Deterministic description

Moreover, from the conservation of receptor molecules, Nrxy is given by 

By differentiating (7.32)-(7.34) and substituting (7.13) we can derive the differential equations that govern the dynamics of Nr i), NrxQ), and NRY(t). After carrying out all of the involved calculations we obtain 

To analyze the above dynamical system we need to find its stationary solutions. It is not hard to prove that the system of differential equations (7.36)-(7.38) has a unique fixed point given by Eqs. (7.24)-(7.28). The stationary value for NRxxit) can then be computed from Eqs. (7.24)-(7.28) and Eq. (7.35), resulting in the expression in Eq. (7.29). 

It is possible to linearize the differentiate equation system (7.36)-(7.38) by making the following change of variables  

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In the 1990s, Bakker and Van den Akker (1994, 1996)—see also R.A. Bakker s PhD thesis (1996)—continued this mechanistic modeling approach by attempting a completely deterministic description of the 3-D small-scale flow field in which the chemical reactions take place at the pace the various species meet. Starting point is a lamellar structure of layers intermittently containing the species involved in the reaction. These authors conceived such small-scale structures as Cylindrical Stretched Vortex (CSV) tubes being strained in the direction of their axis and—as a result—shrinking in size in a plane normal to... 

S. Nicolay, E.-B. Brodie of Brodie, M. Touchon, Y. d Aubenton-Carafa, C. Thermes, and A. Arneodo, Erom scale invariance to deterministic chaos in DNA sequences towards a deterministic description of gene organization in the human genome. Physica A 342,270-280 (2004). 

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. 

The scope of this book is as follows. Chapter 2 gives a general review of different theoretical techniques and methods used for description the chemical reactions in condensed media. We focus attention on three principally different levels of the theory macroscopic, mesoscopic and microscopic the corresponding ways of the transition from deterministic description of the many-particle system to the stochastic one which is necessary for the treatment of density fluctuations are analyzed. In particular, Section 2.3 presents the method of many-point densities of a number of particles which serves us as the basic formalism for the study numerous fluctuation-controlled processes analyzed in this book. 

Shortcomings of the above described approach are self-evident the fluctuations entering equation (2.2.2) are independent of the deterministic motion, the passage from the deterministic description given by equation (2.1.1) to the stochastic one needs a large number of additional phenomenological parameters determining Gij. To define them, the fluctuation-dissipative theorem should be used. 

Since a number of particles involved in any reaction event are small, a change in concentration is of the order of 1 /V. Therefore, we can use for the system with complete particle mixing the asymptotic expansion in this small parameter 1 /V. The corresponding van Kampen [73, 74] procedure (see also [27, 75]) permits us to formulate simple rules for deriving the Fokker-Planck or stochastic differential equations, asymptotically equivalent to the initial master equation (2.2.37). It allows us also to obtain coefficients Gij in the stochastic differential equation (2.2.2) thus liquidating their uncertainty and strengthening the relation between the deterministic description of motion and density fluctuations. 

The discussions, started at the sessions, were continued during the evenings within smaller groups, in particular, among Bohr, Ehrenfest, and Einstein, who, as is well known, was reluctant to renounce the deterministic description. 

In this section we sketch a possible answer to this latter question. Within the framework of the deterministic description adopted throughout... 

At this point we will, briefly, describe some of the fundamental qualitative differences between a quantum mechanical and a classical mechanical description. First of all, a trajectory R(t) is replaced by a wave packet, which implies that a deterministic description is replaced by a probabilistic description. x(R,t) 2 is a probability density, giving the probability of observing the nuclei at the position R at time t. In... 

Kramers idea was to give a more realistic description of the dynamics in the reaction coordinate by including dynamical effects of the solvent. Instead of giving a deterministic description, which is only possible in a large-scale molecular dynamics simulation, he proposed to give a stochastic description of the motion similar to that of the Brownian motion of a heavy particle in a solvent. From the normal coordinate analysis of the activated complex, a reduced mass pi has been associated with the motion in the reaction coordinate, so the proposal is to describe the motion in that coordinate as that of a Brownian particle of mass g in the solvent. 

The problem of Brownian motion relates to the motion of a heavy colloidal particle immersed in a fluid made up of light particles. In Fig. 11.1.1 the trajectory of a Brownian particle is shown. The coordinates of a particle with diameter 2 /xm moving in water are observed every 30 s for 135 min. At the very first step in the argument one renounces an exact deterministic description of the motion and replaces it with a probabilistic description. 

A central concept for the derivation is again the transition probability [47,57] introduced in Eq. (29). Within the framework of the deterministic description, the equalities of Eq. (29) and Eq. (30) is always valid for all chemical reactions, so that all one needs is to evaluate individual chemical reaction rates, say v, and vR. Let ps be the fraction of sites to be occupied by monomer units. The possible bond number is then /M0 ps2. Considering that the typical percolation model is the R-Ay type, and familiar branching reactions are irreversible, it suffices to solve the following equality ... 

The Langevin equation [Eq. (25)] still defines a Markov process, and it is therefore fairly straightforward to show that the corresponding fluctuation-averaged (deterministic) description is given in terms of the space-fractional Fokker-Planck equation (22) [54,60]. In what follows, we solve it with 8-initial condition... 

Here = //(/ -t-di) denotes the fraction of subunits in the activatable state 10 that are activated. The sutecripts (3) and (3,4) indicate that we used p ns), h = 3 and pins + >14), /i = 4 for averaging, respectivel,y. Note that in tlie limit N - oo,jiio 00 equations (11.51), (11.52) reduce to tile well known expre.ssions of the deterministic description. All results in section 11.6 are based on equation (11.52), which can be further simplified by approximating all denominators by 4A due to 4A 1. 

The deterministic description by the proposed model does not capture the stochastic effects that are always present in living systems. 

TURBULENCE is chaotic fluid flow characterized by the appearance of three-dimensional, irregular swirls. These swirls are called eddies, and usually turbulence consists of many different sizes of eddies superimposed on each other. In the presence of turbulence, fluid masses with different properties are mixed rapidly. Atmospheric turbulence usually refers to the small-scale chaotic flow of air in the Earth s atmosphere. This type of turbulence results from vertical wind shear and convection and usually occurs in the atmospheric boundary layer and in clouds. On a horizontal scale of order 1000 km, the disturbances by synoptic weather systems are sometimes referred to as two-dimensional turbulence. Deterministic description of turbulence is difficult because of the chaotic character of turbulence and the large range of scales involved. Consequently, turbulence is treated in terms of statistical quantities. Insight in the physics of atmospheric turbulence is important, for instance, for the construction of buildings and structures, the mixing of air properties, and the dispersion of air pollution. Turbulence also plays an... 

There is an enormous range of scales in atmospheric motion, as indicated in Fig. 1. For many subjects the detailed description of the small-scale turbulent motions is not required. In addition, the randottmess of atmospheric turbulence makes a deterministic description difficult. As such, there is a need to separate the small scales of atmospheric mrbulence from mean motions on the larger scales. Let C denote an atmospheric variable, such as specific humidity. Then C represents a mean or smoothed value of C, typically taken on a horizontal scale of order 10 km or a time scale in the order of 30 min to 1 h. A local or instantaneous value of C would differ from C. Thus, we have... 

Doka, E. Lente, G. Mechanism-Based Chemical Understanding of Chiral Symmetry Breaking in the Soai Reaction. A Combined Probabilistic and Deterministic Description of Chemical Reactions. /. Am. Chem. Soc. 2011,133, 17878-17881. 

Records on earthquake Induced ground motion reveal a highly irregular motion defying any deterministic description. Ground motions due to earthquakes are therefore modeled by stochastic processes. 

In this section we have considered the two simplest examples using the deterministic description of chemical reactions. This approach is adequate but only in the so-called thermodynamic limit when we can neglect the discrete nature of the processes considered, as well as the fluctuations of the reactants. Rigorous consideration of these processes becomes possible within a stochastic approach to the description of chemical reactions (for references, see the excellent review by McQuarrie [20]). For the sequence of monomole-cular reactions in open systems with an arbitrary number of intermediates, the problem has been investigated in depth by Nicolis and Babloyantz [31], Ishida [32] and other authors (see, for references, [33]). The stochastic approach, however, faces serious analytical difficulties for more complex systems (for instance, the bimolecular reaction A BoC). Some unusual properties of this reaction in small volumes, associated with enormously large fluctuations, will be considered in Chapter 3. 

Deterministic description of a closed system Let us take as an example the Schlogl model 1 ]... 

When a nonlinear system ewolwes under far-from-equilibrium conditions in the vicinity of a bifurcation point, a purely deterministic description often proved to be incomplete. The fluctuations of the dynamical variables can play an essential role and obstruct the observation of a transition expected by a deterministic analysis. In the framework of the deterministic approach, the stability of the different states according to the values of the control parameters is studied through a mathematical analysis of the velocity field. In particular, the theory of normal forms leads to the determination of the various kinds of attractors [l,2]. As far as we are concerned with the stochastic approach, the rrLa te.n. equation, has been widely used to analyze bifurcations of homogeneous or spatially ordered steady states or of limit cycles [3,4]. Our aim in the present contribution is to insist on the generality of the method to analyze various kinds of bifurcations in nonlinear nonequilibrium systems. The general procedure proposed to obtain a local description of the probability, which allows us to determine the system s attractors, turns out to display marked analogies with the theory of normal forms. 

Notice that CF is a function of both No and time. Indeed, CV = 1/V at t = 0, and it increases monotonically and without limit as time passes. Assume that (Vo > 1. This means that CV 1 for small times, t < and so the deterministic approximation N t) = Noe provides an accurate description in the early stages of the decaying process. However, for times much larger than the deterministic description is no longer good enough, and in consequence we are obliged to take into account the process stochastic nature. 

Three different simulations of decaying processes, carried out with the algorithm just described and y = 1, are shown in Fig. 4.2. Notice that, as expeeted, all three of them are very much alike when the molecular count is large. As a matter of fact, the system evolution at large n values is well approximated by the deterministic description in terms of the average molecule count. However, this approximation is not good enough for low molecule counts. One fact worth of consideration is... 

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