This section deals with the influence of the systemIz, the components of which partic- ipate in the process of measurement but nominally appear only at the stage of posing the measurement task. Here we have: a measurand ('), th eobject of measurement (o), external conditions ( ). Time–space limitations will be considered separately (in Sec- tion 3.4.4). All these components are uncontrollable (given) within the framework of the systemIz. However, the accuracy of giving them (or more precisely the accuracy of giving the parameters characterizing them) is not infinite. It is determined by the amount of a priori information about them and the degree of its correspondence to present-day reality.
3.3.1 Measurand and object models
The measurand as a quality is given by its definition in the system of quantities and concepts, i.e., on the basis of agreements. The agreements (if they are sufficiently uni- vocal), in principle, can be implemented with complete precision. However, in prac- tice, as applied to a specific situation and a definite object of study, some assumptions giving a concrete expression to the definition of the measurand arise. Let us consider the above for physical quantities, which is the most important case.
In the system of physical quantities a formalized reflection of the quality of each quantity (its definition) is the quantity dimensionality that reflects the relationship be- tween this quantity and the basic quantities of the system. Such definitions are accurate (univocal) information about the quality (kind) of the quantities measured under the condition of the successful choice of the basic quantities. Thus, the velocity of a body movement is determined by the derivative of a distance with respect to time, the density of substance is determined by the ratio of a mass of this substance to its volume, etc.
However, even for the simplest derived quantities of this kind their definitions are abstractions and are related to idealized objects, since any physical quantity is the property that in a qualitative respect is inherent in many objects.
Thus, all quantities of kinematics (for example, displacement, velocity, acceleration, and others) are introduced for an idealized “material point”. The definition of substance density, indicated above, assumes its homogeneity, etc. The matter stands even worse with basic quantities. As a matter of fact, these quantities are indefinable in the system of physical quantities (although there are vague wordings such as “length is a measure of a spatial stretch of bodies”, “mass is a measure of the inertial and gravitational properties of a body”). Such is the dialectics of knowledge. The higher the degree of the concept generality, the more abstract this concept is, i.e., it is separated from concreteness.
Therefore, while formulating the measurement problem and then planning the mea- surement experiment, in addition to understanding the measurand as a quality, it is necessary to detail it as applied to the object under study, having constructed a model of the object analyzed.
It should be added that in constructing such a model, it is necessary to define more exactly the following parameters.
1) To construct a model of the measurand of an object under study, i.e., to de- termine a particularmeasurable parameterof the object uniform with respect to the physical quantity being studied. Usually this parameter is connected with an outside part (surface) of the object. A number of examples of linear dimensions and electri- cal parameters are given in Section 3.4. Below one more example from the field of mechanics is given.
Example 3.4. When measuring the vibration parameters of linearly oscillating me- chanical systems, where the object of study is the oscillation process of a hard body movement for the model of an object serves, as a rule, the harmonic law of material point movement:
x.t /Dxmsin.!tC'0/, (3.9) wherexis the displacement of a definite point on a body surface with respect to the static (equilibrium) position,xm is the maximum value (amplitude) of the displace- ment,!is the frequency of the oscillation movement, and'0is its starting phase.
Usually as the measurand either a length (linear displacement) or acceleration a, connected withx, are used:
a.t /Dd2x=dt2D xm!2cos.!tC'0/D amcos.!tC'0/.
In a particular case, depending on the goal of measurement and reasoning from other considerations regarding the object to be measured either an amplitude of dis- placementxm, or of accelerationam, or an “instantaneous” value of displacementx.t / or accelerationa.t /, or their mean values over a definite interval of time are used.
2) As noted in [37], to perform measurements it is necessary to have a priori infor- mation not only about thequalitativebut also thequantitative characteristicsof the measurand, its dimension. The more precise and consequently more useful the avail- able a priori information about the measurand dimension is, the more forces and means there are for getting more complete a posteriori information for improving the a priori information.
3) A measurement signal entering from the object at an input of a measuring instru- ment in addition to information about the measurand can contain information about some other“noninformational” parametersof the object which influence the read- outs of the measuring instrument. This is especially characteristic for measurements in the dynamic mode. Thus, in Example 3.4 one noninformational parameter of the object is the frequency!at which its oscillations take place.
Section 3.3 Accuracy limitations due to the components of a measurement task 175 4) Although the “informative” and “noninformative” parameters of the object con- sidered above reflect its inherent properties, however, being embodied into parameters of the measurement signal, they experience real transformations in the measurement chain.
In addition any object possesses a variety of other “inherent properties” which do not influence the parameters of the measurement signal, but rather the information about it which allows a correct model of the object (for the given measurement problem) to be constructed, which corresponds as much as possible to reality.
Parameters characterizing such properties of the object which influence the model ideas about this object will be called themodel parametersof the object.
Thus, while carrying out precise measurement of the density or viscosity of a sub- stance it is necessary to know its composition, distribution of admixtures over the whole volume, its temperature, its compressibility, etc.
The fact that the model in Example 3.4 is not ideal usually becomes apparent be- cause of deflection of a real form of oscillation of a point on the surface of a body from the harmonic law (3.9).
To construct a more accurate model it is necessary to know such model parameters of the object as dislocation of the mass center of an oscillating body relative to a point of measurement, elastic properties of the object, etc. A more detailed example illus- trating the influence of imperfection of the object model on the measurement accuracy will be considered in Section 3.4.
Provided that the constructed object model is such that it allows the dependence of a measurand on informative (qio), noninformative (qno), and model (qmo) parameters to be expressed analytically, then the uncertainty of a measurement result caused by these parameters can be determined with the help of the corresponding functions of influence (3.7) and the uncertainties of the values of these parameters (qi0):
Oo'.Iz/D vu uu t
l0
X
i0D1
@f .qi0/
@qi0qi0 2
. (3.10)
(here the index “0” accentuates that the system parameters considered belong to the object).
3.3.2 Physical limitations due to the discontinuity of a substance structure
It would seem that improving the values of an objectqi0 more and more (as knowl- edge is accumulated, i.e., decreasing their uncertaintyqi0) we can gradually, but still continuously, decrease the uncertainty of the measurement result, connected with the parameters0' !0 atqi0 !0 (see equation (3.10)).
However there is a certain type of physical knowledge about objects of the world surrounding us which leads us to speak of theprincipal limits of accuracywith which we can obtain quantitative information about a majority of physical measurands.
The first circle of such knowledge is connected with molecular-kinetic ideas about a substance structure, the theory of which is described in courses on molecular physics, thermodynamics, and statistical physics.
Themolecular-kinetic theoryproceeds from the assumption that a substance exists in the form of its components, i.e., molecules (microscopic particles), moving accord- ing to laws of mechanics. All real physical objects which are met in practice (be it gases, liquids, or hard bodies) have a discrete structure, i.e., they consist of a very large number of particles (molecules, ions, electrons, etc.).
So, for example, one cubic centimeter of metal contains1022 ions and the same numbefr of free electrons; at room temperature and at an atmospheric pressure of 1 cm3 of air contains31019molecules; even in a high vacuum (one million times less than atmospheric pressure) 1 cm3 of air contains1012 molecules. At the same time the particle dimensions are of the order of1010mD108cm.
Although the laws of mechanics allow the behavior of each separate particle in the system to be described precisely, nevertheless a huge number of the system par- ticles offer neither the technical nor physical possibility to describe the behavior of the system as a whole. Under these conditions a single possibility is provided only by aprobabilistic descriptionof a complicated system state (an ensemble of a set of particles).
Statistical physics is based on these ideas and uses the methods and ways of math- ematical statistics to establish thestatistical lawsof the behavior of complicated sys- tems consisting of a great number of particles. Using this as a base, it studies the macroscopic properties of these systems.
Owing to the great number of microparticles composing the objects of a physical study, all their properties considered as random quantities, are averaged. This allows these properties to be characterized with some macroscopic (observable) parameters such as density, pressure, temperature, charge, and others, which are related to the thermodynamicparameters of the system which do not take into account their link with the substance structure.
Such a link, however, becomes apparent when the indicated thermodynamic param- eters are described usingstatistical mean valuesof the corresponding quantities which are introduced according to usual rules of forming a mean:
N 'D
Z
'.'/d', (3.11)
where'is the quantity being averaged and characterizing the system state, and.'/is the probability density function of quantity values'(of the system states) (integration is performed over all system states).