Machlup/Onsager equation

We next develop the Machlup-Onsager equation from Eq. (A.19) by making the familiar assumption that the macroscopic parameters A(f) are slow variables. This assumption is usually justified by the idea that a timescale separation exists between the parameters A(t) and their bath variables, due to the macroscopic nature of the former and microscopic nature of the latter. 

In Eq. (A.17), as before X(f) is the vector of thermodynamic forces while M is a symmetric matrix of phenomenological parameters introduced by Machlup and Onsager [4]. We adopt Eq. (A.17) inasmuch as it is the simplest equation of motion that is consistent with the Machlup-Onsager Eq. (A.24). Notice that Eq. (A.17) is similar in form to Newton s equation of motion for a particle system. Thus, we denote the matrix of phenomenological parameters by M in order to emphasize the analogy to particle masses. The analogy, however, is not perfect because M may be nondiagonal [4]. 

The Machlup-Onsager slow variable equation of thermodynamic relaxation then follows from eqs. (A. 19) and (A.23) as... 

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

We use the sine series since the end points are set to satisfy exactly the three-point expansion [7]. The Fourier series with the pre-specified boundary conditions is complete. Therefore, the above expansion provides a trajectory that can be made exact. In addition to the parameters a, b and c (which are determined by Xq, Xi and X2) we also need to calculate an infinite number of Fourier coefficients - d, . In principle, the way to proceed is to plug the expression for X t) (equation (17)) into the expression for the action S as defined in equation (13), to compute the integral, and optimize the Onsager-Machlup action with respect to all of the path parameters. 

The exponential decay predicted by the Onsager-Machlup theory, and by the Langevin and similar stochastic differential equations, is not consistent with the conductivity data in Fig. 8. This and the earlier figures show a constant value for >.(x) at larger times, rather than an exponential decay. It may be that if the data were extended to significantly larger time scales it would exhibit exponential decay of the predicted type. 

The functional above was used already by Gauss [12] to study classical trajectories (which explains our choice of the action symbol). Onsager and Machlup used path integral formulation to study stochastic trajectories [13]. The origin of their trajectories is different from what we discussed so far, which are mechanical trajectories. However, the functional they derive for the most probable trajectories, O [X (t)] is similar to the equation above ... 

To Start we describe an extension an extension of Eq. (A. 15) due to Machlup and Onsager [4], We begin by writing down a model equation of thermodynamic relaxation patterned after the particle mechanics eq. (2.7). 

Also, this contains the entropy production, and was introduced first by Onsager and Machlup in their work about non-equilibrium fluctuation theory [i ]. With this we can formulate the minimum theorem of the generalized Onsager constitutive theory the OM-function is the non-negative function of the fluxes, forces and intensive parameters. It only becomes zero, which is its minimum, when the material equations of the generalized Onsager constitutive theory are satisfied. The OM-function has crucial importance, because it contains all the important constitutive properties of the linear and generalized constitutive theories these follow from the necessary conditions of the minimum of OM-function ... 

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