Table Of Contenthttp://mmg.tversu.ru ISSN 2311-1275
Mathematical Modelling
G
MM and Geometry
Volume 3, No 1, p.1 – 24 (2015)
Discrete dynamical models: combinatorics,
5
1 statistics and continuum approximations
0
2
Vladimir V. Kornyak
l
u
J
Joint Institute for Nuclear Research, Dubna, Russia
9
1 e-mail: [email protected]
]
h Received 5 January 2015. Published 21 January 2015.
p
-
t Abstract. This essay advocates the view that any problem that has a mean-
n
ingful empirical content, can be formulated in constructive, more definitely, finite
a
u terms. We consider combinatorial models of dynamical systems and approaches
q
to statistical description of such models. We demonstrate that many concepts of
[
continuous physics— suchas continuous symmetries, theprincipleof least action,
2
Lagrangians, deterministic evolution equations — can be obtained from combi-
v
6 natorial structures as a result of the large number approximation. We propose
5 a constructive description of quantum behavior that provides, in particular, a
3
natural explanation of appearance of complex numbers in the formalism of quan-
7
0 tum mechanics. Some approaches to construction of discrete models of quantum
1. evolution that involve gauge connections are discussed.
0
Keywords: combinatorial models, quantum mechanics, finite groups, gauge in-
5
1 variance, statistical descriptions
v: PACS numbers: 02.20.-a, 03.65.Aa, 03.65.Ta
i
X
r
a
The work is supported in part by the Ministry of Education and Science of the Russian
Federation (grant 3003.2014.2) and the Russian Foundation for Basic Research (grant
13-01-00668).
c The author(s) 2015. Published by Tver State University, Tver, Russia
(cid:13)
2 V.V. Kornyak
1. Introduction
Any continuous physical model is empirically equivalent to a certain finite model.
This thesis is widely used in practice: solution of differential equations by the finite
difference method or by using truncated series is typical example.
Itisoftenbelievedthatcontinuousmodelsare“morefundamental”thandiscrete
or finite ones. However, there are many indications that nature is fundamentally
discrete at small (Planck) scales, and is possibly finite.1 Moreover, description of
physical systems by, e.g., differential equations can not be fundamental in principle,
since it is based on approximations of the form f (x) f (x )+ f (x )∆x.
0 0
≈ ∇
This essay advocates the view that finite models provide a more relevant de-
scription of physical reality than continuous models which are only approximations
in the limit of large numbers.2 Using simple combinatorial models, we show how
such concepts as continuous symmetries, the principle of least action, Lagrangians,
deterministic evolution equations, etc. arise from combinatorial structures as a re-
sult of the large number approximation. We also consider some approaches to the
construction of discrete models of quantum behavior and related models describing
evolution of gauge connections.
Any statistical description assumes one or another concept of macrostate. We
definemacrostatesasequivalence classes ofmicrostates. Thisdefinitionisespecially
convenient for models incorporating symmetry groups. We distinguish two types
of statistical models:
1. Isolated system is purely combinatorial object in the sense that “probability”
of a microstate has a priori nature. Namely, all microstates are equiprobable,
so their probabilities are equal to the inverse of their total number. The
macrostates are specified by an equivalence relation on microstates.
2. Open system is obtained from an isolated system by the following modifi-
cation: The macrostates are specified by the same equivalence relation, but
the probabilities of microstates depend on some parameters, which are in-
troduced for approximate description of interaction of the system with the
environment.
Thearchetypalexamplesofisolatedandopensystemsare,respectively, microcanon-
ical (macrostates are defined as collections of microstates with equal energies) and
canonical (macrostates are defined similarly, and interaction with the environment
is parameterized by the temperature) ensembles.
The classical description of a (reversible) dynamical system looks schematically
as follows. There are a set X of states and a group G Sym(X) of transfor-
cl
≤
mations (bijections) of X. Evolutions of X are described by sequences of group
1The total number of binary degrees of freedom in the Universe is about 10122 as estimated
via the holographic principle and the Bekenstein–Hawking formula.
2 −35
Comparing the Planck length, 10 m, with the minimum length observable in experi-
∼
ment, 10−15 m, we may assume that the emergence of the empirically perceived continuous
∼
space is provided by averaging over about 1020 discrete elements.
Discrete dynamical models 3
elements g G parameterized by the continuous time t = [t ,t ] R. The
t cl a b
∈ ∈ T ⊆
observables are functions h : X R.
→
We can “quantize” an arbitrary set X by assigning numbers from a number
system to the elements x X, i.e., by interpreting X as a basis of a module.
F ∈
Thequantumdescriptionofadynamicalsystemassumesthatthemoduleassociated
with the set of classical states X is a Hilbert space over the field of complex
X
H
numbers, i.e., = C; the transformations g and observables h are replaced by
t
F
unitary U and Hermitian H operators on , respectively.
t X
H
To make the quantum description constructive, we propose the following mod-
ifications. We assume that the set X is finite. Operators U belong to the group
t
of unitary transformations of the Hilbert space Aut( ). Using the fact that this
X
H
groupcontains afinitely generated—andhenceresidually finite—dense subgroup,
we can replace Aut( ) by unitary representation of some finite group G that is
X
H
suitable to provide an empirically equivalent description of a particular problem.
A plausible assumption about the nature of quantum amplitudes implies that the
field C can be replaced by an abelian number field1 . This field is a subfield
F
of a certain cyclotomic field Q which in turn is a subfield of the complex field:
m
Q < C. The natural number m, called conductor, is determined by the struc-
m
F ≤
ture of the group G. Note that the fields Q and C provide empirically equivalent
m
descriptions in any applications, because Q is a dense subfield of C for any m 3.
m
≥
In this paper we will assume that the time is discrete and can be represented
T
as a sequence of integers, typically = [0,1,...,T].
T
Notealsothat thesubscript ainthenotation forHilbert spaces isoverloaded
a
H
and can mean, depending on context: dimension of the space, a set on which the
space is spanned, a group whose representation space is , etc.
a
H
2. Scheme of statistical description
For convenience of presentation, let us fix some notation:
U is the full set of states of a system A. The states from U are usually called
•
“microstates” in statistical mechanics.
N = U is the total number of microstates.
• | |
p is the probability (weight) of a microstate u U, p = 1.
u u
• ∈
u∈U
P
is an equivalence relation on the set U.
• ∼
Taking an element u U, we define the macrostate λ as the equivalence class
• ∈
λ = v U v u .
{ ∈ | ∼ }
Λ denotes the set of macrostates.
•
1An abelian number field is an algebraic extension of Q with abelian Galois group. According
to the Kronecker–Weber theorem, any such extension is contained in some cyclotomic field.
4 V.V. Kornyak
The equivalence relation determines the partition U = λ.
• ∼
λ∈Λ
`
K = Λ is the number of macrostates.
• | |
N = λ is the size of a macrostate λ Λ.
λ
• | | ∈
P denotes the probability of an arbitrary microstate from U to belong to the
λ
•
macrostate λ.
Isolated systems. The probability of a microstate of an isolated system is defined
naturally1 as p = 1/N for any u U, and the probability of any microstate
u
∈
from U to belong to the macrostate λ is, respectively, P = N /N. Since the
λ λ
“probabilities” in isolated systems have an a priori nature, such systems are in fact
purely combinatorial objects, and we can talk about the number of combinations
instead of probability.
Open systems interact with the environment. This interaction is parameterized
by assigning, in accordance with some rule, probabilities to all microstates. That
is, the probability of a microstate u λ is a function
∈
p = p (α ,α ,...) (1)
u u 1 2
of some parameters α ,α ,.... These parameters and function (1) are determined
1 2
bythespecificsofaparticularproblem. ForopensystemsP = p (α ,α ,...).
λ u∈λ u 1 2
P
Entropy. One of the central issues of the statistical description is the search for
the most probable macrostates, i.e. the macrostates with the maximum value of
P . Technically, entropy is defined as the logarithm of the number (or probability)
λ
of microstates that belong to a particular macrostate. The concept of entropy is
convenient for two reasons:
1. Since the logarithm is a monotonic function, the logarithm of any function
has the same extrema as the function itself.
2. If a system can be represented as a combination of two independent systems,
A = A′ A′′, thenthemacrostatesofAcanberepresented asλ′ λ′′. So, when
⊗ ⊗
computing entropy of such decompositions, we can replace multiplication by
a simpler operation — addition: log(Pλ′Pλ′′) = logPλ′ +logPλ′′.
1This is “the equal a priori probability postulate” of statistical mechanics [1].
Discrete dynamical models 5
Stirling’s formula is one of the main tools for obtaining continuum approxima-
tions of combinatorial expressions:
1 1 1
lnn! nlnn n+ ln(2πn)+ + (2)
≈ − 2 12n − 360n3 ···
superlinear
logarithmic decreasingterms
| {z }
For our purposes it is sufficient to re|tain{oznly}the|terms w{hzich grow} with n, i.e. the
superlinear and logarithmic terms.
2.1 Examples of isolated and open systems
To illustrate the above, let us give a few examples of isolated and open systems.
Sequences of symbols. Isolated system. Let Σ be an alphabet of size M:
Σ = (σ ,...,σ ,...,σ ). (3)
1 m M
The microstates are sequences of the length T of symbols from Σ:
u = a a a U, a Σ.
1 t T t
··· ··· ∈ ∈
The total number of microstates is N = MT. Let ku = (ku,...,ku,...,ku ) be
1 m M
the vector of multiplicities of symbols σ in a microstate u. It is obvious that
m
ku + +ku = T. We define the equivalence relation as follows
1 ··· M ∼
u v ku = kv k. (4)
∼ ⇐⇒ ≡
The macrostate λ , defined by equivalence (4), consists of all sequences with the
k
multiplicity vector k. The total number of macrostates is
T +M 1
K = − .
M 1
(cid:18) − (cid:19)
The size of the macrostate λ is
k
T!
N = .
λk k !k ! k !
1 2 M
···
Introducingthevector of“frequencies” (f = k /T,...,f = k /T)andapply-
1 1 M M
ing the leading part of Stirling approximation logn! nlogn n to the entropy
≈ −
S = logN of the macrostate λ we obtain S TH(X), where
λk λk k λk ≈
M
H(X) = f logf
i i
−
i=1
X
is the Shannon entropy [2] of a random variable X whose M outcomes have prob-
abilities f ,...,f .
1 M
6 V.V. Kornyak
The model of symmetric random walk [3] is a slight modification of the
above isolated system. Alphabet (3) contains now an even number of symbols
M = 2d, and it is divided into two parts Σ = Σ Σ , where
+ −
Σ = σ Σ 1 i d and Σ = σ` Σ d+1 i M .
+ i − i
{ ∈ | ≤ ≤ } { ∈ | ≤ ≤ }
The elements of Σ and Σ can be interpreted, respectively, as “positive” and
+ −
“negative” unit steps in the d directions of the integer lattice Zd.
The vector of multiplicities of symbols from Σ can be written as k = (k ,k ),
+ −
where k = (k ,...,k ) and k = (k ,...,k ). The equivalence of microstates u
+ 1 d − d+1 2d
and v is defined now as follows
u v ku ku = kv kv. (5)
∼ ⇐⇒ + − − + − −
The partition defined by (5) is a coarsening1 of the partition defined by (4). The
numbers of equivalence classes are figurate numbers of some d-dimensional regular
convex polytopes — d-dimensional analogues of the octahedron. For example, in
the case d = 3 the number of macrostates is equal to the (T +1)th octahedral
number:
2(T +1)2 +1
K = (T +1) .
3
Multinomial distribution. Open system. The microstates are also sequences
of symbols from alphabet (3). But now the presence of an environment is assumed.
The influence of the environment is parameterized by the assumption that any
M
symbolσ Σcomeswithafixedindividual probabilityα , suchthat α = 1.
m m m
∈
m=1
Thus, αk1αk2 αkM p is the probability of a microstate from a macProstate λ ,
1 2 ··· M ≡ λk k
defined by equivalence relation (4). The probability of a microstate from U to
belong to the macrostate λ is described by the multinomial distribution:
k
T!
P = N p = αk1αk2 αkM. (6)
λk λk λk k !k ! k ! 1 2 ··· M
1 2 M
···
Microcanonical ensemble. Isolated system. Theconceptofamicrocanonical
ensemble is based on the classification of microstates by energy. More specifically,
if there is a real-valued function on microstates E : U R, then we can impose
→
the equivalence relation on U:
u v E(u) = E(v) E, u,v U. (7)
∼ ⇐⇒ ≡ ∈
1The partition 2 = Bi ofasetS is acoarsening ofthe partition 1 = Aj ofthe sameset
P { } P { }
if for every subset Aj 1 there is a subset Bi 2 such that Aj Bi. The opposite relation
∈ P ∈ P ⊆
among partitions is called the refinement [2].
Discrete dynamical models 7
Assuming that the energy E takes a finite number of values: E E ,E ,...,E ,
1 2 K
∈ { }
we define the microcanonical ensemble as the macrostate λ which is an equivalence
k
class of relation (7), i.e., the set of microstates with the energy E . More formally:
k
λ = u u U E(u) = E . (8)
k k
{ | ∈ ∧ }
In statistical mechanics [4] the microcanonical ensemble is defined in terms of the
Boltzmann entropy formula
S = k lnN
λk B λk
or, equivalently, via the microcanonical partition function
N = eSλk/kB,
λk
where N = λ and k is the Boltzmann constant (we may assume that k = 1).
λk | k| B B
Canonical ensemble. Open system. A canonical ensemble is an open coun-
terpart of the microcanonical ensemble. Macrostates of the canonical ensemble are
defined by (8). Probability (1), that parameterizes the interaction with the envi-
ronment in our general scheme, is now a function of a single parameter α = T, the
temperature of the environment. Namely, the probability of a microstate u λ is
k
∈
given by the Gibbs formula
1
p (T) = e−Ek/kBT,
λk Z
where the normalization constant
K
Z = N e−Ek/kBT
λk
k=1
X
is called the canonical partition function.
3. Continuum approximations
In this section we show that concepts such as continuous symmetry and the prin-
ciple of least action may be obtained from combinatorial models as a result of the
transition to the limit of large numbers. As an illustration, consider the open sys-
tem described by distribution (6). In the case M = 2 (binomial distribution), all
calculations can be done explicitly. In this case both equivalence relations (4) and
(5) coincide in virtue of the equality k +k = T. Entropy of (6) for M = 2 takes
1 2
the form
S = lnT! lnk ! ln(T k )!+k lnα +(T k )lnα . (9)
1 1 1 1 1 2
− − − −
8 V.V. Kornyak
Applying the growing terms of Stirling’s approximation (2) to this formula we have
S S = S +S , where
Stirling superlin log
≈
k T k
1 1
S = T lnT k ln (T k )ln − (10)
superlin 1 1
− α − − α
(cid:18) 1(cid:19) (cid:18) 2 (cid:19)
and
1 T
S = ln
log
2 2πk (T k )
(cid:18) 1 − 1 (cid:19)
are superlinear and logarithmic parts of S , respectively. Since S and its
Stirling log
derivatives are small for large T and k , the maximum of entropy (9) is close to
1
that of its leading part (10). Thus, the approximate extremum point k∗ = α T is
1 1
obtained by solving the equation
∂S k T k k α
superlin 1 1 1 1
= ln +ln − = 0 = .
∂k − α α ⇐⇒ T k 1 α
1 (cid:18) 1(cid:19) (cid:18) 2 (cid:19) − 1 − 1
Further, we can expand S around the point k∗. Retaining terms up to the
Stirling 1
second order, we obtain the chain of approximations
1∂2S
S S S∗ = S + superlin (k α T)2,
≈ Stirling ≈ log|k1=α1T 2 ∂k2 × 1 − 1
1 (cid:12)k1=α1T
(cid:12)
(cid:12)
which leads to the final formula
(cid:12)
1 1
S S∗ = ln (k α T)2. (11)
1 1
≈ 2πTα α − 2Tα α −
r 1 2 1 2
3.1 On the origin of continuous symmetries
Itiswell known[3]thatthelargenumbersasymptoticoftheprobabilitydistribution
of symmetric random walk on the lattice Zd is the fundamental solution of the heat
equation, also called the heat kernel
1 x2 +x2 + +x2
K(t,~x) = exp 1 2 ··· d . (12)
(4πt)d/2 − 4t
(cid:18) (cid:19)
Here t R and x R are the continual substitutes for T N and for the
+ i
∈ ∈ ∈
difference k k Z, respectively.
i d+i
− ∈
This gives an example of the emergence of continuous symmetries from the
large numbers approximation. The symmetry group of the integer lattice Zd has
the structure of the semidirect product Zd ⋊ G . For simplicity, we can drop the
d
normal subgroup Zd, the subgroup of translations, as inessential for our purposes.
The group G is isomorphic to the semidirect product (Z )d ⋊S or, equivalently,
d 2 d
to the wreath product Z S . The size of G is equal to 2dd!. For example, for
2 d d
≀
the square lattice the group G is the symmetry group of a square — dihedral
d=2
Discrete dynamical models 9
group of order 8. On the other hand, approximate expression (12) is symmetric
with respect to the orthogonal group O(d,R) with cardinality of continuum.
TheLorentz symmetries—atleastin1+1dimensions—canalsobeobtained
in a similar way. Let us consider approximation (11) for the entropy of binomial
distribution. We introduce the following continuous substitutes
x := k k ,
1 2
−
t := T,
v := α α . (13)
1 2
−
Obviously, 1 v 1. With these substitutions, the approximation of binomial
− ≤ ≤
distribution takes the form
2
2 1 x vt
P∗(x,t) = eS∗ = exp − . (14)
sπ(1 v2)t (−2t √1 v2 )
− (cid:18) − (cid:19)
Thecontinuousvariablesx, t,andvmaybecalled, respectively, the“space”,“time”,
and “velocity”.1 Expression (14) is the fundamental solution of the equation
∂P∗(x,t) ∂P∗(x,t) (1 v2)∂2P∗(x,t)
+v = − . (15)
∂t ∂x 2 ∂x2
This equation is called — depending on interpretation of the function P∗(x,t) —
the heat, or diffusion, or Fokker-Plank equation. In the “limit of the speed of light”
v = 1 equation (15) turns into the wave equation
| |
∂P∗(x,t) ∂P∗(x,t)
= 0.
∂t ± ∂x
Let us introduce the change of variables: t = T +t′ and x = vT +x′. If we assume
H H
that t′ T (i.e., T can be thought as a “Hubble time”, and t′ as a “typical time
H H
≪
of observation”), then (14) can be rewritten as
2 1 x′ vt′ 2 t′
P∗(x,t) = exp − +O .
sπ(1−v2)TH (−2TH (cid:18)√1−v2(cid:19) ) (cid:18)TH(cid:19)
The principal part of this expression is “relativistically invariant”.
3.2 The least action principle as the principle of selection of the
most likely configurations
Let us compare the exact probability distributions with their continuum approxi-
mations within individual equivalence classes of relation (5).
1In the paper [5], which is devoted to the “Zitterbewegung” effect in the 1+1 dimensional
Diracequation,a“driftvelocity”isdefined—justlikein(13)—asthe differenceofprobabilities
of steps in opposite directions. It is shown that this definition leads to the relativistic velocity
addition rule: w =(u+v)/(1+uv).
10 V.V. Kornyak
Exact distributions. In the binomial case, an equivalence class of (5) is defined
by fixing the difference k k =: X. We denote the equivalence class of sequences
1 2
−
connecting the space-time points (0,0) and (X,T) by λ . The size of λ is
X,T X,T
equal to
T! T!
N = .
X,T k !k ! ≡ T+X ! T−X !
1 2 2 2
The binomial distribution in terms of the variables X and T and parameter v takes
(cid:0) (cid:1) (cid:0) (cid:1)
the form
T+X T−X
T! 1+v 2 1 v 2
P (X,T) = − .
T+X ! T−X ! 2 2
2 2 (cid:18) (cid:19) (cid:18) (cid:19)
Consider an increasing seq(cid:0)uence(cid:1) o(cid:0)f tim(cid:1)e instants (“times of observations”)
τ = T = 0,...,T ,T ,...,T = T , T < T . (16)
0 i−1 i n i−1 i
{ }
Let us select trajectories that pass through the sequence of spatial points
χ = X = 0,...,X ,X ,...,X = X (17)
0 i−1 i n
{ }
corresponding to sequence of times (16). Admissible trajectories must satisfy the
inequality X X T T — “the light cone restriction”. According to the
i i−1 i i−1
| − | ≤ −
conditional probability rule, the probability of the trajectory (χ,τ) is equal to
n
1
P = P (X X ,T T )
χ,τ i i−1 i i−1
P (X,T) − −
i=1
Y
T+X ! T−X ! n (T T )!
= 2 2 i − i−1 .
(cid:0) (cid:1)T(cid:0)! (cid:1) Yi=1 Ti+2Xi − Ti−1+2Xi−1 ! Ti−2Xi − Ti−1−2Xi−1 !
(cid:16) (cid:17) (cid:16) (cid:17)
For a given sequence of time instants (16) one can formulate the problem of find-
ing trajectories with maximum probability P = maxP . This can be done by
τ χ,τ
χ
searching among all admissible sequences (17). In the general case, there are many
distinct trajectories with the same maximum probability, i.e., we do not have here
a “deterministic” trajectory.
Continuum approximation. To apply our reasoning to approximate distribu-
tion (14), we will consider the time sequence
τ = t = t ,...,t ,t ,...,t = t
0 a i−1 i n b
{ }
together with respective sequences of spacial points
χ = x = x ,...,x ,x ,...,x = x . (18)
0 a i−1 i n b
{ }
We assume that the time points are equidistant: t t = ∆t, and we will use the
i i−1
−
notation ∆x = x x .
i i i−1
−
