Application of the Onsager-Machlup integral in solving dynamic equations in nonequilibrium systems

Abstract

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 exact solution. The new method has an advantage that it allows us to determine the steady state in non-equilibrium system by a variational calculus. We demonstrate this using three examples, (a) simple diffusion problem, (b) capillary problem in a tube with corners, and (c) free boundary problem in liquid coating, for which the kinetic equations are written in second or fourth order partial differential equations..

Onsager's variational principle in active soft matter

“…This subsection deals with some direct variational methods of finding approximate solutions that are obtained directly from above variational principles for the system dynamics in both short and long time scales. 63,68,76,89 Variational principles have been proposed in various fields of physics and several variational methods have been developed † It is interesting to mention that in active matter, the presence of active forces and the breakdown of TRS can significantly change the statistical behaviors of the system such as the distribution of a and barrier crossing kinetics. The dynamics of downhill and uphill processes are very different even for passive systems, and hence must be treated using their respective variational approaches.…”

“…This variational principle is called Onsager-Machlup variational principle (OMVP). 63,68 In contrast to OVP, a local principle that predicts the most probable state in the immediate future, the OMVP is a global principle that can determine the most probable path taking the system to the far future under various constraints. For example, it can be used to determine the long-time behaviors such as the steadystates.…”

“…For example, it can be used to determine the long-time behaviors such as the steadystates. 63,68 It can also be used to locate the most probable transition pathway that takes the system from one free energy minimum to another. 87,88…”

“…61 In recent years, OVP has been widely used as an indispensable and powerful tool for the study of nonlinear and nonequilibrium phenomena of soft matter. 53,58,[62][63][64][65] OVP can be used to derive many transport equations for soft matter dynamics, 53,62 e.g., the Stokes equation for incompressible low-Reynolds-number flows, 58 the diffusion equation, 53,62 the reaction-diffusion equations for (low molecular weight) multicomponent solutions, 51,64,65 the thin film evolution equations, 63,[66][67][68] the phase field model for two-phase hydrodynamics, 58,69 the electrorheological hydrodynamic equation, 70 the two-fluid model for the phase separation dynamics in colloids and polymers, 53,62,63 the dynamics of polymer gels, 39,63 the dynamic equations of lipid membrane, [71][72][73] etc. Moreover, OVP also provides a very convenient way to derive thermodynamically consistent boundary conditions that supplement the transport equations in the bulk region.…”

“…In addition, Doi and his collaborators have recently proposed that OVP can be used as a direct variational method to find approximate solutions for complex soft matter dynamics. 63,68,76 This approximation method has been successfully used to study the evolution of droplets and thin films, 63,67,77,78 the dynamics of the beads-on-string structure of viscoelastic polymer filaments, 63,79 the sedimentation in colloidal suspensions, 80 and the translocation of a vesicle through a narrow hole, 81 etc. Although OVP has been widely applied with great successes in the study of inert soft matter dynamics, it is rarely used in the study of active soft matter dynamics.…”