Application of the Statistical Physics Methods for the …€¦ · · 2016-11-29Application of...
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Applied Mathematical Sciences, Vol. 10, 2016, no. 64, 3143 - 3164
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ams.2016.69248
Application of the Statistical Physics Methods
for the Investigation of Phase Transitions
in the Lotka–Volterra System with
Spatially Distributed Parameters
Yu. V. Bibik
Dorodnicyn Computing Center, Federal Research Center “Computer Science and
Control” of Russian Academy of Sciences
Vavilov str. 40, 119333, Moscow, Russia
Copyright © 2016 Yu. V. Bibik. This article is distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
Abstract
The application of statistical physics methods for the investigation of phase
transitions for a generalization of the Lotka–Volterra system with additional terms
that take into account the spatial distribution of species is considered. The
additional terms are chosen in such a way that the generalized system, as well as
the classical Lotka–Volterra system, is Hamiltonian. Kerner's approach, who was
the first to use statistical mechanics for the analysis of Hamiltonian Volterra-type
systems, is used to investigate the generalized Lotka–Volterra system. In addition,
the Lee–Yang approach is used to calculate zeros of the statistical sum. The
position of these zeros is used to find out whether there are phase transitions in the
model under examination.
Keywords: Lotka–Volterra system, statistical sum, gamma function, Lee–Yang
zeros
1. Introduction
It has been shown (e.g., see [4], [22], [25], [26], [28], [29], [30] [31]) that in some ecological and biological systems processes similar to phase transitions in statistical
3144 Yu. V. Bibik
physics are observed. In such systems, qualitative changes occur similarly to the
statistical and thermodynamic changes that are observed in phase transitions in
statistical physics. Examples of such processes that have not yet found a proper
scientific explanation are as follows:
– Qualitative changes in forest ecosystems in the course of ecological succession
when, e.g., a deciduous forest is replaced by coniferous forest and (or) a
community of herbaceous plants is replaced by a community of woody plants.
– Bursts of mass reproduction of lemmings in tundra zone accompanied by their
migration from the habitat and mass mortality. As a result, the population of
lemmings first significantly decreases and then gradually increases up to the new
burst of mass reproduction.
– Periodically repeating bursts of mass reproduction of forest insects.
– Phase transitions in biological membranes accompanied by the transition from
the liquid crystal state to the solid crystal state and vice versa.
In this paper, we apply methods and approaches of statistical physics for the
analysis of phase transitions in ecological and biological systems. The choice of
statistical physics methods and approaches is explained by the fact that during
many years of the development of statistical physics, a vast amount of
experimental and theoretical data that make it possible to solve complicated
problems have been accumulated.
The theory of phase transitions is in important branch of statistical physics. The
theory of phase transitions and related critical phenomena has a rich history. Many prominent physicists, both theorists and experimenters, contributed to the
development of the theory of phase transitions. Even though the first significant
studies of critical phenomena in various substances appeared as early as in the
second part of the 19th century, the development of this division of the statistical
physics continues.
Studies go on in various divisions of physics, and the accumulated data are
steadily updated. The mathematical methods are elaborated, and advanced devices
are used that improve the accuracy of experiments.
Presently, the modern statistical physics has a variety of methods,
approaches, and theories that make it possible to describe, analyze, and
understand the essence of the processes and critical phenomena that occur
during phase transitions in various systems (see [1], [2], [6], [7], [8], [9], [12],
[17], [20], [21], [23], [27], [43], [48], [49], [50], [52], [54], [55], [56], [57]).
The large number of methods, approaches, and theories is explained by the
complexity and diversity of phase transitions and critical phenomena that cannot
be explained using a unified theory or approach.
The first theory used a hundred years ago to theoretically describe critical
phenomena was the mean field theory.
Among the mean field theories are the Gibbs critical point theory [18], [19], the
Weiss theory of ferromagnetism, and the Landau theory of phase transitions of the
second kind [44], [45].
The mean field theory turned out to be inapplicable when fluctuations become
significant, i.e., in the vicinity of the point of phase transition.
Application of the statistical physics methods for the investigation… 3145
The fluctuation theory of phase transitions of the second kind [53] describes the
interaction of fluctuations, makes it possible to take into account the interaction of
fluctuation phenomena at critical points, and determine the values of critical
parameters.
The microscopic theory of phase transitions makes it possible to investigate the
properties of concrete models that undergo a phase transition. The main difficulty
in the development of the microscopic theory of phase transitions is the absence
of an explicit small parameter.
Recently, quantum phase transitions have been widely studied (see [24], [41],
[42], [71]); transitions of matter from one quantum thermodynamic phase into
another under changes of the external conditions in the case when there are no
thermal fluctuations are investigated.
An important step in the history of the development of phase transition theories
are the theories that appeared in the 1960s and 1970s, which enriched the
statistical physics by the concepts of renormalization group, scale invariance, and
universality (see [10], [11], [13], [14], [15], [16], [33], [34], [35], [36], [64],
[65], [66], [67], [68], [69], [72]).
Using the coarsening techniques, these methods provide a tool for the analysis of
such difficult physical problems as quantum field theory and the theory of critical
phenomena.
The renormgroup divides the ferromagnetic system in the vicinity of a critical
point into cells of which each contains a small number of atomic-level magnets;
the size of cell determines the scale. This method made it possible to describe the
behavior in the vicinity of the critical point and obtain a quantitative estimate of
the properties of the system under examination on a computer.
The scale invariance is the property of the equations describing a physical theory
or process to remain invariant when all the distances and time intervals are
multiplied by the same factor.
A large number of model systems are used in statistical physics to describe
and investigate phase transitions and critical phenomena. Such models include
lattice magnetic models, such as two-dimensional and three-dimensional Ising
models, Heisenberg's model, Matsubara and Matsuda models, Stanley n-
component models, spherical Berlin and Kac model, lattice percolation model
used to investigate complex systems, the lattice gas model for the description of
the liquid–vapor critical point, and others.
Among them, the Ising model, which is widely used for the investigation of phase
transitions, deserves special attention. This model was designed for the
investigation and description of ferromagnetic phenomena. Ising claimed in 1924
(see [32]) that there are no phase transitions in one-dimensional systems and the
one-dimensional lattice does not exhibit ferromagnetism.
In 1942, Onsager obtained an exact solution for the two-dimensional Ising model
[51].
One of the first researchers who applied the methods and approaches of
statistical physics for the investigation of biological models was the American physicist E.H. Kerner. Beginning from 1957, he published a series of papers [37],
3146 Yu. V. Bibik
[38], [39], [40] devoted to the application of statistical mechanics methods for the
analysis of Volterra-type models. Using the features of these systems, Kerner
developed for them analogs of the main thermodynamic parameters, such as
internal energy, entropy, and statistical sum. Kerner explicitly calculated the
statistical sum for the Lotka–Volterra equation. He also discovered a remarkable
feature of the Hamiltonian of the Lotka–Volterra system. This is the fact that the
statistical sum for this system can be calculated explicitly in terms of the gamma
functions up to a factor representable in terms of the elementary functions.
Kerner's studies confirmed the possibility and usefulness of applying the statistical
mechanics methods to the analysis of Volterra-type equations.
In addition to Kerner's studies, the methods of statistical physics were used for the
investigation of biological and ecological models in [3], [5], [58], [59], [60].
In this paper, we want
– to use the result of theoretical and experimental studies of phase transitions and
critical phenomena obtained in statistical physics for the investigation of phase
transitions in biological and ecological systems by finding an appropriate simple
mathematical biology model;
– to find a calculation method and use mathematical transformations and
approximations that would enable us to obtain the result as easily as possible
(practically manually) without using long and complicated computer calculations;
– to obtain an analytical solution that would enable us to determine graphically
whether there is a phase transition in the system.
The aim of this paper is to develop an approach that provides simple means for
determining whether there is a phase transition in the system.
For the investigation of qualitative changes in biological and ecological
systems that resemble phase transitions in physical systems, we use a
generalization of the classical Lotka–Volterra model [47], [61], [62], [63] in
which terms taking into account the spatial distribution of species are added.
As the investigation techniques, we use the approach of Kerner, who was the
first to apply statistical physics methods for the analysis of Volterra systems,
and the Lee–Yang approach [46], [70], which allows one to detect the existence
of phase transitions from the position of zeros on the complex plane.
The thermodynamic state and properties of the system are studied using the
statistical sum, which is represented in terms of a simple combination of
elementary and gamma functions. The existence and nonexistence of phase
transitions is determined based on the position of the zeros of the statistical sum
on the complex plane.
The paper is organized as follows:
1. Introduction.
2. The description of the investigation techniques and features of the model.
3. Calculation of the statistical sum for the model under examination.
4. Investigation of the phase transition in the model.
5. Conclusions.
Application of the statistical physics methods for the investigation… 3147
2. The description of the Investigation Techniques and Features of
the Model
2.1. The Investigation Technique
We use the approach of Kerner [3], [38], [39], [40], who was the first to apply
statistical mechanics methods for the analysis of Volterra systems. Kerner used a
change of variables that allowed him to calculate the statistical sum in terms of the
gamma functions accurate to elementary functions. Kerner applied the change of
variable directly to the whole statistical sum. In this paper, we cannot apply
Kerner’s approach directly because in our case we have extra terms added to the
classical Lotka–Volterra system that take into account the spatial distribution of
species. This considerably complicated the equations under examination. To
overcome this difficulty, we expand the statistical sum into Taylor’s series and
apply Kerner’s change of variable to each term of this series, which has a proper
form. In this case, each term is represented by a combination of the gamma
functions and elementary functions. Then, the series is summed to obtain the final
statistical sum.
The existence or nonexistence of phase transitions in the system is determined
graphically judging by the position of the zeros of the statistical sum on the
complex plane.
2.2. A Description of the Model Features
The classical Lotka–Volterra system has the form
dt
d,
dt
d. (2.2.1)
Here is the population of prey, and is the population of predators.
The classical Lotka–Volterra system does not take into account the dependence of
the species of predators and prey on spatial coordinates. Taking into account this
dependence can considerably improve the analytical power of the model. To take
the spatial dependence into account, we denote the flows of predators and prey by
1j and 2j , respectively. The generalization of the Lotka–Volterra system with
account for the spatial dependence of the predator and prey population has the
form
1divjdt
d
, (2.2.2)
2divjdt
d
. (2.2.3)
3148 Yu. V. Bibik
The contribution of the predator and prey population to time derivatives is
determined by the divergences of the flows 1j and 2j .
We assume that the flow 1j is equal to the gradient of the predator population
with the opposite sign:
1j . (2.2.4)
The interpretation is that the prey tend to the place where the population of
predators is lower. Similarly, we assume that the flow of predator population 2j
equals the gradient of the prey population :
2j . (2.2.5)
This means that the predators tend to the place where the prey is more numerous.
Under these assumptions, the generalization of the Lotka–Volterra system takes
the form
dt
d, (2.2.6)
dt
d. (2.2.7)
To be able to apply the statistical mechanics techniques to the above
generalization of the Lotka–Volterra system with account of the spatial
dependence, we transform Eqs. (2.2.6) and (2.2.7) to Hamiltonian form. Make the
change of variables lnq , lnp .
In the new variables, Eqs. (2.2.6) and (2.2.7) take the form
22 )()(1 y
p
yy
p
x
p
xx
pp pepepepeedt
dq , (2.2.8)
22 )()(1 y
q
yy
q
x
q
xx
qq qeqeqeqeedt
dp . (2.2.9)
We reduce the terms containing spatial derivatives appearing in Eqs. (2.2.8) and
(2.29) to Hamiltonian form. We assume that the values of the spatial derivatives
are small and, therefore, we can neglect the quadratic terms. In addition, the same
assumptions imply that the coefficients multiplying pe and qe may be replaced
by constants. Using appropriate linear transformations of the variables x and y ,
Application of the statistical physics methods for the investigation… 3149
we can make these constants equal to unity. Then, Eqs. (2.2.8) and (2.2.9) take the
form
yyxx
p ppedt
dq1 , (2.2.10)
yyxx
q qqedt
dp 1 . (2.2.11)
Equations (2.2.10) and (2.2.11) are Hamiltonian equations with the Hamiltonian
)2
1
2
1
2
1
2
1( 2222
yxyx
pq qqpppeqedxdyH . (2.2.12)
Unfortunately, the form of Hamiltonian (2.2.12) does not allow one to use
Kerner's approach because the statistical sum cannot be explicitly calculated in
terms of the gamma functions accurate to elementary functions. To resolve this
problem, we select new interaction terms that are similar to the interaction terms
in Hamiltonian (2.2.12). These new interaction terms should enable us to calculate
the statistical sum in terms of the gamma and elementary functions. We select a
replacement for the terms 2
xp and 2
yp in Hamiltonian (2.2.12) using linear and
exponential functions. An appropriate function to replace 2x is xe x . Instead of
the variable x , we will use xp and yp .
Upon the transformations, Hamiltonian (2.2.12) takes the form
)( yx
qqq
yx
ppp qqeeqeppeepedxdyH yxyx . (2.2.13)
After the integration, the total derivatives xp , yp ,
xq , and yq disappear. The
resulting expression for the Hamiltonian is
)( yxyxqqqppp eeqeeepedxdyH . (2.2.14)
For this Hamiltonian, the Hamiltonian equations take the form
yy
p
xx
pp pepeedt
dqxx 1 , (2.2.15)
yy
q
xx
qq qeqeedt
dpxx 1 . (2.2.16)
3150 Yu. V. Bibik
Due to the above assumption that xp , yp ,
xq , and yq are small, the quantities
,xpe yp
e and xqe , yq
e are close to unity. The Hamiltonian equations (2.2.15) and
(2.2.16) are close to the Hamiltonian equations (2.2.10), (2.2.11). This confirms
the fortunate selection of the new spatial terms. In the next section, we calculate
the statistical sum for Hamiltonian (2.2.14).
3. Calculation of the Statistical Sum for the Model under
Examination
In this section, we calculate the statistical sum for Hamiltonian (2.2.14) with the
new spatial terms. To this end, we replace the continuum of the points yx, with
the discrete set of points on which the periodic boundary conditions are given. It
suffices to select three points on each axis x and y . As a result, a square grid
consisting of nine points will be used for the calculations. The derivatives of the
dependent variables p and q with respect to x and y will be replaced by finite
differences. For simplicity, the grid size is set to unity. For convenience, we
rename q to and p to . Note that, due to the additivity of Hamiltonian
(2.2.14), the statistical sum to be calculated is represented as the product of two
identical statistical sums
21ZZZ . (3.1)
The statistical sum 1Z has the form
)1,,(
,
),1,(
,
,, )(
1 ][
jiji
ji
jiji
ji
jiji eee
eedZ
. (3.2)
The statistical sum 2Z has the same form as 1Z .
Following Kerner's method, we try to represent the expression for the statistical
sum by a combination of the gamma and elementary functions. However, the term
in (3.2) containing )( ,1, jijie prevents the representation of the integral of interest
as Euler's integral, which is the integral expression for the gamma function. To
overcome this difficulty, we make the following transformations in the expression
ji
jijijiji eeeE ,
1,,,1, )()(
(3.3)
in (3.2). Assume that the major contribution to the statistical sum is made by the
small values ji , . Then, we have the following heuristic consideration for the
transformation of jijie ,1, . Note that both 1, jie
and 1,1 jie
are small.
Therefore, it holds that
Application of the statistical physics methods for the investigation… 3151
....)1()1(1
)]1(1)][1(1[
,1,
,1,,1,,1,
jiji
jijijijijiji
ee
eeeee
. (3.4)
Next, we replace ji ,1 by 1,1 jie
to obtain
ji
jiji
jijiji ee eeeeee,1
,,1
,,1, )1(
. (3.5)
Then, formula (3.3) can be replaced by
ji ji
jieji
ji
jijijji eeeee
ee , ,
,,
,
1,,,1, 22)()(
. (3.6)
Upon these transformations, the expression for the statistical sum 1Z takes the
form
ji ji
jieji
ji
jiji eeee
i eedZ , ,
,,
,
,, 22)(
1 ][
. (3.7)
The last term in the exponent in (3.7) prevents one from representing integral
(3.7) by Euler's integrals. To overcome this difficulty, we expand the last
exponential function in this formula in Taylor's series:
)......))()((6
)2(
))((2
)2()2(1(][
, ,
3
,,
2
,
2
1
,
1
,,
,,,
,
,
,
,,
ji qp
e
lk
ee
ekji
e
ji
eee
qpplkji
ekjiji
ji
ji
ji
jiji
eeee
eee
eeeedZ
.
(3.8)
Now, each term in Taylor's series can be represented by Euler's integral, which
has the form
dtetz tz
0
1)( . (3.9)
The typical integral in the series is
BeAedI . (3.10)
It is reduced to Euler's integral by the change of variables Bet :
3152 Yu. V. Bibik
A
Be
A
A
A
BeAA
B
Ae
B
BeBed
B
eB
Be
BedI
)()()(
)( 1
. (3.11)
Thus, we can reduce the series to a combination of the gamma and elementary
functions:
....)]))2((
)([]
)1)2((
)([639]
))2((
)([
)1)2((
)(
)2)2(
)(81
]))2(
)([
)3)2((
)(9(
6
)2(]
))2((
)([]
)1)2(
)([72
])2((
)([
)2)2((
)(9(
2
)2(]
))2((
)([
)1)2((
)(9)2(]
))2((
)([
637
83
72
82
89
1
e
eeZ
(3.12)
In series (3.12), the combination of the gamma and elementary functions
9]))2((
)([
can be factored out, and the remaining terms are represented by
elementary functions:
....).)
))2(
11(
639
))2(
11()
)2(
21(
81
))2(
31(
9(
6
)2(
)
))2(
11(
172
))2(
21(
9(
2
)2()
))2(
11(
1(9)2(1(]
))2((
)([
3
3
2
29
1
e
eeZ
(3.13)
It is seen from (3.13) that all terms of the series consisting of elementary functions
include the same expression
)
)2((1(
n. For the case 0 , it is equal to
unity:
1)))2(
(1( 0
n. (3.14)
A remarkable property of this expression is that it tends to a constant as .
This constant is
2)))2(
(1(
n
en
. (3.15)
Application of the statistical physics methods for the investigation… 3153
Only the linear (in ) term in (3.13) contains a single expression
)
)2(
1(1(
. The coefficients of the other terms with each power of
consist of combinations of one or several such expressions. However, the
structure of the right-hand side of formula (3.15) is such that all the combinations
of exponential functions multiplying have the same exponent for each
term of the series, which coincides with the index of the series term. This allows
us to individually sum up the terms in (3.13) as 0 and and
extrapolate the statistical sum using two values of the series. There is a more
rigorous approach. The major contribution to the terms of series (3.13) is made by
the powers of the function
)
)2(
11
1(
. The other terms exhibit similar
behavior for each n . For this reason, we expand the function
n)
)2(
11
1(
and its powers into the sum of expressions for each of which series (3.13) can be
summed separately. The desired formula has the form
2)1()
)2(
11
1(
nn eee nn . (3.16)
In (3.16), we need to determine the parameters n . Note that the right-hand side
of (3.16) provides the correct value for the power n of the function
)
)2(
11
1(
as 0 and .
It remains to require the left- and right-hand sides of formula (3.16) be equal at a
certain intermediate value of . Set 1 , and consider the case when 1 .
Then, the left-hand side of (3.16) is
nn )4
3()
)2(
11
1(
. (3.17)
Upon our transformations, formula (3.16) has the form
3)1()4
3(
nn eee nn
. (3.18)
3154 Yu. V. Bibik
Now, (3.18) implies a formula for n :
.)716.0(1
)954.01()75.0(
)716.0(1
)75.0
716.0(1
)75.0()716.0(1
)716.0()75.0(
)(1
)()4
3(
3
1
3
1
n
nn
n
n
n
n
nn
n
nn
e
e
e n
(3.19)
We transform (3.19) to a form that can be used for the summation of series (3.13):
161.0)75.0(284.0
046.0)75.0(
))284.01(1(
))046.01(1()75.0( nn
n
nn
n
ne n
. (3.20)
Formula (3.20) gives the final expression for n . Now we turn to determining the
parameter ne
. To this end, we raise the left- and right-hand sides of Eq. (3.20)
to the power and introduce the parameters )161.0ln(1 and
)75.0ln(1 . Then we obtain
11)161.0ln()75.0ln( nn eeeee n
. (3.21)
After these transformations, we can write series (3.13) as
....).))1((6
)18()1((
2
)18(
))1()(18(1(]))3((
)([
1333
3
22
2
3
19
1
1111112
11
1111
eeee
eeee
eeeeZ
(3.22)
Series (3.22) can be divided into three series of which each is just a series for the
exponential function with the corresponding exponent. The first series
corresponds to 11 nee , the second series to 3
n
e
, and the third series to
113 n
n
eee
. Each of these series is easily summed, which yields the formula
).(])3((
)([
)1(1(])3((
)([
13/21
3/211
13/21
3/2111
1818189
1818189
1
eeeee
eeeee
eeeee
eeeeeeeZ
(3.23)
Application of the statistical physics methods for the investigation… 3155
The statistical sum (3.23) interpolates the behavior of series (3.13). This fact can
be verified in the following way.
As 0 , the second and third terms in (3.23) cancel out, and only the first term
remains, which takes the form ee 18 .
As 0 , the same behavior is exhibited by formula (3.13), which confirms the
validity of our approximation.
This completes the calculation of the statistical sum for our model. Formula (3.23)
gives an approximation of the statistical sum for the generalized Lotka–Volterra
system with spatial terms. The derivation of this formula is an achievement
because the calculation of the statistical sum for the majority of models is a very
difficult task.
We managed to calculate the statistical sum analytically due to the following
trick. The last exponential function in (3.7) was expanded in Taylor's series, and
Kerner's change of variables was made in each term of this series. As a result,
each term of the series was represented by a combination of the gamma and
elementary functions. This allowed us to perform the inverse operation, i.e., sum
up the remaining series and represent it as a combination of exponential functions
multiplied by a term containing the gamma function. This is an advantage of the
proposed method because such a combination is most convenient for the further
analysis.
In the next section, we will investigate the phase transition in the model using the
expression of the statistical sum obtained above.
4. Investigation of the phase transition in the original model
The existence or nonexistence of phase transitions in the system is determined
analogously to the Lee–Yang approach [46], [70] by the position of the zeros of
the statistical sum on the complex plane. For the analysis of the position of zeros
of the statistical sum, only the expression
)(13/2
13/21
1 181818
eeeee eeeee in (3.23) is used because the
expression in brackets makes no contribution to the zeros of the statistical sum.
For this expression, a simple computer program for determining the absolute
value of this expression was developed. This absolute value was plotted using the
program Surfer. Figures 1 and 2 show the level lines of the absolute value, which
can be used to visually determine the position of zeros of the statistical sum. For
the construction of the level lines, the expression
)1(3/213/2
13/21
1 18181818 eeeeee eeee
was used, which makes it
possible to view the surface structure in the domain of the small terms for (3.23).
It is seen from Fig. 1 that the generalized Lotka–Volterra system with spatial
terms has a phase transition. This is seen from the behavior of two zeros of the
statistical sum, which are symmetric about the real axis. These two zeros are in
the right half-plane of the complex plane near the real axis. The distance from the
zeros to the real axis is 0.19. The line passing through these zeros intersects the
real axis at the point 0.034.
3156 Yu. V. Bibik
Figure 1 was constructed for the additional spatial terms )( yxpp
eed in
Hamiltonian (2.2.14) for the coefficient 1d . This implies that the phase
transition point is 0.034.
To verify this result, the level lines for the same model with the coefficient 3d
were constructed (Fig. 2). The zeros for this case are clearly seen; they testify that
the phase transition also exists in this case. The distance from two symmetric
(about the real axis) zeros from the real axis is small—it is equal to 0.13. The line
passing through these zeros intersects the real axis at the point 0.03. This implies
that the phase transition point is 0.03.
The third (rightmost) zero in Figs. 1 and 2, which lies on the real axis, has nothing
to do with phase transitions. This zero appears due to small errors in the
calculation of the approximate statistical sum (3.23) as a result of cancelling out
small terms consisting of exponential functions with large negative exponents.
Thus, the data above imply that the proposed generalization of the Lotka–Volterra
system with spatially distributed parameters has a phase transition.
0.00 0.10 0.20 0.30
-0.30
-0.20
-0.10
-0.00
0.10
0.20
0.30
Y
X
Fig. 1
Roots of statistical sum for d=1
Application of the statistical physics methods for the investigation… 3157
0.00 0.10 0.20 0.30
-0.30
-0.20
-0.10
-0.00
0.10
0.20
0.30
Y
X
Fig. 2
Roots of statistical sum for d=3
5. Conclusions
In statistical physics, there are a lot of techniques for the analysis of phase
transitions. For each problem, an appropriate most convenient approach can be
found. The Lee–Yang method [46], [70] makes it possible to investigate the phase
transitions judging by the position of the zeros of the statistical sum on the
complex plane. To apply this method, one must know how to calculate the
statistical sum, which is a difficult problem. In this paper, the case when this sum
can be calculated was considered. A generalization of the Lotka–Volterra system
that takes into account the spatial variables was studied. The Lotka–Volterra
system has a specific Hamiltonian that allows one to calculate the statistical sum
in terms of gamma and elementary functions. A special choice of the spatial terms
3158 Yu. V. Bibik
makes it possible to preserve this property for the generalized system, which was
done in the present paper.
The addition of spatial dependence of the population of species to the classical
Lotka–Volterra system makes it possible to study more complex behavior
patterns, including the emergence of phase transitions.
The Hamiltonian equations (2.2.15), (2.2.16) for the system under examination
are nothing more nor less than nonlinear wave equations that have soliton-like
solutions. In this system, as in all systems with phase transitions, there are two
counteracting factors. The first factor "wants" to order the system in the form of
soliton-like solutions. The second factor, in the form of temperature, strives to
destroy the ordering. At a certain relation of these factors, a phase transition
occurs.
The proper choice of the functions that take into account the spatial dependence of
the population of species and their ultimate critical population made it possible to
perform all the calculations analytically. As a result, the position of zeros of the
statistical sum on the complex plane was analyzed analytically, and the condition
of the occurrence of the phase transition was obtained judging by this position.
It seems that the study should be continued to obtain a greater number of zeros
using a grid with a greater number of points. Since this considerably complicates
the mathematical calculations, this issue is left out of the scope of the present
paper.
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Received: October 12, 2016; Published: November 28, 2016