Vol.3, No 12, 2002 pp. 469-474
LETTERS TO EDITOR
ASYMPTOTIC CORRESPONDENCE OF PHYSICAL
THEORIES
Igor Adrianov
INTRODUCTION
Almost any physical theories, formulated in mathematical terms in a
general way, are extremely complicate. Thus both in creating a theory and
in its further development, of paramount importance are the simplest limiting
cases that admit an analytical solution. Commonly, for them that the number
of equations is decreased, the equations' order be-comes smaller (is reduced),
the non-linear equations are replaced by the linear ones, the original
system is subjected to a kind of averaging, etc. Behind the above idealizations,
however diverse they may seem, lies a high degree of symmetry inherent
in a mathematical model of a phenomenon at issue in its limiting situation.
An asymptotic approach to a com-plex, "insoluble" problem consists basically
of treating an original insufficiently symmetric - system as approximating
to a given symmetric one. It is basically important that deter-mining corrections
allow for deviations from a limiting cases is much simpler than a direct
study of the original system. At first sight, the potentialities of such
an approach are limited by a narrow range of system parameter variations.
Experience gained in the study of various physical problems has shown,
however, that in the case of system parameters varying con-siderably and
the system itself departing from one limiting symmetric pattern, another
lim-iting system, often with a less pronounced symmetry generally exists
and a perturbed solu-tion can now be formed for the latter one. This enables
the system's behavior defined over the entire range of parameter variations
using a finite number of limiting cases. Such an approach uses the most
of the physical intuition and contributes to its further development, while
leading to the forming of new physical concepts. Thus the boundary layer,
an impor-tant concept in the fluid mechanics, is of pronounced asymptotic
nature and is related to the localization at the boundaries of a streamlined
body in the zone where fluid viscosity cannot be neglected. In the mechanics
of deformable rigid body and in the electricity theory, simi-lar phenomena
are termed the "edge effect" and "skin effect" respectively. These questions
were clarified in [1-3].
ASYMPTOTIC APPROACHES AND PROGRESS IN MODERN PHYSICS
Frequently the progress in modern physics appears to be closely connected
with typical (small or large) asymptotic parameters. In particular, smallness
of famous "thin structure constant" a = e2/h (e is electron charge, h is
Plank constant, c is velocity of light) gives us possibility to investigate
the photons' and electrons' interaction in the quantum electrody-namics
with high precision. This non-dimensional parameter defines the intensity
of elec-tromagnetic interactions. All fundamental results of quantum electrodynamics,
describing experimental data with surprising accuracy, have been obtained
due to the use of perturba-tion theory [4]. In this theory the solutions
are represented by expansions in powers of a. Similar parameters for the
case of strong interaction particles - hadrons (for example, neu-trons
and protons) exceed an in value multifold. This is the main cause of essential
difficul-ties on the way of development of strong interaction theory. Only
the discovery of quantum quark structure of hadrons and of the phenomenon
of "asymptotic freedom" [5] (the essence of this phenomena is in relaxation
of interaction between quarks and interconnective gluons at the short distances)
immediately changed situation and gave birth of new theory of strong interaction
- quantum chromodynamics. Often the possibilities and ways of using small
or large parameters, natural for some physical theory, may be completely
realized only in a long run. For example, dependence of rigid characteristics
in some point of elastic body upon its position (nonhomogeneity) or chosen
direction (anisotropy) arises serious difficul-ties for investigators.
Many methods of solution of isotropic media problems, based on spe-cific
symmetries of such media, can't be used for anisotropic or nonhomogeneous
case. But in the case of strong anisotropy (or nonhomogeneity) one can
invent the set of small or large parameters (as a rule, ratio of rigidity
in the various directions)and, using suitable as-ymptotic methods, achieve
the fast development of such media theory. Sometimes the sim-plified equations
of anisotropic media appear to be less - complicated than ones of isotropic
and homogeneous model [6]. Let us consider another example. It is known
that the difficul-ties of analysis of molecular systems grow fast with
increase of size and mass of molecules. Meantime, in the case of macromolecules
of polymers, there are natural small and large parameters. The first evident
large parameter for such systems is the number of atoms in the chain, N.
This parameter allows to investigate even the single polymer molecule as
macro-scopic system, to apply effective averaging procedure commonly used
in statistical physics. One of the main problems of modern physics of polymers
- study of asymptotic behavior of polymer systems for N ? ?. In particular,
such fundamental characteristic as average size of polymer ball during
dissolving or melting of polymer, b is defined by relations r ~ N ?, where
exponent ? depends upon physical conditions of polymer system [7]. Furthemore,
polymer systems possess a set of small parameters, subjected to the interaction
hierarchy. Covalent interaction (canonical bond) of atoms along the chain
is stronger than any other ("physical") kind of interaction. It gives us
possibility to assume atom sequence along the chain to be fixed. Intensity
of physical interactions of other types also differs significantly. The
simplest of asymptotics corresponds to neglecting of all physical interactions
(assum-ing the lengths of bonds fixed). Next, anyone can take into account
the physical interaction between links of the polymer chain, depicting
chain resistance to bending and torsion (as earlier, lengths of bonds are
supposed to be fixed). At last, interaction between spatially adjacent
(but not neighboring along the chain!) link of twisted polymer chain may
be taken into account.
ASYMPTOTIC CORRESPONDENCE
Of no less importance is the fact the asymptotic method assists in
relating different physical theories with one another. Albert Einstein
would point out that "the happiest lot of a physical theory is to serve
as a basis for a more general theory while remaining a limiting case thereof'.
Naturally, most impressive and rich in content examples of the asymptotic
correspondence appear in revealing the relations among the fundamental
physical theories. Each new theory, brought in by advancement of science,
used to be considered as the negation of the preceding one, i.e. the incompatibility
of the old and the new ideas and concepts, that had come to replace them,
was pushed into the foreground. Only the formulation of Bohr's correspondence
principle and creation of quantum me-chanics brought the successiveness
of physical theories under thorough investigation of physicists and philosophers.
Despite the variety of viewpoints, even of alternative ones, upon the correlation
of successive physical theories, the mathematical link between these theories
may be easily revealed. Such link may be expressed in the terms of asymptotic
correspondence, which becomes apparent in various, very often in implicit
forms. There are, so to speak, various types of 'limit transitions' from
a new theory to the old one, as a rule, for zero or infinite value of some
parameters or variables. New theory may be con-sidered as generalization
of the previous one (one can recall Einstein words cited in), but this
generalization is not only qualitative, but also a quantitative one. That's
why new theory includes possibilities totally unpredictable before. Often
such possibilities become explicitly apparent in the contrarily limiting
cases, when parameters, presumed small, become large and vice versa. The
effects, formally principal, become negligible, and new contents of physical
theory shows out in explicit form. We'll try to retrace this correspon-dence
for various physical theories.
FROM ARISTOTLE TO NEWTON
At first, in this context let's try to discuss the transition from
Aristotelian forced mo-tion theory to Newton's mechanics, which is a good
example of radical change of scien-tific conceptions, views and methods
predominating during long time. Meantime even in such a case the asymptotic
relation comes to light showing the applicability of Aristotle's model
for high friction resistant motion. It seems not so surprising because
Aristotles' reasoning had relied on the intuitive representations following
from everyday observation of moving objects in the restricted range of
external conditions and certainly containing a seed of truth [8]. The investigations
of psychologists confirm that even some our contem-poraries, being not
acquinted enough with modern theory, easily come to Aristotle's kind of
conclusions and explanations [9,10]. There are conceptions of force as
a cause of mo-tion, of arrest of movement due to running out of driving
force - "impetus", of vertical downfall of body thrown from horizontally
moving object, at last - of different downfall time of the bodies with
different weight. In this investigations the surprising similarity between
views of antique or medieval philosophers and our contemporaries' ones
was found. As a rule, in this investigations the special emphasis is laid
on the unconsistency of Aristotelian views with Newton mechanics. Meantime
in the field of usual human ex-perience, namely under the earth conditions,
the endurance of these representations may be explained from Newton's mechanics
point of view, namely, by asymptotic relation mentioned above. This relation
may be put up despite the deepest ideological distinctions between old
and new theories and essential contradictions in philosophical concepts
which it originate from. To confirm above mentioned let us consider the
motion of body subjected to the constant force F through the media with
friction coefficient h. Aristotle had not considered the friction force
itself; he considered it as natural and unremovable attribute of the motion.
He had not also formulated the motion law in the mathematical language
also. Aristotle's "motion law" (in the case of linear dependence of resistance
force upon the velocity) may be written as
F = h v
If the force is constant the velocity will be constant too. The increase
of the force gives rise to the increase of the velocity. These conclusions
in general are in the accor-dance with observations of motion under the
Earthly conditions if the resistance is large enough. According to Newton
the resistance force is the external one and the motion law for a point-line
particle under the conditions mentioned above has a form:
M dv/dt = F ? hv
In the absence of initial velocity one has
V = (F/h)(1 ? exp(?ht/m))
For the case of high frictional resistance, the second (transitional)
term diminishes rapidly, "switching off the force" (presuming long enough
observation period).The re-maining term corresponds to Aristotle's mechanics.
Obviously, observations of motion at small friction levels should show
at once the essential variability of velocity and its slow approach to
the stabilization value. The everyday experience of ancient Greeks appar-ently
lacked such kind of observations. Only 2000 years later Galilei made the
idealized mental experiment and came to understanding of the inertial motion
as one of the main initial concepts of the New Times' physics [8]. From
the physical point of view the Aris-totle's approach preserves its importance
as asymptotics of the motion for the long time span, the larger friction
the earlier time point at which this approximation will be appli-cable.
From mathematical point of view we meet with singular perturbations here:
the velocity v increases not smoothly but infinitely quickly. In such a
case the additional as-ymptotics exists which may be easily found studying
if the behavior of exact solution at small values of exponent power:
V = Ft/m
It describes the nonuniform motion of the body with the constant acceleration
induced by the constant force in the absence of resistance. This solution
is correct for any friction level, if observation time is small enough.
Smaller the friction, wider the solution's appli-cability (and the later
is the time-point of reaching Aristotle's asymptotics). The equation of
motion corresponding to the small times
Mdv/dt = F
represents the mathematical note of the well-known Newton's law. Here
we enter in the field of the conservative, or Hamiltonian systems (the
constancy of the mechanical en-ergy is valid), which allows the forms of
motion absolutely alien for Aristotle's mechan-ics: vibrations, periodic
rotations. Theory of the conservative systems is the most impor-tant part
of the Newton's mechanics because the motion modes described by it (in
par-ticular, periodic and nearly periodic ones) for many physical systems
are the very good approximations to the reality. Nevertheless, Aristotle's
approximation has its own field of applicability, when the friction becomes
large enough, as, for example, for the motion of the polymeric molecules
in the solutions. Such systems are called overdamped and corre-sponding
dynamic processes relaxation, that is transition to equilibrium.
NEWTON'S MECHANICS AND PARTIAL THEORY OF RELATIVITY
The creation of the theory of relativity broke the notions, considered
the only possible ones, deeply implanted and of Newton's mechanics - about,
independence of space and time, about absolute time and so on. But Newton's
mechanics, as it should be, was not rejected by the partial theory of relativity
and had become its asymptotic limit [11]. As-ymptotic relation between
these two theories may be illustrated by the example of the particle with
mass m which 0 undergoes the action of constant force F beginning from
the time t = 0. It can be easily shown, in the partial theory of relativity
the particle's velocity in unmovable coordinate system will be
v = V/C
where V = Ft/m, C is square root from R, R is sum 1 and square of V/c.
The solution in the framework of Newton's mechanics corresponds to
asymptotics for the small times or velocities (V/c << 1). First correction
to this solution is very small:
v = V(1 ? 0.5R)
In the theory of relativity there is an additional asymptotics of "the
large times", ir-relevant to Newton's mechanics. Indeed, taking into account
the first correction, expres-sion for velocity yields:
v = c(1 ? 0.5/R)
It brings us new concepts of simultanuity, absence of absolutely rigid
bodies and so on, re-flecting the deepness of ideological revolution accomplished
by the theory of relativity.
GEOMETRIC AND WAVE OPTICS
The study of relationship between wave and geometric optics is of importance
both it-self and for understanding of the relationship between classical
and quantum mechanics. For a long time it was accepted for good that the
elementary geometric constructions form the basis of geometric optics.
After discovering of light diffraction the wave theory became generally
accepted, while geometrical optics, seemed kind of homespun prescription,
which doesn't reflect fundamental laws of the Nature. Only in the twenties
of our century it was clearly determined that transition from wave to geometric
optics is connected with small wavelength l ? 0. Since l = 10?7 m for visible
light geometric optics is the good approxi-mation in many cases [12-14].
From mathematical viewpoint transition to geometric optics is performed
in the framework of so called WKB-method (named after Wentzel, Kramers
and Brillouin). In the space point with coordinates (x,y,z) every characteristic
of electro-magnetic field in the light wave U(x,y,z) is represented in
the form:
U =A(x,y,z,l) exp(q(x,y,z)/l),
where A is wave amplitude and q is wave phase. Then A and q are represented
in the form of power series of 1/l. After substitution of this expression
into wave equation and sorting off the terms with equal powers of 1, the
nonlinear differential equation for phase ? may be obtained (eikonal equation).
Namely, this one corresponds to geometrical optics. For de-termination
of the expansion coefficients the recurrent sequence of linear differential
equa-tions may be obtained (so called transfer equations). In the geometrical
optics it is supposed that light rays propagate along certain curves. The
edge of the beam seems very sharp, but in reality the intensity of light
boundary changes although quickly but continuously in boundary layer which
thickness has the order of wave length l. Asymptotics describing purely
wave phenomenon of diffraction can be constructed using boundary layer
concept.
CLASSICAL AND QUANTUM MECHANICS
The relationship between classical and quantum mechanics in certain
sense is similar to that which exist between geometric and wave optics.
In quantum mechanics the wave func-tion W of quasiclassical, that is almost
classical physical system may be represented in the form W = A exp(S/h),
where S is so called action. The small parameter here is the ratio h/S.
Transition from quantum to classical mechanics formally is described by
the WKB-method at h tends to 0. The essence of such transition is that
the center of localized wave packet which is the initial probability distribution
of the particle coordinates moves then in accor-dance with laws of classical
mechanics. But the quasiclassical approach loses its sense for very small
particle momentums. This occurs, for example, in the vicinity of "turning
points", where presuming classical mechanics is valid the particle should
stop and reverse its movement. But in the quantum mechanics the principally
nonclassical phenomenon be-comes possible "tunneling" of the particle over
the potential barrier. This phenomenon can be described by the asymptotics
using just the smallness of the momentum. In the process of creation of
quantum mechanics heuristic role of asymptotic correspondence was mani-fested
outstandingly. This role increases especially nowadays when the attempts
of con-struction of the theory uniting all fundamental interactions are
performed. In the framework of the such theory concepts of electromagnetic,
weak, strong and gravitational interactions themselves have to be asymptotic,
having the sense only for small energy levels. Let's em-phasize that constructive
role of asymptotic relationship between classical and quantum mechanics
reflected in - the famous Bohr's "correspondence principle" in the special
physi-cal and philosophical literature. Meantime we include this example
in our topic for the sake of completeness of the picture.
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