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. 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