Vladimir Aristov. Multidimensional space and time: a relational statistical approach and nonequilibrium phenomena kinetic description

Vladimir Vladimirovich Aristov—Doctor of Physical and Mathematical Sciences, Chief Researcher of the Dorodnicyn Computing Center of Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, professor of the Department of Mathematical Modeling of Complex Systems and Optimization of the Moscow Institute of Physics and Technology, as well as the Department of Higher Mathematics, Faculty of Cybernetics, MIREA. Research interests: relational statistical models of space-time, methods for solving the Boltzmann equation and other kinetic equations, the dilute gas dynamics, the nonequilibrium open systems theory. Author and co-author of many articles and several monographs, in particular, V.V. Aristov. Direct methods for solving the Boltzmann equation and study of nonequilibrium flows. Dordrecht. Kluwer Academic Publishers. 2001; В.В. Aristov, S.A. Zabelok, A.A. Frolova. Nonequilibrium structures simulation by kinetic methods. M.: Physmathkniga, 2017.

He is a writer also, author of 12 published poetry books, two novels, plays, short stories, essays, translations. Literary Prizes named after Andrei Bely, Aleksei Kruchyonykh, “Razlichie”.

Modern physics problems determine the search and construction of an extended space and time theoretical model. Wherein, physics considers time inextricably linked with four-dimensional space-time. How do you consider the time phenomenon?

We are following a tradition already established, according to which time and space are not independent substances or some a priori abstract entities but are constructed on the basis of the greater generality concepts. Moreover, they are interconnected because of the creation of theoretical statistical models of fundamental measuring instruments: clocks and rulers. Thus, a more general description than the usual one, defining various geometric interpretations, is possible. The relational statistical approach that we follow defines the direct links between space and the system mass configuration, as well as time and space. In a certain sense, this is an attempt to develop the special and general relativity theoretical views. Yu.S. Vladimirov and his followers develop a relational statistical approach that is close in its physical ideas.

From the time theory development viewpoint, an important aspect of our approach is that time is a multidimensional concept, it is described in terms of N parameters (more precisely 3N, if three-dimensionality of space is accepted)—the spatial coordinates of the system under consideration with N elements (particles). The transition to the regular one-dimensional time occurs as a statistical averaging result.

In the conventional physical description, there is no concept of time as a state (a point on the numerical axis of time does not bear any indication of this particular moment’s accentuality—shifts along the time axis do not affect the processes’ nature). The main interest, as a rule, is focused on determining time intervals; therefore, the physics dynamic equations are written in terms of time derivatives.

In our description, we introduce a new concept of time as a state, which can lead to the time irreversibility interpretation. Such a transition is permissible for a time as a flow and, accordingly, for a time interval that is specified by a metric associated with the mean-square change in all spatial coordinates of the system. The use of link quadratic forms is due to several reasons. Namely, by directly obtaining the kinetic energy conservation relation and then (considering the invariance of model time for shear transformations) the momentum conservation. Besides, it is possible to obtain analogues of the Riemannian geometry relations, which defines equations for the metric that are similar to those adopted in general relativity (GR), and corresponding generalizations.

This is where the “the time moment uniqueness” idea formalization appears, intuitively correlated with the impressions’ multiplicity. All coordinates attribution to one moment should occur according to some rule, in the physical description a single picture is set by a light signal when using a hypothetical ideal camera.

In our conception, movement is represented not only by a single “particle” but by the assembly, the sum of all moving parts of the system, for which time is introduced. The “sum” word involves both a general and a quantitative meaning, which is deciphered in a particular equation. Such watches reproduce well-known time properties.

An individual particle can stop but there will always be moving particles in a sufficiently large particles’ system, therefore the time flow continuity property is satisfied. Unidirectionality is explained by the model statistical character. The uniformity is determined by the model statistical nature also: individual particles can move randomly, but when averaging, random errors are compensated, and the assembly movement turns out to be uniform. To detect the relationships with known physical relations according to the correspondence principle the constructions begin mainly by reproducing the time metric properties.

They talk in traditional physics about the processes’ irreversibility (or reversibility), and the time irreversibility concept (the time arrow) is discussed also. What is your proposed theory development for simulating irreversible time?

We directly study the time irreversibility concept since this model allows you to constructively set a moment in time that has physical meaning. It was shown already as a set of spatial radii-vectors related to one “light section” of the “camera” location.

We note that “returns in time” are possible in GR, closed time-like curves are known for some solutions, but without determining the moment, this is a formal concept. We have to introduce various restrictions to avoid known contradictions with reality when returning in time, such as the “grandfather’s paradox.”

The “return of time” in the relational statistical model makes sense as conversion to zero of a certain metric, which is associated with the conditions for the coincidence of the elements’ coordinates set at two points in time. Fulfilment of them would mean all elements’ return to the previous positions (which is unlikely). This model confirms Hawking’s hypothesis of “chronology protection conjecture”.

The irreversible time interval is introduced, its difference is found from the “unidirectional” time interval, measured by an ordinary clock. From a mathematical point of view, the following formal generalization can be noted in comparison with the previous model: the use of three rather than two points in time to formulate irreversibility.

What new fundamental instruments for measuring time do you propose?

With the use of more complex multidimensional geometry, the opportunity is opened for introducing new fundamental theoretical and then acceptable materialized instruments’ models.

New concepts are introduced for an accurate description—“photography” and “temporometer” (as a theoretical generalization of clocks). The temporometer is equipped with an “ideal camera” that allows you to find the spatial positions of all particles of the system in question. A set of such spatial coordinates of the particles in a separate ideal photograph determines the system state, which corresponds to the “point in time” concept.

The temporometer implies the storage device presence also that allows you to save any number of photos, and the photo combines a visual image and coordinate values’ set for a given moment. The time interval must be expressed directly through spatial intervals. Such a connection is contained in any ordinary clock: the arrow shows exactly the spatial changes between two dimensions. But this property, as a rule, is “untrue,” since the clock hand shows precisely “time”—this is how they indicate implicitly a special distinguished movement character. We are trying to connect these concepts that are so different in their common notions. This reveals the deep connection between space and time, which is the theory basis.

Since space and time are now constructive entities, it is permissible to construct other models of fundamental instruments. The previously noted models are “tuned” to the connection with existing relations and equations according to the correspondence principle. But new models in the case of construction will allow other patterns. Thus, reality can be described more fully. For example, we can consider the sum, where, in contrast to the statistical sum, weights appear, for example, equal to the radius vector modulus reciprocal value, thereby reducing the distant particles’ contribution to the sum. It is possible then to obtain new equations that require other instruments for compiling new sums in statistical time.

This approach means a more complete and detailed description that overcomes, in a sense, the quantum mechanics indeterminism. Here essentially an answer is offered to the question: are hidden parameters contained in the existing physical theory (primarily in quantum mechanics)? There are no hidden parameters in the traditional physical theory. But we can introduce “open parameters” for a more detailed description by introducing new space and time models, which is associated with the construction of new clocks’ statistical models.

How does the new concept fit well-known proven theories?

An equation is postulated where an infinitesimal time interval is associated with the quadratic average of the spatial intervals of the particles in the system (the corresponding dimensional factor appears in the equation, which turns out to be equal to the reciprocal value of the speed of light in a vacuum). Based on the defined equation, the known kinematic and dynamic relations are derived: analogues of the Galilean transformations, permanence relations, and motion equations are obtained. When generalizing, Lorentz transformations and dynamic equations of special relativity (SR) are derived.

It is required to introduce a relational space model to set the equation for the force. A connection is established between the distance concept and the particles configuration represented by a ruler. The space metric properties are always determined by a measuring device – a scale ruler consisting of atoms. But, like a clock, not every atoms’ configuration is suitable to become a medium for the ruler “manufacturing”, but the one satisfying a symmetrical mutual disposition only. In fact, there is the averaging when obtaining a homogeneous medium from particles.

In the formalization, we present a geometric scheme that differs from the traditional one: through two points (associated with the particles) one can draw a non-single line. A line segment is understood as a minimum length line segment, where the distance is determined by counting the particles through which the line passes. The discrete non-Euclidean geometry formalism produces Euclidean geometry at large distances, where the line becomes the only one in a certain sense—this is the transition to Euclidean geometry.

The relational concept allows us to approach the motion indeterminism problem at small distances (“measurable by the atoms’ number of the line”) and the quantum phenomena description. It is possible to derive the Schrödinger equation and Heisenberg’s uncertainty principle analogues.

Two sums of many dimensionless quantities are compared in joint consideration of the space and time model. These sums are compared based on the probability theory limit theorems and it is possible to derive an expression for the Newtonian gravitational potential. The connection is made here between the physics axioms and the mathematics axioms based on the “number-particle-space-time” scheme.

Thus, generalizations are obtained for quantum and gravitational phenomena joint description. It relates to the multidimensional nature of this model. The ability to describe quantum effects at microscopic scales. And gravitational ones on a macroscopic scale.

“Key experiment” may be the discovery of facts that were not predicted by previous theories. In the proposed concept, it is possible to obtain conformity with equations of GR. Relational statistical theory is non-field, so the goal is not to get the Einstein equations for the gravitational field here. But the constructed space and time allow us to describe the space-time curvature. One can obtain the Schwarzschild metric analogues directly and all known experimental effects accordingly. But in the next approximation (the gravitational radius relation to the distance to the body under study), we can expect deviations from the GR predictions. Although to detect this, the current experiments’ accuracy should be improved significantly.

There are several other hypothetical effects that, in principle, are subject to verification. They have a small statistical fluctuation character and are beyond the modern experiments’ accuracy. These are differences in the masses of the same types of elementary particles and the principle of equivalence violation.

How does time behave at particle speeds close to the speed of light?

The possibility of correlating the clock readings with the corresponding sum is related to the proximity of the mathematical expectation with the tests’ average, which is expressed in the law of large numbers, and the necessary limit theorem must be satisfied to ensure the similarity of these values. Under this condition, the time interval for the physical clock differs from the model statistical clock by a small amount associated with the N number of particles in the system. Other statistical laws associated with the violation of the law of large numbers and limit theorems in the presence of individual terms that significantly exceed the value of other terms in the sum under consideration are admissible. It leads to the kinematics and dynamics equations’ change.

Then the difference from the traditional description of motion at high speeds is noticeable. Such deviations can occur when the particle velocity differs from the speed of light by a relative order of magnitude equal to (N)^(-1/2), where N = 1080 is the Eddington number (the number of nucleons in the Metagalaxy). We understand clearly that the achievement of such speeds and energies is far beyond the modern accelerators’ capabilities.

The maximum speed is limited to a value lower than the speed of light, although it differs little from it. Consequently, both momentum and energy are limited. This resolves a paradox that is rarely paid attention to: in the SR, an unlimited increase in the particle energy is possible when its speed tends to the speed of light.

What are the possible spacetime generalizations in a relational concept?

The introduced time is associated with the particle movements multiplicity. Therefore, one can consider many separate times also (the usual one-dimensional time is obtained as a result of a single averaging). Each separate system has its temporal characteristics. Global or world time (the course of which is described by an ordinary clock) can be considered as the time of an extremely large system in a hierarchy of systems nested into each other. It is proposed to solve the problem of introducing the system internal time: biological, geological, economic, historical time can be set by the movement of the elements that make up this system.

How can the “returns in time” problems be interpreted?

It is easy to imagine this clearly for a system with a small number of particles: the return of particles to their previous spatial positions can be interpreted as “returning to the same moment in time.” It is clear that for a large number of elements such a task implementation is significantly complicated.

How can the psychological effects of time perception be interpreted?

It is known that biological processes slow down with age, which means that for an “objective” time unit, measured by ordinary hours, the average motions will be smaller than before. The internal time interval will also be less, in other words, per unit of internal time that determines the internal sense of self, there is a large amount of “objective time” according to ordinary hours now, which can be interpreted as “time acceleration.” From a psychological point of view, one can imagine the “psychological time” interval as the occurring events average using the same spatial (in the conventional sense) measure. The event’s novelty decreases with age. This can be interpreted as conditional movements “slowdown,” which are included in the amount for the model psychological time. The internal psychological time unit will correspond to the greater time in physical hours, which can be interpreted as the ordinary time acceleration also.

From a relational point of view, ideas about space and time grow out of simple notions (and the concepts of time, space, a particle, a number turn out to be  interpenetrating at this level).

The time points sequence can be correlated operationally (a rough analogy is pressing the stopclock button) with the set theory axioms, in particular, with the axioms of order.

The relational model constructs a physical discrete metric geometry. The topological dimension is not assumed: analogues of planes, volumes, etc. are built inductively in this case. We can hypothesize that the space three-dimensionality arises as to the minimum necessary description of the relational ratio of bodies. Three relationships: “I” (subject)—“not me” (object)—“rest of the world”—“I” define the space three-dimensionality idea. Then we need to construct the space parametric and topological dimensionality with possible generalizations. An important conclusion is that space three-dimensionality is not determined by physical interaction. We have shown that the dependence of the potential in the first degree is related to the fact that the distance is linearly dependent on the mass, which in turn correlates with the relational design of the rulers. That is the potential type does not depend on space dimensionality.

Here you can recall the mechanics of time and the “temporal field” of the science fiction writer A. Azimov.

Do you mean Asimov’s “The End of Eternity”? But the “field” concept does not correspond to relational views. It is convenient to address the global time by association with Leibniz’s views, in which relational representations are expressed (though without any mathematical constructions). It is worth mentioning important general concepts here. A differential can act an infinitesimal time interval as the Leibniz monad (which does not have windows). The time interval infinitesimal value “opens windows to the whole world” in our approach—it is associated with all spatial intervals—thereby establishing a direct relationship between micro- and macro-levels. Spatial cosmos is reflected by a “stopped instant” (we recall once again that in modern physics, an instant is an extremely mathematically poor concept represented by a point on the numerical axis). One can relate this to the generalized Mach’s principle who was another relationalist.

The time globality appears in the elapsed times’ sum, where any past instant is understood as its world image: by dividing the time flow, seeing each part separately, we can connect its various parts.

While preserving the entire memory (“photos” “snapshots”), we can not only return moments (“cinematic analogy”) but also create more complex images. There is a hint of the possibility of returns in time (the light imprint, as in a backward moving film, convinced in it), of the real element-wise coincidence of the previous form—the light gluing of crashed, destroyed things.

You considered problems with nonequilibrium distribution functions at the boundaries for the Boltzmann kinetic equation. What consequences can be drawn from this?

This made it possible to simulate nonequilibrium supersonic and subsonic flows and transport processes for such flows. The possibility of the so-called anomalous transfer is determined, in which the heat flux and temperature gradient signs, as well as the nonequilibrium stress tensor and velocity gradients corresponding components, coincide. This denotes that modes with anomalous heat transfer in spatial zones are revealed, which means that the heat flows from cold regions to hot regions (heat transfer occurs traditionally in the formulation with equilibrium conditions).

The nonequilibrium processes described by the kinetic theory and nonequilibrium thermodynamics methods with the usual assumption of local equilibrium and the alleged relationships between fluxes and gradients can lead to different results. The transition to a macroscopic description with known transfer relations in kinetic theory is based on the Chapman-Enskog method for small Knudsen numbers. When the Knudsen number is of the order of unity, as in the problems we are studying, the laws of transfer are admissible not only quantitatively, but qualitatively different also.

For example, we study the nonequilibrium boundary conditions effect on the temperature and heat flux behaviour in the problem of heat transfer between two surfaces in the absence of convection, i.e. with zero mass flow. The problem is studied at various Knudsen numbers with the identification of anomalous heat transfer zones.

An experimental verification of such effects, which seems important, faces various technical problems, and above all, the creation of nonequilibrium distributions that are stationary retained in the spatial region. In principle, it is possible to create nonequilibrium at the free boundary now using the so-called optical arrays or magnetic traps. One can use the molecule groups’ emission at different speeds, for example, for a solid surface. The distribution function determination in a nonequilibrium flow can be based on diagnostics using an electron beam.

What methods exist for simulation of biological structures as open nonequilibrium systems?

There are new aims to study nonequilibrium boundary conditions at the boundary, and the nonequilibrium state (distribution) is gradually transformed into an equilibrium state downstream. The spatial nonuniformity part has a scale depending on the substance transfer rate in an open system and the metabolism characteristic time.

In the proposed approximation, the molecular motion internal energy is much less than the translation energy; in other, more “realistic” terms: the kinetic energy of the average blood speed is significantly higher than the energy of the random movement of particles in the blood.

Such a problem of relaxation in space simulates a living system since it compares the thermodynamic nonequilibrium and nonuniform parts. The entropy flux Sx (calculated by the well-known statistical formula) in the system under study increases downstream, which corresponds to the general ideas of E. Schrödinger that the living system “feeds” on negentropy (the negentropy flux Hx = – Sx).

The results are compared with empirical data, in particular for mammals (the greater is the animals’ size, the lower is the metabolism specific energy). It is reproduced in the proposed kinetic model since the nonequilibrium part dimensions turn out to be larger in the system where the reaction rate is lower, or in terms of the kinetic approach, the greater the relaxation time of the typical interaction between the molecules. The approach is used also to determine the characteristics of the individual organ of a living system, namely the green leaf.

The aging problems are considered as the open nonequilibrium system degradation. The analogy is related to structure: for a closed system, there is a tendency to the equilibrium of the same molecules’ structure, in an open system, there is a transition to the equilibrium of particles, the configuration of which changes due to metabolism (structural distribution function).

Therefore, two essentially different time scales are distinguished, the ratio of which is approximately constant for different animal species. Assuming these two timescales exist, the kinetic equation splits into two equations that describe the metabolic (stationary) and “degradation” (non-stationary) parts of the process.

Interview: Ivan Stepanyan

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