External fluctuation

A further refinement of the harmonic oscillator model is possible, in which the lattice is put into contact with a heat bath at temperature T and remains in contact with the heat bath, so that the initial correlations decay not only through mutual interactions but also through random collisions with an external fluctuating field. Although it might appear that such a case would contain features of both the independent particle case and the harmonic oscillator model just analyzed, the resulting formalism is much closer to that required for the latter, and the results differ only in detail. The model to be discussed is specified by the equations of motion... 

Stratonoviclr first studied the influence of external fluctuations via a vacuum tube oscillator. He noticed a phenomenological behavior reminiscent of that of physical systems far from equilibrium. His pioneer work showed that the use of electric circuits is a simple means of shedding light on general problems, thereby stimulating further experimental work of this kind. 

Iwamoto and Seno (1980) proposed a three dimensional mathematical model and analyzed it under the influence of external fluctuation terms. In (1982) Iwamoto et al. gave a three dimensional model to explain small and large amplitude oscillations observed in the reactions. 

Another problem arises from temperature changes due to gas expansion or due to external fluctuations. If only a small quantity of the sample is available, as usual for nanostructured materials, the error in the evaluation of the hydrogen storage capacity due to temperature changes can be of the same order of magnitude as the measured value. Temperature fluctuations are even more serious if the measurements are executed at 11K. 

Equation (75) expresses the partition function of the many-polymer system in terms of the partition functions of single polymers subjected to external fluctuating fields. The self-consistent field theory approximates this functional integral over the fields by the value of the integrand evaluated at those values of the fields, and Wb, that minimize the functional F[h, 4, vb]. From the definition of F it follows that these functions satisfy the self-consistent equations... 

While internal fluctuations are self-generated in the system, and they can occur in closed and open systems as well, external fluctuations (or noise) are determined by the environment of the system. A characteristic property of internal fluctuations is that they scale with the system size, i.e. they tend to vanish in the thermodynamic limit (except in the vicinity of critical points). Since external noise reflects the random character of the environment, the measure of fluctuations is completely independent of the system size. A natural way to introduce external noise is to assume that the control parameters (i.e. the rate constants for isothermal pure chemical systems) are not strictly constants, but that it is better to consider them as stationary stochastic processes. 

The influence of external fluctuations on two-dimensional oscillatory systems has been studied by Ebeling Engel-Herbert (1980). The stochastic counterpart of the appearence of a limit cycle is strongly connected with the formation of a probability crater on the stationary probability surface. They studied particular cases when the system has the form... 

It is a quite natural endeavour to search for a unified mathematical framework of describing internal and external fluctuations. Internal fluctuations used to be described by the Markovian master equation. Sancho San Miguel (1984) offered two equivalent techniques for a unified theory, at least for single-variable systems, when internal fluctuations were modelled specifically by a one-step Markovian master equation, and external noise was considered by dichotomous noise. 

To have a stochastic differential equation model for both the internal and external fluctuations in complex chemical reactions would have advantages other than providing the framework for estimation. As it turned out from the analysis of Chapter 1 it would be necessary to have a common model of macroscopic phenomena that does include stochasticity, spatial processes (such as diffusion) and sources and sinks (such as reactions). These topics will be further analysed in Section 6.3. 

Models of population growth in random environment have been constructed by inserting white noise fluctuations in the deterministic growth equations (May, 1972 Capocielli Ricciardi, 1974 Ricciardi, 1977 Nobile Ricciardi, 1984a, b). The structure of these models is similar to those described in Section 5.8 ( external fluctuations ) and we shall not discuss them here. 

Ebeling, W. Engel-Herbert, H. (1980). The influence of external fluctuations on self-sustained temporal oscillations. Physica, 104A, 378-86. 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

In multiphase microfiuidics, transient phenomena can be divided into fluctuations that are flow induced (inside the channel) and induced by external fluctuations. Kraus et al. [69] measured statistical properties (distribution of liquid slug and gas bubble lengths) in segmented gas-liquid flow and documented the sensitivity to external disturbances (e.g. syringe pump pressure fluctuations). Van Steijn et al. [87] investigated such fluctuations in detail for low-velocity flow in a short... 

Stochastically driven systems exhibit a variety of interesting nonequilibrium effects. These have been recently reviewed by HORSTHEMKE and LEFEVER [1] and also addressed by other authors in this workshop. In this contribution we focus our attention on the role played by the internal fluctuations of a system driven by an external noise [2,3,4]. External noise effects are usually studied in the thermodynamic limit in which internal fluctuations become negligible. This procedure assumes that the external driving noise completely dominates the fluctuations in the system. Nevertheless, a framework in which internal and external fluctuations are simultaneously considered is necessary to calculate finite size effects. Within such a framework a better understanding of the physical contents of "noise induced transition" phenomena [1] is obtained by investigating how changes in a stationary distribution induced by external noise are smoothed out by internal fluctuations. A major novel outcome of the unified theory of internal and external fluctuations presented here is the existence of "crossed-fluctuation" contributions which couple the two independent sources of randomness in the system. 

The mean value is independent of V. The first and second terms in the r.h.s. of (3.2) are the contributions from internal and external fluctuations respectively. The third term is the "crossed-fluctuation" contribution discussed earlier in general. 

We finally note that similar conclusions about the interplay of internal and external fluctuations are obtained when considering fluctuations in the source (o() or creation (<) parameters. Also in these cases a transition is smeared out by internal fluctuations. In these two cases the probability distribution in the thermodynamic limit is defined for values of x larger than a boundary value. A change of behavior at this boundary is found for a critical value of X. Fluctuations of o and X turn out to have qualitatively the same effects. They are qualitatively different from the... 

But if due to external fluctuations, e.g. By thermal noise, side-branches rj are permanently excited, a drift in r towards r will be observed. This is due to the nonzero component of the... 

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