Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space
We consider limit cycle oscillators in terms of canonical-dissipative systems that exhibit a short-range interaction in velocity and position space as described by the Dirac delta function. We derive analytical expressions for stationary distribution functions in phase space and energy space and pro...
Saved in:
| Published in: | Condensed Matter Physics |
|---|---|
| Date: | 2010 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут фізики конденсованих систем НАН України
2010
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/32038 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space / S. Mongkolsakulvong, T.D. Frank // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13001: 1-18. — Бібліогр.: 49 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860149389472825344 |
|---|---|
| author | Mongkolsakulvong, S. Frank, T.D. |
| author_facet | Mongkolsakulvong, S. Frank, T.D. |
| citation_txt | Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space / S. Mongkolsakulvong, T.D. Frank // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13001: 1-18. — Бібліогр.: 49 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | We consider limit cycle oscillators in terms of canonical-dissipative systems that exhibit a short-range interaction in velocity and position space as described by the Dirac delta function. We derive analytical expressions for stationary distribution functions in phase space and energy space and propose a numerical simulation scheme for the simulation of the many body system as well. We show that the short-range interaction squeezes or stretches energy distribution functions depending on whether the interaction can be regarded as attractive or repulsive. In addition to the interaction effect, we show that energy distribution functions become narrower when limit cycle attractors become stronger. Finally, energy distributions become broader when the pumping energy is increased. The latter effect however disappears in the high energy domain.
Розглянуто граничні циклічні осцилятори в термінах канонічно-дисипативних систем, які демонструють короткосяжну взаємодію у швидкісному та позиційному просторі, що описаний дельта-функцією Дірака. Виведено аналітичні вирази для стаціонарних функцій розподілу у фазовому просторі та енергетичному просторі та запропоновано числову симуляційну схему для моделювання багаточастинкової системи. Показано, що короткосяжна взаємодія стискає чи розтягує енергетичні функції розподілу залежно від того, чи взаємодію можна вважати притягальною, чи відштовхувальною. Крім впливу взаємодії, показано, що енергетичні функції розподілу стають вужчими, якщо межа циклічних атракторів стає сильнішою. Накінець, енергетичні розподіли стають ширшими з ростом енергії накачки. Цей ефект, проте, зникає у високоенергетичній області.
|
| first_indexed | 2025-12-07T17:51:12Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 1, 13001: 1–18
http://www.icmp.lviv.ua/journal
Canonical-dissipative limit cycle oscillators with a
short-range interaction in phase space
S. Mongkolsakulvong1, T.D. Frank2∗
1 Faculty of Science, Department of Physics, Kasetsart University, Bangkok 10900, Thailand
2 Center for the Ecological Study of Perception and Action, Department of Psychology, University of
Connecticut, 406 Babbidge Road, Unit 1020, Storrs, CT 06269, USA
Received July 16, 2009, in final form December 16, 2009
We consider limit cycle oscillators in terms of canonical-dissipative systems that exhibit a short-range inter-
action in velocity and position space as described by the Dirac delta function. We derive analytical expres-
sions for stationary distribution functions in phase space and energy space and propose a numerical simu-
lation scheme for the simulation of the many body system as well. We show that the short-range interaction
squeezes or stretches energy distribution functions depending on whether the interaction can be regarded as
attractive or repulsive. In addition to the interaction effect, we show that energy distribution functions become
narrower when limit cycle attractors become stronger. Finally, energy distributions become broader when the
pumping energy is increased. The latter effect however disappears in the high energy domain.
Key words: canonical-dissipative systems, Haissinski equation, Tsallis entropy, nonequilibrium systems,
nonlinear Fokker-Planck equations
PACS: 05.40.-a, 05.45.Tp, 29.27.Bd, 41.75.-i
1. Introduction
Classical Newtonian and Hamiltonian mechanics is a fundamental topic in physics. With the
increasing number of interdisciplinary applications of physics the question arises to what extent
the concepts of classical mechanics can be generalized for more general classes of systems as ad-
dressed by classical mechanics. In this context, the concept of canonical-dissipative systems has
been developed to address self-propagating systems, self-excited systems, and in particular limit-
cycle oscillators that are supplied by energy sources [1–5]. Several studies have demonstrated that
the canonical-dissipative approach can yield helpful insights into this kind of open systems [6–18].
Let us briefly review some of the key properties of a typical canonical-dissipative system [1–5]
that will be essential for our present study. While the transient dynamics of a canonical-dissipative
system differs qualitatively from the dynamics of a classical Newtonian and Hamiltonian system, in
the long term limit the trajectory of a canonical-dissipative system converges to the trajectory of
a classical Newtonian and Hamiltonian system. The reason for this behavior is that the invariants
of motions of the corresponding classical, dissipation-free Hamiltonian system are not constant for
the canonical-dissipative system. Rather, these invariants evolve as functions of time during the
aforementioned transient period and only in the long term limit converge to particular stationary
values. The evolution of the invariants is caused by several possible environmental impacts in terms
of energy, matter, or information supply. Taking a dynamic systems perspective, these impacts
may be simply classified into two categories: damping and pumping. That is, we may say that the
evolution of the invariants is determined by the interplay of the damping and pumping forces that
reflect the interaction of the system under consideration with its environment. In the long time limit,
a balance between damping and pumping is established, the invariants become constant and the
canonical-dissipative system evolves on a trajectory which is consistent with a classical Hamiltonian
system. The canonical-dissipative approach is not only concerned with the deterministic evolution
∗
E-mail: till.frank@uconn.edu
c© S. Mongkolsakulvong, T.D. Frank 13001-1
http://www.icmp.lviv.ua/journal
S. Mongkolsakulvong, T.D. Frank
of open dynamic systems but also is concerned with the relationship between damping forces
and fluctuating forces. In doing so, stochastic system properties can be addressed. For example,
distribution functions of the Hamiltonian energy of canonical-dissipative systems in general exhibits
one or several peaks at energy values larger than the minimal energy [1, 4, 5, 8, 10, 17, 19–23].
The objective of the present study is to examine such energy distributions for oscillatory
canonical-dissipative systems that are subjected to a short-range interaction described by the
Dirac delta function. Such a short-range interaction has been proposed by Haissinski [24, 25] and
has been studied extensively in accelerator physics [26–38]. Note that recently it has been shown
that the Tsallis entropy [39, 40] is also closely related to this type of short-range interaction [41].
From previous research it is known that due to the short-range interaction, systems can become
more concentrated or less concentrated at the points in phase space that are most probably oc-
cupied. In other words, the short-range interaction is known to be attractive or repulsive and is
known to squeeze or stretch distributions [24, 34]. In section 2.1 we define an ensemble of canonical-
dissipative oscillators subjected to the aforementioned short-range interaction. Analytical results
for the distribution function of the oscillators will be derived in sections 2.2 and 2.3. They will
be compared with numerical results in section 3. The impact of the interaction and the model
parameters related to pumping and damping will be discussed in section 4. Some generalizations
will be addressed in section 5.
2. Analytical results
2.1. The oscillator model
To begin with, we consider a single oscillator. The oscillator is assumed to evolve in one spatial
dimension and exhibits a Hamiltonian function H(x, v), where x is the position (or elongation)
and v the velocity of the oscillator. The Hamiltonian oscillator equations read
d
dt
x =
∂
∂v
H = Ix, (1)
d
dt
v = − ∂
∂x
H = Iv , (2)
where I = (Ix, Iv) denotes the conservative drift vector and t is time. Let P (x, v, t) denote the
probability density of the oscillator. Then, P satisfies the Liouville equation
∂
∂t
P = −∇ · IP (3)
with ∇ = (∂/∂x, ∂/∂v). Taking dissipation and fluctuations into account we consider a free energy
functional F [P ] that determines the evolution of the probability density P (x, v, t) such that the
Liouville equation becomes a Fokker-Planck equation of the form [5, 42]
∂
∂t
P = −∇ · IP + γ
∂
∂v
P
∂
∂v
δF
δP
(4)
with γ > 0. The expression δF/δP denotes the variational derivative of the functional F with
respect to the probability density P . We assume that the free energy is bounded from below. It
can be shown that in this case the probability density P as a function of time t converges to
a stationary function. That is, the underlying Markov process related to equation (4) becomes
stationary (H-theorem) [5, 43, 44]. For the free energy functional F we may put
F = 〈H〉 − TSBGS[P ], (5)
where SBGS denotes the Boltzmann-Gibbs-Shannon entropy or information measure defined by
SBGS = −〈ln P 〉 (here and in what follows we put the Boltzmann constant equal to unity) [45].
For oscillators in thermodynamic equilibrium, T reflects the oscillator temperature. In our context
of an oscillator operating far from thermodynamic equilibrium, we consider T as an effective
13001-2
Interacting canonical-dissipative oscillators
oscillator temperature related to the strength of an external fluctuating force acting on the oscillator
(we return to this issue below). Substituting equation (5) into equation (4), we can show that
trajectories x(t), v(t) are determined by the Langevin equation
d
dt
x =
∂
∂v
H, (6)
d
dt
v = − ∂
∂x
H − γ
∂
∂v
H +
√
γT Γ(t). (7)
The function Γ(t) is a Langevin force normalized like 〈Γ(t)Γ(t′)〉 = 2δ(t − t′), where δ(·) denotes
the Dirac delta function. We obtain Brownian motion for the special case of a free particle with
H = v2/2 (and unit mass). Due to damping, the Hamiltonian energy of the oscillator decays as
a function of time. In order to study a self-excited oscillator exhibiting a limit cycle, we need to
consider an oscillator pumped by an energy source. We study energy-uptake in the framework of
canonical dissipative oscillators and replace equation (5) by
F =
1
2
〈
(H − E)2
〉
− TSBGS[P ], (8)
where E is a constant. In this case, the underlying Markov process exhibiting the free energy (8)
is defined by the Langevin equation
d
dt
x =
∂
∂v
H, (9)
d
dt
v = − ∂
∂x
H − γ
∂H
∂v
(H − E) +
√
γT Γ(t), (10)
which can be shown by substituting equation (8) into equation (4). In the deterministic case
(T = 0), from equations (9) and (10) it follows that
d
dt
H = −γ
(
∂H
∂v
)2
(H − E). (11)
Consequently, for E > 0 the Hamiltonian H as a function of time t converges to the constant E.
We refer to E as oscillator fixed point energy. In the stochastic case (i.e. for T > 0) the stationary
distribution of the Hamiltonian energy Pst(H) exhibits a maximum at the fixed point energy E.
Next we consider an ensemble of identical canonical-dissipative oscillators. We label individual
oscillators by the index k. That is, we assume that equations (9) and (10) describe the evolution
of oscillators k = 1, . . . , N when they are isolated from each other:
d
dt
xk =
∂
∂vk
H(xk , vk), (12)
d
dt
vk = − ∂
∂x
H(xk , vk) − γ
∂H(xk, vk)
∂vk
(H(xk , vk) − E) +
√
γT Γk(t). (13)
The oscillator distribution ρ is formally defined by ρ(x, v) = N−1
∑
k δ(x−xk)δ(v−vk). We assume
that there is an interaction between the oscillators that can be expressed in terms of the energy
κ̃
2
J (14)
with
J [ρ] =
∫
ρ2 dx dv. (15)
For κ̃ > 0 the expression κ̃J/2 can be interpreted as a heat energy with J corresponding to the
Tsallis entropy ST,2 [39] with nonextensivity index 2: J = ST,2 − 1. In general (i.e. for κ̃ < 0 and
κ̃ > 0), the term κ̃J/2 can be interpreted as mean field energy
κ̃
2
J =
1
2
∫
ρ(x, v)W (x, x′, v, v′)ρ(x′, v′) dx dx′ dv dv′ (16)
13001-3
S. Mongkolsakulvong, T.D. Frank
emerging from a local (i.e. short-range) interaction W = κ̃δ(x−x′)δ(v−v′) in the two-dimensional
phase space spanned by x and v. As mentioned in the introduction, this type of interaction has been
proposed by Haissinski [25]. In the original Haissinski model the parameter κ̃ describes attractive
interactions for κ̃ < 0 (distributions are squeezed) and repulsive interactions for κ̃ > 0 (distributions
are stretched) [24, 33]. In line with the mean field theory [46–48], we consider a large ensemble
of oscillators (i.e. the limiting case N → ∞) and replace the oscillator distribution ρ by the
distribution P of an individual representative oscillator such that equation (15) becomes
J [P ] =
∫
P 2 dx dv. (17)
Taking the interaction terms (14) and (17) into account, the free energy (8) of an individual
oscillator becomes
F =
1
2
〈
(H − E)2
〉
+
κ̃
2
J [P ] − TSBGS[P ]. (18)
Substitution of equation (18) into the free energy Fokker-Planck equation (4) yields a closed de-
scription for the canonical-dissipative oscillators under consideration. We obtain
∂
∂t
P = −∇ · IP + γ
∂
∂v
{
∂H
∂v
(H − E)P
}
+ κ
∂
∂v
{
∂P
∂v
P
}
+ Q
∂2
∂v2
P (19)
with κ = γκ̃ and Q = γT . Since the interaction term J is quadratic in P , the closed description
given by equation (19) is nonlinear with respect to P and exhibits the form of a so-called nonlinear
Fokker-Planck equation [5, 42, 46, 47]. The oscillator trajectories are determined by a self-consistent
Langevin equation [5]
d
dt
x =
∂
∂v
H, (20)
d
dt
v = − ∂
∂x
H − γ
∂H
∂v
(H − E) − κ
∂
∂v
P
∣
∣
∣
∣
x=x(t),v=v(t)
+
√
QΓ(t). (21)
We see that Q can be interpreted as strength of the external fluctuating force acting on the
oscillator. In the literature, the parameter Q is also referred to as noise amplitude.
In what follows, we will study in detail the interacting oscillators when H corresponds to the
Hamiltonian of a harmonic oscillator with angular oscillation frequency ω and unit mass:
H =
v2
2
+
ω2x2
2
. (22)
In this case, the nonlinear Fokker-Planck equation (19) reads
∂
∂t
P = − ∂
∂x
vP +
∂
∂v
ω2xP + γ
∂
∂v
{v(H − E)P} + κ
∂
∂v
{
∂P
∂v
P
}
+ Q
∂2
∂v2
P (23)
and the Langevin equations (20) and (21) read
d
dt
x = v, (24)
d
dt
v = −ω2x − γv(H − E) − κ
∂
∂v
P
∣
∣
∣
∣
x=x(t),v=v(t)
+
√
QΓ(t). (25)
We will briefly address Hamiltonian functions different from equation (22) in section 5.
2.2. Derivation of the stationary probability density
2.2.1. Stationary probability density in the phase space
In the stationary case ∂Pst(x, v)/∂t = 0, we let Pst(x, v) = f(H(x, v)) = f(H). We have (see
equation (23))
− v
∂
∂x
f(H) +
∂
∂v
{
ω2x + γv[H(x, v) − E] + κ
∂
∂v
f(H)
}
f(H) + Q
∂2
∂v2
f(H) = 0. (26)
13001-4
Interacting canonical-dissipative oscillators
Further, we try to solve the equation (26). Expanding the second term by taking partial derivative
with respect to v the equation becomes
− v
∂
∂x
f(H) + ω2x
∂
∂v
f(H) +
∂
∂v
{
γv[H(x, v) − E] + κ
∂
∂v
f(H)
}
f(H) + Q
∂2
∂v2
f(H) = 0. (27)
Considering the partial derivative of f(H) with respect to x and v, we have
∂
∂x
f(H) =
∂f
∂H
∂H
∂x
= ω2x
∂f
∂H
(28)
and
∂
∂v
f(H) =
∂f
∂H
∂H
∂v
= v
∂f
∂H
. (29)
We put these terms in the equation (27) and rearrange, so we have
∂
∂v
{(
γv(H − E) + κ
∂
∂v
f
)
f + Q
∂f
∂v
}
= 0. (30)
After that, we integrate both sides and choose the constant of integration equal to zero (because
of natural boundary conditions [44]). Then equation (30) becomes
{
γv(H − E) + κ
∂f
∂v
}
f + Q
∂f
∂v
= 0. (31)
We expand the equation (31) and use equation (29). Then, we get the equation that depends on
H as follows:
γ(H − E)f + κf
∂f
∂H
+ Q
∂f
∂H
= 0. (32)
Let H̃ = H − E so that f(H) = f̃(H̃). Equation (32) becomes
[
κ
γ
f̃ +
Q
γ
]
∂f̃
∂H
= −H̃f̃ . (33)
Comparing equation (33) with the classical Haissinski equation [31, 33], the solution of equa-
tion (33) can be found in the form
f̃(H̃) =
Q
κ
LW
κg
(
H̃
)
zQ
, (34)
where the function g(H̃) is
g(H̃) = exp
[
−γH̃2
2Q
]
. (35)
In equation (34) the function LW(·) denotes the Lambert-W function [49] and z > 0 corresponds
to an integration constant. From f̃(H̃) we obtain the original function f(H) as
f(H) =
Q
κ
LW
(
κ
zQ
exp
{
− γ
2Q
(H − E)2
})
. (36)
After back substitution in terms of x,v,H ,and E, we get
Pst(x, v) =
Q
κ
LW
[
κg(H − E)
zQ
]
(37)
and
g(H − E) = exp
{
−γ[H(x, v) − E]2
2Q
}
, (38)
13001-5
S. Mongkolsakulvong, T.D. Frank
where the constant z satisfies
∞∫
−∞
∞∫
−∞
Pst(x, v) dx dv = 1. (39)
That is, we will interpret z in what follows as a normalization constant. In a more concise form,
the stationary probability density Pst(x, v) defined by equations (37) and (38) can be written as
Pst(x, v) =
Q
κ
LW
(
κ
zQ
exp
{
− γ
2Q
(H(x, v) − E)2
})
. (40)
2.2.2. Stationary probability density of the Hamiltonian energy
From the probability density Pst(x, v) we can compute the probability densityPst(H) which
satisfies
∫
∞
0 Pst(H)dH = 1. We have
Pst(H) =
∞∫
−∞
∞∫
−∞
δ
(
H − v2 + ω2x2
2
)
Pst(x, v)dx dv. (41)
We put y = ωx and dx = dy/ω:
Pst(H) =
1
ω
∞∫
−∞
∞∫
−∞
δ
(
H − v2 + y2
2
)
f
(
v2 + y2
2
)
dy dv. (42)
Next we introduce ỹ = y/
√
2 and ṽ = v/
√
2 and transform ỹ and ṽ into polar coordinates r and
ϕ. We obtain (v2 + y2)/2 = ṽ2 + ỹ2 = r2 , dydv = 2dỹdṽ = 2rdrdϕ and
Pst(H) =
2
ω
2π∫
0
∞∫
0
δ(H − r2)f(r2) r dr dϕ, (43)
Pst(H) =
4π
ω
∞∫
0
δ(H − r2) f(r2) r dr. (44)
We use s = r2, ds = 2rdr and obtain
Pst(H) =
2π
ω
∞∫
0
δ(H − s) f(s) ds =
2π
ω
f(H). (45)
That is, Pst(H) reads explicitly
Pst(H) =
2πQ
ωκ
LW
(
κ
zQ
exp
{
− γ
2Q
(H − E)2
})
. (46)
2.2.3. Limiting case of vanishing oscillator-oscillator interaction
Recall that the Lambert-W function LW(·) is implicitly defined by LW(x) exp{LW(x)} = x
and we have LW(0) = 0, see [49]. Differentiating the implicit definition of LW with respect to x
yields d LW(x)/dx[1 + LW(x)] exp{LW(x)} = 1. This implies that at x = 0 the slope equals 1:
d LW(0)/dx = 1. Having said that we can consider the limiting case of a vanishing interaction:
κ → 0. In this case, the numerator and denominator in equations (40) and (46) vanish. Using the
rule of de l’Hopital, we differentiate numerator and denominator with respect to κ and consider
13001-6
Interacting canonical-dissipative oscillators
the limiting case κ → 0 again. For example, for equation (40) we calculate the limiting case of
vanishing oscillator-oscillator interaction as follows:
lim
κ→0
Pst(x, v)
de l’Hopital
= Q lim
κ→0
d
dκ
LW
(
κ
zQ
exp
{
− γ
2Q
(H(x, v) − E)2
})
=
(
1
z
exp
{
− γ
2Q
(H(x, v) − E)2
})
lim
ξ→0
d
dξ
LW(ξ)
=
1
z
exp
{
− γ
2Q
(H(x, v) − E)2
}
. (47)
Likewise, for Pst(H) we obtain
lim
κ→0
Pst(H) =
1
z′
exp
{
− γ
2Q
(H − E)2
}
(48)
with z′ = ωz/(2π). Equations (47) and (48) recover the well-known results for canonical-oscillators
for J = 0 (i.e., without interaction) [1–5]. However, these limiting cases not only hold for non-
interacting oscillators. They also hold when the fluctuating force dominates the interaction: κ/Q →
0. That is, repeating the same calculations as before, we obtain
lim
κ/Q→0
Pst(x, v) =
1
z
exp
{
− γ
2Q
(H(x, v) − E)2
}
(49)
and
lim
κ/Q→0
Pst(H) =
1
z′
exp
{
− γ
2Q
(H − E)2
}
. (50)
Note that these limiting cases include the cases considered in equations (47) and (48) as special
cases but hold for more general situations. For example, we may consider oscillator systems with
γ → ∞, Q = c1γ, and κ = c2 ln γ with c1, c2 > 0. Then κ/Q = (c2 ln γ)/(c1γ) = 0 for γ → ∞ but
the amount of interaction between the oscillators as measured by κ does not vanish.
2.3. Boundaries for the critical value of the parameter κ
For systems involving attractive interactions (i.e., interaction parameters κ < 0), it is known
that stationary probability densities exist only up to a critical parameter value κc equation [31,
34, 35, 38, 41]. In this section, we determine an interval in which this critical value can be found.
As we will see, the precise value of the critical parameter satisfies an implicit equation that has to
be solved numerically.
In this section, we consider the oscillator system from the perspective of the Hamiltonian energy
H , again. We start from equation (32), which we write in the form:
[Q + κf(H)]
dHf(H)
dH
= −γ(H − E)f(H). (51)
We need to consider the case κ < 0, for which equation (51) reads
[Q − |κ|f(H)]
dHf(H)
dH
= −γ(H − E)f(H). (52)
As shown in [31, 33–35, 41], positive definite, continuous and normalizable solutions of equations of
the form (52) only exist if Q−|κ|f(H) > 0 holds for all H . Furthermore, in this case f is maximal
at H = E. Let fM = f(E) occur at H = E. Then the inequality Q − |κ|f(H) > 0 holds for all H
if fM satisfies
Q − |κ|fM > 0. (53)
Consequently, we define the critical parameter κc by
Q − |κc|fM = 0, (54)
13001-7
S. Mongkolsakulvong, T.D. Frank
which implies fM = Q/|κc|. By separating variables and using fM = Q/|κc|, equation (52) for
κ = κc can be written in the form
Q
[
1
f
− 1
fM
]
df = −γ(H − E)dH. (55)
Integrating the right hand side from E to an arbitrary value H and accordingly the left hand side
from fM to f(H), we get
Q
(
ln
f
fM
− f
fM
+ 1
)
= −γ
H−E∫
0
H̃ dH̃ = −1
2
γ(H − E)2. (56)
From equation (56) we obtain
H − E = ±
√
√
√
√−2Q
γ
(
ln
f̃
f̃M
− f̃
f̃M
+ 1
)
. (57)
The minus sign holds for the monotonously increasing part of the graph f(H) on the interval
[0, E], where we have H −E 6 0. Likewise, the plus sign holds for the decreasing part of f(H) for
H ∈ [E,∞), where we have H−E > 0. Substituting equation (57) into equation (55), we eliminate
the bracket (H − E) in equation (55) and obtain
dH = ±
√
Q
2γ
1
f̃M
− 1
f̃
√
−(ln f̃
f̃M
− f̃
f̃M
+ 1)
df. (58)
Next, we exploit the normalization condition
∫
∞
0
Pst(H) dH = 1, which can alternatively be ex-
pressed as
2π
ω
∞∫
0
f(H) dH = 1. (59)
We split the integral into two parts
2π
ω
E∗
∫
0
f(H) dH +
∞∫
E∗
f(H) dH
= 1 (60)
with E∗ = E for E > 0 and E∗ = 0 otherwise. That is, for E 6 0 the first integral vanishes.
Substituting dH (see equation (58)) into equation (60), yields
√
Q
2γ
2π
ω
−
fM∫
f(0)
( f
fM
− 1)
√
−(ln f̃
f̃M
− f̃
f̃M
+ 1)
df +
0∫
fM
( f
fM
− 1)
√
−(ln f̃
f̃M
− f̃
f̃M
+ 1)
df
= 1. (61)
Note that for E 6 0 we have fM = f(0) (see section 4.1) and the first integral vanishes (which is
consistent with the comment on equation (60) made above). Note also that for the first integral we
have used the minus sign occurring in equation (58) because the first integral concerns the interval
[0, E∗]. Let us define y = f/fM, which implies df = fMdy. Then equation (61) becomes
√
2Q
γ
π
ω
fM
1∫
f(0)/fM
1 − y√
y − ln y − 1
dy +
0∫
1
y − 1√
y − ln y − 1
dy
= 1 (62)
13001-8
Interacting canonical-dissipative oscillators
or
√
2Q
γ
π
ω
fM
1∫
f(0)/fM
1 − y√
y − ln y − 1
dy +
1∫
0
1 − y√
y − ln y − 1
dy
= 1. (63)
Using fM = Q/|κc|, we find
|κc| =
π
ω
√
2Q
γ
1∫
f(0)/fM
1 − y√
y − ln y − 1
dy +
1∫
0
1 − y√
y − ln y − 1
dy
. (64)
Both integrals yield positive numbers. Recall that the critical parameter is negative. In the limit
E → ∞ we have f(0) → 0 and consequently both integrals approach the same value. In this case,
we have the smallest possible critical value
κc,min = 2κ∗ (65)
with
κ∗ = −π
ω
√
2Q
γ
1∫
0
1 − y√
y − ln y − 1
dy. (66)
In contrast, for E 6 0 the first integral vanishes and we have the maximal critical value
κc,max = κ∗. (67)
For E > 0 we therefore can identify the interval in which the critical value of the parameter κ
can be found as κc ∈ [2κ∗, κ∗]. Alternatively, we say that the interval [2κ∗, κ∗] contains all critical
values for a fixed set of parameters ω, γ, and Q when all possible values E ∈ lR are considered. The
precise value of κc for a given energy E can be determined numerically by solving equation (64)
when taking into account that both f(0) and fM depend on κc. That is, κc satisfies the implicit
equation
κc = −π
ω
√
2Q
γ
1∫
f(0,κc)/fM,κc
1 − y√
y − ln y − 1
dy +
1∫
0
1− y√
y − ln y − 1
dy
. (68)
As mentioned above, for large enough parameters E the probability to find energies H = 0 becomes
negligibly small and κc equals 2κ∗. A numerical value for the integral in equation (66) is 1.0964
such that for large parameters E we have κc ≈ −2.2πω−1
√
2Q3/γ.
3. Numerical simulation scheme for the stationary case
In order to solve equations (24) and (25) numerically for the stationary case, we first realize
that in the stationary case the probability density Pst(x, v) becomes a function of the Hamiltonian
H . Consequently, we have
− κ
∂
∂v
Pst(x, v) = −κv
d
dH
Pst(H). (69)
Substituting this result into equations (24) and (25), we obtain
d
dt
x = v, (70)
d
dt
v = −ω2x − γv(H(t) − E) − κv
d
dH
Pst(H)
∣
∣
∣
∣
H=H(t)
+
√
QΓ(t) (71)
13001-9
S. Mongkolsakulvong, T.D. Frank
with H(t) = [v2(t) + ω2x2(t)]/2. The Langevin equation involves the gradient of the probability
density Pst(H). We compute this gradient analytically rather than numerically. From our discussion
in section 2.2 it follows that
− γ(H − E)Pst = (κPst + Q)
d
dH
Pst(H). (72)
Consequently, we compute the derivative like
d
dH
Pst(H) = −γ(H − E)Pst
κPst + Q
. (73)
We discretize time on a grid with time step ∆t. Using an Euler forward method [44] that also holds
for self-consistent Langevin equations [5], we transform equations (70, . . . , 72) into
xi+1 = xi + ∆t vi , (74)
vi+1 = vi − ∆t
[
ω2xi + γvi(Hi − E)
]
+ ∆t
κγ(Hi − E)Pi
κPi + Q
+
√
∆t Q ei (75)
with
Hi =
v2
i
2
+
ω2x2
i
2
, (76)
Pi = Pst(H)|H=Hi
. (77)
In the simulation scheme above, ei is a Gaussian random number with variance 2. As indicated
in equation (77), the value Pi denotes the function value of the probability density of Pst(H) at
H = Hi. In order to determine Pst(H) numerically, we compute simultaneously an ensemble of N
oscillators. That is, we solve equations (74, . . . , 77) simultaneously for N subsystems. At every
iteration step we compute from the N observed Hi values the probability density Pst(H) using
the MATLAB kernel density estimator. This implies for example that the velocities vi+1 of all
N oscillators depend on the same function Pst(H). However, for every individual oscillator the
function Pst(H) is evaluated at the appropriated, individual Hamiltonian value of that oscillator.
We checked the consistency of the numerical simulation scheme with the analytical result derived
in section 2.2. To this end, we plotted the Pst(H) as computed from equation (46) and as obtained
numerically in the same figure. Figures 1 and 2 show the results obtained for attractive (κ < 0)
and repulsive (κ > 0) interactions between the oscillators.
0 2 4 6 8 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Hamiltonian H [a.u.]
P
ro
ba
bi
lit
y
de
ns
ity
0 2 4 6 8 10
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Hamiltonian H [a.u.]
P
ro
ba
bi
lit
y
de
ns
ity
Figure 1. Analytical (solid line) versus nu-
merical (circles) results for Pst(H). Parame-
ters: E = 5, Q = 2, γ = 1, κ = 6, ω = 3.
Simulation parameters: ∆t = 0.01, N = 7000,
number of iterations in time: 600.
Figure 2. Analytical (solid line) versus nu-
merical (circles) results for Pst(H). Parame-
ters as in figure 1 except for κ. In figure 2 we
have κ = −6.
13001-10
Interacting canonical-dissipative oscillators
4. Impacts of model parameters
We will study the impact of the model parameters on the distribution Pst(H) of the Hamiltonian
energy H as defined by equation (46). The Lambert-W-function LW(·) is a monotonously increasing
function. Consequently, from equation (46) it follows that Pst(H) is unimodal (i.e. exhibits only
one maximum or peak). Let Hpeak and Pmax denote the position and the height of this peak,
respectively. At issue is to study how the model parameters affect the peak position and height. To
this end, we consider the functions Hpeak = Hpeak(E, γ, Q, κ, ω) and Pmax = Pmax(E, γ, Q, κ, ω).
From equation (46) it is obvious that the distribution function Pst(H) only depends on the ratios
γ/Q and κ/Q such that it is sufficient to consider the functions Hpeak = Hpeak(E, γ/Q, κ/Q, ω)
and Pmax = Pmax(E, γ/Q, κ/Q, ω).
4.1. Impact of the net pumping force described by the fixed point energy E
For E > 0 the pumping force exceeds the friction force such that the net pumping force is
positive. For E < 0 the friction dominates the pumping. From equation (46) it follows that the
peak position depends exclusively on the fixed point energy E. We have
Hpeak(E) =
{
E, E > 0,
0, E < 0.
(78)
This relationship is illustrated in figure 3. In addition, figure 4 shows two distributions Pst(H)
computed from equation (46) for E < 0 and attractive (κ < 0) as well as repulsive (κ > 0)
interaction between the oscillators. Figure 4 clearly demonstrates that for E < 0 the maximum
can be found at the minimal possible energy H = 0.
−5 0 5 10
0
2
4
6
8
10
E [a.u.]
P
os
iti
on
o
f M
ax
im
um
0 0.5 1 1.5 2
0
1
2
3
4
5
Hamiltonian H [a.u.]
P
ro
ba
bi
lit
y
de
ns
ity
κ=1
κ=−1
Figure 3. Hpeak as a function of E drawn from
equation (78).
Figure 4. Distribution Pst(H) computed from
equation (46) for negative fixed point energy
E. Parameters: E = −5, Q = 2, γ = 1, κ =
±1, ω = 3.
The probability maximum Pmax in general depends on the fixed point energy E. Since Pst(H)
can be cast into the form Pst(H) = r(H − E), where r(·) is a function of the difference H − E,
we can regard E as a parameter that shifts the coordinate system of H . If E increases then
we shift the distribution from the left to the right. When considering Pst(H) as a mathematical
function (not as a probability density) then we can plot Pst(H) on the whole real line H ∈ lR.
However, when considering Pst(H) as a probability density, then the area under Pst(H) for H >
0 must be equal to 1. If we increase E and shift the mathematical function Pst(H) for fixed
normalization constant z from the left to the right along the coordinate axis H , then the area
under the mathematical functions on the semi-line H > 0 increases. Consequently, if we increase
E and adjust the normalization constant z such that we are dealing with a normalized probability
13001-11
S. Mongkolsakulvong, T.D. Frank
density defined on H > 0, then the maximum of the Pst(H) decreases. We have
∂
∂E
Pmax(E, γ/Q, κ/Q, ω) < 0. (79)
For relatively large values of E we are dealing with functions as illustrated in figures 1 and 2. The
probability to find oscillators with H = 0 is relatively small. In the limit E → ∞ this probability
can be neglected and a shift of E does not change the normalization constant z. That is, we have
∂z/∂E = 0 for E → ∞ and
lim
E→∞
∂
∂E
Pmax(E, γ/Q, κ/Q, ω) = 0. (80)
4.2. The relative interaction parameter κ/Q
The ratio κ/Q can be interpreted as relative interaction parameter. It measures the strength
of the interaction relative to the strength of the fluctuating force. The ratio κ/Q assumes positive
values for repulsive interactions and negative values for attractive interactions. As mentioned above
the peak position Hpeak does not depend on κ/Q. However, κ/Q crucially affects the concentration
of the oscillators energies H around the most probable energy Hpeak. That is, Pmax depends on
κ/Q. Figure 5 illustrates the function Pmax(·, κ/Q, ·) for fixed values of E, γ/Q and ω. We see
that Pmax decreases monotonously. In other words, for repulsive interaction (κ > 0) an increase of
the relative interaction strength makes the distribution of H broader and results in a decrease of
the peak value Pmax. In contrast, for attractive interactions (κ < 0), increasing the amount of the
relative interaction strength |κ|/Q implies that the attractive interaction becomes stronger and the
oscillator energies become more concentrated at Hpeak. The peak height Pmax increases and the
energy distribution Pst(H) becomes narrower.
−5 0 5 10
0.2
0.3
0.4
0.5
0.6
0.7
κ/Q
P
m
ax
γ/Q=0.5
γ/Q=1.5
Figure 5. Peak height Pmax = Pst(H)|
H=E
as a function of κ/Q computed by means of equa-
tion (46). Fixed parameters: E = 5, Q = 2, ω = 3. For the graph with γ/Q = 0.5 (γ = 1)
the parameter κ was varied in the range of −9, . . . , 25 such that κ/Q varied in the range of
−4.5, . . . , 12.5. For the graph γ/Q = 1.5 (γ = 3) the parameter κ was varied in the range of
−2.5, . . . , 12.5.
4.3. The parameter γrel = γ/Q
From equation (11) it is clear that in the deterministic case the parameter γ determines how
fast the Hamiltonian energy H relaxes to the fixed point energy E. Therefore, we may interpret
γ as a measure for the attractor strength of the limit cycle of an individual canonical-dissipative
oscillator. Note that the relaxation time depends not only on the parameter γ but also on other
parameters involved in the function H . However, if we divide both sides of equation (11) with γ
and substitute t with a new time variable t̃ = γt̃ then we see that γ defines a time scale relevant
for the relaxation dynamics. Therefore, it is sensible to consider γ as a key relaxation parameter.
13001-12
Interacting canonical-dissipative oscillators
Furthermore, recall that attractor strength usually is defined in terms of relaxation times towards
attractors. In our case, it is plausible to develop the notion that the attractor strength of the limit
cycle of an individual oscillator is proportional to γ. The larger γ, the stronger is the limit cycle
attractor. The ratio γ/Q describes the relative limit cycle attractor strength and can be used to
answer the question: how strong is the attraction towards the limit-cycles relative to the erratic
perturbations that push the oscillator away from the limit cycle and are due to the fluctuating
force with amplitude Q?
As discussed in section 4.1 the parameter γrel = γ/Q does not affect the peak position Hpeak.
From equation (46) it follows that if γ/Q increases then the width of distribution becomes smaller.
Consequently, the peak value Pmax increases monotonously as a function of γrel = γ/Q:
∂
∂γrel
Pmax(E, κ/Q, γrel, ω) > 0. (81)
This relationship is illustrated in figure 6. Note also that figure 5 illustrates that the peak value
increases as a function of γrel. For a fixed value of κ/Q we can compare in figure 5 the values of
Pmax for γrel = 0.5 and γrel = 1.5. We see that Pmax(γrel = 1.5, κ/Q) > Pmax(γrel = 0.5, κ/Q) for
fixed values of ω and E.
2 4 6 8 10 12
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
γ/Q
P
m
ax
κ/Q=0.5
κ/Q=1.5
Figure 6. Peak height Pmax = Pst(H)|
H=E
as a function of γ/Q computed by means of equa-
tion (46). Fixed parameters: E = 5, Q = 2, ω = 3. For both graphs κ/Q = 0.5 (κ = 1) and
κ/Q = 1.5 (κ = 3) the parameter κ was varied in the range of 1, . . . , 25 such that κ/Q varied in
the range of 0.5, . . . , 12.5.
5. Alternative Hamiltonian functions
5.1. Stationary phase space probability densities
In section 2.2 we derived analytical expressions for the stationary probability densities Pst(x, v)
and Pst(H) for canonical-dissipative oscillators involving the Hamiltonian function (22) of a har-
monic oscillator. We consider next Hamiltonian functions that involve a potential function V (x)
which is continuous and globally attractive V (x → ±∞) → ∞ but does not necessarily correspond
to a parabolic potential. That is, we put
H =
v2
2
+ V (x). (82)
A detailed analysis of the analytical steps carried out in section 2.2 shows that the derivation
of Pst(x, v) holds for all kinds of potentials V (x) and does not exploit the explicit form V (x) =
ω2x2/2 considered in section 2.2. Consequently, Pst(x, v) can be expressed in terms of the function
f(H) (defined by equation (36)): Pst(x, v) = f(H(x, v)). In particular, the stationary distribution
Pst(x, v) for oscillators involving the Hamiltonian function (82) is again defined by equation (40).
13001-13
S. Mongkolsakulvong, T.D. Frank
5.2. Stationary distributions of Hamiltonian energy
In order to derive the stationary distribution Pst(H) of the Hamiltonian energy H , we need to
determine the proportionality factor β that relates Pst(H) to f(H). For the harmonic potential
V (x) = ω2x2/2 we found in section 2.2 that β = 2π/ω and Pst(H) = 2πf(H)/ω. That is, we found
that for parabolic potentials β is independent of H . However, in general the factor β can depend
on the Hamiltonian energy H .
In order to determine β(H) and Pst(H), we proceed as in section 2.2. We substitute H =
v2/2 + V into f(H) (defined by equation (36)) such that f = f(v2/2 + V (x)) and subsequently
solve the integral
Pst(H) =
∫∫
δ
(
H − v2
2
− V (x)
)
f
(
v2
2
− V (x)
)
dx dv. (83)
We first evaluate the integration with respect to v. To this end, we define the integral
I(x, H) =
∫
δ
(
H − v2
2
− V (x)
)
f
(
v2
2
− V (x)
)
dv. (84)
Recall that for the Dirac delta function that involves an arbitrary continuously differentiable func-
tion u the relation
δ(u(v)) =
∑
i
δ(v − vi)
|du/dvi|
(85)
holds, where vi are the roots of u: u(vi) = 0. From equation (84) it follows that u = H − v2/2− V
with roots v1 =
√
2[H − V (x)] and v2 = −
√
2[H − V (x)]. Both roots exist only for H > V (x).
Furthermore, we have |du/dvi| = |vi| =
√
2[H − V (x)]. Consequently, equation (84) becomes
I(x, H) =
∫
δ(v − v1) + δ(v − v2)
√
2[H − V (x)]
f
(
v2
2
− V (x)
)
dv, V (x) 6 H,
0, V (x) > H
(86)
which can equivalently be expressed as
I(x, H) = 2w(x, H)f(H) (87)
with
w(x, H) =
1
√
2[H − V (x)]
, V (x) 6 H,
0, V (x) > H.
(88)
From equations (83), (84), and (87) we conclude that
Pst(H) = 2
∫
w(x, H) dx
︸ ︷︷ ︸
β(H)
f(H) . (89)
As indicated in equation (89), the integral with respect to x over the function w essentially defines
the factor β. Let am and bm with m = 1, . . . , M denote the positions at which the potential energy
V equals H (i.e. we require V (am) = H and V (bm) = H) with a1 < b1 < a2 < b2 < . . . . Then the
integration with respect to x is only carried out over the intervals [am, bm]:
β(H) = 2
M∑
m=1
bm∫
am
dx
√
2[H − V (x)]
. (90)
In the context of classical Hamiltonian systems, the integral
∫ bm
am
(
√
2[H − V (x)])−1dx is well-
known. Recall that from the so-called first integral v = dx/dt = ±
√
2([H − V (x)] we obtain
13001-14
Interacting canonical-dissipative oscillators
dt = (
√
2[H − V (x)])−1dx with the plus sign in the case of a motion for which x(t) increases as
function of time (i.e. for v > 0). Therefore, the integral
∫ bm
am
(
√
2[H − V (x)])−1dx corresponds to
the duration Tam→bm
of a particle with unit mass that travels from am to bm and is subjected to
a classical Hamiltonian dynamics. We define
Tam→bm
(H) =
bm∫
am
dx
√
2[H − V (x)]
. (91)
The round-trip time form am to bm and back to am is 2 times the duration for the one-way
travel because classical Hamiltonian dynamics is time reversible. Let Tam↔bm
(H) = 2Tam→bm
(H)
denote the round-trip time. Note that the round-trip time corresponds to the oscillation period of
oscillations around a potential minimum (or several potential minima) of V (x) located in [am, bm].
Using Tam↔bm
(H), we can express the equations (89, . . . , 91) as follows:
Pst(H) = β(H)f(H), (92)
β(H) = 2
M∑
m=1
bm∫
am
dx
√
2[H − V (x)]
=
M∑
m=1
Tam↔bm
(H). (93)
From equation (93) it is clear that the factor β reflects the sum of all oscillation periods (round-
trip times) for oscillations around those potential minima of V (x) that exhibit potential energies
smaller than H . Each individual oscillation period and the sum of all periods depend in general on
H . Consequently, as anticipated in the beginning of this section, the factor β in general depends
on H .
In closing this section, let us return to the harmonic oscillator potential V (x) discussed in sec-
tion 2.2. For V = ω2x2/2 we have only one potential minimum (i.e. we put M = 1 in equation (93))
and the oscillation period T (a ↔ b) is independent of H and simply corresponds to the inverse of
the oscillation frequency fosc = ω/(2π). We have T (a ↔ b) = 2π/ω. Consequently, the factor β is
independent of H and we recover the result β = 2π/ω derived in section 2.2.
6. Conclusions
Limit cycle oscillators are at the heart of nonlinear physics and play a key role in nonlinear
sciences. Unfortunately, the derivation of analytical results for stochastic limit cycle oscillators is
usually mathematically involved. Canonical-dissipative limit cycle oscillators provide an exception
to this rule because they frequently can be studied analytically. In view of this property of canonical-
dissipative oscillators, we proposed to derive an analytical expression for the distribution function
of canonical-dissipative limit cycle oscillators that are coupled with each other by means of a
short-range interaction in phase space.
We discussed canonical-dissipative oscillators involving a parabolic potential in detail. We were
able to determine the impacts of various forces by studying how the distribution function of the
Hamiltonian energy is affected by these forces. That is, we demonstrated that stochastic proper-
ties of the canonical-dissipative oscillators exhibiting short-range interaction can conveniently be
studied when we consider the distribution function of the Hamiltonian energy rather than the dis-
tribution function in the phase space spanned by position and velocity. We found that the peaks of
energy distribution functions become more pronounced and consequently that energy distribution
functions become narrower when limit cycle attractor forces and interaction forces become stronger.
This holds for attractive interactions between oscillators. However, a peculiarity of the interaction
considered in our study (and motivated by early work of Haissinski) is that the oscillator-oscillator
interaction cannot only be attractive but also repulsive. For a repulsive interaction, we found that
peaks become less pronounced and energy distributions become broader when the magnitude of
the interaction is increased. The relative impact of the limit cycle attractor as indexed by the
13001-15
S. Mongkolsakulvong, T.D. Frank
parameter γrel is the same for attractive and repulsive inter-oscillator interactions. From this ob-
servation we conclude that repulsive inter-oscillator interactions can be counter-balanced by the
intrinsic (within-oscillator) attractor forces of limit cycle attractors. In other words, when the rela-
tive magnitude κ/Q of the repulsive inter-oscillator interactions is increased and at the same time
the relative limit cycle attractivity γrel = γ/Q is increased appropriately, then the impacts of the
two manipulations may cancel out each other such that there is a zero net effect.
In section 3, we presented a numerical simulation scheme which can be used to compute tra-
jectories of individual oscillators of the oscillator ensemble. We showed that the results of the
simulation scheme are consistent with the analytically derived results. At this stage, however, we
should note that the simulation scheme can also be used to numerically compute the second-order
statistical properties (e.g. autocorrelation functions, mean square displacements, and so on) which
we have not addressed in our theoretical considerations. Consequently, the proposed model pro-
vides a complete description of the oscillator systems under consideration. That is, all statistical
quantities of interest are well defined and can at least be computed numerically.
In section 5, we briefly illustrated that our considerations can be generalized to oscillator
systems that involve potentials V different from parabolic potentials. We would like to point out
that the simulation scheme proposed in section 3 can be modified such that we can deal with this
more general situation. One can show that for systems addressed in section 5 involving potentials
V (x) equations (74, . . . , 77) become
xi+1 = xi + ∆t vi , (94)
vi+1 = vi + ∆t
[
dV
dxi
+ γvi(Hi − E)
]
+ ∆t
κγ(Hi − E)Pi
κPi + Q
+
√
∆t Q ei (95)
with
Hi =
v2
i
2
+ V (xi), (96)
Pi = Pst(H)|H=Hi
. (97)
The realm of canonical-dissipative systems is much broader than the systems addressed in
our study [1–4]. The key quantity of canonical-dissipative systems are the invariants of classical
mechanical systems featuring a Newtonian or Hamiltonian dynamics. The Hamiltonian energy is
one of the most important invariants but other invariants may play important roles as well in
certain applications. Therefore, the present study should consider a first step in exploring the
properties of canonical-dissipative systems under various types of short-range interactions. The
results summarized above and the analytical techniques developed in our work may be used as
guidelines in investigating canonical-dissipative systems that focus on invariants different from the
Hamiltonian energy.
References
1. Haken H., Z. Physik, 1973, 263, 267.
2. Graham R., in Order and fluctuations in equilibrium and nonequilibrium statistical mechanics, ed. by
G. Nicolis, G. Dewel, J. W. Turner. John Wiley and Sons, New York, 1981, 235–288.
3. Feistel R., Ebeling W., Evolution of complex system: self-organization, entropy and development. VEB
Verlag, Berlin, 1989.
4. Ebeling W., Sokolov I.M., Statistical thermodynamics and stochastic theory of nonequilibrium systems.
World Scientific, Singapore, 2004.
5. Frank T.D., Nonlinear Fokker-Planck equations: Fundamentals and applications. Springer, Berlin,
2005.
6. Schweitzer F., Ebeling W., Tilch B., Phys. Rev. Lett., 1998, 80, 5044.
7. Schweitzer F., Schimansky-Geier L., Physica A, 1994, 206, 359.
8. Schweitzer F., Brownian agents and active particles. Springer, Berlin, 2003.
9. Erdmann U., Ebeling W., Schimansky-Geier L., Schweitzer F., Eur. Phys. J. B. 2000, 15, 105.
10. Schweitzer F., Ebeling W., Tilch B., Phys. Rev. E, 2001, 64, 021110.
13001-16
Interacting canonical-dissipative oscillators
11. Erdmann U., Ebeling W., Mikhailov A., Phys. Rev. E, 2005, 71, 051904.
12. Ebeling W., Eur. Phys. J. B, 2000, 15, 105.
13. Ebeling W., Fluctuation and Noise Letters, 2003, 3, L137.
14. Ebeling W., Schimansky-Geier L., Eur. Phys. J. ST, 2008, 157, 17.
15. Lindner B., Nicola E.M., Eur. Phys. J. ST, 2008, 157, 43.
16. Ebeling W., Condens. Matter Phys., 2000, 3, 285.
17. Ebeling W., Gudowska-Nowak E., Fiasconaro A., Acta Physica Polonica B, 2008, 39, 1251.
18. Bödeker H.U., Frank T.D., Friedrich R., Purwins H.G. – In: Anomalous Fluctuation Phenomena in
Complex Systems: Plasmas, Fluids and Financial Markets, ed. by C. Riccardi and H. E. Roman.
Research Signpost, Kerala, 2008, p. 145–184.
19. Ebeling W., Engel-Herbert H., Physica A, 1980, 104, 378.
20. Ebeling W. – In: Stochastic Processes in Physics, Chemistry and Biology, edited by J. A. Freund and
T. Pöschel. Springer, Berlin, 2000, p. 392–399.
21. Ebeling W., Schimansky-Geier L. – In: Noise in Nonlinear Dynamical Systems, Vol. 1, edited by F.
Moss and P. V. E. McClintock. Cambridge University Press, Cambridge, 1989, p. 279–306.
22. Frank T.D., Daffertshofer A., Physica A, 2001, 292, 392.
23. Frank T.D., Phys. Lett. A, 2005, 337, 224.
24. Chao A.W., Physics of Collective Instabilities in High Energy Accelerators. John Wiley and Sons, New
York, 1993.
25. Häıssinski J., Nuovo Cimento Soc. Ital. Fis. B, 1973, 18, 72.
26. Heifets S., Phys. Rev. E, 1996, 54, 2889.
27. Stupakov G.V., Breizman B.N., Pekker M.S., Phys. Rev. E, 1997, 55, 5976.
28. Heifets S., Podobedov B., Phys. Rev. ST-AB, 1999, 2, 044402.
29. Heifets S., Phys. Rev. ST-AB, 2001, 4, 044401.
30. Thomas C., Bartonlini R., Botman J.I.M., Dattoli G., Mezi L., Migliorati M., Europhys. Lett., 2002,
60, 66.
31. Venturini M., Phys. Rev. ST-AB, 2002, 5, 054403.
32. Heifets S., Phys. Rev. ST-AB, 2003, 6, 080701.
33. Frank T.D., Phys. Lett. A, 2003, 319, 173.
34. Frank T.D., Phys. Rev. ST-AB, 2006, 9, 084401.
35. Shobuda Y., Hirata K., Part. Accel., 1999, 62, 165.
36. Shobuda Y., Hirata K., Phys. Rev. E, 2001, 64, 067501.
37. Petracca S., Demma T., Hirata K., Phys. Rev. ST-AB, 2005, 8, 074401.
38. Frank T.D., Mongkolsakulvong S., Physica D, 2009, 238, 1186.
39. Tsallis C., J. Stat. Phys., 1988, 52, 479.
40. Abe S., Okamoto Y., Nonextensive Statistical Mechanics and its Applications. Springer, Berlin, 2001.
41. Frank T.D., Mongkolsakulvong S., Physica A, 2008, 387, 4828.
42. Frank T.D. – In: Encyclopedia of Complexity and Systems Science, edited by R. A. Meyers. Springer,
Berlin, 2009, p. Entry 764.
43. Shiino M., Phys. Rev. A, 1987, 36, 2393.
44. Risken H., The Fokker-Planck Equation – Methods of Solution and Applications. Springer, Berlin,
1989.
45. Haken H., Synergetics: Introduction and Advanced Topics. Springer, Berlin, 2004.
46. Kuramoto Y. Chemical Oscillations, Waves, and Turbulence. Springer, Berlin, 1984.
47. Tass P.A., Phase Resetting in Medicine and Biology – Stochastic Modelling and Data Analysis. Spri-
nger, Berlin, 1999.
48. Mongkolsakulvong S., Frank T.D., Tang I.M., Phase Transit., 2007, 80, 967.
49. Weisstein E.W., CRC Concise Encyclopedia of Mathematics. Chapman and Hall/CRC, Boca Raton,
1998.
13001-17
S. Mongkolsakulvong, T.D. Frank
Канонiчно-дисипативнi граничнi циклiчнi осцилятори з
короткосяжною взаємодiєю у фазовому просторi
С. Монгколсакулвонг1, Т.Д. Франк2
1 Унiверситет Касетсарт, Бангкок 10900, Таїланд
2 Унiверситет Коннектикут, Сторрс CT 06269, США
Ми розглядаємо граничнi циклiчнi осцилятори в термiнах канонiчно-дисипативних систем, якi де-
монструють короткосяжну взаємодiю у швидкiсному i позицiйному просторi, що описаний дельта-
функцiєю Дiрака. Ми виводимо аналiтичнi вирази для стацiонарних функцiй розподiлу у фазовому
просторi та енергетичному просторi i також пропонуємо числову симуляцiйну схему для моделю-
вання багаточастинкової системи. Ми показуємо, що короткосяжна взаємодiя стискає чи розтягує
енергетичнi функцiї розподiлу в залежностi вiд того, чи взаємодiю можна вважати притягальною,
чи вiдштовхувальною. Крiм впливу взаємодiї, ми показуємо, що енергетичнi функцiї розподiлу ста-
ють вужчими, якщо границя циклiчних атракторiв стає сильнiшою. Накiнець, енергетичнi розподiли
стають ширшими з ростом енергiї накачки. Цей ефект, проте, зникає у високоенергетичнiй областi.
Ключовi слова: канонiчно-дисипативнi системи, рiвняння Гайссiнскi, ентропiя Цалiса, нерiвноважнi
системи, нелiнiйнi рiвняння Фоккера-Планка
13001-18
Introduction
Analytical results
The oscillator model
Derivation of the stationary probability density
Stationary probability density in the phase space
Stationary probability density of the Hamiltonian energy
Limiting case of vanishing oscillator-oscillator interaction
Boundaries for the critical value of the parameter
Numerical simulation scheme for the stationary case
Impacts of model parameters
Impact of the net pumping force described by the fixed point energy E
The relative interaction parameter /Q
The parameter rel=/Q
Alternative Hamiltonian functions
Stationary phase space probability densities
Stationary distributions of Hamiltonian energy
Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-32038 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T17:51:12Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Mongkolsakulvong, S. Frank, T.D. 2012-04-06T17:16:53Z 2012-04-06T17:16:53Z 2010 Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space / S. Mongkolsakulvong, T.D. Frank // Condensed Matter Physics. — 2010. — Т. 13, № 1. — С. 13001: 1-18. — Бібліогр.: 49 назв. — англ. 1607-324X PACS: 05.40.-a, 05.45.Tp, 29.27.Bd, 41.75.-i https://nasplib.isofts.kiev.ua/handle/123456789/32038 We consider limit cycle oscillators in terms of canonical-dissipative systems that exhibit a short-range interaction in velocity and position space as described by the Dirac delta function. We derive analytical expressions for stationary distribution functions in phase space and energy space and propose a numerical simulation scheme for the simulation of the many body system as well. We show that the short-range interaction squeezes or stretches energy distribution functions depending on whether the interaction can be regarded as attractive or repulsive. In addition to the interaction effect, we show that energy distribution functions become narrower when limit cycle attractors become stronger. Finally, energy distributions become broader when the pumping energy is increased. The latter effect however disappears in the high energy domain. Розглянуто граничні циклічні осцилятори в термінах канонічно-дисипативних систем, які демонструють короткосяжну взаємодію у швидкісному та позиційному просторі, що описаний дельта-функцією Дірака. Виведено аналітичні вирази для стаціонарних функцій розподілу у фазовому просторі та енергетичному просторі та запропоновано числову симуляційну схему для моделювання багаточастинкової системи. Показано, що короткосяжна взаємодія стискає чи розтягує енергетичні функції розподілу залежно від того, чи взаємодію можна вважати притягальною, чи відштовхувальною. Крім впливу взаємодії, показано, що енергетичні функції розподілу стають вужчими, якщо межа циклічних атракторів стає сильнішою. Накінець, енергетичні розподіли стають ширшими з ростом енергії накачки. Цей ефект, проте, зникає у високоенергетичній області. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space Канонічно-дисипативні граничні циклічні осцилятори з короткосяжною взаємодією у фазовому просторі Article published earlier |
| spellingShingle | Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space Mongkolsakulvong, S. Frank, T.D. |
| title | Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space |
| title_alt | Канонічно-дисипативні граничні циклічні осцилятори з короткосяжною взаємодією у фазовому просторі |
| title_full | Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space |
| title_fullStr | Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space |
| title_full_unstemmed | Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space |
| title_short | Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space |
| title_sort | canonical-dissipative limit cycle oscillators with a short-range interaction in phase space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32038 |
| work_keys_str_mv | AT mongkolsakulvongs canonicaldissipativelimitcycleoscillatorswithashortrangeinteractioninphasespace AT franktd canonicaldissipativelimitcycleoscillatorswithashortrangeinteractioninphasespace AT mongkolsakulvongs kanoníčnodisipativnígraničníciklíčníoscilâtorizkorotkosâžnoûvzaêmodíêûufazovomuprostorí AT franktd kanoníčnodisipativnígraničníciklíčníoscilâtorizkorotkosâžnoûvzaêmodíêûufazovomuprostorí |