Geometric models of statistical physics: billiard in a symmetric phase space
The billiard problem of statistical physics is considered in a new geometric approach with a symmetric phase space. The structure and topological features of typical billiard phase portrait are defined. The connection between geometric, dynamic and statistic properties of smooth billiard is establis...
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| Zitieren: | Geometric models of statistical physics: billiard in a symmetric phase space / S.V. Naydenov, V.V. Yanovsky // Вопросы атомной науки и техники. — 2001. — № 6. — С. 218-222. — Бібліогр.: 15 назв. — англ. |
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| author | Naydenov, S.V. Yanovsky, V.V. |
| author_facet | Naydenov, S.V. Yanovsky, V.V. |
| citation_txt | Geometric models of statistical physics: billiard in a symmetric phase space / S.V. Naydenov, V.V. Yanovsky // Вопросы атомной науки и техники. — 2001. — № 6. — С. 218-222. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The billiard problem of statistical physics is considered in a new geometric approach with a symmetric phase space. The structure and topological features of typical billiard phase portrait are defined. The connection between geometric, dynamic and statistic properties of smooth billiard is established.
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| first_indexed | 2025-12-07T15:17:54Z |
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GEOMETRIC MODELS OF STATISTICAL PHYSICS:
BILLIARD IN A SYMMETRIC PHASE SPACE
S.V. Naydenov*, V.V. Yanovsky
Institute for Single Crystals of National Academy of Science of Ukraine
Kharkov, Ukraine
* e-mail: naydenov@isc.kharkov.com
The billiard problem of statistical physics is considered in a new geometric approach with a symmetric phase
space. The structure and topological features of typical billiard phase portrait are defined. The connection between
geometric, dynamic and statistic properties of smooth billiard is established.
PACS: 05.45-a; 05.45.Ac; 05.45.Pq; 05.45.Mt.
1. INTRODUCTION
Billiard is one of the most important models of
statistical physics and chaotic dynamics. G.D. Birkhov
suggested regarding billiard as a typical conservative
system [1]. A.N. Krylov based his explanation of solid
spheres gas statistic properties on exponential
divergence of its “billiard” trajectories [2]. In the works
by Ya.G. Sinai [3] and L.A. Bunimovich [4] on phase
trajectories mixing in scattering and defocusing billiards
Boltzmann’s hypothesis of molecular chaos found its
further grounding. Now billiard became a paradigm of
deterministic chaos [5] of classical systems and is often
applied [6, 7] for the research of their quantum “twins”.
A lot of applied physics problems can be reduced to a
billiard problem [8-12].
A classical billiard problem is in studying its
character and distribution of its trajectories. Among the
typical billiard motions one can point out the following:
periodical, quasiperiodical (integrable) and irregular
(chaotic) motions. Compound billiard dynamics appears
in the phase portrait structure of the corresponding map.
The latter is plotted using different geometric methods
or Poincare sections. For the specification of a billiard
ray it’s usual to choose local Birkhov coordinates:
natural parameter l in the reflection point on the border
of the billiard ∂ Ω and the incidence angle θ in the
same point. They stand for canonical variables – the
coordinate and the moment for Hamiltonian description
of the system. Many important properties at this choice
of phase space coordinates stay unnoticed. Let us choose
another unifying approach. It identifies billiards with
reversible map (with projective involution) in a
symmetrical phase space. In its framework one can join
together geometric, dynamic and statistic properties of
billiards.
2. SYMMETRIC COORDINATES
Let us describe geometric propagation of the rays of
billiard as a reversible map B of the phase space Z
with symmetric coordinates 1 2( , )z z . The pair of these
coordinates defines two successive reflections of a
billiard ray from ∂ Ω . At the same time, each of the
coordinates corresponds to some parameterization of the
billiard border, ( ) ( ) ( )( )zy,zxzrr ==Ω∂
. The following
topological construction appears: Z ∂ ∂∝ Ω × Ω . For a
closed planar billiard one can accept 1z S∈ (circle) or
[0,1]z I∈ = the periodicity being ( ) ( 1)r z r z= +
. So
we’ll have a phase space as a torus 2 1 1Z T S S= = × or
its unfolding I IΠ = × on the plane. After each
reflection of an arbitrary (incoming) billiard ray with the
coordinates 1 2( , )z z , we have (reflected) ray with new
coordinates 1 2( , )z z′ ′ . As a result, the evolution of these
successive reflections is described with a cascade
1 2 1 2( , ) ( , )z z z z′ ′→ [12]
1 2
1 2 2 1
2 1 2
: ; ( ( , ), )
( , )
z z
B f f z z z z
z f z z
′ =
= = ′ =
;
2 1 2 2
( ) ( )( , ) ( ( , ), )); ( , )
( ) ( )
a z z b zg f z g R z z z R z z
b z z a z
′ ′+′= =
′ ′− (1)
with the involution 1 2( , )f f z z= (on the first argument
1z ), which is defined by implicit dependence on the
corresponding fractional rational involution R ,
( ( , ), )R R z z z z′ ′ = . The coefficients
( ) ( ) ( )znznza 2
y
2
x −= , ( ) ( ) ( )znz2nzb 2
y
2
x= are
expressed with (Cartesian) components of the exterior
normal field ( ) extn z n ∂ Ω=
on the border ∂ Ω ,
( ; ) ( ( ); ( ))x yn n n y z x z′ ′= = −
(the stroke marks
differentiation). Function g depends on the form of
∂ Ω , [ ] [ ]1 2 2 1 1 2 1 2( , ) ( , ) ( ) ( ) / ( ) ( )g z z g z z x z x z y z y z= = − − .
The map (1) is invariant to the substitution,
1 2 2 1: ;S z z z z= → → , of the incoming ray to the
reflected one, i.e. B S S B=o o for the composition of
transformations. This means reversibility of the
constructed maps. The physical reason of this is
reversibility of the system to the changes of the time
sign (the direction of the motion). This is a global
property. In the billiard cascade, phase trajectories with
opposite directions of the motion or with opposite-
directional initial rays 10 20( , )z z and 20 10( , )z z are
218 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 218-222.
present simultaneously. This requirement of local
reversibility is stronger. The inverse of the ray reflected
with its successive reflection makes the initial incoming
ray. Mathematically it leads to the appearance of a
involution f in the map (1). The symmetry
(reversibility) leads to the symmetry of the phase space
and the phase portrait of the maps (1). For every element
Z there is one symmetrical to it relative to the diagonal
({ }1 2 1 2( , )z z Z z z∆ = ∈ = . For every function
1 2 2 1( , ) ( , )z z z zχ χ= . (2)
That’s why it’s natural to regard the coordinates of Z as
symmetrical.
In the research of periodical trajectories of the
billiard the powers of billiard map kB are also used
{ }1 1 2 1 2: ( , ); ( , )k
k kB z f z z z f z z−′ ′= = = ;
1 2 2 1 2 1 1 2( , ) ( ( , ), ( , ))k k kf z z f f z z f z z− −= . (3)
They include billiard “compositions” kf , where
0,1, 2,k = K ; 1f f= ; 0 2f z= ; 1 1f z− = . They lose the
property of involution, but preserve reductibility to
fractional rational transformations. The maps (3)
describe “pruned” billiard trajectories with the omission
of a set of ( 1)k − successive reflections.
3. BILLIARD GEOMETRY: INVOLUTION
PROPERTIES
All the geometric properties of a billiard are
established in the specialization of maps (1). They are
concretized in the features of the involution f . In the
appropriate (local) coordinates it can be reduced to
fractional rational involution R . Projective
transformations are described with fractional rational
functions. Billiard is one of those transformations. In
every reflection point incoming and reflected rays are
joined together by a harmonic transformation G . For G
projective invariant (a complex relation of four rays,
incoming i , reflected r , normal n and tangent t ) is
equal to ( , , , ) 1i r n t = − . In geometric terms
( ; , )r G i n t= ; G G Id=o , (4)
where Id is an identical transformation. Let us
emphasize the locality of the projective property of the
billiard. The concrete form of G depends on the
guiding-lines of the normal in the point of reflection,
that is, on the form ∂ Ω . Harmonic map (4) is an
involution and changes the sequence order of the
ordered projective elements to the opposite. The
monotony of f on the first argument is the
consequence of this. This monotony of piece-wise
continuous f (involution can stand discontinuity) is
true for every billiard. Using the correlations (1), the
involution can be laced of local branches of the form
1f g R g−= o o . From that the property of monotony
immediately follows
1 2 1( , ) / 0f z z z∂ ∂ < . (5)
Fractional rational functions are dense everywhere in
the space of continuous functions. In fact, this means the
possibility of arbitrary precise approximation of
different physical systems with their billiard models.
This fact is used, for instance, in the analysis of
energetic spectra of multi-particle systems (nuclei,
molecules and so on), and for description of kinetic
properties of continuums (Lorenz gas model) etc. If
sinus and cosine have physically appeared from the
problem of oscillator, then fractional rational functions
can be generated by billiard.
The reflection of ray beams from the border of the
billiard can be of diffractive, focusing and neutral
character. This depends on the curvature of ∂ Ω . The
representation (1) gives the following property
{ }1 2
2
2
( , ) ˆ ( )f z zsign sign K z
z
∂
∂
=
, (6)
where K̂ is oriented curvature in the point of reflection.
For the convex border ˆ 0K > involution appears to be a
monotonous function on both arguments. For instance,
for a circle, 1 2 2 1( , ) 2 (mod 1)f z z z z= − . On a torus,
2Z T= , ∂ Ω such involution has no breaks (Unlike
dispersive billiard, ˆ 0K < , with lacunas in phase space.)
Involutivity and projectivity are the main geometric
properties of a billiard. The geometry (form) of its
border defines the explicit form of involution. At the
same time, it also defines the dynamics of the billiard.
4. BILLIARD DYNAMICS:
THE SYMMETRIC PHASE SPACE
Let us analyze the structure of symmetric phase
space (Figure) of a typical billiard. This principally
solves the question of the types of dynamics and
stability. For high-quality research of phase portrait of
the maps and its local bifurcations normal Poincare
forms are especially useful [13]. In the symmetric
approach the theory of normal billiard forms appears to
be the most advanced. This is connected with the
flexibility (a wider class of allowable variables) of
reversible systems. Any changes of variables in
Hamiltonian approach are to preserve the conservation
character of the map with the Jacobian 1J = (canonic
changes). Whereas the map (1) doesn’t demand it. It
Jacobian 1 2 1( , ) / 0J f z z z∂ ∂= − > can take arbitrary
values, 0 1; 1J J< ≤ ≥ . As a weak limitation, the
demand for the map (1) to preserve measure remains.
This means that ( ) ( ) / ( )J J z z Bzρ ρ= =
, where
1 2 2 1 2( , ); ( , ( , ))z z z Bz z f z z= =
should be true. The
proof uses the equation of Frobenius–Perron for the
density ρ of invariant measure (see further) and the
symmetry (2) for it, 1 2 2 1( , ) ( , )z z z zρ ρ= . This limitation
can always be met preserving the main property of
involution f f id=o in new coordinates.
Omitting the details, let us present the expression for
normal billiard form in symmetric coordinates. It is true
in the neighbourhood of an arbitrary cycle of p order
(periodic trajectory of period p )
219
1 1 1 1 2 1 1 2
2 1 2 2 1 2
( )
:
( )
p pp
p p
z z z z P z z
NB
z z z z Q z z
µ ν
µ ν
− −′ = − + +
= ′ = − + +
, (7)
where the coefficients of the linear part are defined by
the expansion of “compositions” 1pf − and pf (see
Eq. (3)) in the initial point neighbourhood of the cycle
under consideration. They constitute the matrix of L̂
linear part. Homogeneous polynomials ,P Q (without
absolute terms) define nonlinear additives. Their explicit
form depends on the involution of billiard f , that is, on
the form of ∂ Ω .
The character of the cycle depends on the size of
trace ˆtrL . For an elliptic cycle tr ^
L <2, for a
hyperbolic one tr
^
L >2. In the neutral case, for
instance, for a billiard in a circle, ˆ 2trL = . It can be
shown that for any cycle, corresponding to a periodic
trajectory, passing through a concave section with
concavity 2
ˆ ( ) 0K z < , ˆ 2trL < − will be true. That’s why
the trajectories near such cycles always are unstable and
exponentially diverge from one another. Near elliptic
cycles, including 2-cycles, regions of regular motion
form. With the loss of ellipticity they are ruined, first
forming stochastic layers and then, when the latter are
covered, a chaotic sea. Normal forms (7) let us trace
typical properties of such bifurcations, taking place
when the billiard border is deformed.
The diagonal ∆ 1 2( )z z= contains all fixed points of
the billiard, B∆ = ∆ . This follows from the diagonal
property ( , )f z z z= of billiard involution, resulting
from its coordinate expression (1). For a convex billiard
in the neighbourhood of the phase space diagonal,
normal form (7) can be reduced to the map of a turn, i.e.
a particular case of a billiard in a circle. Here the
structure of elliptic and hyperbolic cycles of arbitrary
high order is shown. The motion stays regular. The
appearing of negative curvature ruins this situation.
There is no unified transformation (or the integral of
motion) near the diagonal because of appearing breaks
of billiard involution.
Analytic research of the symmetric phase space
structure can be continued using geometric methods. In
addition to regular and chaotic components of motion
the phase portrait can contain regions of forbidden
motion – “lacunas” L and regions of degenerated
motion – “discriminants” D .
Lacunas (Fig.) appear in the billiards with regions of
negative curvature. They occupy the phase space part,
the points of which correspond to the rays lying outside
of the billiard region Ω . The coordinates of these rays
meet the condition 1 2( ) ( )r z r z− ∉ Ω
. This condition
defines the inner region of lacuna L in Z . The form of
the lacuna is defined by its border
(1 2 1 2: {( , ) ( )}L z z Z z z∂ λ= ∈ = . (There is another
parameterization 1
2 1( )z zλ −= . In this case functions
( )zλ and 1( )zλ − specify the same simple closed curve,
but passable in different directions.) The border L∂
comes to the diagonal ∆ transversally and crosses it
twice in the points with coordinates 0 0( , )z z ,
corresponding to the points of inflexion, 0
ˆ ( ) 0K z = .
The forbidden billiard rays (points of lacuna) lie in
classically inaccessible region – geometric shade,
generated by the regions ˆ 0K < . The number of lacunas
(on a torus 2Z T= ) is equal to the number of negative
curvature components ∂ Ω . Every lacuna is a simply
connected set. The contrary would mean non-closed
character of ∂ Ω . With the appearance of lacunas a part
of diagonal ∆ is cut out. The corresponding fixed points
disappear. For Sinai billiard, lacuna absorbs all the
diagonal and the map (1) will lack all fixed points. At a
special configuration of such a border ∂ Ω one can cut
out cycles of higher order, 2p ≥ .
The schematic form of symmetric phase space with
elliptic zones of regular motion R , a chaotic region C
, the diagonal ∆ , lacunas L , discriminants D
In a topological way one can glue up the lacuna on
the torus with a two-dimensional manifold. According to
the rule of L∂ bypass, it can be only a piece of a
projective plane. This is directly connected with the
projectivity of the billiard. On a projective plane, metric
conceptions “inside” and “outside” of a closed region
lose their sense. (For example, a closed curve and a right
line that doesn’t cross it on a plane may have common
points after central projection onto the other plane.)
That’s why “forbidden” rays turn out to be involved into
the general billiard flow. Such global motion takes place
on non-oriented manifold.
On the projective plane the initial involution f also
rules the motion of the rays. Almost every such ray
(exceptional cases are of measure null) is continued to
an ordinary billiard ray, further dynamics of which is
known. As a result of further evolution, this ray after
some time will return to the section of negative
curvature ∂ Ω , corresponding to the lacuna under
consideration. This is specified by the mentioned
hyperbolicity of cycles that contain points on the
concave border. Being continued then to a classically
inaccessible region (preserving the direction of motion),
it would give a new position of the initial ray (phase
point in the lacuna). A recurrent map appears. It is
defines by one of the “compositions” kf , included in the
equation (3), the order k always depending on the
coordinates of the initial ray (the initial point of the
lacuna). The lacuna plays the role of a secant for the
Poincaré section of the billiard flow. Similar evolution
220
also takes place with other points of all lacunas. Only
phase trajectories of ordinary billiard rays remain in this
case “visible”.
The condition of “connecting” for the inner rays, that
are tangent to the concave region in the point 3( )r z
and
that cross ∂ Ω in the points 1 2( ) , ( )r z r z
outside of it,
defines the border of the lacuna
( )1 2 3( ) ( ), ( ) 0r z r z n z− =
;
1 2 1,2 3 3
ˆ( , ) | ; ( , ) ; ( ) 0z z Z L z z L K z∂∈ ∈ < , (8)
where (.,.) is the scalar product of the vectors. Solving
the equation (8) according to the theorem about an
implicit function, we have 1 1 3 2 2 3( ); ( )z z z zλ λ= = .
Excluding 3z , we come to the desired equation
1
1 2 1 2( );z zλ λ λ λ −= = o .
The discriminants D correspond to the zone of
“stuck-together” trajectories or “non-continuable”
trajectories that cross special (corner) points of ∂ Ω .
That’s why they appear in the billiards with straight
regions of ∂ Ω , ˆ 0K = . Their border D∂ is defined by
( )1 2 2 1 2( ) ( ), ( ) 0;( , ) |r z r z n z z z Z L− = ∈
; 2
ˆ ( ) 0K z = . (9)
It also can have explicit form 1 2( )z zµ= . The
discriminants (Fig.) have shape squares with a diagonal,
which coincide with a part of ∆ in the region
corresponding to the straight-line component ∂ Ω .
Lacunas and discriminants make a principal property
of a symmetric phase space. In fact, they are filled with
the rays of the billiard that fell out of its ordinary
dynamics. (At them passing it’s easy to show that the
billiard involution f breaks (on the first and the second
arguments), that are different from the factor of
periodicity (mod 1) and are not removed when passing
to a torus, 2 2Z I Z T= → = .) There are no such non-
local elements in the phase space of Hamiltonian
approach. At the same time, these hidden “topological”
obstacles for the billiard flow to flow around, and the
diagonal ∆ , on which they arise, play an important role
in the chaotic dynamics and must be included into the
full description.
5. BILLIARD KINETICS: INVARIANT
DISTRIBUTIONS
The geometry of phase space structural elements
depends on the form of the border ∂ Ω and (or)
involution f . Let us show that in the symmetrical
approach not only dynamics but also kinetics of the
billiard is connected with these characteristics. The
kinetics becomes apparent in the case of chaotic billiard,
whose deterministic trajectories have all the properties
of random sequences in the asymptotic limit of infinitely
large number of reflections. That requires statistic
description of (two-dimensional) dynamic system in the
manner of deterministic chaos [14].
In a typical billiard both integrable and ergodic (as a
rule, with mixing) types of motion are present.
Absolutely continuous distributions are of the greatest
physical interest. From the operator equation Bρ ρ=
for an invariant measure after transformations using
piece-wise monotony (5) we have
( )1 2
1
( , )
1 2 2 1 2( , ) ( , ( , )) f z z
zz z z f z z ∂
∂ρ ρ= − . (10)
Geometrically, ρ is a two-point density; it depends
on the coordinates of two points on the border ∂ Ω . The
topology of the direct product Z ∂ ∂∝ Ω × Ω causes one
to choose a special factorized solution,
1 2 1 2( , ) ( ) ( )z z z zρ ω ω= . Instead of the expression (10)
we get a functional equation for one-point plane ( )zω
( ) ( ) ( , ) ( ) ( )z f J z z z dz f dfω ω ω ω′= ⇔ = − , (11)
written in total differentials. The factorization is
coordinated with the symmetry of ρ and preserves its
normalization ( )∫ =ω=ω
1
0
1dzz .
The physical sense of ( )zω is an asymptotic plane
of billiard flow reflection points (with coordinates
( ) ,r z z I∂∈ Ω ∈
). This is a truncate distribution in the
sense that the dimension falls twice. It will be very
useful in the description of physical characteristics in
different billiard problems, for instance, the
“probability” of ray escaping from a fixed place of
resonator, wave-guide or detector. Besides, it is directly
connected with the involution and geometry of the
billiard. After integrating the differential relation (11)
for ω we have
0
0 1
2 1 2( ) ( ) ; ( , )
zf
z z
z dz C z f f z zω
− = =
∫ ∫ , (12)
where 0z is an arbitrary initial point on ∂ Ω ; ( )C z is
the function to define. With different character of border
∂ Ω ( )C z has different forms. For everywhere convex
billiard it’s one can just use the diagonal condition
( , )f z z z= , so
0
( ) 2 ( )
z
z
C z z dzω ′ ′= ∫ . In the general case
the border 0∂ ∂ ∂ ∂+ −Ω = Ω Ω ΩU U contains regions of
positive, ∂ +Ω , negative, _∂ Ω , and zero, 0∂ Ω ,
curvature. During the defining of ( )C z the solutions in
symmetric “halves” of phase space over and under the
diagonal ∆ , that is, in the involutionally connected
regions with coordinates 1 2( , )z z and 1 2 2( ( , ), )f z z z are
laced. In the presence of _∂ Ω and 0∂ Ω components
connecting takes place on the borders of corresponding
lacunas and discriminants. Summing it up, let us set the
border ∂ Σ , that divides different symmetric components
of Σ (outside special zones)
2 1 2
1 2 2 1 2
2 1 2
, ( , )
( ) ( ), ( , )
( ), ( , )
z z z
z z z z z L
z z z D
λ ∂
µ ∂
∈ ∆
= Λ = ∈
∈
(13)
with known dependencies in the cases of lacunas and
discriminants (see above). Let us note that in each half
221
of the phase space ( )zΛ is a multi-valued function (the
number of branches doesn’t exceed the doubled number
of _∂ Ω and 0∂ Ω components, but self-intersections and
multiple connection ∂ Σ are forbidden by the uniqueness
of the flow. ( )zΛ is the functional of ∂ Ω . Connecting
on the border ∂ Σ gives us
0
0
( ( ), )
( )
( ) ( )
zf z z
z z
z dz C zω
Λ
Λ
′ ′− = ⇒
∫ ∫
1 2 2 2
2 1
( , ) ( ( ), )
( )
( ) 0
f z z f z z
z z
z dzω
Λ
Λ
− =
∫ ∫ . (14)
The dependence of the initial point 0z , as would be
expected, falls out. The equation obtained lets one to
restore billiard involution f on the one-point billiard
distribution function ω and vice versa. At the same
time both functions are connected with the equation of
border ∂ Ω by the expressions (13) and (1). The billiard
problem takes on a single meaning from dynamic,
statistic and geometric points of view.
Direct dependence of ω on f can be obtained by
differentiation of Eq. (11)
2
1 2 1 2
1 2 1 1 2 2
( , ) /ln ( )
( ( , ) / )( ( , ) / )
f z z z zd f
df f z z z f z z z
ω ∂ ∂ ∂= −
∂ ∂ ∂ ∂
. (15)
In the equations (10) and (15) the densities ρ and
ω are uniquely defined by the involution of billiard f .
The latter is uniquely defined by the border ∂ Ω
equation, according to the representation (2). The
invariant measures of the billiard become its individual
characteristics. In a chaotic billiard they acquire the
character of equilibrium statistic distributions. So, on the
whole billiard analysis in symmetrical coordinates
shows that its main characteristics are uniquely
connected with one another
1 2 1 2( , ) ( , ) ( )f z z z z z∂ ρ ω ∂Ω → ↔ ↔ → Ω . (16)
One of the most designing and old problems of
statistical physics is finding out the transition from the
reversibility of deterministic motion equations to
irreversibility of statistic ones, see [15]. Generally
accepted point of view is that irreversibility appears at
roughening in the macroscopic description of the system
on the kinetic stage of evolution and is connected with
the fundamental principle of correlations unlinking.
Here usually the problem of distribution functions
calculation with given Hamiltonian (the equations of
motion) is posed. In physical applications the inverse
problem may also appear: to restore the dynamic law for
a chaotic system (not necessarily of mechanic origin)
with known statistic characteristics. It can be of special
actuality for the system with a small number of freedom
degrees. Statistic irreversibility prevents the reverse of
the “time arrow”, but doesn’t necessarily break the
feedback of kinetic and dynamic. A remarkable
peculiarity of the billiard is the possibility to solve direct
as well as indirect problems. The form of the border
∂ Ω defines involution, on which the invariant measure
is calculated. And vice versa: the involution (that is, the
dynamic of the billiard) is restored from the one-point
distribution of reflections on the border. The border of
the billiard can be restored by its involution [7]. Such
closure is the consequence of geometric (projective)
nature of the billiard.
Symmetric approach allows direct generalization on
the multiply connected, multidimensional and other
cases of different billiard border topology. The
peculiarities named here preserve their key role.
ACKNOWLEDGMENT
The authors feel great pleasure to express their deep
gratitude to Academician of NAS of Ukraine
S.V. Peletminsky and to Professor Yu.L. Bolotin for the
attention paid to this paper.
REFERENCES
1. G.D. Birkhov. Dynamical Systems. New
York, American Mathematical Society, 1927.
2. N.S. Krylov. Works on the Foundation on
Statistical Physics. Princeton: Princeton Univ.
Press, 1979 (English translation).
3. Ya.G. Sinai. Dynamical systems with elastic
reflections // Russ. Math. Serv. 1970, v. 25, №2,
p. 141-192.
4. L.A. Bunimovich. On ergodic properties of
some billiards // Funct. Anal. Appl. 1974, v. 8, №3,
p. 254-255-257.
5. Proc. of the Intern. Conf. on Classical and
Quantum Billiards // J. Stat. Phys. 1996, v. 83, №1-
2.
6. V.F. Lazutkin. KAM Theory and
Semiclassical Approximations to Eigenfunctions.
Berlin-Heidelberg: Springer-Verlag, 1993.
7. M.G. Gutzwiller. Chaos in Classical and
Quantum mechanics. New York: “Springer-Verlag”,
1990.
8. C.M. Marcus, A.J. Rimberg et al.
Conductance fluctuations and chaotic scattering in
ballistic microstructures // Phys. Rev. Lett. 1992,
v. 69, p. 506-510.
9. C. Ellegaard, T. Ghur et al. Spectral statistics
of acoustic resonances in aluminum blocs // Phys.
Rev. Lett. 1995, v. 75, p. 1546-1550.
10. H. Alt, H.D. Graf et al. Chaotic dynamics in
a three-dimensional superconductiving microwave
billiard // Phys.Rev. E. 1996, v. E54, p. 2303-2307.
11. J.U. Nocel, A.D. Stone. Ray and wave
chaos in assymmetric resonant optical cavities //
Nature. 1997, v. 385, p. 45-47.
12. S.V. Naydenov, V.V. Yanovsky. The
geometric-dynamic approach to billiard systems //
Theor. Math. Phys. 2001, v. 127, №1, p. 500-512.
13. V.I. Arnold. Dopolnitelnye glavy teorii
obyknovennych differenzcialnych uravneniy.
Moskva: “Nauka”, 1978, 304 p. (in Russian).
222
14. H.G. Shuster. Deterministic Chaos.
Heidelberg: Springer, 1982.
15. A.I. Akhiezer, S.V. Peletminsky. Metody
Statisticheskoy Fiziki. Moskva: “Nauka”, 1977,
368 p. (in Russian).
223
THE SYMMETRIC PHASE SPACE
ACKNOWLEDGMENT
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79892 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:17:54Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Naydenov, S.V. Yanovsky, V.V. 2015-04-06T16:25:25Z 2015-04-06T16:25:25Z 2001 Geometric models of statistical physics: billiard in a symmetric phase space / S.V. Naydenov, V.V. Yanovsky // Вопросы атомной науки и техники. — 2001. — № 6. — С. 218-222. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 05.45-a; 05.45.Ac; 05.45.Pq; 05.45.Mt. https://nasplib.isofts.kiev.ua/handle/123456789/79892 The billiard problem of statistical physics is considered in a new geometric approach with a symmetric phase space. The structure and topological features of typical billiard phase portrait are defined. The connection between geometric, dynamic and statistic properties of smooth billiard is established. The authors feel great pleasure to express their deep gratitude to Academician of NAS of Ukraine S.V. Peletminsky and to Professor Yu.L. Bolotin for the attention paid to this paper. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Anomalous diffusion, fractals, and chaos Geometric models of statistical physics: billiard in a symmetric phase space Геометрические модели статистической физики: биллиард в симметричном фазовом пространстве Article published earlier |
| spellingShingle | Geometric models of statistical physics: billiard in a symmetric phase space Naydenov, S.V. Yanovsky, V.V. Anomalous diffusion, fractals, and chaos |
| title | Geometric models of statistical physics: billiard in a symmetric phase space |
| title_alt | Геометрические модели статистической физики: биллиард в симметричном фазовом пространстве |
| title_full | Geometric models of statistical physics: billiard in a symmetric phase space |
| title_fullStr | Geometric models of statistical physics: billiard in a symmetric phase space |
| title_full_unstemmed | Geometric models of statistical physics: billiard in a symmetric phase space |
| title_short | Geometric models of statistical physics: billiard in a symmetric phase space |
| title_sort | geometric models of statistical physics: billiard in a symmetric phase space |
| topic | Anomalous diffusion, fractals, and chaos |
| topic_facet | Anomalous diffusion, fractals, and chaos |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79892 |
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