Machlup

Onsager L and Machlup S 1953 Fluctuations and irreversible processes Rhys. Rev. 91 1505... 

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 ... 

The trajectories with the highest probability are those for which the Onsager-Machlup action... 

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 square brackets denote a vector, and [ ] a transposed vector. The exact expression for the Onsager-Machlup action is now approximated by... 

Fig. 1. Optimization of the Onsager-Machlup action for the two dimensional harmonic oscillator. The potential energy is U(x,y) = 25i/ ), the mass is 1...

To compute the above expression, short molecular dynamics runs (with a small time step) are calculated and serve as exact trajectories. Using the exact trajectory as an initial guess for path optimization (with a large time step) we optimize a discrete Onsager-Machlup path. The variation of the action with respect to the optimal trajectory is computed and used in the above formula. 

On a related point, there have been other variational principles enunciated as a basis for nonequilibrium thermodynamics. Hashitsume [47], Gyarmati [48, 49], and Bochkov and Kuzovlev [50] all assert that in the steady state the rate of first entropy production is an extremum, and all invoke a function identical to that underlying the Onsager-Machlup functional [32]. As mentioned earlier, Prigogine [11] (and workers in the broader sciences) [13-18] variously asserts that the rate of first entropy production is a maximum or a minimum and invokes the same two functions for the optimum rate of first entropy production that were used by Onsager and Machlup [32] (see Section HE). 

Figure 8. The dimensionless thermal conductivity, b1tljcjljX,(t), at p = 0.8 and T0 = 2. The symbols are the simulation data, with the triangles using the instantaneous velocity at the end of the interval, X/(t), Eq. (277), and the circles using the coarse velocity over the interval, Eq. (278). The solid line is the second entropy asymptote, essentially Eq. (229), and the dotted curve is the Onsager-Machlup expression om(t)> Eq. (280). (Data from Ref. 6.)...

Onsager and Machlup [32] gave an expression for the probability of a path of a macrostate, p[x]. The exponent may be maximized with respect to the path for fixed end points, and what remains is conceptually equivalent to the constrained second entropy used here, although it differs in mathematical detail. The Onsager-Machlup functional predicts a most likely terminal velocity that is exponentially decaying [6, 42] ... 

Consequently, the time correlation function given by Onsager-Machlup theory... 

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. 

Figure 9. Simulated thermal conductivity X/(t) for a Lennard-Jones fluid. The density in the center of the system is p = 0.8 and the zeroth temperature is To = 2. (a) A fluid confined between walls, with the numbers referring to the width of the fluid phase. (From Ref. 6.) (b) The case I, — 11.2 compared to the Markov (dashed) and the Onsager-Machlup (dotted) prediction.

The patent, a form of IP, "confers the right to secure the enforcement power of the state in excluding unauthorized persons, for a specified number of years, from making commercial use of a clearly identified invention" (Machlup 1958). As with all forms of IP, patents exist only because government says they do. Unlike real property, a house, a chair, a book, IP is non-rivalrous. There is no natural limit to the number of users of IP and no natural economic value except the production cost of the physical support. Thus, patents do "not arise out of the scarcity of the objects which become appropriated" (Plant 1934). 

Machlup, F. 1958. Am Economic Review of the Patent System Study of the Subcommittee on Patents, Trademarks and Copyrights of the Committee of the Judiciary, US Senate, 85 Congress, 2 Session, Study Number 15, Washington, DC US Government Printing Office. 

Machlup, F. and E. T. Penrose. 1950. The Patent Controversy in the Nineteenth Century. Journal of Economic History 10 1-29. 

Astumian presents a pedagodic discussion on FRs and their treatment in terms of Onsager-Machlup theory close to equilibrium. 

Golinelli and Mallik have examined FRs in mathematical models that have been developed for studying general properties of nonequilibrium systems, and which are analytically tractable. Tanaiguchi and Cohen consider the relationship between Onsager-Machlup theory and the FRs. 

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 ... 

---