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Entropy Decrease during Excitation of Sustained Oscillations.

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A N N A L E N D E R PHYSIK
7. Folge. Band 44. 1987. Heft 6, S. 393-472
Entropy Decrease during Excitation of Sustained Oscillations
By H. ENGEL-HERBERT
and M. SCHUMA"
Sektion Physik der Humboldt-Universitlt Berlin
Abstract. The behaviour of the entropy is studied during a nonequilibrium phase transition.
It is shown that for the generation of sustained oscillations the entropy decreases monotonously if
the average oscillator energy remains fixed. This result is valid both for hard and soft excitation of
oscillations.
Entropieverringerung bei der Anfachung selbsterregter Schwingungen
Inhaltsubersicht. Das Verhalten der Entropie wahrend eines Nichtgleichgewichtsphasenubergangs wird untersucht. Es wird gezeigt, daR die Entropie bei der Anfachung selbsterregter
Schwingungen monoton fLllt, wenn wahrend des fibergangs die mittlere Energie der Oszillationen
fixiert wird. Dieses Resultat gilt sowohl im Fall des harten als auch des weichen Einsatzes der Schwingungen.
1. Introduction
If the distance from equilibrium is increased beyond a certain threshold some physical, chemical or biological systems show spontaneous formation of structure [I]. Well
known examples are hydrodynamic instabilities (Benard convection, turbulence), current instabilities in semiconductors, biological clocks and pattern formation in morphogenesis.
I n this connedtion the question arises how to determine the degree of order in stationary nonequilibrium states. Recently [2, 3, 7-91 it was suggested to use the entropy
given by Boltzmann formula
S ( u ) = -k,
J P ( X ;u)In P ( X ; u)d X
(1.1)
(X = {Xl, ..., Xm} - set of variables determining the state of the system,
u = {ul,.. , uk}- set of external parameters, Po - stationary distribution function).
Klimontovich developed the idea that with increasing distance from equilibrium S
decreases provide states of the same average energy are compared, i.e.
.
Sea> Xtb > S,, if Eea= Etb = E d 3 .
(1.2)
Sea,S,,, S , and Eeq,Etb,Ed, denote the entropy and the energy of the system in equilibrium, near equilibrium (thermodynamic branch) and jn the stationary nonequilibrium
state (dissipative structure) respectively. I n situations where thi.s statement holds we
may conclude that judging from the lowering of entropy the far from equilibrium structures are more ordered than the equilibrium structures of the same energy.
Up to now the statement, was proven to hold for soft excitztion of sustained oscillations [2, 8, 91, for the onset of turbulence in Couette and Poiseuille flows [4, 5, 7, 81
and for the transition of an oscillator with nonlinear delay to chaotic motion due to
394
Ann. Physik Leipzig 44 (1987)G
period doubling bifurcation in accordance with Feigenbauin’a law [6]. I n all three cases
the distance from equilibrium was defined by one external parameter: the level of feedback of the oscillator and the Reynolds number of the flow respectively. I n order to
satisfy the condition of fixed energy during the nonequilibrium transition the intensity
of the fluctuations acting on the oscillator or the temperature of the flow was varied
appropriat,ely.
In this paper we present results for the entropy behaviour during the transition to a
stationary nonequilibrium state governed by two independent external parameters.
Studying a certain class of excitable nonlinear oscillators in the presence of additive
short correlated noise it will be shown, that the generation of sustained oscillations at
fixed average oscillator energy lowers the entropy even if the noise intensity remains
constant.
2. The Model
We consider a n excitable oscillator with nonlinear damping subjected to additive
Gaussian white noise of intensity D according to
q
+ Ey(a,a; u ) + wtq = (0 a t ) , ( W >= 0, ( a t +
7)
5(0> = 4 7 ) .
(2.1)
q denotes the coordinate, the overdot, as usual, represents differentiation with respect
to time and E is a small parameter. The common feature of all such oscillators is that they
use positive feedback which must be strong enough to amplify and to sustain the initial
perturbations. To realize for both soft and hard mechanism of excitation the damping
function y is choosen to be
Eq. (2.1) and (2.2) describe for example the operation of a vacuum tube oscillator in the
presence of noise; it is convenient t o approximate the tube characteristic by a polynomial [lo].
If E is sufficiently small E < 1 well separated time scales occur in the system. The
oscillator energy H = q2/2
wtq2/2 becomes a slowly varying quantity (compared with
q and q) with characteristic time of order 1/e which is much larger than the period of
harmonic oscillations To = 2n/00 performed if dissipation and noise are neglected ( E =
0). It may be shown that the distribution function of the energy P ( H , t : u ) obeys the
Fokker-Planck equation [ll,121
+
(2.3)
We are interested in the stationary solution which under natural boundary conditions
is found to be
(ulH
N
+ u,H2 + u 5 P )
where
W
N
=
0
[
1
D
d H exp - - (uIH
is the normalization constant.
+ u3H2 + us@)
1
(2.8)
H. EXGEL-HERBERT
and 11. SCHUMANN,
Entropy Decrease st Sustained Oscillations
395
Using (2.4) we obtain for the entropy from (1.1)
(in the following we put k, = 1)
S(U)= 111N
1
+(u,(H)+ U3(H2)+ u 5 ( H 3 ) ) .
D
In order to compare the entropy of states with the same energy we assume that the average oscillator energy
ca
(H)-=
J
d H H P ( H ;U )
(2.7)
0
remains constant during the transition.
From the local extrema of the stationary energy distribution we determine a nonoscillatory region (ul> 0, ~3 > -f1/3u,u5) and two different oscillatory regions in the
parameter plane spanned by u1 over u5 and by u3over u5.In the first oscillatory regime
(ul< 0) one stable limit cycle contains an unstable focus or node inside whereas the
second oscillatory region (ul> 0, u3< -(1/3u,u,) is characterized by the coexistence
of two limit cycles, one being stable and surrounding the other which is unstable. This
is shown on Fig. l., the typical shape of the distribution function of the energy in each
caw is sketched too. If u3is positive and the feedback is increased beyond the excitation
threshold u1= 0 the transition from nonoscillatory to oscillatory behaviour corresponds
t o soft excitation of sustained oscillations (Hopf bifurbation). If changing the parameters
the parabola 1/3(~~/u,)~
= u1/u5
is crossed a t negative values of u, suddenly sustained
oscillations of finite amplitude are generated due to hard excitation.
We remark that the bifurcation set obtained from the stationary energy distribution
coincides with the threshold values in the absence of noise [lo]. The reason is that we
deal with additive noise. Noise induced shifts of the deterininistic threshold values due
t o multiplicative noise are studied in [12].
L- 2
Fig. 1. Bifurcation dingram for the nonlinear oscillator defined by (4.1) and (2.2)
Ann. Physik Leipzig 44 (1967)6
396
3. Soft Excitation of Sustained Oscillations
The entropy lowering during soft excitation of sustained oscillations was studied in
[2, 7, ... 91 .In the following we present some results concerning the variation of the
entropy with the excitation parameter ul(ul < 0 iii the region of generation) at constant
noise int,ensity D under the condition of fixed average energy of the oscillator.
Without loss of generality we put u5 = 0, u3 > 0 so that
U
s = In N + 2
(H)+%(~2>,
D
(3.1)
with
1 "
1
(Hn)=-/
d H H % e x p [-Tii(ulH+
No
u&2)],
n=O,1,
....
(3.2)
For the derivative of the entropy
(3.3)
we obtain from (3.1, 3.2)
where
2
=-
uLand
/
bJ
y(z) = 2 exp (3)
2 1/%D
exp ( - t 2 ) dt.
(3.6)
t
We consider the three most interesting situations :
1. 111the stationary state without feedback (ul> 0) the nonlinearity may be neglected (u3= 0). With
( H n ) = - F1 d H H n e x p
No
from (3.1) and ( 3 . 4 ) follows
(3.7)
2. The excitation threshold (ul= 0).
111 thiA
case we have
and therefore from (3.1) and (3.4)
(3.9)
H. ExuI~I,-HERBERT and 31. ScHunrAxs, Entropy Decrease a t Sustained Osci&dions
< 1) we can
3. In the case of fully developed oscillat~ions(ul < 0, 2u,D/u~
Gaussian approximation
397
use a
W
(an)
=
$ d H HnPo(H; u
) ~
- w
(3.10)
L
to obtain froin (3.1) and (3.4)
J
u3
(3.ii)
bezause for well pronounced oscillations t l k parameter
6 3 ( (6H)2)/(H)2= is much smaller than one.
Summarizing we conclude that in the considered stationary states the entropy is a
monotonously increasing function (compare Fig. 2.). Starting in a nonoscillatory stationary state with uI> 0 if u1 is decreased beyond the threshold u1 = 0 the generation of
oscillations a t constant noise intensity is connected with entropy lowering so far as
during the transition the average energy of the oscillator is fixed.
\
I
-15
-10
-5
5
10
15
U1
Fig. 2. Entropy loweriag during soft excitation of sustained oscillations at constant noise intensity.
Numerical results from (3.1) m d (3.4) €or l/D = 0.05 and ( H ) = 0.86
If the last condition i s dropped the opposite behaviour for the entropy is found:
for zero feedback (ul> 0) and small
nonlinearity,
1
=j_
1-
1
__
2 InDu,
-0
<0
a t the excitation threshold,
for fully developed oscillations.
(3.12)
398
Ann. Physik Leipzig 44 (1987) 6
The entropy increases during the transition, for example
In contrary a t fixed average energy if
(3.14)
we get
(3.15)
4. Hard Excitation of Sustained Oscillations
First we take u1 positive t o study the hard mechanism of excitation along path 1
on Fig. 1. I n this cam the average energy and the entropy depend on the noise intensity
D and on the parameter us. For a given positive value of u1 we determined numerically
the values of D and u3which satisfy the condition of constant average oscillator energy
(2.7). In the oscillatory regime the effective noise intensity becomes smaller compared
with the nonoscillatory region (Fig. 3.). For the derivative of the entropy with respect
to u3 we get the following expression
as
u,D
+
( m ) u , , u 3 (&)u,,(H)
1.
- - -( ( H 2 0 2
8D
( H ) 2 )(UJ
+u p+
UsH3))
+1
(4.1
Fig. 3. The noise intensity as function of t h s parameter u3for hard excitation of sustained oscillations
according to path 1in Fig. 1 (ul= 16). A'nmericel calculation for ug = 80 and (H) = 0.7
H. ENOEL-HERBERT
end M. SCHUMANS,Entropy Decrease at Sustained Oscillations
399
which contains the moments of the energy distribution up to sixth order. Numerical
calculations on the basis of (4.1) show that this derivative remains positive during the
transition indicating the decrease of the entropy for hard excitation of sustained oscillations. The entropy calculated from (2.6) is in the oscillatory states smaller than in the
norioscillatory states of the same average energy (compare Fig. 4.). If we dont fix the
average energy during the transit.ion the opposite behaviour is observed, (i3S'/&3)ul,D
for example is negative.
I
/-
O09to5
V
008
j
007 40.4
006-
-03
005OO4..O2
003\
0.01-.
threshold
-30
-25
-20
-15
-10
-5
5
"9
Fig. 4. Decrease of entropy along path 1 (ul= 15) determined numerically from (2.6) and (4.1) for
ug= 20 and ( H ) = 0.7
Quite similar results for the entropy lowering we obtained entering the oscillatory
region along path 2 on Fig. 1. The energy distribution is presented for some values of
D and u1 on Fig. 5 . Because u3 = const now, the condition (2.7) determines the noise
intensity D as function of the excitation parameter u1as shown on Fig. 6. The resolts for
S are obtained from (2.6), the derivative was calculated from the expression
Again the entropy decreases during the transition t o sustained oscillations (see Fjg. 7.).
The disappearance of the unstable limit cycle a t u1 = 0 by an inverted Hopf bifurcation
does'nt affect the entropy behaviour.
Finally let us consider the hard mechanism of generation at constant noise intensity.
The condition (2.7) determines a u3(uI)dependence (path 3 on Fig. 1)which intersects
both hard and soft branches of the bifurcation set. Due to this special choice we expect
a maximum of the entropy at some values ul,u3which belong to the nonoscillatory region.
0.5DL
0.1
1
3
2
H
Fig. 5. Energy distribution for some values of u1 ;md of IJwhich correspond to path 2 (u3= -30,
u6 = 20, (H)= 1.0)
a) u1 = 30, 1/D = 0.0106 nonoscillatoryregion, b) u1 = 16, 1/D = 0.0138 threshold for hard excitation, c) u1 = 5, 1/11= 0.0181 coexistcncc of two limit cycles, d) u1 = 0, 1/D = 0.0221 threshold for
soft excitation, e ) u, = -5, 1/D = 0.0298 regime with one stable limit cycle
\
1
\
-0.015
c
-5
5
Yo
15
20
U1
Fig. 6. Soise intensity I ) against excitation pir,zmc.ter ulfor thc: transition to sustained oscillations
at u3 = -30 for u5 = 20 and (11) = 1.0
H. EXCEL-HERBERT
and 31. SVHUYANX,
Entropy Decrezzse at Sustained Oscillations
40 1
The numerical calculations of S froiii (2.6) and of the derivative from
1
- (H2>( H ) )
+i9
+
u5((H4>
- (@> (H))1 ((H3>- ( H 2 > ( H > )
(H2>- (H>2
(4.3)
confirm this assumption (see Fig. 8.).
Summarizing both soft and hard excit,ation of sustained oscillations are connected with
entropy lowering if during the transition the averge energy of the oscillator is kept fixed
I
-1 0
-2
‘ 2
c
1’0
“1
Fig. 7. Entropy lowering along path 2 on Fig. 1 according to (2.6) and (4.2). Parameter values:
US = -30, ~g = 20, ( H ) = 1.0
Fig. 8. Entropy behaviour during the genera.tionof sustained oscillations at constant noise intensity
(path 3 on Fig. 1)calculated from (2.6) and (4.3) for ug = 20, 1/D = 0.05 and ( H ) = 0.8%
5. Conclusions
Changing a suitable external parameter and thereby crossing the excitation threshold
into the region of oscillations the entropy decreases monotonously provided states with
equal energy are compared. This condition of constant energy can be ensured by appropriately varying the noise intensity D or a second external parameter.
Ann. Physik Leipzig 44 (1987) 6
402
We emphasize that the system under consideration is in thermodynamic equilibrium
with a heat bath of temperature T if
u1 = D/k,T
and u3 = u5 = 0 .
For nonvanishing u3and u5the fluctuations c ( t )in (2.1)’are determined by nonequilibrium
processes. The system evolves to thermodynamic equilibrium only if the noise intensity
D becomes state dependent. The unknown function D(q)may be obtained from the condition that the stationary solution of the corresponding Fokker-Planck equation i s the
canonical distribution [3].
In general it is difficult to decide from Fig. 1. which of two different’ nonoscillatory
states is more distant from thermodynamic equilibrium. I n open systems a reasonable
measiire for the distance from thermodynamic equilibrium is the energy dissipated in
the stationary nonequilibrium state, i. e. the entropy production P. Indeed, under
stationary conditions the energy dissipation characterizes the amount of energy pumped
into the system. For the generation of sustained oscillations P increases if the feedback
level is intensified. For turbulent flows the entropy production j s an increasing function
of the Reynolds number. This will be studied in greater detail in a subsequent paper.
Thus it is possible t o measure the degree of order due to the generation of sustained
oscillations (which represent a typical nonequilibrium state in a thermodynamically
open, pumped system) by the corresponding values of the entropy and of the entropy
production.
The authors thank W. Ebeling and H. Herzel for fruitful discussions.
References
R. : PPhjsi‘k c‘er Srlbstorgani~ationund Evolution, Berlin: Akadrmic
[l] EBELING,
W. ; €EISTEL,
Verlag 1982.
p2] KLIXONTOVICH,
Yu. L.: Pis’ma Zh. Tekh. Fiz. 9 (1983) 1412.
[:3] KLIMONTOVICH,
Yu. L. : Statistical Physics (in Russian), Moscow: Nauka 1982.
[4] KLIJIONTOVICH,
Yu. L.: Pis’ma Zh. Tekh. Fiz. 10 (1984) 50.
[5] KLLXOWFOVICH,
Yu. L. : The Role of Fluctuations for Self-Organization in Physical Systems
(an Exemplary Case of Transition from a Laminar to a Turbulent Flow. In: Self-Organization
Autowaves and Structures far from Equilibrium. Springer Series in Synergetics vol. 28 ed. by
V. I. KRINSKY,77. Berlin: Springer-Verlag 1984.
[t;] ANISHENRO,
V. S. ; KLIMONTOVICH,
Yu. L. : Pis’ma Zh. Tekh. Fiz. 10 (1984) 876.
[7] EBELINC,
W.; KLIMONTOVICH,
Yu. L.: Selforganization and Turbulence in Liquids, Leipzig:
Teubner-Verlag 1984.
[8] EBELING,
W. ; ENQEL-HERBERT,
H. ; HERZEL,
H. : Ann. Phys. 48 (1986) 187.
[9] EBELING,
W.; HERZEL,H.; ENQEL-HREBERT,
H.: Z. Phys. Chem. (Leipzig) 866 (1985) 253.
[lo] ANDRONOV,
A. A. ; WITT,A. A. ; CHAIKIN,S. E. : Theorie der Schwingungen, Bd. 2, Berlin:
Akademie-Verlag 1969.
[ll]STRATONOVICII,
R. I..: Topics in the Theory of Random Noise, Vol. 2. New York: Gordon and
Breach 1967.
[12] EBELING,
W.; ENQEL-HERBERT,
H.: Physica 104 A (1980) 378; Adv. Mechanics 5 (1982) 41.
Bei der Redoktion eingegangen om 17. Juni 1985.
Anschr. d. Verf. :Dr. H. ENQEL-HERBERT
Humboldt-Universitkt Berlin
Sektion Physik 04
Invalidenstr. 42, Berlin, DDR-1040
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