蔽时的重入现象。在较高的粒子密度下，静电排斥导致完全抑制有利于空间非均匀液晶的有序。
Entropy-Driven Initiation and Cellular Uptake Mediated by Viscoelastic Cytoskeleton: A Kinetic Phase Diagram from Onsager Variational Principle
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Zhejiang Key Laboratory of Soft Matter Biomedical Materials, Wenzhou Institute, University of Chinese Academy of Sciences（温州研究院，中国科学院大学，浙江省软物质生物医用材料重点实验室）
Neural optimization of the most probable paths of 3D active Brownian particles
Wenzhou Institute, University of Chinese Academy of Sciences（中国科学院大学温州研究院）；Zhejiang University（浙江大学）；Xiangshan Hospital of Wenzhou Medical University（温州医科大学象山县医院）；Xiamen University（厦门大学）；Guangdong Technion – Israel Institute of Technology（广东以色列理工学院）；Technion - Israel Institute of Technology（以色列理工学院）；Research Institute for Mathematical Sciences, Kyoto University（京都大学数理科学研究科研究所）；Department of Mathematics, Kyoto University（京都大学数学系）
Entropy-Driven Initiation and Cytoskeletal Viscoelasticity in Endocytosis: An Onsager Variational Framework
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Zhejiang Key Laboratory of Soft Matter Biomedical Materials, Wenzhou Institute, University of Chinese Academy of Sciences（温州研究院，中国科学院大学，浙江省软物质生物医用材料重点实验室）
In this paper, we proved that by choosing the proper variational function and variables, the variational approach proposed by M. Doi in soft matter physics was equivalent to the Conservation-Dissipation Formalism. To illustrate the correspondence between these two theories, several novel examples in soft matter physics, including particle diffusion in dilute solutions, polymer phase separation dynamics and nematic liquid crystal flows, were carefully examined. Based on our work, a deep connection among the generalized Gibbs relation, the second law of thermodynamics and the variational princip…
Variational methods have been widely used in soft matter physics for both static and dynamic problems. These methods are mostly based on two variational principles: the variational principle of minimum free energy (MFEVP) and Onsager's variational principle (OVP). Our interests lie in the applications of these variational methods to active matter physics. In our former work [Soft Matter, 2021, 17, 3634], we have explored the applications of OVP-based variational methods for the modeling of active matter dynamics. In the present work, we explore variational (or energy) methods that are based on…
The cellular uptake of self-propelled nanoparticles (NPs) or viruses, usually nonspherical, by cell membrane is crucial in many biological processes. In this study, using Onsager variational principle, we obtain a general wrapping equation for nonspherical self-propelled nanoparticles. Two analytical critical conditions are theoretically derived, one for the continuous full uptake of prolate particles and the other for snapthrough full wrapping of oblate particles. They capture considerably well the full uptake critical boundaries in the phase diagrams constructed in terms of active force, asp…
The properties of a macroscopic assembly of weakly-repulsive bosons at zero temperature are well described by Gross-Pitaevskii mean-field theory. According to this formalism the system exhibits a quantum transition from superfluid to cluster supersolid as a function of pressure. We develop a thermodynamically rigorous treatment of the different phases of the system by adopting a variational formulation of the condensate wave function --- represented as a sum of Gaussians --- that is amenable to exact manipulations. Not only is this description quantitatively accurate, but it is also capable to…
The variational principle of minimum free energy (MFEVP) has been widely used in the study of soft matter statics. MFEVP can be used not only to derive equilibrium equations (including both bulk equations and boundary conditions), but also to develop direct variational methods (such as Ritz method) to find approximate solutions to these equilibrium equations. In this work, we applied these variational methods to study long-range force transmission in nonlinear elastic biopolymer gels. We showed that the slow decay of cell-induced displacements measured experimentally for fibroblast spheroids i…
Utilising Onsager's variational formulation, we derive dynamical equations for the relaxation of a fluid membrane tube in the limit of small deformation, allowing for a contrast of solvent viscosity across the membrane and variations in surface tension due to membrane incompressibility. We compute the relaxation rates, recovering known results in the case of purely axis-symmetric perturbations and making new predictions for higher order (azimuthal) $m$-modes. We analyse the long and short wavelength limits of these modes by making use of various asymptotic arguments. We incorporate stochastic …
