Effective temperature of self-similar time series
Within slightly non-extensive statistics and the related numerical model, a picture is elaborated to treat self-similar time series as a thermodynamic system. Thermodynamic-type characteristics relevant to temperature, pressure, entropy, internal and free energies are introduced and tested. The...
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| Цитувати: | Effective temperature of self-similar time series / A. Olemskoi, S. Kokhan // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 761–772. — Бібліогр.: 10 назв. — англ. |
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irk-123456789-1210502017-06-14T03:04:46Z Effective temperature of self-similar time series Olemskoi, A. Kokhan, S. Within slightly non-extensive statistics and the related numerical model, a picture is elaborated to treat self-similar time series as a thermodynamic system. Thermodynamic-type characteristics relevant to temperature, pressure, entropy, internal and free energies are introduced and tested. The statistics developed is shown to be governed by the effective temperature being exponential measure of the fractal dimension of the time series. Testing of the analytical consideration is based on the numerical scheme of non-extensive random walk. Effective temperature is found numerically to show that its value is reduced to averaged energy per one degree of freedom. В рамках слабо неекстенсивної статистики та пов’язаної числової моделі, розробляється картина розгляду самоподібних часових наборів як термодинамічної системи. Вводяться та тестуються характеристики термодинамічного типу, що відповідають температурі, тиску, ентропії, внутрішній та вільній енергіям. Показано, що розвинута статистика підкоряється ефективній температурі, будучи експонентною мірою фрактальної розмірності часових наборів. Тестування аналітичного розгляду базується на числовій схемі неекстенсивних випадкових блукань. Чисельно знаходиться ефективна температура з метою показати, що її значення приводить до усередненої енергії на один ступінь вільності. 2005 Article Effective temperature of self-similar time series / A. Olemskoi, S. Kokhan // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 761–772. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 05.90.+m, 05.45.Tp, 05.40.Fb DOI:10.5488/CMP.8.4.761 http://dspace.nbuv.gov.ua/handle/123456789/121050 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
Within slightly non-extensive statistics and the related numerical model, a
picture is elaborated to treat self-similar time series as a thermodynamic
system. Thermodynamic-type characteristics relevant to temperature,
pressure, entropy, internal and free energies are introduced and tested.
The statistics developed is shown to be governed by the effective temperature
being exponential measure of the fractal dimension of the time series.
Testing of the analytical consideration is based on the numerical scheme
of non-extensive random walk. Effective temperature is found numerically
to show that its value is reduced to averaged energy per one degree of
freedom. |
| format |
Article |
| author |
Olemskoi, A. Kokhan, S. |
| spellingShingle |
Olemskoi, A. Kokhan, S. Effective temperature of self-similar time series Condensed Matter Physics |
| author_facet |
Olemskoi, A. Kokhan, S. |
| author_sort |
Olemskoi, A. |
| title |
Effective temperature of self-similar time series |
| title_short |
Effective temperature of self-similar time series |
| title_full |
Effective temperature of self-similar time series |
| title_fullStr |
Effective temperature of self-similar time series |
| title_full_unstemmed |
Effective temperature of self-similar time series |
| title_sort |
effective temperature of self-similar time series |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121050 |
| citation_txt |
Effective temperature of self-similar time series / A. Olemskoi, S. Kokhan // Condensed Matter Physics. — 2005. — Т. 8, № 4(44). — С. 761–772. — Бібліогр.: 10 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT olemskoia effectivetemperatureofselfsimilartimeseries AT kokhans effectivetemperatureofselfsimilartimeseries |
| first_indexed |
2025-07-08T19:06:22Z |
| last_indexed |
2025-07-08T19:06:22Z |
| _version_ |
1837106815135907840 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 4(44), pp. 761–772
Effective temperature of self-similar
time series
A.Olemskoi, S.Kokhan
Sumy State University, Ukraine
Received July 18, 2005, in final form October 17, 2005
Within slightly non-extensive statistics and the related numerical model, a
picture is elaborated to treat self-similar time series as a thermodynam-
ic system. Thermodynamic-type characteristics relevant to temperature,
pressure, entropy, internal and free energies are introduced and tested.
The statistics developed is shown to be governed by the effective tempera-
ture being exponential measure of the fractal dimension of the time series.
Testing of the analytical consideration is based on the numerical scheme
of non-extensive random walk. Effective temperature is found numerically
to show that its value is reduced to averaged energy per one degree of
freedom.
Key words: time series, non-extensive statistics, simulations
PACS: 05.90.+m, 05.45.Tp, 05.40.Fb
1. Introduction
Time series analysis allows one to elaborate and verify macroscopic models of
complex system evolution based on the data analysis [1]. This analysis is known
to be focused on numerical calculations of the correlation sum for delay vectors
which make it possible to find principal characteristics of the time series. Being
traditionally a branch of the theory of statistics, time series analysis is based on
the class of harmonic oscillator models which are related to the simplest case of the
Gaussian random process. However, a well known real time series is relevant to the
Lévy stable processes, rather than to the Gaussian ones being a very special case [2].
Since the former processes are invariant with respect to dilatation transformation,
the problem is reduced to considering the self-similar stochastic processes.
The simplest characteristic of a time series is known to be a set of the Lyapunov
exponents whose largest positive value yields a predictability domain in the system
behaviour. The degree of complexity in such a behaviour is determined using the
Kolmogorov-Sinai entropy that is equal to the sum over positive magnitudes of the
whole set of the Lyapunov exponents and can be reduced to the usual Shannon value
in the information theory. Transferring to the nonlinear system, the probability pn
c© A.Olemskoi, S.Kokhan 761
A.Olemskoi, S.Kokhan
of n-th scenario of the system behaviour is transformed into the power function pq
n
determined by index q 6 1, so that the Kolmogorov-Sinai entropy should be replaced
by the Renyi entropy
Kq ≡
1
1 − q
ln
∑
n
pq
n . (1)
Correspondingly, the master equation dp/dε = −βpq, β = const > 0 describes the
probability variation with energy ε = εn to derive the Tsallis distribution p(ε) ∝
[1 − (1 − q)βε](1−q)−1
[3]. In the limit q → 1, this distribution assumes the usual
Boltzmann form p(ε) ∝ exp(−βε) decaying exponentially fast contrary to the power
asymptotic of the Tsallis exponent. Physically, such a behaviour is caused by a
self-similarity of the non-extensive system [4].
As a result, the problem appears to study the self-similar time series that present
the processes corresponding to Zipf-Mandelbrot power-law distribution. This work
is devoted to both analytical and numerical considerations of this type of time series
as a thermodynamic system. It is worth stressing that our approach is principal-
ly different to the pseudo-thermodynamic formalism developed for considering the
multifractal type objects [6]. Indeed, if within the latter formalism, the role of pa-
rameters of state is played by the related multifractal indices, we introduce a set of
effective parameters of state, whose meaning is of true thermodynamic type (hence,
an effective temperature is a measure of data scattering related to the fractal di-
mension of the time series).
The paper consists of two main sections 2 and 3, the first of which is devoted
to the analytical consideration and the second one – to numerical study. We start
the analytical consideration with the elaboration of a model which allows us to
address a self-similar time series as slightly non-extensive thermodynamic system.
Then, we calculate the entropy, internal energy and temperature of the time series.
We show that a temperature governing the time series statistics is an exponential
measure of a self-similarity index related to the fractal dimension. The testing of
the analytical consideration is based on numerical scheme of non-extensive random
walk [7] whose stochastic equation and its solutions are treated in section 3. As
a result, we obtain non-trivial time series whose form is governed by the friction
coefficient that fixes the related fractal dimension. We introduce a statistical scheme
that allows us to consider the modelled time series as a grand canonical ensemble for
which we calculate the entropy and the internal energy as functions of the particle
number. We find the numerically effective temperature and show that its value is
reduced to the averaged kinetic energy per one particle of the ideal gas.
2. Analytical study of self-similar time series as a slightly non-
extensive ideal gas
The principal peculiarity of time series is that it evolves during a large enough
but finite time interval T < ∞. On the other hand, the self-similarity means that a
762
Effective temperature of self-similar time series
series is related to a fractal manifold characterized by the fractal dimension D. We
show hereinafter that the both properties pointed out are taken into account in a
natural way if we use non-extensive statistical mechanics where a simple combination
of quantities T and D fixes the non-extensivity exponent q.
We start with considering the d-dimensional time series x(ti) related to the set
{xi} of consequent values xi ≡ x(ti) of the principle variable x(t) taken at a discrete
time instant ti ≡ iτ that we obtain as a result of dividing the whole time series
length T ≡ N0τ by N0 equal intervals τ . Following the ergodic hypothesis we shall
imitate the time series x(ti) by a set of generalized coordinates {xn} supplemented
by a conjugated set {vn} of velocities that show the jumping rates of coordinates
xi with the time variation. It is principally important to take into account that the
mapping of the time series x(ti) into the manifold {xn} is not one-to-one correspon-
dence because different terms x(ti) and x(tj) may be equal (similarly, this occurs
at mapping the velocity time series v(ti) into the related manifold {vn}). This pe-
culiarity is displayed first of all in that the number N0 of specimens of the time
series is much greater than the total number N of the terms of the related manifold.
Hereinafter, just the latter number N plays the role of the particle number of the
statistical system. Obviously, this system should be considered as grand canonical
ensemble with variable number Nn of particles in a state n.
Along the line of the ergodic hypothesis, the paradigm of our approach is to
address the time series as a physical system defined by an effective Hamiltonian
H = H{xn,vn} based on which the statistical characteristics of this series could
be found. If one proposes that the series terms xn related to different n are not
associated, then the effective Hamiltonian is additive:
H =
N
∑
n=1
εnNn, εn ≡ ε(xn,vn). (2)
Physically, this means that the series under consideration is relevant to an ideal gas
comprising N identical particles with energy εn and number Nn in state n. Further,
we suppose different terms of the time series to be statistically identical, so that the
effective particle energy does not depend on the coordinate xn: ε(xn,vn) → ε(vn).
Moreover, since this energy does not vary with the inversion of the coordinate jumps
xn − xn−1, the function ε(vn) should be even. We use the simplest square form
εn =
1
2
v2
n , (3)
which is reduced to the usual kinetic energy for a particle with mass 1. In the
simplest case of Markovian consequence, the velocity is defined as follows:
vn ≡ xn − xn−1
τ
. (4)
To study the behaviour of the time series as a whole one needs to fulfill the
summation over a set of states given by the manifold {xn,vn} that is relevant to the
763
A.Olemskoi, S.Kokhan
system phase space. In so doing, it is convenient to pass to the related integrations
as follows:
∑
{xn,vn}
⇒
∫∫ N
∏
n=1
dxndvn
N !∆
= N−1
N
∏
n=1
∫∫
dyndun . (5)
Here, the factorial takes into account the statistical identity of the time series terms,
∆ is the effective Planck constant that determines a minimal volume of the phase
space per a particle related to a term. The inverted factor
N ≡ N !
(
X2
τ∆
)−dN
'
[
eX2d
N(τ∆)d
]−N
(6)
is caused by the change of variables yn ≡ xn/X, un ≡ τvn/X rescaled with respect
to macroscopic length X chosen to guarantee the conditions
∫
dyn = 1,
∫
dun = 1.
As explained in the introduction, the self-similarity condition leads to an appli-
cation of Tsallis statistics. The last one is characterized by distribution pn whose
tail decays as a power-law function. The distribution obeys the condition
∑
n
pq
n ≡ 〈1〉q 6= 1 (7)
so that the definition of the internal energy reads:
E =
∑
n
εnp
q
n
〈1〉q
. (8)
To take into account these constraints, an escort distribution was proposed to use [8]
Pn ≡ pq
n
〈1〉q
. (9)
In the explicit form it reads as follows:
Pq{yn,un} =
1
Z
[
1 − (1 − q)H{yn,un}−E
〈1〉qTs
]
q
1−q
at (1 − q)H{yn,un}−E
〈1〉qTs
< 1,
0 otherwise.
(10)
Here, the partition function is defined by the condition
Z ≡ N−1
N
∏
n=1
∫∫
[
1 − (1 − q)
H{yn,un} − E
〈1〉q Ts
]
q
1−q
dyndun , (11)
where 0 < q < 1 is a parameter of non-extensivity, Ts is energy scale. Internal energy
E is determined by the equality
E ≡ N−1
N
∏
n=1
∫∫
H{yn,un}Pq{yn,un}dyndun , (12)
764
Effective temperature of self-similar time series
and normalization parameter 〈1〉q = Z1−q is expressed by the partition function (11)
in accordance with normalization condition.
To check the statistical scheme proposed let us first address the trivial case of
time series xn = const. Here, the particle energy ε is a constant as well, so that the
Hamiltonian is H = Nε. The partition function Z = N−1 and the normalization
parameter 〈1〉q = N−(1−q) are given by the inverted normalization factor (6), while
the internal energy E = Nε is reduced to the Hamiltonian. Then, the entropy
H = −a lnN , a = 1/2 · (1 − q)dN 6= 0 is reduced to zero if only the normalization
factor takes the value N = 1. As a result, we find the effective Planck constant:
∆ =
( e
N
)
1
d X2
τ
. (13)
Our future consideration is based on the assumption that the volume V ≡ Xd
of d-dimensional domain of the xn coordinate variation depending on the particle
number N is governed by Lévy-type law
Xd = xdN
1
z . (14)
Here, x is a microscopic scale and a dynamic exponent z is reduced to the fractal
dimension D of self-similar manifold [5]
z = D. (15)
Then, one obtains the following scaling relation for the phase space volume per a
term of the time series:
∆d = e
(
x2
τ
)d
N
2
D
−1. (16)
In the case of Gaussian scattering, when D = 2, the minimal volume ∆d of the phase
space does not depend on the number N of the particles. Such a condition approves
of our choice of the relation (14) for the whole volume V ≡ Xd as the function of
the number N of time series terms.
Now we consider the main thermodynamic quantities of a time series represented
as an ideal gas. In so doing, we are based on the expressions for the partition function
(11) and the internal energy (12), whose explicit form has been obtained in [8–10].
The principal point of our approach is the entropy definition
H ≡ aKq = a ln Z =
dN
2
ln 〈1〉q , a ≡ 1
2
(1 − q)dN, (17)
that is chosen to guarantee the main thermodynamic relations. Within the limits
1 − q � 2/d, N � 1 (18)
one has
H ' Na
2(1 − a)
ln
[
e2+d
(
2πTs
∆2
)d (
Xd
N
)2
]
. (19)
765
A.Olemskoi, S.Kokhan
Taking into consideration the scaling relation (14), this expression takes the usual
form
H = N
D − 1
D
ln
(
G
N
)
, G ≡ (2πeTs)
dD
2
(x
τ
)−dD
(20)
if the dynamic exponent is determined as
z ≡ D = (1 − a)−1. (21)
Respectively, the internal energy and the normalization parameter read:
E =
dN
2
(
G
N
)
2a
d
Ts, 〈1〉q =
(
G
N
)
2a
d
, a ≡ D − 1
D
. (22)
The temperature is defined as follows [9]
T ≡ 〈1〉q Ts =
(
G
N
)
2a
d
Ts , (23)
where the last equality takes into account the second of the relations (22). This
definition guarantees the equipartition law
E = CT, C ≡ cN, c ≡ d/2 , (24)
where the quantity C = ∂E/∂T is the specific heat. It is easy to prove that equations
(20)–(23) arrive at standard thermodynamic relation ∂H/∂E ≡ T−1.
Hereinabove we have considered the simplest model which makes it possible to
examine analytically a self-similar time series in a standard statistical manner. A
peculiarity of the related equalities is a scale invariance with respect to the varia-
tion of the non-extensivity parameter 1 − q which is contained everywhere through
the parameter a, given the last equation (17), that is related to the dynamic expo-
nent z and the fractal dimension D of the time series according to equation (21).
This invariance is clear to have been caused by self-similarity of the system under
consideration.
The main progress in our consideration is that the time series statistics is gov-
erned completely by the temperature (23). Taking into consideration the second of
the equalities (20) and rescaling the temperature unit Ts into Tsc ≡ (2πe)aTs we
arrive at the expression
T
Tsc
=
[
( τ
X
)2
Tsc
]
a
1−a
. (25)
Being independent of the number of terms N , the time series temperature shows the
exponential dependence on the index a related to the fractal dimension according
to equation (21). To establish the character of the power dependence on the ratio of
the range X of the principle variable to the time interval τ , it is natural to choose
766
Effective temperature of self-similar time series
the measure units of the temperature in the following manner: Tsc ≡ e (X/τ)2,
Ts ≡ e1−a(2π)−a (X/τ)2. Then, the expression (25) for the time series temperature
takes the simplest form
T =
(
X
τ
)2
eD, D ≡ 1
1 − a
(26)
according to which the value T is the exponential measure of the fractal dimension
D of the self-similar time series.
3. Numerical study of time series
The purpose of this section is to numerically verify the definitions based on
expressions (4)–(3), (10)–(12), (20)–(26). As the above analysis has shown, one of
the peculiarities of the statistical system under consideration is that it is slightly non-
extensive due to condition (18). Thus, we can put q = 1 in the following numerical
consideration.
We shall follow the numerical scheme of non-extensive random walk based on
the discrete stochastic equation [7]
xi+1 =
√
τζi +
[
(1 − γτ) +
√
τξi
]
xi . (27)
Here, discrete time ti = iτ is fixed by integers i = 0, 1, . . . , N0 and minimal interval τ ;
ζi and ξi are additive and multiplicative stochastic sources normed with white-noise
conditions 〈ζiζj〉 = 〈ξiξj〉 = δij; friction coefficient γ determines the parameter ν =
γ/(1+γ) which fixes, in accordance with stationary distribution, the non-extensivity
parameter q ≡ (2−ν)−1 = (1+γ)/(2+γ). Making use of iteration procedure (27) we
arrive at stochastic time series, which are explicitly relevant to the Lèvy flights at
friction coefficient −1 < γ, when the parameter ν is negative. Transferring to positive
parameters γ, ν, their growth arrives at gradual transformation of superdiffusion
process into Brownian diffusion that is related to the magnitudes γ = ∞, ν = 1.
As ν increases, the relevant fractal dimension D grows as well. Making use of the
origin time series x(t), it is easy to obtain the velocity time dependencies v(t), in
accordance with definition (4).
To study the statistical properties of the above time series we keep to the ergodic
hypothesis that assumes the identity of averaging over both velocity time series v(ti),
i = 1, 2, . . . , N0 and statistical ensemble {vn}, n = 1, 2, . . . , N , which describes the
velocity scattering in the phase space. We determine such an ensemble dividing
maximum interval of the velocity variation domain into N � 1 zones n = 1, 2, . . . , N ,
within which the velocities have the mean value vn and small variation δvn. Then,
the probability pn to hit the interval [vn − δvn/2, vn + δvn/2] and the relevant
probability density function π(vn) are determined as follows:
pn ≡ νn
N0
, π(vn) ≡ N−1
0
νn
δvn
, (28)
767
A.Olemskoi, S.Kokhan
Figure 1. The probability density function of the velocity distribution at
different fractal dimensions (curves 1 – 5 correspond to parameters ν =
0.85; 0.50; 0.35; 0.99; − 0.10; respectively, fractal dimensions are D =
1.81; 1.58; 1.45; 2.57; 1.45).
where νn is a number of the time series specimens with the mean value vn. According
to figure 1 the probability density function π(v) has the usual bell-shaped form
centered at the velocity v = 0.
Since the distribution (28) corresponds to the grand canonical ensemble, the
related thermodynamic functions are determined by the n-state number
Nn ≡ Npn =
N
N0
νn (29)
and energy (3). Thus, the total energy of the ideal gas reads:
E ≡
N
∑
n=1
εnNn =
N
N0
N
∑
n=1
v2
n
2
νn . (30)
The plots of the dependencies E(N) of energy (30) accompanied with the related
dependencies H(N) of entropy (17) are shown in figures 2 for different values of frac-
tal dimensions D. It is principally important, that the energy E should be directly
proportional to the particle number N , while the entropy H should increase much
more slowly with N . As for the velocity distribution π(v), non-monotonic variations
of both energy and entropy as the functions of the fractal dimension take place, but
the main tendency is the decrease of E and the increase of H with the growth of D.
The peculiarity of the self-similar system under consideration is that it is finite
and is characterized by the following effective temperatures:
Θ−1 ≡ ∂H
∂E
∣
∣
∣
∣
∣
D
, T−1 ≡ N
∂H
∂E
∣
∣
∣
∣
∣
N
. (31)
768
Effective temperature of self-similar time series
Figure 2. Dependencies of energy E and entropy H on the number N
of effective particles (curves 1 – 4 correspond to fractal dimensions D =
1.50; 1.45; 1.58; 1.81).
Being determined at a constant value of the fractal dimension D, the first of these
is a function of the particle number N , whereas the second one depends on the
magnitude D to be determined at a fixed value N . According to the calculations,
the above temperatures are associated by the following relation:
T =
Θ
N
[
1 − 1
(1 − q)D2
(
H
D − 1
+
D − 1
D2
N ln N
)−1
]−1
. (32)
Taking into account the definition (20) where Ts is rescaled by factor N , one obtains
T =
Θ
N
1 −
[
D ln
(
NT
Ts
)
+
2
d
(D − 1)2
D
ln N
]−1
−1
. (33)
Making use of the data presented in figure 2 we receive the dependence Θ(N) for
the first temperature (31) shown in the left panel of figure 3. This temperature takes
very large values which increase directly proportionally to the N number growth.
Such a behaviour displays an intermediate character of the quantity Θ that plays
the role of an effective energy complementary to the energy (30). On the other
hand, using the relation (33) we arrive at the dependencies of the temperature
T (N) shown in the right panel of figure 3. It is seen that with the increase of the
fractal dimension to the values D which are close to D = 2, the dependencies T (N)
approach the constant magnitudes. These dimensions relate to large values of the
parameter ν 6 1 which is connected with non-extensivity index q according to the
equality q = (2−ν)−1. Thus, we find that the temperature T takes the values which
are non-dependent on the particle number N in the region q ≈ 1 that is relevant to
a slightly non-extensive limit where the above analytical consideration is valid. Due
to such a behaviour we can conclude that the temperature T has a usual physical
sense.
769
A.Olemskoi, S.Kokhan
Figure 3. Dependencies of effective Θ and physical T temperatures on the
number N of effective particles (curves 1 – 4 correspond to fractal dimensions
D = 1.50; 1.45; 1.58; 1.81).
To confirm this conclusion we examine the equipartition law (24) rewritten using
the averaged kinetic energy (3):
T = 〈ε〉 ≡ 〈v2〉
2
. (34)
To this end we compare the values 〈ε〉 and T at different magnitudes of the fractal
dimension D in figure 4. We can see that the difference between the above values
Figure 4. Dependencies of the averaged kinetic energy 〈ε〉 and the physical tem-
perature T on the fractal dimension D.
does not exceed 10% if one does not take into account the points related to the three
smallest magnitudes of D where our analysis is not applicable.
770
Effective temperature of self-similar time series
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771
A.Olemskoi, S.Kokhan
Ефективна температура самоподібних часових
наборів
О.Олємской, С.Кохан
Сумський державний університет, Україна
Отримано 18 липня 2005 р., в остаточному вигляді –
17 жовтня 2005 р.
В рамках слабо неекстенсивної статистики та пов’язаної числової
моделі, розробляється картина розгляду самоподібних часових
наборів як термодинамічної системи. Вводяться та тестуються
характеристики термодинамічного типу, що відповідають темпе-
ратурі, тиску, ентропії, внутрішній та вільній енергіям. Показано,
що розвинута статистика підкоряється ефективній температурі,
будучи експонентною мірою фрактальної розмірності часових на-
борів. Тестування аналітичного розгляду базується на числовій
схемі неекстенсивних випадкових блукань. Чисельно знаходить-
ся ефективна температура з метою показати, що її значення
приводить до усередненої енергії на один ступінь вільності.
Ключові слова: часові набори, неекстенсивна статистика,
симуляції
PACS: 05.90.+m, 05.45.Tp, 05.40.Fb
772
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