Deterministic equation

An algorithm for performing a constant-pressure molecular dynamics simulation that resolves some unphysical observations in the extended system (Andersen s) method and Berendsen s methods was developed by Feller et al. [29]. This approach replaces the deterministic equations of motion with the piston degree of freedom added to the Langevin equations of motion. This eliminates the unphysical fluctuation of the volume associated with the piston mass. In addition, Klein and coworkers [30] present an advanced constant-pressure method to overcome an unphysical dependence of the choice of lattice in generated trajectories. 

Sometimes the theoretical or computational approach to description of molecular structure, properties, and reactivity cannot be based on deterministic equations that can be solved by analytical or computational methods. The properties of a molecule or assembly of molecules may be known or describable only in a statistical sense. Molecules and assemblies of molecules exist in distributions of configuration, composition, momentum, and energy. Sometimes, this statistical character is best captured and studied by computer experiments molecular dynamics, Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. Interaction potentials based on quantum mechanics, classical particle mechanics, continuum mechanics, or empiricism are specified and the evolution of the system is then followed in time by simulation of motions resulting from these direct... 

By functional differentiation, Equation 4.22 leads us to the Euler-Lagrange deterministic equation for the electron density, viz.,... 

In parallel with the studies described above, which concern perfectly deterministic equations of evolution, it appeared necessary to complete the theory by studying the spontaneous fluctuations. Near equilibrium, any deviation is rapidly damped but near a bifurcation point, a fluctuation may may lead the system across the barrier. The fluctuation is then stabilized, or even amplified this is the origin of the phenomenon which Prigogine liked calling creation of order through fluctuations. More specifically, one witnesses in this way a step toward self-organization. 

As mentioned in the previous section, the exact master equation cannot be reduced to the Schrodinger equation for a transformed wave function that is a deterministic equation in time. However, it can reduce to a stochastic Schrodinger equation [18]. 

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

Statistical models based on data correlations and on Markov chains are being actively developed and their fidelity evaluated by several research groups. Photochemical-diffusion models based on deterministic equations are also being developed, but because of their complexity will probably be used only as research tools for some time. 

JThermal fluctuations could also be taken into account through the addition of noise, leading to Langevin-type equations like those studied by Bartelt, et al. , but in the applications we consider here, the systems are far from equilibrium, either from initial conditions or because of an explicit driving force, so the simpler deterministic equations will prove adequate. 

Fig. 1. Numerical solutions of deterministic equations at steady state. Steady-state N02 concentration is given by percent absorption and plotted versus incident laser light intensity for various values of the boundary temperature Te (degrees Kelvin).

Fig. 2. Plot similar to Fig. 1. According to the deterministic equations on increasing the light intensity, the curve AFBCD is traced on decreasing the light intensity, the curve DCEFA is traced.

These three steps constitute what I call the Langevin approach . It is widely utilized, also when the fluctuations are not due to thermal motion or to discreteness of particles, and even when they are of unspecified or unknown provenance. The applications in many branches of physics, chemistry and biology are far too numerous to list. The approach has been highly successful whenever the deterministic equations are linear, but for nonlinear cases it leads to difficulties, which are analyzed in this section. The purpose is to shield the reader from the labyrinthine literature and to convince him of the necessity of the more firmly based treatment in the next chapter. 

External noise denotes fluctuations created in an otherwise deterministic system by the application of a random force, whose stochastic properties are supposed to be known. Examples are a noise generator inserted into an electric circuit, a random signal fed into a transmission line, the growth of a species under influence of the weather, random loading of a bridge, and most other stochastic problems that occur in engineering. In all these cases clearly (4.5) holds if one inserts for A(y) the deterministic equation of motion for the isolated system, while L(t) is approximately but never completely white. Thus for external noise the Stratonovich result (4.8) and (4.9) applies, in which A(y) represents the dynamics of the system with the noise turned off. 

Internal noise is described by a master equation. When this equation cannot be solved exactly it is necessary to have a systematic approximation method - rather than the naive Fokker-Planck and Langevin approximations. Such a method will now be developed in the form of a power series expansion in a parameter Q. In lowest order it reproduces the macroscopic equation and thereby demonstrates how a deterministic equation emerges from the stochastic description. 

This Ansatz is the essential step. The -expansion is not just one out of a plethora of approximation schemes, to be judged by comparison with experimental or numerical results 0. It is a systematic expansion in and is the basis for the existence of a macroscopic deterministic description of systems that are intrinsically stochastic. It justifies as a first approximation the standard treatment in terms of a deterministic equation with noise added, as in the Langevin approach. It will appear that in the lowest approximation the noise is Gaussian, as is commonly postulated. In addition, however, it opens up the possibility of adding higher approximations. 

These results lead to the following physical interpretation of the two terms in (4.7). The first term alone is the Liouville operator belonging to the deterministic equation... 

Master equations of diffusion type were characterized by the property that the lowest non-vanishing term in their -expansion is not a macroscopic deterministic equation but a Fokker-Planck equation. One may ask whether it is still possible to obtain an approximation in the form of a deterministic equation, although Q is no longer available as an expansion parameter. The naive device of omitting from the Fokker-Planck equation the term involving the second order derivatives is, of course, wrong the result would depend on which of the various equivalent forms (4.1), (4.7), (4.17), (4.18) one chooses to mutilate in this way. 

This is the deterministic equation that emerges in the limit of low temperature. Do not confuse with the macroscopic equation , which is the deterministic equation that emerged in the limit oo for systems not of diffusion type. Next, let x(t) be a solution of (5.2) and define a new variable by... 

In the case of the quasilinear Fokker-Planck equation (2.4), the free energy U defined in terms of the stationary solution by (2.6) is identical with the potential in the deterministic equation (5.2). That identity is often taken for granted when time-dependent solutions have to be constructed for systems of which only the equilibrium distribution is known. We shall now show, however, that it holds only for systems of diffusion type whose Fokker-Planck equation is quasilinear, i.e., of the form (2.4). 

Exercise. The deterministic equation obtained by the naive device of dropping the second term from the Fokker-Planck equation is not invariant for nonlinear transformations of x. 

Warning. The idea of using a nonlinear Fokker-Planck equation as a general framework for describing fluctuating systems has attracted many authors. Detailed balance, in its extended form, was a useful aid, but the link with the deterministic equation caused difficulties. It may therefore be helpful to emphasize three caveats. 

For convenience we restrict the discussion to the quasilinear Fokker-Planck equation (1.8). The corresponding results for the one-step process will be left for Exercises. Our U(x) has the shape of fig. 36b the deterministic equation has the same form as a10 in fig. 34. 

For y = 0 the equation (6.1) reduces to the Liouville equation for particles moving acccording to the deterministic equations... 

Remark. It is easily seen that the second term of (5.2) by itself causes the norm of if/ to change. In order that this is compensated by the fluctuating term the two terms must be linked, as is done by the relation U = V V. This resembles the classical fluctuation-dissipation theorem, which links both terms by the requirement that the fluctuations compensate the energy loss so as to establish the equilibrium. The difference is that the latter requirement involves the temperature T of the environment that makes it possible to suppress the fluctuations by taking T = 0 without losing the damping. This is the reason why in classical theory deterministic equations with damping exist, see XI.5. 

One of the simplest methods to generalize formal kinetics is to treat reactant concentrations as continuous stochastic functions of time, which results in a transformation of deterministic equations (2.1.1), (2.1.40) into stochastic differential equations. In a system with completely mixed particles the macroscopic concentration n (t) turns out to be the average of the stochastic function Cj(<)... 

Its obvious peculiarity as compared with the standard chemical kinetics, equation (2.1.10), is the emergence of the fluctuational second term in r.h.s. The stochastic reaction description by means of equation (2.2.37) permits us to obtain the equation for dispersions crjj which, however, contains higher-order momenta. It leads to the distinctive infinite set of deterministic equations describing various average quantities, characterizing the fluctuational spectrum. 

Since these characteristics are time-dependent, let us assume particle birth-death and migration to be the Markov stochastic processes. Note that making use of the stochastic models, we discuss below in detail, does not contradict the deterministic equations employed for these processes. Say, the equations for nv t), Xu(r,t), Y(r,t) given in Section 2.3.1 are deterministic since both the concentrations and joint correlation functions are defined by equations (2.3.2), (2.3.4) just as ensemble average quantities. Note that the... 

Note here that the relation between mesoscopic and microscopic approaches is not trivial. In fact, the former is closer to the macroscopic treatment (Section 2.1.1) which neglects the structural characteristics of a system. Passing from the micro- to meso- and, finally, to macroscopic level we loose also the initial statement of a stochastic model of the Markov process. Indeed, the disadvantages of deterministic equations used for rather simplified treatment of bimolecular kinetics (Section 2.1) lead to the macro- and mesoscopic models (Section 2.2) where the stochasticity is kept either by adding the stochastic external forces (Section 2.2.1) or by postulating the master equation itself for the relevant Markov process (Section 2.2.2). In the former case the fluctuation source is assumed to be external, whereas in the latter kinetics of bimolecular reaction and fluctuations are coupled and mutually related. Section 2.3.1.2 is aimed to consider the relation between these three levels as well as to discuss problem of how determinicity and stochasticity can coexist. 

Monte Carlo (MC) methods can address the time gap problem of MD. The basis of MC methods is that the deterministic equations of the MD method are replaced by stochastic transitions for the slow processes in the system.3 MC methods are stochastic algorithms for exploring the system phase space although their implementation for equilibrium and non-equilibrium calculations presents some differences. 

Therefore, the trajectory calculations with the same initial states, i, j, b, <[>, v, may lead to different trajectories, as shown in Fig. 4.1.1, when the phases of the internal degrees of freedom are different. It therefore makes sense to speak about a transition probability from an initial state to a final state, even in a case where the dynamics is governed by a deterministic equation, because of the indefiniteness of the phase of the internal degrees of freedom. In order to determine this probability, we need to make many trajectory calculations, with the given initial state, for different choices of the phases of the internal degrees of freedom. We may report the results of the trajectory calculation in Fig.4.1.1, by counting the number, dfVC(/) C) c+dt,c)(fl,fl + [Pg.54]

Most published work refers to the treatment of model pure solvents, with a limited number of species the electron and the hole, and their daughter species. Interference of a solute is more conveniently treated using, at least in part, deterministic equations. 

This equation is Schrodinger s wave equation, where h is Planck s constant and H is the Hamiltonian of the system to be investigated. The Schrodinger equation is a deterministic wave equation. This means that once ip t = 0) is given, ip t) can be calculated uniquely. Prom a conceptual point of view the situation is now completely analogous with classical mechanics, where chaos occurs in the deterministic equations of motion. If there is any deterministic quantum chaos, it must be found in the wave function ip. 

An(t)/n, the transiently induced birefringence. By this means, deterministic equations of motion, without stochastic terms, can be used via computer simulation to produce spectral features. As we have seen, a stochastic equation such as Eq. (1) is based on assumptions which are supported neither by spectral analysis nor by computer simulation of free molecular diffusion. The field-on simulation allows us the direct use of more realistic fimctions for the description of intermolecular interaction than any diffusional equation which uses stochastically generated intermolecular force fields. 

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