Why the application of models affects the limits of applicability. Causality and interaction in physics

The purpose of the lesson

Continue the discussion of wave diffraction, consider the problem of the limits of applicability of geometric optics, develop skills in the qualitative and quantitative description of the diffraction pattern, consider the practical applications of light diffraction.

This material is usually considered in passing within the framework of the study of the topic "Diffraction of light" due to lack of time. But, in our opinion, it must be considered for a deeper understanding of the phenomenon of diffraction, understanding that any theory describing physical processes has limits of applicability. Therefore, this lesson can be conducted in base classes instead of a problem solving lesson, since the mathematical apparatus for solving problems on this topic is quite complicated.

No. p / p	Lesson stages	Time, min	Techniques and methods
1	Organizing time	2
2	Repetition of the studied material	6	Frontal survey
3	Explanation of the new material on the topic "Limits of applicability of geometric optics"	15	Lecture
4	Consolidation of the studied material using a computer model	15	Working on a computer with worksheets. Model "Diffraction limit of resolution"
5	Analysis of the work done	5	Frontal conversation
6	Homework explanation	2

Repetition of the studied material

Frontally repeat questions on the topic "Diffraction of light".

Explanation of the new material

Limits of applicability of geometric optics

All physical theories reflect the processes occurring in nature approximately. For any theory, certain limits of its applicability can be indicated. Whether a given theory can be applied in a particular case or not depends not only on the accuracy that the theory provides, but also on what accuracy is required when solving a particular practical problem. The boundaries of a theory can only be established after a more general theory covering the same phenomena has been constructed.

All these general propositions also apply to geometric optics. This theory is approximate. It is unable to explain the phenomena of interference and diffraction of light. A more general and more accurate theory is wave optics. The law of rectilinear propagation of light and other laws of geometric optics are satisfied accurately only if the dimensions of the obstacles in the path of light propagation are much greater than the wavelength of light. But they are definitely never fulfilled.

The operation of optical devices is described by the laws of geometric optics. According to these laws, we can distinguish arbitrarily small details of an object with a microscope; using a telescope, one can establish the existence of two stars at any, arbitrarily small, angular distances between them. However, in reality this is not so, and only the wave theory of light makes it possible to understand the reasons for the limiting resolution of optical instruments.

Resolution of the microscope and telescope.

The wave nature of light limits the ability to distinguish the details of an object or very small objects when observed with a microscope. Diffraction does not make it possible to obtain distinct images of small objects, since light does not propagate in a strictly straight line, but bends around objects. Because of this, the images are "blurry". This happens when the linear dimensions of objects are comparable to the wavelength of light.

Diffraction also imposes a limit on the resolving power of a telescope. Due to the diffraction of waves, the image of the star will not be a point, but a system of light and dark rings. If two stars are at a small angular distance from each other, then these rings are superimposed on each other and the eye is not able to distinguish whether there are two luminous points or one. The limiting angular distance between the luminous points at which they can be distinguished is determined by the ratio of the wavelength to the lens diameter.

This example shows that diffraction always occurs, on any obstacles. It cannot be neglected for very fine observations even for obstacles much larger in size than the wavelength.

The diffraction of light determines the limits of applicability of geometric optics. Light bending around obstacles imposes a limit on the resolution of the most important optical instruments - the telescope and microscope.

"Diffraction Limit of Resolution"

Worksheet for the lesson

Sample Answers
"Diffraction of Light"

Last name, first name, class ______________________________________________

Set the hole diameter to 2 cm, the angular distance between the light sources 4.5 ∙ 10 -5 rad . By changing the wavelength, determine from what wavelength the image of two light sources will be impossible to distinguish, and they will be perceived as one.

Answer: from about 720 nm and longer.

How does the resolution limit of an optical instrument depend on the wavelength of the observed objects?

Answer: the longer the wave, the lower the resolution limit.

Which binary stars - blue or red - can we detect at a greater distance with modern optical telescopes?

Answer: blue.

Set the minimum wavelength without changing the distance between the light sources. At what aperture diameter will the image of two light sources be impossible to distinguish, and they will be perceived as one?

Answer: 1.0 cm or less.

Repeat the experiment with the maximum wavelength.

Answer: about 2 cm or less.

How does the resolution limit of optical instruments depend on the diameter of the hole through which light passes?

Answer: the smaller the hole diameter, the lower the resolution limit.

What telescope - with a lens of a larger diameter or a smaller one - will allow you to consider two nearby stars?

Answer: with larger lens.

Find experimentally at what minimum distance from each other (in angular value - radians) can the image of two light sources be distinguished in this computer model?

Answer: 1.4∙10 -5 rad.

Why can't molecules or atoms of matter be seen with an optical microscope?

Answer: if the linear dimensions of the observed objects are comparable to the wavelength of the light, then diffraction will not make it possible to obtain their distinct images in a microscope, since the light does not propagate strictly rectilinearly, but bends around the objects. Because of this, the images are "blurry".

Give examples when it is necessary to take into account the diffraction nature of images.

Answer: for all observations through a microscope or telescope, when the dimensions of the observed objects are comparable to the wavelength of light, for small sizes of the inlet of telescopes, for observations in the long red wavelength range of objects located at small angular distances from each other.

The disclosure of the content and the concretization of concepts should be based on one or another specific model of the interconnection of concepts. The model, objectively reflecting a certain side of communication, has limits of applicability, beyond which its use leads to false conclusions, but within the limits of its applicability, it should not only be figurative, visual and specific, but also have heuristic value.

The variety of manifestations of causal relationships in the material world has led to the existence of several models of causal relationships. Historically, any model of these relationships can be reduced to one of the two main types of models, or a combination of both.

a) Models based on a temporal approach (evolutionary models). Here the main attention is focused on the temporal side of the cause-and-effect relationship. One event - the "cause" - gives rise to another event - the "effect", which lags behind the cause in time (late). Delay is a hallmark of the evolutionary approach. Cause and effect are interdependent. However, the reference to the generation of an effect by a cause (genesis), although legitimate, is introduced into the definition of a causal relationship, as if from the outside, from the outside. It fixes the external side of this connection without capturing deep essence.

The evolutionary approach was developed by F. Bacon, J. Millem and others. Hume's position was the extreme polar point of the evolutionary approach. Hume ignored genesis, denying the objective nature of causality, and reduced causation to a simple regularity of events.

b) Models based on the concept of "interaction" (structural or dialectical models). We will find out the meaning of the names later. The focus here is on interaction as a source of cause-and-effect relationships. The cause is the interaction itself. Kant paid much attention to this approach, but the dialectical approach to causality acquired its clearest form in the works of Hegel. Of the modern Soviet philosophers, this approach was developed by G.A. Svechnikov, who sought to give a materialistic interpretation of one of the structural models of causation.

Existing and currently used models reveal the mechanism of cause-and-effect relationships in various ways, which leads to disagreements and creates the basis for philosophical discussions. The sharpness of the discussion and the polar nature of points of view testify to their relevance.

Let's highlight some of the issues discussed.

a) The problem of simultaneity of cause and effect. This is the main problem. Are cause and effect simultaneous or separated by a time interval? If cause and effect are simultaneous, then why does the cause give rise to the effect, and not vice versa? If cause and effect are not simultaneous, can there be a "pure" cause, i.e. a cause without an effect that has not yet occurred, and a “pure” effect, when the effect of the cause has ended, but the effect is still ongoing? What happens in the interval between cause and effect if they are separated in time, etc.?

b) The problem of uniqueness of cause-and-effect relationships. Does the same cause give rise to the same effect, or can one cause give rise to any effect from several potential ones? Can the same effect be produced by any one of several causes?

c) The problem of the reciprocal effect of the effect on its cause.

d) The problem of the connection between cause, occasion and conditions. Can, under certain circumstances, the cause and condition change roles: the cause becomes a condition, and the condition becomes a cause? What is the objective relationship and distinguishing features of cause, occasion and condition?

The solution to these problems depends on the chosen model, i.e. largely on what content will be put into the original categories of "cause" and "effect". The definitional nature of many difficulties is manifested, for example, already in the fact that there is no single answer to the question of what should be understood by "cause". Some researchers think of a material object as a cause, others think of a phenomenon, others think of a change in state, others think of an interaction, and so on.

The solution of the problem does not lead to attempts to go beyond the framework of the model representation and give a general, universal definition of a causal relationship. As an example, the following definition can be given: “Causality is such a genetic connection of phenomena in which one phenomenon, called a cause, under certain conditions, inevitably generates, causes, brings to life another phenomenon, called a consequence.” This definition is formally valid for most models, but, without relying on the model, it cannot solve the problems posed (for example, the problem of simultaneity) and therefore has limited epistemological value.

Solving the problems mentioned above, most authors tend to proceed from the modern physical picture of the world and, as a rule, pay somewhat less attention to epistemology. Meanwhile, in our opinion, there are two problems here that are of great importance: the problem of removing elements of anthropomorphism from the concept of causality and the problem of non-causal relationships in natural science. The essence of the first problem is that causality as an objective philosophical category must have an objective character, independent of the cognizing subject and his activity. The essence of the second problem: should we recognize causal connections in natural science as universal and universal, or consider that such connections are of a limited nature and there are connections of a non-causal type that deny causality and limit the limits of applicability of the principle of causality? We believe that the principle of causality is universal and objective and its application knows no limits.

So, two types of models, objectively reflecting some important aspects and features of causal relationships, are to a certain extent in conflict, since they solve the problems of simultaneity, unambiguity, etc. in different ways, but at the same time, objectively reflecting some aspects of causal relationships , they must be related. Our first task is to identify this connection and refine the models.

Limit of applicability of models

Let us try to establish the limit of applicability of models of evolutionary type. Causal chains that satisfy evolutionary models tend to have the property of transitivity. If event A is the cause of event B (B is the effect of A), if, in turn, event B is the cause of event C, then event A is the cause of event C. If A → B and B → C, then A → C. Thus the simplest cause-and-effect chains are compiled in a way. Event B may be the cause in one case and the effect in the other. This regularity was noted by F. Engels: “... cause and effect are ideas that matter, as such, only when applied to a given individual case: but as soon as we consider this individual case in general connection with the entire world whole, these representations converge and intertwine in the representation of a universal interaction in which cause and effect are constantly changing places; what here or now is the cause becomes there or then the effect, and vice versa” (vol. 20, p. 22).

The property of transitivity allows a detailed analysis of the causal chain. It consists in the division of the final chain into simpler causal links. If A, then A → B 1 , B 1 → B 2 ,..., B n → C. But does a finite causal chain have the property of infinite divisibility? Can the number of links of a finite chain N tend to infinity?

Based on the law of the transition of quantitative changes into qualitative ones, it can be argued that when dismembering the final causal chain, we will encounter such a content of individual links in the chain when further division becomes meaningless. Note that infinite divisibility, which denies the law of the transition of quantitative changes into qualitative ones, Hegel called "bad infinity"

The transition of quantitative changes to qualitative ones occurs, for example, when a piece of graphite is divided. When the molecules are separated up to the formation of a monatomic gas, the chemical composition does not change. Further division of matter without changing its chemical composition is no longer possible, since the next stage is the splitting of carbon atoms. Here, from a physicochemical point of view, quantitative changes lead to qualitative ones.

In the above statement of F. Engels, the idea is clearly traced that the cause-and-effect relationships are based not on spontaneous will, not on a whim of chance and not on a divine finger, but on universal interaction. In nature, there is no spontaneous emergence and destruction of motion, there are mutual transitions of some forms of motion of matter into others, from one material object to another, and these transitions cannot occur otherwise than through the interaction of material objects. Such transitions, caused by interaction, give rise to new phenomena, changing the state of interacting objects.

Interaction is universal and forms the basis of causality. As Hegel rightly noted, "interaction is a causal relationship posited in its full development." F. Engels formulated this idea even more clearly: “Interaction is the first thing that comes before us when we consider moving matter as a whole from the point of view of modern natural science ... Thus, natural science confirms that ... that interaction is a true causa finalis of things. We cannot go beyond the knowledge of this interaction precisely because there is nothing more to know behind it” (vol. 20, p. 546).

Since interaction is the basis of causality, let us consider the interaction of two material objects, the scheme of which is shown in Fig. 1. This example does not violate the generality of reasoning, since the interaction of several objects is reduced to pair interactions and can be considered in a similar way.

It is easy to see that during the interaction both objects simultaneously act on each other (reciprocity of action). In this case, the state of each of the interacting objects changes. No interaction - no state change. Therefore, a change in the state of any one of the interacting objects can be considered as a particular consequence of the cause - interaction. A change in the states of all objects in their totality will constitute a complete consequence.

Obviously, such a cause-and-effect model of an elementary link in an evolutionary model belongs to the class of structural (dialectical) ones. It should be emphasized that this model is not limited to the approach developed by G.A. Svechnikov, because under investigation by G.A. Svechnikov, according to V.G. Ivanov, understood "... a change in one or all interacting objects or a change in the nature of the interaction itself, up to its disintegration or transformation" . As for the change of states, this change is G.A. Svechnikov attributed to the non-causal type of connection.

So, we have established that evolutionary models as an elementary, primary link contain a structural (dialectical) model based on interaction and change of states. Somewhat later, we will return to the analysis of the relationship between these models and the study of the properties of the evolutionary model. Here we would like to note that, in full accordance with the point of view of F. Engels, the change of phenomena in evolutionary models that reflect objective reality occurs not due to the simple regularity of events (as in D. Hume), but due to the conditionality generated by the interaction (genesis ). Therefore, although references to generation (genesis) are introduced into the definition of causal relationships in evolutionary models, they reflect the objective nature of these relationships and have a legal basis.

Fig. 2. Structural (dialectical) model of causality

Let's return to the structural model. In its structure and meaning, it is in excellent agreement with the first law of dialectics - the law of unity and struggle of opposites, if interpreted:

– unity– as the existence of objects in their mutual connection (interaction);

– opposites- as mutually exclusive tendencies and characteristics of states, due to interaction;

– fight- as an interaction;

– development– as a change in the state of each of the interacting material objects.

Therefore, a structural model based on interaction as a cause can also be called a dialectical model of causality. From the analogy of the structural model and the first law of dialectics, it follows that causality acts as a reflection of objective dialectical contradictions in nature itself, in contrast to subjective dialectical contradictions that arise in human consciousness. The structural model of causality is a reflection of the objective dialectic of nature.

Consider an example illustrating the application of the structural model of cause-and-effect relationships. Such examples, which are explained using this model, can be found quite a lot in the natural sciences (physics, chemistry, etc.), since the concept of "interaction" is fundamental in natural science.

Let's take as an example an elastic collision of two balls: a moving ball A and a stationary ball B. Before the collision, the state of each of the balls was determined by a set of attributes Ca and Cb (momentum, kinetic energy, etc.). After the collision (interaction), the states of these balls have changed. Let us denote the new states C "a and C" b. The reason for the change in states (Ca → C "a and Cb → C" b) was the interaction of the balls (collision); the consequence of this collision was a change in the state of each ball.

As already mentioned, the evolutionary model is of little use in this case, since we are not dealing with a causal chain, but with an elementary causal link, the structure of which cannot be reduced to an evolutionary model. To show this, let's illustrate this example with an explanation from the point of view of the evolutionary model: "Before the collision, ball A was at rest, so the reason for its movement is the ball B that hit it." Here ball B is the cause, and the movement of ball A is the effect. But from the same positions, the following explanation can be given: “Before the collision, ball B moved uniformly along a rectilinear trajectory. If it were not for ball A, then the nature of the movement of ball B would not change. Here the cause is already ball A, and the effect is the state of ball B. The above example shows:

a) a certain subjectivity that arises when applying the evolutionary model beyond the limits of its applicability: the cause can be either ball A or ball B; this situation is due to the fact that the evolutionary model snatches out one particular branch of the investigation and is limited to its interpretation;

b) a typical epistemological error. In the above explanations from the position of the evolutionary model, one of the material objects of the same type acts as an "active", and the other - as a "passive" beginning. It turns out that one of the balls is endowed (in comparison with the other) with “activity”, “will”, “desire”, like a person. Therefore, it is only thanks to this "will" that we have a causal relationship. Such an epistemological error is determined not only by the model of causality, but also by the imagery inherent in living human speech, and by the typical psychological transfer of properties characteristic of complex causality (we will talk about it below) to a simple causal link. And such errors are very typical when using the evolutionary model beyond the limits of its applicability. They occur in some definitions of causality. For example: “So, causality is defined as such an impact of one object on another, in which a change in the first object (cause) precedes a change in another object and in a necessary, unambiguous way generates a change in another object (consequence)”. It is difficult to agree with such a definition, since it is completely unclear why, in the course of interaction (mutual action!) Objects should be deformed not simultaneously, but one after another? Which of the objects should be deformed first and which should be deformed second (priority problem)?

Model Qualities

Let us now consider what qualities the structural model of causality holds in itself. We note the following among them: objectivity, universality, consistency, unambiguity.

Objectivity causality is manifested in the fact that interaction acts as an objective cause, in relation to which interacting objects are equal. There is no room for anthropomorphic interpretation here. Versatility due to the fact that the basis of causality is always interaction. Causality is universal, just as interaction itself is universal. Consistency due to the fact that, although cause and effect (interaction and change of states) coincide in time, they reflect various parties causal relationships. Interaction involves a spatial connection of objects, a change in state - a connection of the states of each of the interacting objects in time.

In addition, the structural model establishes unequivocal connection in causal relationships, regardless of the method of mathematical description of the interaction. Moreover, the structural model, being objective and universal, does not prescribe natural science restrictions on the nature of interactions. Within the framework of this model, both instantaneous long-range or short-range interaction and interaction with any finite velocities are valid. The appearance of such a limitation in the definition of cause-and-effect relationships would be a typical metaphysical dogma, once and for all postulating the nature of the interaction of any systems, imposing a natural philosophical framework on the part of philosophy on physics and other sciences, or limiting the limits of applicability of the model to such an extent that the benefit of such a model would be very modest.

Here it would be appropriate to dwell on questions related to the finiteness of the propagation velocity of interactions. Consider an example. Let there be two fixed charges. If one of the charges began to move with acceleration, then the electromagnetic wave will approach the second charge with a delay. Doesn't this example contradict the structural model and, in particular, the property of reciprocity of action, since in such an interaction the charges are in an unequal position? No, it doesn't contradict. This example describes not a simple interaction, but a complex causal chain in which three different links can be distinguished.

1. Interaction of the first charge with an object that causes its acceleration. The result of this interaction is a change in the state of the source that acted on the charge, and in particular, the loss of part of the energy by this source, a change in the state of the first charge (acceleration) and the appearance of an electromagnetic wave that was emitted by the first charge during its accelerated movement.

2. The process of propagation of an electromagnetic wave emitted by the first charge.

3. The process of interaction of the second charge with an electromagnetic wave. The result of the interaction is the acceleration of the second charge, the scattering of the primary electromagnetic wave and the radiation of the electromagnetic wave by the second charge.

In this example, we have two different interactions, each of which fits into the structural model of causality. Thus, the structural model is in excellent agreement with both classical and relativistic theories, and the finite speed of propagation of interactions is not fundamentally necessary for the structural model of causality.

Concerning the structural model of causality, we note that it does not contradict the decay reactions u. object synthesis. In this case, a relatively stable connection between objects is either destroyed as a special type of interaction, or such a connection is formed as a result of interaction.

Since quantum theories (as well as classical ones) widely use the categories of "interaction" and "state", the structural model is fundamentally applicable in this area of natural science. The difficulties that sometimes occur are due, in our opinion, to the fact that, having a well-developed mathematical formalism, quantum theories are not yet fully developed and refined in terms of conceptual interpretation.

Mario Bunge writes, for example, about the interpretation of the f-function:
“Some attribute the function ψ to some individual system, others to some actual or potential statistical ensemble of identical systems, others consider the ψ-function as a measure of our information, or the degree of confidence in relation to some individual complex consisting of a macrosystem and an instrument, or, finally, , simply as a catalog of measurements made on many identically prepared microsystems. Such a variety of options for interpreting the ψ-function makes it difficult to rigorously causally interpret the phenomena of the microworld.

This is one of the evidence that quantum theories are in the process of formation and development and have not reached the level of internal completeness characteristic of classical theories.

But the problems of the formation of quantum theories are evidenced not only by the interpretation of the ψ-function. Although relativistic mechanics and electrodynamics at first glance seem to be complete theories, a deeper analysis shows that for a number of reasons these theories also did not avoid contradictions and internal difficulties. For example, in electrodynamics, there is the problem of electromagnetic mass, the problem of the reaction of charge radiation, etc. Failures in attempts to resolve these problems within the framework of the theories themselves in the past and the rapid development of theories of the microcosm gave rise to the hope that the development of quantum theories will help eliminate difficulties. Until then, they should be perceived as an inevitable "evil", with which one has to put up with one way or another and expect success from quantum theories.

At the same time, quantum theories themselves faced many problems and contradictions. It is curious to note that some of these difficulties are of a "classical" nature, i.e. inherited from classical theories and is due to their internal incompleteness. It turns out a "vicious circle": we assign the resolution of contradictions of classical theories to quantum theories, and the difficulties of quantum ones are determined by the contradictions of classical ones.

Over time, hope for the ability of quantum theories to eliminate contradictions and difficulties in classical theories began to fade, but until now, interest in resolving the contradictions of classical theories within their own framework still remains in the background.

Thus, the difficulties sometimes encountered in explaining the phenomena of the microworld from the standpoint of causality have an objective origin and are explained by the peculiarities of the formation of quantum theories, but they are not fundamental, forbidding or limiting the application of the principle of causality in the microworld, in particular, the application of the structural model of causality.

Causality and interaction are always interconnected. If the interaction has the properties of universality, universality and objectivity, then cause-and-effect relationships and relationships are just as universal, universal and objective. Therefore, in principle, it is impossible to agree with Bohm's statements that when describing the phenomena of the microworld, one can rely on philosophical indeterminism in some cases, and adhere to the principle of causality in others. We consider V.Ya. Perminov that “the concept of complementarity indicates way reconciliation(our italics - VK.) determinism and indeterminism”, regardless of whether this idea refers to the philosophy of natural science or to a specific theory of natural science. The way of reconciling the materialist point of view with the position of modern agnosticism on this question is eclecticism, it is the negation of objective dialectics. IN AND. Lenin emphasized that "the question of causality is of particular importance for determining the philosophical line of this or that newest 'ism'..." (vol. 18, p. 157). And the path of formation of quantum theories lies not through denial or limitation, but through the affirmation of causality in the microcosm.

Two sides of scientific theories

The structure of scientific theories of natural science and the functions of scientific theories are directly or indirectly related to the causal explanation of the phenomena of the material world. If we turn to the structural model of causality, we can identify two characteristic moments, two important aspects that are somehow connected with the functions of scientific theories.

The first concerns the description of causal relationships and answers the question: how, in what sequence? It corresponds to any branch of a particular consequence that connects the conditioned states. It gives not only a description of the transition of an object from one state to another, but describes and covers the entire causal chain as a sequence of connected and conditioned states, without going deep into the essence, into the source of the change in the states of the links in the chain.

The second side answers the question: why, for what reason? On the contrary, it splits the causal chain into separate elementary links and gives an explanation for the change in state, based on the interaction. This is the explanatory side.

These two aspects are directly related to two important functions of scientific theory: explanatory and descriptive. Since the principle of causality has been and will be at the basis of any natural science theory, the theory will always perform these two functions: description and explanation.

However, the methodological function of the principle of causality is manifested not only in this. The internal structuring of the theory itself is also related to this principle. Take, for example, classical mechanics with its three traditional divisions: kinematics, dynamics and statics. In kinematics, force interactions are not considered, but there is a description (physical and mathematical) of the types of movement of material points and material objects. Interaction is implied, but it fades into the background, leaving priority to the description of complex connected movements through the characteristics of their states. Of course, this fact cannot serve as a reason for classifying kinematics as a non-causal method of description, since kinematics reflects the evolutionary side of the cause-and-effect relationships linking different states.

Dynamics is a theoretical section that includes a complete causal description and explanation based on a structural model of causal relationships. In this sense, kinematics can be considered a subsection of dynamics.

Of particular interest from the point of view of causality is statics, in which the investigative chains are degenerate (absent), and we are dealing only with connections and interactions of a static nature. In contrast to the phenomena of objective reality, where absolutely stable systems do not exist, static problems are an idealization or a limiting case that is acceptable in particular scientific theories. But the principle of causality is also valid here, since it is impossible not only to solve static problems, but also to understand the essence of statics without applying the “principle of virtual displacements” or principles related to it. "Virtual displacements" are directly related to the change of states in the vicinity of the equilibrium state, i.e. ultimately with a causal relationship.

Consider now electrodynamics. Sometimes it is identified only with Maxwell's equations. This is not true, since Maxwell's equations describe the behavior of waves (radiation, propagation, diffraction, etc.) under given boundary and initial conditions. They do not include a description of the interaction as a reciprocal action. The principle of causality is introduced together with boundary and initial conditions (retarded potentials). This is a kind of "kinematics" of wave processes, if such a comparison is permissible. "Dynamics", and with it causality, is introduced by the Lorentz equation of motion, which takes into account the reaction of charge radiation. It is the connection between Maxwell's equations and Lorentz's equation of motion that provides a fairly complete cause-and-effect description of the phenomena of electromagnetism. Similar examples could be continued. But even the above is enough to make sure that causality and its structural model are reflected in the structure and functions of scientific theories.

If at the beginning of our work we went from the evolutionary model of causality to the structural one, now we have to go back from the structural model to the evolutionary one. This is necessary in order to correctly assess the interconnection and distinctive features of the evolutionary model.

Already in an unbranched linear causal chain, we are forced to abandon the full description of all causal relationships, i.e. we do not take into account some particular consequences. The structural model allows unbranched linear cause-and-effect chains to be reduced to two main types.

a) Object causal chain. It is formed when we select any material object and follow the change in its state over time. An example would be observations of the state of a Brownian particle, or the evolution of a spacecraft, or the propagation of an electromagnetic wave from a transmitter antenna to a receiver antenna.

b) Information causal chain. It appears when we follow not the state of a material object, but some informing phenomenon, which, in the process of interactions of various material objects, is connected successively in time with various objects. An example is the transmission of oral information using a relay race, etc.

All linear unbranched causal chains are reduced to one of these two types or to a combination of them. Such chains are described using the evolutionary model of causality. In the evolutionary description, interaction remains in the background, and a material object or an indicator of its state comes to the fore. Because of this, the main attention is focused on the description of the sequence of events in time. Therefore, this model is called evolutionary.

A linear unbranched causal chain is relatively easy to analyze by reducing it to a set of elementary links and analyzing them through a structural model. But such an analysis is not always possible.

There are complex causal networks in which simple causal chains intersect, branch, and intersect again. This leads to the fact that the use of a structural model makes the analysis cumbersome and sometimes technically impossible.

In addition, we are often not interested in the internal process itself and the description of internal cause-and-effect relationships, but in the initial impact and its final result. A similar situation is often encountered in the analysis of the behavior of complex systems (biological, cybernetic, etc.). In such cases, the detailing of internal processes in their entirety turns out to be redundant, unnecessary for practical purposes, and clutters up the analysis. All this led to a number of features in the description of cause-and-effect relationships using evolutionary models. Let's list these features.

1. In the evolutionary description of the causal network, the complete causal network is coarsened. The main chains are highlighted, and the non-essential ones are cut off and ignored. This greatly simplifies the description, but such simplification is achieved at the cost of losing part of the information, at the cost of losing the unambiguity of the description.

2. In order to preserve unambiguity and bring the description closer to objective reality, cut off branches and causal chains are replaced by a set of conditions. The completeness, unambiguity and objectivity of the causal description and analysis depend on how correctly the main causal chain is identified and how fully the conditions that compensate for coarsening are taken into account.

3. The choice of one or another causal chain as the main one is largely determined by the researcher's goals, i.e. between which phenomena he wants to analyze the connection. It is the target setting that makes us look for the main cause-and-effect chains, and replace the cut-off ones with conditions. This leads to the fact that for some settings, the main role is played by some circuits, while others are replaced by conditions. With other settings, these chains can become conditions, and the role of the main ones will be played by those that were previously secondary. Thus, causes and conditions reverse roles.

Conditions play an important role, linking objective cause and effect. Under different conditions affecting the main causal chain, the consequences will be different. The conditions, as it were, create the channel along which the chain of historical events or the development of phenomena in time flows. Therefore, to identify deep, essential cause-and-effect relationships, a thorough analysis is necessary, taking into account the influence of all external and internal factors, all conditions affecting the development of the main causal chain, and assessing the degree of influence.

4. The evolutionary description focuses not on interaction, but on the connection of events or phenomena in time. Therefore, the content of the concepts of "cause" and "effect" changes, and this is very important to take into account. If in the structural model the interaction is the true causa finalis - the final cause, then in the evolutionary - the effective cause (causa activa) becomes a phenomenon or event.

The investigation also changes its content. Instead of a connection between the states of a material object during its interaction with another, some event or phenomenon acts as a consequence, closing the causal chain. Because of this, the cause in the evolutionary model always precedes the effect.

5. In the above sense, cause and effect in the evolutionary model can act as one-qualitative phenomena, closing the chain of cause and effect from both sides. The consequence of one chain can be the cause and the beginning of another chain, following the first one in time. This circumstance determines the property of transitivity of evolutionary models of causality.

We have touched here only on the main features and distinguishing features of the evolutionary model.

Conclusion

The structural model of causality can be successfully used for relatively simple causal chains and systems. In real practice, one has to deal with complex systems. The question of a causal description of the behavior of complex systems is almost always based on the evolutionary model of causality.

So, we have considered two types of models that reflect cause-and-effect relationships in nature, analyzed the interconnection of these models, the limits of their applicability and some features. The manifestation of causality in nature is diverse both in form and content. It is likely that these models do not exhaust the entire arsenal of forms of cause-and-effect relationships. But no matter how diverse these forms may be, causality will always have the properties of objectivity, generality and universality. Because of this, the principle of causality has performed and will always perform the most important philosophical and methodological functions in modern natural science and the philosophy of natural science. The variety of forms of manifestation of cause-and-effect relationships cannot serve as a reason for rejecting the materialistic principle of causality or assertions about its limited applicability.

Information sources:

Svechnikov G.A. Causality and connection of states in physics. M., 1971.
Svechnikov G.A. Dialectical-materialistic concept of causality // Modern determinism: Laws of nature. M., 1973.
Tyukhtin V.S. Reflection, systems, cybernetics. M., 1972
Uemov A.I., Ostapenko S.V. Causality and time // Modern determinism: Laws of nature.
Orudzhev Z.M., Akhundov M.D. Temporal structure of causality // Philos. Sciences. 1969. No. 6.
Zharov A.M. Time relationship of cause and effect and uncertainty. 1984. No. 3.
Kuznetsov I.V. Selected works on the methodology of physics. M., 1975.
Materialist dialectics: In 5 vols. Vol. 1: Objective dialectics / Under the general. ed. F.V. Konstantinov and V.G. Marakhova; Rep. ed. F.F. Vyakkerev. M., 1981.
Naletov N.3. Causality and the theory of knowledge. M., 1975.
Hegel G.W.F. Encyclopedia of Philosophical Sciences: In 3 vols. Vol. 1: Science of Logic. M., 1974.
Starzhinsky V.P. The concept of "state" and its methodological role in physics. Minsk, 1979.
Ivanov V.G. Causality and determinism. L., 1974.
materialist dialectics. T. 1. S. 213.
Bunge M. Philosophy of Physics. M., 1975. S. 99.
Bohm D. Causality and randomness in modern physics. M., 1959.
Perminov V.Ya. The problem of causality in philosophy and natural sciences. M., 1979. S. 209.
Nikitin E.P. Explanation is the function of science. M., 1970.

Kuligin V.A. Causality and interaction in physics. Collection of Voronezh State University: "Determinism in modern science". Voronezh, 1987.

The question that naturally arises in the study of any science is to assess the prospects for the practical applicability of its conclusions: is it possible, on the basis of this theory, to formulate a sufficiently accurate prediction of the behavior of the object under study? Given that economic theory is concerned with "the choices people make with limited resources to satisfy their desires" 1 , the question posed will be about predicting people's behavior in situations of choice. The dominant strand of economic theory, main stream economics, claims to be able to accurately describe the behavior of individuals making any choice in any situation with limited resources. The subject of the choice, the external conditions for making the choice, the historical epoch in which the choice is made, do not play a special role. The analytical model of neoclassicism remains unchanged, whether it is about buying fruit on the market, about the "choice" of the patron by the overlord in the feudal era, or about choosing a life partner.

One of the first to question the universality claims of classical economics was J.M. Keynes. His main thesis is this: "The postulates of the classical theory are applicable not to the general, but only to the special case, since the economic situation that it considers is only the limiting case of possible equilibrium states" 2 . More precisely, the classical postulates are true only in conditions of full employment of available resources and lose their analytical value as the market moves away from a situation of full employment of resources. Are there other restrictions on the application of the neoclassical model?

Completeness of information

The neoclassical model suggests completeness of information which individuals have at the moment of making a choice. Is this condition reached automatically and is it always achievable? One of the postulates of the neoclassical theory says that all the necessary information about the state of the market is contained in prices, the possession of information about equilibrium prices and allows the participants in the exchange to make transactions in accordance with their interests. L. Walras speaks of the existence of a certain "auctioneer" (commisaire-priseur) in the market, which accepts "bids" from buyers and "offers" from sellers. Comparison of the aggregate demand and aggregate supply obtained on their basis lies at the basis of the "groping" (tatonnement) of the equilibrium price 3 . However, as Oskar Lange showed back in the 1930s in his model of market socialism, in reality the function of an auctioneer can and should be best performed by the planning body, the central planning bureau. The paradox of Lange's argument is that it is in the existence of a planning body that he sees the main prerequisite for the functioning of the neoclassical model of the market 4 .

An alternative to the socialist centralization of pricing can only be a local market model. It is under the condition that transactions are limited to a certain circle of persons or a certain territory, all participants in the exchange can be provided with complete information about transactions planned and made on the market. Medieval fairs are an example of a local market from history: a constant circle of participants and their limited number allowed all traders to have a clear idea of the market situation and build reliable assumptions about its change. Even if the merchants did not have full information about the transaction ex ante the personal reputation of each of them served as the best guarantee of the absence of deceit and the use of additional information by someone to the detriment of the others 5 . Despite the seeming paradox, modern exchanges and individual markets (for example, the diamond market) also operate on the basis of the principles of the local market. Although transactions here are made on a global or at least national scale, the circle of their participants is limited. We are talking about a kind of communities of merchants living on the basis of the personal reputation of each of them 6 . Let's summarize the above: the completeness of information is achievable only in two cases - centralized pricing or the local market.

Perfect Competition

Another requirement of the neoclassical market model is minimum interdependence of participants in transactions: a situation where decisions about the choice of one individual do not depend on the decisions of other individuals and do not affect them. The minimum interdependence in decision-making is achieved only within a certain market structure, i.e. when making transactions on a perfectly competitive market. For a market to meet the criteria of perfect competition, the following conditions must be met:

The presence of a large, potentially infinite number of participants in transactions (sellers and buyers), and the share of each of them is insignificant in the total volume of transactions;

The exchange is carried out by standardized and homogeneous products;

Buyers have complete information about the products they are interested in;

There is a possibility of free entry and exit from the market, and its participants have no incentives for mergers 7 .

Under conditions of perfect competition, the resources that are the object of economic choice become non-specific those. it is easy for them to find an equivalent replacement, and the result from their use will be the same. However, here again it is worth mentioning the Keynesian limitation of the sphere in which the neoclassical analysis remains true. N. Kaldor sees the existence of monopolistic competition as one of the main reasons for underemployment and, consequently, the unattainability of neoclassical equilibrium in the market. "The natural framework for Keynesian macroeconomics is the microeconomics of monopolistic competition" 8 . Thus, the second factor that determines the limits of applicability of the neoclassical model is the structure of the market.

Homo oeconomicus

Another prerequisite for the applicability of neoclassical models to the analysis of real markets is conformity of people who make choices to the ideal of homo oeconomicus. Although the neoclassicists themselves pay insufficient attention to this issue, limiting themselves to references to rationality and to the identification of a person with a perfect calculator, the neoclassical model assumes a very specific type of human behavior. Interest in the behavior of participants in transactions in the market is already characteristic of the founder of classical economic theory, Adam Smith, who is the author of not only the "Study on the Nature and Causes of the Wealth of Nations" (1776), but also the "Theory of Moral Sentiments" (1759). What is the portrait of an ideal participant in transactions in the neoclassical market?

First, he must be purposeful. Following Max Weber, purposeful rational behavior is understood as "the expectation of a certain behavior of objects of the external world and other people and the use of this expectation as "conditions" and "means" to achieve one's rationally set and thought-out goal" 9 . A goal-oriented person is free to choose both goals and means to achieve them.

Secondly, the behavior of homo oeconomicus must be utilitarian. In other words, his actions should be subordinated to the task of maximizing pleasure, utility. It is utility that becomes the basis of human happiness 10 . Two forms of utilitarianism should be distinguished - simple and complex. In the first case, a person is simply aimed at the task of maximizing his pleasure, while in the second he associates the amount of utility received with his own activity. It is the awareness of the connection between utility and activity that characterizes the ideal participant in market exchange.

Thirdly, he must feel empathy in relation to other participants in the transaction, i.e. he must be able to put himself in their place and look at the ongoing exchange from their point of view. “Since no direct observation is able to acquaint us with what other people feel, we cannot form an idea of their feelings otherwise than by imagining ourselves in their position” 11. Moreover, empathy is distinguished from emotionally colored sympathy by impartiality and neutrality: we must be able to put ourselves in the place of a person who can be personally unpleasant.

Fourth, between the participants in transactions in the market there must be confidence. No, even the most elementary transaction in the market can be carried out without at least a minimum of trust between its participants. It is in the existence of trust that the prerequisite for the predictability of the behavior of the counterparty, the formation of more or less stable expectations regarding the situation on the market lies. "I trust another if I think that he will not deceive my expectations about his intentions and about the conditions of the transaction being made." For example, any transaction with prepayment 12 is built on the basis of the buyer's confidence in the seller's fulfillment of its obligations after making advance payments to them. Without mutual trust, the deal will seem irrational and never get done.

Finally, participants in transactions in the market must have the ability to interpretative rationality, which is a kind of synthesis of the above four elements. Interpretative rationality includes, on the one hand, the ability of an individual to form correct expectations about the actions of another, that is, to correctly interpret the intentions and plans of the latter. At the same time, a symmetrical requirement is presented to the individual: to make it easier for others to understand his own intentions and actions 13 . Why is interpretative rationality important in the market? Without it, it is impossible for exchange participants to find the optimal solution in situations like the "prisoners' dilemma" that always arise when transactions involve the production and distribution of public goods.

The prerequisites for interpretive rationality are the existence focal points, options spontaneously chosen by all individuals, and agreements, well-known variants of behavior of individuals1 4 . Spontaneous selection of the same options from a certain set of alternatives is possible only within socially homogeneous groups or within the same culture. Indeed, focal points are associated with the presence of common reference points in actions and assessments, common associations. An example of a focal point is a common meeting point in a city or building. As far as agreements are concerned, generally accepted behavior in any given situation. The presence of agreements allows individuals to behave as others expect it to, and vice versa. The agreement regulates, for example, the communication of random fellow travelers on the train. It determines the topics of conversation, the degree of openness allowed, the degree of respect for the interests of another (in matters of noise, light), etc.

focal point- spontaneously chosen everyone individuals who find themselves in a given situation have a variant of behavior.

Agreement– regularity R in the behavior of a group of individuals P in a frequently occurring situation S if the following six conditions are met:

1) everyone obeys R;

2) everyone thinks that everyone else obeys R;

3) belief that others are following orders R, is for the individual the main incentive to fulfill it too;

4) everyone prefers full compliance R partial compliance;

5) R is not the only regularity in behavior that satisfies conditions 4 and 5;

6) conditions from the 1st to the 5th are well-known (common knowledge).

Findings. Summing up the discussion of the limits of applicability of neoclassical market models, let us recall the main ones. The market structure is close to perfectly competitive; pricing in the market is either centralized or local in nature, because only in this case all information circulates freely on the market and it is available to all participants in transactions; all participants in transactions are close in their behavior to homo oeconomicus. Drawing a conclusion about a significant reduction in the scope of applicability of neoclassical models, it is easy to notice another, more serious problem. The above requirements contradict each other. Thus, the local market model contradicts the requirement of a sufficiently large, potentially unlimited number of participants in transactions (the condition of perfect competition). If we take the case of centralized pricing, then it undermines mutual trust between the participants in the transaction themselves. The main thing here is not trust at the "horizontal" level, but "vertical" trust in the auctioneer, in whatever form he may exist 15 . Further, the requirement of minimum dependence of participants in transactions contradicts the norm of empathy and interpretive rationality: by taking the point of view of the counterparty, we partially give up our autonomy and self-sufficiency in decision-making. This series of contradictions can be continued. Consequently, interest in such factors as the organization of the market, the behavior of people in the market, not only limits the scope of applicability of the neoclassical model, but also calls it into question. There is a need for a new theory that can not only explain the existence of these limitations, but also take them into account when building a market model.

Lecture number 2. INSTITUTIONAL THEORY: "OLD" AND "NEW" INSTITUTIONALISM

Institutionalism is a theory focused on building a market model, taking into account these limitations. As the name suggests, this theory focuses on institutions, "man-made frameworks that structure political, economic, and social interactions" 16 . Before proceeding to the actual discussion of the postulates of institutional theory, we need to determine the criteria by which we will evaluate the degree of its novelty in relation to the neoclassical approach. Is it really about a new theory, or are we dealing with a modified version of neoclassicism, an expansion of the neoclassical model into a new area of analysis, institutions?

Neoclassical paradigm

Let us use the scheme of epistemological* analysis of the theory proposed by Imre Lakatos (Fig. 2.1) 17 . According to him, any theory includes two components - "hard core" (hard core) and "protective shell" (protective belt). The statements that make up the "hard core" of the theory must remain unchanged in the course of any modifications and refinements that accompany the development of the theory. They form a research paradigm, those principles that any researcher who consistently applies the theory cannot refuse, no matter how sharp the criticism of opponents may be. On the contrary, the statements that make up the "protective shell" of the theory are subject to constant adjustments as the theory develops. The theory is criticized, new elements are included in its subject of study - all these processes contribute to the constant change of the "protective shell".

Rice. 2.1

*Epistemology is a theory of knowledge.

The following three statements form the "hard core" of neoclassicism - no neoclassical model can be built without them.

The "hard core" of neoclassicism:

Equilibrium in the market always exists, it is unique and coincides with the Pareto optimum (Walras-Arrow-Debre 18 model);

Individuals choose rationally (rational choice model);

The preferences of individuals are stable and are exogenous in nature, that is, they are not influenced by external factors.

The "protective shell" of neoclassicism also includes three elements.

"Protective shell" of neoclassicism:

Private ownership of resources is an absolute precondition for exchange in the market;

There are no costs for obtaining information, and individuals have all the information about the transaction;

The limits of economic exchange are determined on the basis of the principle of diminishing utility, taking into account the initial distribution of resources between the participants in the interaction 19 . There are no exchange costs, and the only type of cost that is considered in theory is production costs.

2.2. "Tree" of institutionalism

Now we can turn directly to the analysis of the directions of institutional analysis. Let's depict the institutional theory in the form of a tree that grows from two roots - "old" institutionalism and neoclassicism (Fig. 2.2).

Let's start with the roots that feed the "tree" of institutionalism. Let us add only two points to what has already been said about neoclassical theory. The first concerns the methodology of analysis, methodological individualism. It consists in explaining institutions in terms of the interests and behavior of individuals who use them to coordinate their actions. It is the individual who becomes the starting point in the analysis of institutions. For example, the characteristics of the state are derived from the interests and behavior of its citizens. A continuation of the principle of methodological individualism was a special view of the neoclassical on the process of the emergence of institutions, the concept spontaneous evolution of institutions. This concept proceeds from the assumption that institutions arise as a result of people's actions, but not necessarily as a result of their desires, i.e. spontaneously. According to F. Hayek, analysis should be aimed at explaining "unplanned results of people's conscious activity" 20 .

Rice. 2.2

Similarly, the "old" institutionalism uses the methodology holism, in which the starting point in the analysis is not individuals, but institutions. In other words, the characteristics of individuals are inferred from the characteristics of institutions, and not vice versa. The institutions themselves are explained through the functions they perform in the reproduction of the system of relations at the macro level 21 . It is no longer citizens who "deserve" their government, but the government contributes to the formation of a certain type of citizen. Further, the concept of spontaneous evolution is opposed by the thesis institutional determinism:institutions are considered as the main obstacle to spontaneity of development, "old" institutionalists see them as an important stabilizing factor. Institutions are "the result of processes that took place in the past, they are adapted to the circumstances of the past [and therefore are] a factor of social inertia, psychological inertia" 22 . Thus, institutions set the "framework" for all subsequent development.

Methodological individualism - explanation of institutions through the need of individuals for the existence of a framework that structures their interactions in various areas. Individuals are primary, institutions are secondary.

Holism- explanation of the behavior and interests of individuals through the characteristics of institutions that predetermine their interactions. Institutions are primary, individuals are secondary.

2.3. "Old" institutionalism

To give a more complete picture of the "old" institutionalism, let's turn to the most prominent representatives of this scientific direction: K. Marx, T. Veblen, K. Polanyi and J.K. Galbraith 23 . Marx in Capital (1867) widely used both the method of holism and the thesis of institutional determinism. His theory of the factory, as well as the theory of primitive accumulation of capital, are most illustrative from this point of view. In his analysis of the emergence of machine production, Marx draws attention to the influence that organizational forms have on the process of production and exchange. The system of relations between the capitalist and the wage-worker is determined by the organizational form that the division of labor 24 takes: natural division of labor –> cooperation –> manufacture and production of absolute surplus value –> appearance of a partial worker –> appearance of machines –> factory –> production of relative surplus value.

Similarly, in the analysis of initial accumulation, one can see an institutional approach 25 , or rather, one of the variants of institutional determinism, legal determinism. It was with the adoption of a number of legislative acts - the acts of Kings Henry VII and VIII, Charles I on the usurpation of public and church lands, laws against vagrancy, laws against wage increases - that the wage labor market and the capitalist hiring system began to take shape. The same idea is developed by Carl Polanyi, who argues that it was the intervention of the state that underlay the formation of national (as opposed to local) resource markets and the labor market. "The internal market was created everywhere in Western Europe through state intervention", its emergence was not the result of the natural evolution of local markets 26 . This conclusion is especially interesting in connection with our own analysis, which showed a deep gulf separating the local market and the market with centralized pricing 27 .

T. Veblen in his "Theory of the Leisure Class" (1899) gives an example of the application of the methodology of holism to the analysis of the role of habits. Habits are one of the institutions that set the framework for the behavior of individuals in the market, in the political sphere, in the family. So, the behavior of modern people is derived by Veblen from two very ancient habits, which he calls the instinct of competition (the desire to get ahead of others, to stand out from the general background) and the instinct of mastery (a predisposition to conscientious and efficient work). The instinct of competition underlies, according to this author, the basis of property and competition in the market 28 . The same instinct explains the so-called "conspicuous consumption", when an individual is guided in his choice not by maximizing his own utility, but by maximizing his prestige in the eyes of others. For example, the choice of a car is often subject to the following logic: the consumer pays attention not so much to the price and technical characteristics, but to the prestige that provides the possession of a certain brand of car.

Finally, J.K. Galbraith and his theory of technostructure, set out in the books The New Industrial Society (1967) and Economic Theories and Society's Goals (1973). As in our analysis of the limits of applicability of the neoclassical approach, Galbraith starts with questions of information and its distribution among the participants in the exchange. His main thesis is that in today's market no one has complete information, everyone's knowledge is specialized and partial. The completeness of information is achieved only by combining this partial knowledge within an organization or, as Galbraith calls it, a technostructure 29 . "Power has shifted from individuals to organizations with a group personality" 30 . And then follows an analysis of the influence that the technostructure has on the behavior of individuals, i.e. characteristics of individuals are considered as a function of the institutional environment. For example, consumer demand is derived from the growth interests of corporations that actively use advertising to convince consumers, and not from their exogenous preferences 31 .

Activation and use of mental mechanisms as the essence of Erickson's approach; how to calm the patient, "radiating" approval and support

Interaction analysis in various theoretical approaches

Ticket 25. Preparation for a crime and the limits of criminal liability. Distinguishing preparation for a crime from attempted crime

Ticket 27. The totality of crimes, its types. The order and limits of sentencing for the totality of crimes

Bull H. International Relations Theory: An Example of a Classical Approach

What is the principle of a systematic approach to management?

Darcy's law is valid under the following conditions:

a) the porous medium is fine-grained and the pore channels are rather narrow;

b) filtration rate and pressure gradient are low;

c) change in filtration rate and pressure gradient are small.

With an increase in the velocity of the liquid, Darcy's law is violated due to an increase in pressure losses due to the effects associated with inertial forces: the formation of vortices, zones of flow separation from the surface of particles, hydraulic shock against particles, etc. This so-called upper bound . Darcy's law can also be violated at very low filtration rates in the process of the beginning of fluid movement due to the manifestation of non-Newtonian rheological properties of the fluid and its interaction with the solid skeleton of a porous medium. This is bottom line.

Upper border. The criterion for the upper limit of the validity of Darcy's law is usually a comparison of the Reynolds number Re=war/h with its critical Re cr, after which the linear relationship between head loss and flow is broken. In the expression for the number Re:

w- characteristic flow velocity:

a- characteristic geometric size of the porous medium;

r is the density of the liquid.

There are a number of representations of the Reynolds numbers obtained by various authors with one or another substantiation of the characteristic parameters. Here are some of these dependencies most used in underground hydromechanics:

a) Pavlovsky

Critical Reynolds number Re kr \u003d 7.5-9.

b) Shchelkachev

(1.31)

Critical Reynolds number Re cr =1-12.

c) Millionshchikov

(1.32)

Critical Reynolds number Re kr \u003d 0.022-0.29.

Filtration speed u cr, at which Darcy's law is violated is called critical filtration rate . Violation of the filtration rate does not mean a transition from laminar to turbulent motion, but is caused by the fact that the inertial forces arising in the liquid due to the tortuosity of the channels and the change in the cross-sectional area become at u>u cr comparable to frictional forces.

When processing experimental data to determine the critical speed, dimensionless Darcy parameter:

, (1.33)

representing the ratio of viscous friction forces to the pressure force. In the range of Darcy's law, this parameter is equal to 1 and decreases when the number is exceeded Re critical value.

Bottom line. At very low speeds, with increasing pressure gradient (change in pressure with depth), the increase in filtration rate occurs more rapidly than according to Darcy's law. This phenomenon is explained by the fact that at low velocities, the force interaction between the solid skeleton and the liquid becomes significant due to the formation of anomalous, non-Newtonian systems, e.g. stable colloidal solutions in the form of gelatinous films that block pores and break down at a certain pressure gradient t n, called the initial and depending on the proportion of clay material and the value of the residual water saturation. There are many rheological models for non-Newtonian fluids, the simplest of which is the limit gradient model

(1.34)

1.3.1.4. Filtration laws for Re > Re cr

The reliability of well survey data and the determination of reservoir parameters depend on the accuracy of the filtering law used. In this regard, in the area of violation of the Darcy law, it is necessary to introduce more general, nonlinear filtration laws. These laws are divided into one-member and two-member.

Viktor Kuligin

Let's highlight some of the issues discussed.

c) The problem of the reciprocal effect of the effect on its cause.

Limit of applicability of models

The property of transitivity allows a detailed analysis of the causal chain. It consists in the division of the final chain into simpler causal links. If A, then A → B1, B1 → B2,..., Bn → C. But does a finite causal chain have the property of infinite divisibility? Can the number of links of a finite chain N tend to infinity?

Fig. 2. Structural (dialectical) model of causality

Let's return to the structural model. In its structure and meaning, it is in excellent agreement with the first law of dialectics - the law of unity and struggle of opposites, if interpreted:

- unity - as the existence of objects in their mutual connection (interaction);

- opposites - as mutually exclusive tendencies and characteristics of states due to interaction;

- struggle - as interaction;

- development - as a change in the state of each of the interacting material objects.

Model Qualities

Let us now consider what qualities the structural model of causality holds in itself. Let us note the following among them: objectivity, universality, consistency, unambiguity.

The objectivity of causality is manifested in the fact that the interaction acts as an objective cause, in relation to which the interacting objects are equal. There is no room for anthropomorphic interpretation here. Universality is due to the fact that the basis of causality is always interaction. Causality is universal, just as interaction itself is universal. Consistency is due to the fact that, although cause and effect (interaction and change of states) coincide in time, they reflect different aspects of cause-and-effect relationships. Interaction involves a spatial connection of objects, a change in state - a connection of the states of each of the interacting objects in time.

In addition, the structural model establishes an unambiguous relationship in cause-and-effect relationships, regardless of the method of mathematical description of the interaction. Moreover, the structural model, being objective and universal, does not prescribe natural science restrictions on the nature of interactions. Within the framework of this model, both instantaneous long-range or short-range interaction and interaction with any finite velocities are valid. The appearance of such a limitation in the definition of cause-and-effect relationships would be a typical metaphysical dogma, once and for all postulating the nature of the interaction of any systems, imposing a natural philosophical framework on the part of philosophy on physics and other sciences, or limiting the limits of applicability of the model to such an extent that the benefit of such a model would be very modest.

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