Randomness in physical systems

Physical observables as random variables 1.5.1 Origin of randomness in physical systems... 

Whitney, C.A., Random Process in Physical Systems. New York John Wiley, 1990. 

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. 

This is a fairly reasonable way to describe man-made amorphous polymers, which had not been given time to anneal. For polymers that form very quickly, a quick Monte Carlo search on addition can insert an amount of nonoptimal randomness, as is expected in the physical system. 

This, more physical model that visualizes failure to result from random "shocks," was specialized from the more general model of Marshall and Olkin (1967) by Vesely (1977) for sparse data for the ATWS problem. It treats these shocks as binomially distributed with parameters m and p (equation 2.4-9). The BFR model like the MGL and BPM models distinguish the number of multiple unit failures in a system with more than two units, from the Beta Factor model,... 

Consider a physical system with a set of states a, each of which has an energy Hio). If the system is at some finite temperature T, random thermal fluctuations will cause a and therefore H a) to vary. While a system might initially be driven towards one direction (decreasing H, for example) during some transient period immediately following its preparation, as time increases, it eventually fluctuates around a constant average value. When a system has reached this state, it is said to be in thermal equilibrium. A fundamental principle from thermodynamics states that when a system is in thermal equilibrium, each of its states a occurs with a probability equal to the Boltzman distribution P(a) ... 

I have come to believe that under certain conditions the manipulative power of consciousness moves beyond the body and into the world. The world then obeys the will of consciousness to the degree that the inertia of pre-existing physical laws can be overcome. This inertia is overcome by consciousness determining the outcome of the normally random, micro-physical events. Over time the deflection of micro-events from randomness is cumulative so that eventually the effects of such deflections is to shift the course of events in larger physical systems as well. Apparently, when want-ing wishes to come true, patience is everything. 

The number of independent variables required to specify the state of a mechanical or thermodynamic system. Degrees of freedom arise from the possible motions of molecules or particles in a system. (The term generalized coordinates is also used in physics to designate the minimal number of coordinates needed to specify the state of a mechanical system.) 2. The number of independent or unrestricted random variables constituting a statistic. See Statistics (A Primer)... 

Within non-equilibrium thermodynamics, the driving force for relaxation is provided by deviations in the local chemical potential from it s equilibrium value. The rate at which such deviations relax is determined by the dominant kinetics in the physical system of interest. In addition, the thermal noise in the system randomly generates fluctuations. We thus describe the dynamics of a step edge by the equation. 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

There is, however, one obvious difference between a mathematical model and a physical model (or the real system itself). The response of the former to the same set of conditions is always identical. In physical experiments, where results are measured rather than calculated, there are inevitably random errors which may be appreciable. As already pointed out, mathematical models are usually to some extent imperfect in other words, they do contain systematic errors. The important point is that these imperfections are always reproduced in the same way, even though their ultimate source may have been random errors in data on which the model was based. This point has been stressed because it is important to recognize that only partial use of methods from statistical treatments of design of experiments is involved in what follows. The use of these methods here is only for the purpose of studying the geometry of response with respect to the controllable variables. No consideration of probability or of error enters into the discussion. 

We begin by considering some straightforward results from statistics. Suppose that we have two boxes and a machine that will randomly throw marbles into one or the other. This physical system is depicted in Fig. 8.3. We label the bottom box j = 0 it has an area (which we are going to denote go) of 2 arbitrary units. The other box we label j = 1, and its area gi is taken to be 1. We would like to know the expected probability that a single throw of the marble lands in box j = 0 versus j = 1. Our intuition says that it is twice as likely the marble will land in box 0 than in box 1, and indeed that is the correct result. The likelihood is proportional to g,. 

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