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Non-equilibrium transitions and stationary probability distributions of stochastic processes
Volume 68A, number 5, 6
PHYSICS LETTERS
30 October 1978
NON-EQUILIBRIUM TRANSITIONS AND STATIONARY PROBABILITY DISTRIBUTIONS OF STOCHASTIC PROCESSES W. EBELING Sektion Physik der Wilhelm-Pjeck Universitàt Rostock, Rostock, DDR Received 24 August 1978
Examples for transitions in one-dimensional enzyme-reaction systems and in two-dimensional limit-cycle systems are discussed. Considering the stationary probability distributions as a kind of potential, concepts of catastrophe theory are applied to the study of possible types of transitions.
The transition phenomena in non-equilibrium systems have become a subject of widespread interest in the last time. Evidently it is to be expected that fluctuations play an important role in their onset [1—61. In the following we consider the possible qualitative changes of the shape of stationary probability distributions, e.g. changes in the number of extrema. It is well known that for a wide class of physical systems and external conditions stationary distributions P(x, u) over the state space X and the control space C, where x = {x 1 ...Xf} EX and u = {u1 Up.} E C, exist which are the target for the evolution in time [71 Part of the control variables may be macroscopic external parameters and part of them may be parameters of the noise spectrum. In recent work Haken [2] Engel-Herbert and this author [7,8] proposed the application of some concepts of the catastrophe theory [9] which were originally apphed to gradient systems, now to stationary probability distributions. The basic idea is the following [7,8] : Let us considerP(x, u) as a potential function which defines a smooth mapping Rf X R’~-+ R. The most important qualitative characteristics of P(x, u) are the modality (number of maxima) and the structural stability (non-existence of degenerate extrema). We define the hypersurface M of the states of extremal probability aP(x, u)/ax1 = 0. The probability distribution will be called structurally unstable if P(x, u) possesses degenerate extrema or turningpointsx ...
.
,
o = 0. (1) aP(x, u)/ax.I‘ X o 0, a2P(x, u)/ax.ax.I I J X Now we define the transition set K as the set of points in the parameter space C, where the number of location of the extremal states of the system undergoes a sudden change (i.e. a discontinuity) as the parameters vary. As a first result we may claim, following Thom’s well-known theorem, that there exist in the case r = dim C ~ 4 precisely seven essentially different elementary transitions of the probability distributions. Let us discuss now several examples. First we consider processes in a one-dimensional state space and natural boundary conditions satisfying in the stationary case the one time integrated Fokker—Planck equation P(x, u)(d/dx)[V(x, u) + 4(dldx) ED(x,
—
aD(x, u)J
u)P(x, u)]
=
Here D(x, u) is the diffusion coefficient and further we introduced for generality a coefficient a which has the value a = 0 if the underlying stochastic d.e. is interpreted as an Ito-equation and a = 1/4 if the stochastic d.e. is interpreted as a Stratonovich-equation [4,5] The solution of eq. (2) is easily found by integration. For a qualitative discussion we need only the set M corresponding to the states of extremal probability. From eq. (2) follows M: V’(x, u) + (-~— a)D’(x, u) = 0. (3) Therefore the auxiliary function V(x, u) = V(x, u)
430
(2)
0.
Volume 68A, number 5, 6 +
(! — 2
PHYSICS LETTERS
a)D(x, u) possesses the same extrema as
x1x2,
.
P(x, u) (minima of V correspond to maxima ofF) and therefore may serve us as a kind of stochastic potential. As an example which shows all types of elementary catastrophes we consider now the kinetic equation for an enzyme-controlled strongly inhibited reaction [10] : = —V’(x, u) (4) 2 +U5X 4 ) —l —U0—U1X+~U2X+U~~X)~l+X+U4X
3
•
1/2
x~=—g—x1—px2—6x2+D
~(t).
(7)
For p < 0 the system shows stable limit cycles. Assuming constant diffusion the Fokker—Planck equation reads 0=—(a/ax1)(x2P)
(8) +(a/ax2)[(g’ +x1 1-jix2 + ~x~)F+~D aF/ax2].
In. the . case . ~ = 0 one finds easily that the canonical distribution [2] is the exact stationary solution
—
(x is the substrate concentration). For an appropriate choice of parameters this reaction shows the 4 possible elementary transitions for the one-dimensional case: fold, cusp, swallow tail and butterfly. In the case D = const. the stationary distribution shows exactly the same transitions and has the same transition set, The situation changes if we consider an external gaussian white noise of the coefficients, e.g. let us assume that u2 fluctuates with the mean U2 and the variance u~.CalculatingD by the standard procedure [4,5] we find then the stochastic potential ~(x,u 0...u5,u2)=V(x 2x2~1+ x + uu0 ...u5) + u x4’—2 ‘ ‘ + ~! — a’ 102 ~ 4 5 ‘ ‘
Since the additional contribution in eq. (5) produces a new maximum of V, the properties of V can be cornpletely different in comparison to V. Starting, e.g., from a distribution with 3 maxima at 02 = 0, one finds with increasing 02 transitions to a bimodal and fmally to a monomodal system. Therefore external noise may have an essential influence on the type and the location of non-equilibrium transitions [5] Similar effects were observed for internal fluctuations [3] Now we shall consider the more complicated case of a non-gradient system showing a Hopf-bifurcation. We consider as an example the one-dimensional motion of a violin string of unit mass and frequency under influence of the dry friction force D(u) produced by the motion of the bow [7]. The equation of motion is assumed to be x1 = —x1 g (x1) + D(x1) + D 1‘~ ~(t), (6) .
.
-•
30 October 1978
•
—
where g(x1) is the potential of small anharmonic forces and where ~(t) is a gaussian white noise. Modelling the by theequations simple ansatz 3 wefriction get thehere dynamic (x D(u) = —pu — ~u 2= = v)
i
2
1
2
P’Nexp[_(2p/D)(g(x1)+~x +~x)].
(9)
The case where ~ is a small positive number we treated by using the transition to an amplitude-phase formulation described by Stratonovich [6] . Avoiding the derivation we give here only the result r 2 P(x1 x2, p, ~,D) = N exp (g(x1) + ~ + ~x~) (10) 3~ 2 —~ (x~+ x~) ,
[—--i
(10) By substitution is an approximate into eq.solution (8) one for testssmall easily ~ and thatg.eq. Inspection of eq. (10) shows that the Hopf-bifurcation of the deterministic picture at p = 0 corresponds to a cusp catastrophe of the probability surface. The two branches of the transition set are p = 0, ~ > 0 and p = —[9g’(x 26, ~ ~ 0, where x 1)/4] 1 is the root of g’(x1) + (2x1/3) = 0. The edge of the cusp is located at p = 0, ~ = 0. Inside the cusp we find a probability distribution in the form of a crater with a minimum in the center and a maximum as well as a saddlepoint on the crest. The deterministic limit cycle trajectory corresponds to the crest of the crater. Finally we want to mention that the stationary probability surface contains only a limited information on the dynamic motion e.g. the knowledge of eq. (10) alone does not allow a decision whether we have to do with a limit cycle or a special gradient system. Nevertheless the study of the transitions of the probability . distribution can be very. useful. for a better under. systems standing of dynamic including stochastic effects. The method explained here can be applied also for the study of transitions during the evolution of the probability in time if the time is considered as an additional parameter. 431
Volume 68A, number 5, 6
PHYSICS LETTERS
This work has been performed during a stay at the Université Libre de Bruxelles and the Instituts Internationaux de Physique et de Chimie. The author would like to express his gratitude to Professor I Prigogine for his hospitality and interest in this work and to thank Professor G. Nicolis and Drs. W. Horsthemke, M. Malek-Mansour and J.W. Turner for stimulating discussions. References [1] G. Nicolis and I. Prigogine, Selforganization in non-equllibrium systems (Wiley, New York, 1977). [2] H. Haken, Synergetics, an introduction (Springer, Berlin, 1977); H. Haken, ed., Synergetics, a workshop (Teubner, Stuttgart, 1977). [3] G. Nicolis and R. Lefever, Phys. Lett. 62A (1977) 469;
432
30 October 1978
G. Nicolis and J.W. Turner, Effect of fluctuations on bifurcation phenomena, Proc. Conf. Bifurc. theory (New York, 1977). [4] (1976) W. Horsthemke and M. Malek-Mansour, Z. Phys. B24 307. [5] L. Arnold, W. Horsthemke and R. Lefever, Z. Phys. B29 ~1978) 367. [6] R.L. Stratonovich, Topics in the theory of random noise andStrukturbildung Breach, New York, 1967). [7] (Gordon W. Ebeling, bei irreversiblen Prozessen (Teubner, Leipzig, 1976); appendix on the stochastic theory in the russian version (Mir, Moscow, 1979), to be published. [8] W. Ebeling and H. Engel-Herbert, Struktureile Instab~~~t~ten nichtlinearer irreversibler Prozesse, in: Rostocker Physikalische Manuskripte 2(1977) 23. [9] R. Thom, Structural stability and morphogenesis (Benjamin, New York, 1972). [10] G. Czajkowski and W. Ebeling, J. Nonequilibrium Thermodyn. 2(1977)1; Studia Biophys. 60(1976) 201.
PHYSICS LETTERS
30 October 1978
NON-EQUILIBRIUM TRANSITIONS AND STATIONARY PROBABILITY DISTRIBUTIONS OF STOCHASTIC PROCESSES W. EBELING Sektion Physik der Wilhelm-Pjeck Universitàt Rostock, Rostock, DDR Received 24 August 1978
Examples for transitions in one-dimensional enzyme-reaction systems and in two-dimensional limit-cycle systems are discussed. Considering the stationary probability distributions as a kind of potential, concepts of catastrophe theory are applied to the study of possible types of transitions.
The transition phenomena in non-equilibrium systems have become a subject of widespread interest in the last time. Evidently it is to be expected that fluctuations play an important role in their onset [1—61. In the following we consider the possible qualitative changes of the shape of stationary probability distributions, e.g. changes in the number of extrema. It is well known that for a wide class of physical systems and external conditions stationary distributions P(x, u) over the state space X and the control space C, where x = {x 1 ...Xf} EX and u = {u1 Up.} E C, exist which are the target for the evolution in time [71 Part of the control variables may be macroscopic external parameters and part of them may be parameters of the noise spectrum. In recent work Haken [2] Engel-Herbert and this author [7,8] proposed the application of some concepts of the catastrophe theory [9] which were originally apphed to gradient systems, now to stationary probability distributions. The basic idea is the following [7,8] : Let us considerP(x, u) as a potential function which defines a smooth mapping Rf X R’~-+ R. The most important qualitative characteristics of P(x, u) are the modality (number of maxima) and the structural stability (non-existence of degenerate extrema). We define the hypersurface M of the states of extremal probability aP(x, u)/ax1 = 0. The probability distribution will be called structurally unstable if P(x, u) possesses degenerate extrema or turningpointsx ...
.
,
o = 0. (1) aP(x, u)/ax.I‘ X o 0, a2P(x, u)/ax.ax.I I J X Now we define the transition set K as the set of points in the parameter space C, where the number of location of the extremal states of the system undergoes a sudden change (i.e. a discontinuity) as the parameters vary. As a first result we may claim, following Thom’s well-known theorem, that there exist in the case r = dim C ~ 4 precisely seven essentially different elementary transitions of the probability distributions. Let us discuss now several examples. First we consider processes in a one-dimensional state space and natural boundary conditions satisfying in the stationary case the one time integrated Fokker—Planck equation P(x, u)(d/dx)[V(x, u) + 4(dldx) ED(x,
—
aD(x, u)J
u)P(x, u)]
=
Here D(x, u) is the diffusion coefficient and further we introduced for generality a coefficient a which has the value a = 0 if the underlying stochastic d.e. is interpreted as an Ito-equation and a = 1/4 if the stochastic d.e. is interpreted as a Stratonovich-equation [4,5] The solution of eq. (2) is easily found by integration. For a qualitative discussion we need only the set M corresponding to the states of extremal probability. From eq. (2) follows M: V’(x, u) + (-~— a)D’(x, u) = 0. (3) Therefore the auxiliary function V(x, u) = V(x, u)
430
(2)
0.
Volume 68A, number 5, 6 +
(! — 2
PHYSICS LETTERS
a)D(x, u) possesses the same extrema as
x1x2,
.
P(x, u) (minima of V correspond to maxima ofF) and therefore may serve us as a kind of stochastic potential. As an example which shows all types of elementary catastrophes we consider now the kinetic equation for an enzyme-controlled strongly inhibited reaction [10] : = —V’(x, u) (4) 2 +U5X 4 ) —l —U0—U1X+~U2X+U~~X)~l+X+U4X
3
•
1/2
x~=—g—x1—px2—6x2+D
~(t).
(7)
For p < 0 the system shows stable limit cycles. Assuming constant diffusion the Fokker—Planck equation reads 0=—(a/ax1)(x2P)
(8) +(a/ax2)[(g’ +x1 1-jix2 + ~x~)F+~D aF/ax2].
In. the . case . ~ = 0 one finds easily that the canonical distribution [2] is the exact stationary solution
—
(x is the substrate concentration). For an appropriate choice of parameters this reaction shows the 4 possible elementary transitions for the one-dimensional case: fold, cusp, swallow tail and butterfly. In the case D = const. the stationary distribution shows exactly the same transitions and has the same transition set, The situation changes if we consider an external gaussian white noise of the coefficients, e.g. let us assume that u2 fluctuates with the mean U2 and the variance u~.CalculatingD by the standard procedure [4,5] we find then the stochastic potential ~(x,u 0...u5,u2)=V(x 2x2~1+ x + uu0 ...u5) + u x4’—2 ‘ ‘ + ~! — a’ 102 ~ 4 5 ‘ ‘
Since the additional contribution in eq. (5) produces a new maximum of V, the properties of V can be cornpletely different in comparison to V. Starting, e.g., from a distribution with 3 maxima at 02 = 0, one finds with increasing 02 transitions to a bimodal and fmally to a monomodal system. Therefore external noise may have an essential influence on the type and the location of non-equilibrium transitions [5] Similar effects were observed for internal fluctuations [3] Now we shall consider the more complicated case of a non-gradient system showing a Hopf-bifurcation. We consider as an example the one-dimensional motion of a violin string of unit mass and frequency under influence of the dry friction force D(u) produced by the motion of the bow [7]. The equation of motion is assumed to be x1 = —x1 g (x1) + D(x1) + D 1‘~ ~(t), (6) .
.
-•
30 October 1978
•
—
where g(x1) is the potential of small anharmonic forces and where ~(t) is a gaussian white noise. Modelling the by theequations simple ansatz 3 wefriction get thehere dynamic (x D(u) = —pu — ~u 2= = v)
i
2
1
2
P’Nexp[_(2p/D)(g(x1)+~x +~x)].
(9)
The case where ~ is a small positive number we treated by using the transition to an amplitude-phase formulation described by Stratonovich [6] . Avoiding the derivation we give here only the result r 2 P(x1 x2, p, ~,D) = N exp (g(x1) + ~ + ~x~) (10) 3~ 2 —~ (x~+ x~) ,
[—--i
(10) By substitution is an approximate into eq.solution (8) one for testssmall easily ~ and thatg.eq. Inspection of eq. (10) shows that the Hopf-bifurcation of the deterministic picture at p = 0 corresponds to a cusp catastrophe of the probability surface. The two branches of the transition set are p = 0, ~ > 0 and p = —[9g’(x 26, ~ ~ 0, where x 1)/4] 1 is the root of g’(x1) + (2x1/3) = 0. The edge of the cusp is located at p = 0, ~ = 0. Inside the cusp we find a probability distribution in the form of a crater with a minimum in the center and a maximum as well as a saddlepoint on the crest. The deterministic limit cycle trajectory corresponds to the crest of the crater. Finally we want to mention that the stationary probability surface contains only a limited information on the dynamic motion e.g. the knowledge of eq. (10) alone does not allow a decision whether we have to do with a limit cycle or a special gradient system. Nevertheless the study of the transitions of the probability . distribution can be very. useful. for a better under. systems standing of dynamic including stochastic effects. The method explained here can be applied also for the study of transitions during the evolution of the probability in time if the time is considered as an additional parameter. 431
Volume 68A, number 5, 6
PHYSICS LETTERS
This work has been performed during a stay at the Université Libre de Bruxelles and the Instituts Internationaux de Physique et de Chimie. The author would like to express his gratitude to Professor I Prigogine for his hospitality and interest in this work and to thank Professor G. Nicolis and Drs. W. Horsthemke, M. Malek-Mansour and J.W. Turner for stimulating discussions. References [1] G. Nicolis and I. Prigogine, Selforganization in non-equllibrium systems (Wiley, New York, 1977). [2] H. Haken, Synergetics, an introduction (Springer, Berlin, 1977); H. Haken, ed., Synergetics, a workshop (Teubner, Stuttgart, 1977). [3] G. Nicolis and R. Lefever, Phys. Lett. 62A (1977) 469;
432
30 October 1978
G. Nicolis and J.W. Turner, Effect of fluctuations on bifurcation phenomena, Proc. Conf. Bifurc. theory (New York, 1977). [4] (1976) W. Horsthemke and M. Malek-Mansour, Z. Phys. B24 307. [5] L. Arnold, W. Horsthemke and R. Lefever, Z. Phys. B29 ~1978) 367. [6] R.L. Stratonovich, Topics in the theory of random noise andStrukturbildung Breach, New York, 1967). [7] (Gordon W. Ebeling, bei irreversiblen Prozessen (Teubner, Leipzig, 1976); appendix on the stochastic theory in the russian version (Mir, Moscow, 1979), to be published. [8] W. Ebeling and H. Engel-Herbert, Struktureile Instab~~~t~ten nichtlinearer irreversibler Prozesse, in: Rostocker Physikalische Manuskripte 2(1977) 23. [9] R. Thom, Structural stability and morphogenesis (Benjamin, New York, 1972). [10] G. Czajkowski and W. Ebeling, J. Nonequilibrium Thermodyn. 2(1977)1; Studia Biophys. 60(1976) 201.