Appl. Phys. A57, 111-115 (1993)

Applied Ph icsA Surfaces .n. © Springer-Verlag 1993

Basic Concepts of Synergetics H. Haken Institute for Theoretical Physics and Synergetics, Pfaffenwaldring 57/4, D-70550 Stuttgart, Germany (Fax: +49-7l 1/6854909) Received 22 February 1993/Accepted 21 April 1993

Abstract. The paper presents a brief outline of microscopic as well as of macroscopic synergetics. In microscopic synergetics we start from evolution equations for microscopic variables or densities in which fluctuating forces and control parameters are included. When control parameters are changed, the systems are studied close to instability points. The concepts of order parameters, enslaving, critical fluctuations, and critical slowing down are presented. In macroscopic synergetics unbiased estimates on distribution functions and underlying processes are made based on observed moments or correlation functions. In such a case, a FokkerPlanck equation or a corresponding Langevin equation may be derived. PACS: 01.55, 05.00, 64.90. +b

In many disciplines of physics we have to deal with systems that are composed of very many parts, in particular atoms or molecules. In many cases, thermodynamics appears as the adequate tool for the treatment of such systems. Here the familiar concepts of entropy, temperature, specific heat, etc. come in. The enormous importance of thermodynamics stems from the fact that it holds independently of the nature of the systems and their parts under consideration. Thermodynamics applies equally well, say, to solids or to chemical reactions. On the other hand, thermodynamics is based on well-defined assumptions which, however, are occasionally overlooked. It requires that the system is in thermal equilibrium. This is quite obvious from the thermodynamic definition of entropy S by means of the formula d S = dQ, rev/T, where the absolute temperature T enters. Statistical physics, at least in many cases, is considered as a tool to give the laws of thermodynamics a microscopic foundation. It is then shown that in thermal equilibrium the equipartition theorem holds, i.e. each degree of freedom has the same average thermal energy. In addition, thermodynamics requires that the processes go on infinitely slowly.

1 Limits of Thermodynamics and Irreversible Thermodynamics Let us consider a simple example, namely that of a water wave on a lake. It is certainly true that we can attribute to the water a temperature which can be easily measured. But the water wave represents a macroscopic state in which a single degree of freedom is highly excited. On the other hand, its entropy contributes only a tiny fraction to the total entropy. A way out of this dilemma seems to be provided by irreversible thermodynamics. But here again concepts, such as local entropy, temperature, etc. enter and there is no space for highly excited states. Over the past two or three decades, it has become apparent that there are numerous examples of physical processes to which the concepts of thermodynamics, irreversible thermodynamics, or statistical physics of thermal equilibrium cannot be applied. Rather new types of approaches had to be developed. One natural direction of these new approaches is provided by synergetics, where two lines of approach have been followed up: the microscopic and the macroscopic.

2 Microscopic Synergetics In this approach [1, 2] we study systems composed of very many subsystems, where the variables of the subsystems are lumped together into a state vector

q = (q~, q2,..., q N ) .

(1)

Let us mention a few examples. In the laser, the quantum mechanical polarization and inversion of each atom as well as the individual cavity mode amplitudes represent the components of the state vector. In a fluid, we may either start from the microscopic level by describing the individual molecular positions and velocities as components of the state vector or by means of a mesoscopic approach. In this latter picture, we treat individual volume elements to which a local temperature or density of molecules can be attributed. Other examples may be provided by the densities of electrons and holes

112

H. Haken

in semiconductors. The densities may also refer to those of atoms in the liquid state and to those in the solid state. In this way, crystallization processes can be included. We then study the temporal change of the state vector q which is determined by so-called evolution equations

il = N ( q , V , cO + F ( t ) .

(2)

In them the rate of change of q is determined by a non-linear function of the present state vector, and if the state vector depends continuously on space, by spatial gradients of q. In many important cases the system is controlled from the outside by means of so-called control parameters c~. They may be temperature, but also an energy input into the system, etc. Finally, the system is subject to external or internal fluctuating forces. Such forces enter, for instance, when the system is coupled to one or several heat baths. In synergetics, we focus our attention on systems far from thermal equilibrium. To maintain systems in that state, it is required that the system is coupled to heat baths kept at different temperatures. We may study also transients, where a system starts from a state away from thermal equilibrium and is coupled to a single heat bath only. In any case, when the heat bath's variables are eliminated¢damping forces as well as random forces arise. Damping and fluctuations stemming from the individual heat baths are in each case determined by a fluctuation-dissipation theorem. This may actually be used as a tool to determine the correlation functions of the fluctuating forces F j ( t ) in the following forms

(FAt))

(3)

= o,

(Fj(t)Fk(t')}

= Qjk~(t

- ~') .

of simplicity and because of lack of space, we shall present here only the case of the simple time-independent reference solution; for the more general cases the reader is referred to [2]. When we change the control parameter or some of them, q0 may retain its form, e.g. the fluid may remain quiescent. However, eventually, we may reach a control parameter value, where this solution becomes unstable. To study the instability, we perform a linear stability analysis by making the hypothesis q = qo + w ,

(5)

where to is a small space- and time-dependent function. We then insert (5) into (2), neglect the fluctuating forces, and under the assumption that to is small, linearize the resulting equation. We then obtain (6)

fv = L t o ,

where L is a matrix acting on the vector w. L may also depend on spatial derivatives. On quite general grounds, the solution to of (6) can be written in the form

w=

X~tvu(x), ea~tvs(x) ,

{e

Re)~u _> 0 ReAs<0.

In the upper case the real part of )~ is non-negative. In such a case we shall speak of unstable modes. If the real part of )~ is negative, we shall speak of damped modes. The linearized equations (6) allow us to determine the spatial dependence of vu and v , under given boundary conditions. In general, a whole set of modes that are distinguished by the indices u and s will be found. In order to solve the fully nonlinear and stochastic equations (2), we make the hypothesis

(4)

The evolution equations (2) are still of a very general nature and at first sight it appears hopeless to solve them in a general manner. However, for a large class of cases a possible method of solution can be guided by means of experimental facts. A well-known example is provided by a fluid heated from below. If the temperature difference between the lower and upper surface is below a critical value, the fluid remains macroscopically at rest. Heat is transported from the lower to the upper surface by microscopic heat conduction. On the other hand, when the temperature difference surpasses a critical value, suddenly a macroscopic motion in the form of up-welling and down-streaming rolls sets in. In other words, beyond a critical value of a control parameter the former quiescent state of the liquid has become unstable and is replaced by a macroscopic pattern of motion. This simple experimental result suggests the following mathematical procedure: We start from a solution of the nonlinear equations (2) for a certain set of control parameters, in which this solution is known. This may be in the simplest case a timeindependent solution, but also time-periodic, quasi-periodic, or chaotic solutions may serve as a reference state. For sake

where the sum runs over all mode indices u and s and where the mode amplitudes ~u and ~s are still unknown time-dependent functions. In order to solve (2) by means of (8), we insert (8) into (2) and project the resulting equation on the modes by means of their adjoint vectors. We then obtain equations of the form ~,~ = A ~

+ N ~ , ( ~ , ~s) + F~,(t)

(9)

~s = As(s + Ns(~u, ~s) + Fs(t) .

(10)

and

In principle, the difficulty of solving (9) and (10) is as big as that of solving (2). However, when we are close to an instability point, we may assume that the real part of ~ is a small quantity compared to the real part of ),8. In such a case, the slaving principle of synergetics holds. According to this principle, we may express ~ by means of ~, at the same time. The explicit time-dependence of ~s in es = ~s(~, t ) ,

(11)

Basic Concepts of Synergetics

113

results from the action of the fluctuating forces. In most cases of general interest, {s is a low-dimensional polynomial in {~,, for instance a polynomial up to the second order in {.~. (11) allows us to eliminate {s from (9) so that we finally obtain equations of the form ~ = A~

+ 2V~({~) + F~,(t) ,.

(12)

This has a very crucial importance, namely close to instability points the dynamics of even a very complex system is governed by the, in general, few mode amplitudes {~. The {~, are called order parameters. The slaving principle and the concept of order parameters allow for an enormous reduction of information. Instead of determining the behavior of the very numerous amplitudes of the enslaved modes {s, it is sufficient to consider the behavior of the order parameters alone. Once the order parameters are determined from (12), we may construct the wanted solution (8) by invoking (11) and the form of the modes v~,, v~ that result from the linearized equations (6). If q0 is time-periodic, time-quasiperiodic, then also specific phase angles must be introduced and equations of the form ¢.~ = N¢,,,({~,, ¢~) + F e , ~ ( t ) ,

(13)

must be added to those of (12), and N~ in (12) will depend also on ¢u. A number of important concepts may be discussed when we consider (12) in the case of a single order parameter. Here the order parameter equation up to its leading terms reads ~ = A ~ ( ~ + a e2~ - cg~ + F ~ ( t ) .

(14)

If inversion symmetry of the system holds, the quadratic term on the right-hand side vanishes and we are left with an order parameter of the form =

- c~ + F~(t).

(15)

The solution of this equation can be easily visualized, if we interpret this equation as that of a particle undergoing an overdamped motion in a potential V(~,) under the additional impact of a fluctuating force F~. If A~ is negative, there is only one valley of the potential landscape, i.e. only one stable solution that is pushed backwards and forwards by means of the fluctuating forces around its stability value. For A~, = 0, the valley flattens, the random fluctuations may act rather heavily because the restoring force is zero. Close to the equilibrium point, we are dealing with the phenomenon of critical fluctuations. Also the relaxation of the system is slow, i.e. we are dealing with critical slowing down. These concepts are well-known from the theory of phase transitions of systems in thermal equilibrium. However, here we note that our approach allows the treatment of systems away from thermal equilibrium. Finally, for positive A~, the one minimum is replaced by two minima; the system has to decide between one of these two states. We are dealing here with a symmetry-breaking instability. Note that a small initial microscopic fluctuation may determine the macroscopic final

state by providing the system with an initial push into the one or the other valley. Because of the close analogy of these phenomena with those of systems in thermal equilibrium at phase transitions, the transition is called a nonequilibrium phase transition. If the impact of fluctuations is ignored, we may say that the initial state is replaced by two states when the parameter A~, goes from its negative to its positive value. In such a case one speaks of a bifurcation. Note, however, that the concept of an nonequilibrium phase transition is more general, because it deals with fluctuations and it takes into account relaxation processes close to instability points that are ignored in bifurcation theory. An interesting delicate point occurs, if the eigenvalues A~ and A~ belong to a continuous spectrum. Then the required separation between the eigenvalues of undamped and damped modes, which is the basis for the slaving principle, is no more secured. In such a case one may form, however, a wave paket in the form ~u,ko ( X, t) =

f

ko+Ak

~u,k eikX vu,k ( x )dn k ,

(16)

Jko-Ak

where the order parameter becomes now not only time- but also space-dependent as is indicated in (16). It is then possible to derive equations that are generalizations of (12) in the form [2] 4u,ko = A~(ko + V){u,ko + Nu,ko(~,,k' o) + Fu,ko •

(17)

Here the eigenvalue A~ has become an operator acting on the spatial variable occuring in {~. Because of their formal analogy with the Ginzburg-Landau equations of superconductivity, I have called these equations "generalized GinzburgLandau equations". In a number of cases and under simplifying conditions, the equations (17) can be still further simplified, e.g. to the type ~(X, t) = (/~0 4- V2)2~ -- ~3 + F(/~) .

(18)

They are called "Swift-Hohenberg-equations" [3] or if a quadratic term is included "Swifl-Hohenberg-Hakenequations". (18) allows for a variety of spatio-telnporal solutions. Equations (12) may be considered as Langevin-type equations that may be converted into the Fokker-Planck equation. In a number of cases, this conversion may be useful, if the stochastic properties must be investigated.

3

Macroscopic Synergetics

In a number of cases it is not possible or not desirable to identify the individual variables of the very numerous subsystems or/and to determine the corresponding equations of motion. On the other hand, one may make inferences on the systems dynamics by means of observed quantities. One important quantity to be guessed is the distribution function f of the state variable q. To this end, we employ the principle of maximum information [4] that is occasionally also called "principle of maximum entropy", though information is the

114

H. Haken

more adequate notion, because the expression used below is that of Shannon information rather than of Boltzmann entropy. We define the information by

In order to make adequate guesses for the transition probabilities P by means of measured data, we introduce the constraints

i = - f f(q)In f(q)dNq.

fl,e

(19)

J

According to the maximum information principle, (19) must be maximized under the constraint of normalization

f(q)dNq = 1

(20)

and a set of given constraints

= (qg,j+l)qj = / qg,j+lP(qj+l, tj+l I qj, tj)dN qj+l , (26)

f2,g,k = (qg,j+lqk,j+l)qj •

(27)

That means, we consider the state variables at a later time under the condition that they are known at the previous time step. Invoking again the maximum information principle, we may determine the conditional probability in the form /

J f(q)h~(q)dNq

h(k) ,

(21)

that result from measured quantities. It is well-known [4] that the Boltzmann distribution function and similar distribution functions can be derived be means of this principle when (21) refers to total energy, total number of particles, etc. In systems far from thermal equilibrium close to instability points, quite different constraints must be used, namely in general moments of the state vector up to the fourth order [5]. In such a case, the functions hk are polynomials in the state variables. According to this principle, the distribution function is given by

P(qj+l

qj) = exp ( - A - ~

Agqg,j+l

\

- ~ke Aktqk,j+lqt,j+l) ,

(28)

where ),e and Ake are Lagrange parameters which depend on the state vector qj. Using the abbreviations

A = (Ake) = (1Gke(qj)) ,

(29)

(30)

f(q) = exp

- E

'

and

where Ak are Lagrangian multipliers. Invoking the concept of order parameters, one may show after some extended analysis that (22) can be split into a product P({~, {s) = P({s I {~)P({~),,

(23)

where P({u) is a probability distribution of the order parameters, while P({s I {~) is the conditional probability for the values of the enslaved amplitudes under given order parameter values {u. Inserting (23) into (19), one may show that (19) may be split into two parts, namely one referring to the entropy of the order parameters and one into the entropy stemming from the enslaved modes averaged over the distribution function of the order parameters. Lack of space does not allow me to present further details so that I have to refer the reader to [5]. The method just outlined can be generalized to timedependent processes, where a path the system takes during times to, tl, ..., tN is given by the joint probability

P(q?¢, t~¢; qN-1, tN-1,..., qo, to)

1 -1 gA A= qj + •K(qj) ,

(31)

in the limit r --* 0 one may show that (28) obeys a FokkerPlanck equation of the form

= - V q [K(q)f] + ~

{[G-l(q)]}e f } . (32)

To this Fokker-Planck equation an Ito-Stratonovich equation is attached

dqe(t) = K~[q(t)]dt + ~ gt,~[q(t)]dw~(t) ,

= H P(qj+l, tj+l [ qj, tj)P(qo, to).

(24)

j=O Under the assumption of a Markov process, (24) can be split into conditional probabilities as indicated by (24) , where 8

(25)

(33)

m

where the Wiener process is defined by

(dwm) = 0

(34)

and

(dw,~(t)dwe(t)) = ~e~dt ,

N--1

P(qj+l, tj+l ] q j, tJ)dN qJ+l = 1

AN

(22)

k

(35)

and the function 9em can be determined from

½Gk~ = ~-]~.~gk~ge.~ .

(36)

We believe that the formulations of macroscopic synergetics allow for a number of important applications including those in solid-state physics.

Basic Concepts of Synergetics In conclusion, I mention a second method of macroscopic synergetics. As we have seen above, close to instability points the dynamics of a system is, in general, governed by few order parameters that obey a specific order parameter equations in which, in general, the amplitudes ~ are small so that it is sufficient to retain only low powers of ~u in N~. This knowledge may be used to model macroscopically phenomena close to instability points. A major task is, of course, the identification of the order parameters. As we have seen in Sect. 2, order parameters are characterized as the slow variables of a system in contrast to the quickly adapting enslaved variables. Thus a careful consideration of time scales may help - at least in a number of cases - to identify the adequate macroscopic order parameters.

4 Concluding Remarks In my article I have tried to present the reader a short outline of concepts and mathematical methods used in synergetics.

115 To be sure, what could have been presented here was only the tip of an iceberg; readers interested in more details must be referred to my books quoted under [1,2, 5]. There also numerous explit examples may be found.

References 1. H. Haken: Synergetics. An Introduction, 3rd edn., Springer Ser. Syn., Vol. 1 (Springer, Berlin, Heidelberg 1983) 2. H. Haken: Advanced Synergetics, 2nd edn., Springer Ser. Syn., Vol. 20 (Springer, Berlin, Heidelberg 1987) 3. J. Swift, P.C. Hohenberg: Phys. Rev. A15, 319 (1977) 4. E.T. Jaynes: Phys. Rev. 106(4), 620 (1957) E.T. Janes: In Complex Systems. OperationalApproaches, ed. by H. Haken, Springer Ser. Syn., Vol. 31 (Springer, Berlin, Heidelberg 1985) 5. H. Haken: Information and Self-Organization, Springer Ser. Syn., Vol. 40 (Springer, Berlin, Heidelberg 1989)