RELAXATION TIME SCALES OF CRITICAL STOCHASTIC MODELS

Transcription

1 Bnef Reviews Modern Physics Letters B, Vol. 5, No. 27 {1991} World Scientific Publishing Company RELAXATION TIME SCALES OF CRITICAL STOCHASTIC MODELS V. PRIVMAN ' Department of Physics, Theoretical Physics, University of Oxford, 1, Keble Road, Oxford OX1 3NP, UK t on leave of absence from Department of Physics, Clarkson University, Potsdam, NY 13699, USA and B. D. LUBACHEVSKyt t AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA Received 26 October 1991 We review recent results on the discrete density and time behavior of single-variable stochastic models. New applications of scaling ideas and numerical simulations in studies of time scales of the discrete evolution near critical fixed points are emphasized. The emergence of macroscopic equations from microscopic stochastic dynamics is of interest in many fields ranging from evolution of epidemics to chemical reactions and percolation models. The problems addressed are technically and conceptually difficult, requiring diverse theoretical techniques ranging from rather advanced mathematical methods to numerics and phenomenological scaling ideas which have developed more recently. Some of the latter advances will be surveyed here, including several new results. Due to the generality of the problem as posed, it is quite difficult to review all the existing theoretical work. The book by van Kampen l covers a wide range of topics and in a sense defines the field with its terminology and scope of subjects. In many applications, single-variable stochastic dynamics models are useful l - ll ; they are the most studied in the literature. Limited results for more general models are available in some cases.l-2.l2-14 Our present survey will be limited to single variable models and to scaling and numerical results following largely the recent works in Refs as well as Ref. 1. The emphasis will be on critical models, to be defined below, for which the fluctuations are particularly significant and for which recent scaling and numerical simulation studies have yielded new results. PACS Nos.: Ak, m. 1773

2 1774 V. Privman f1 B. D. Lubachevsky One of the simplest "hydrodynamic" evolution equations is dt = F(p) (1) which describes the variation with time of the average density variable p (t), where o ~ p ~ 1. As mentioned, such equations emerge in numerous applications. Typically, p represents the average of the fluctuating "microscopic" density p. There are several ways in which the discrete-density and possibly discrete-time microscopic stochastic dynamics for p can be defined to yield (1) or some other differentialequation or mapping-type macroscopic evolution law. Here we consider the density variable discretized in steps of lin, so that p can take on values n p= - N (n = 0,1,2,..., N). It is also convenient to have discrete time steps (2) 1 f:l.t= N2 (3) where the proportionality constant was absorbed in the time units. The interpretation of the microscopic variables nand N varies according to the context: n may be the number of sick individuals in the total population N, the number of wet sites in the percolation model on a finite, highly connected lattice of N sites, the number of unreacted particles in a chemical-reaction application, etc. 1-9 With time steps chosen as small as in (3) it turns out 1,7-11 that the "locap' transitions n -> n, n ± 1 suffice to represent the general evolution Eq. (1) and furthermore, the single-step changes n -. n ± 1 are rather slow. The transition probabilities can be defined by Prob (n -> n - 1) = N- 1 A(n, N), (4) Prob (n -> n + 1) = N- 1 B(n, N), Prob (n -> n) = 1 - N- 1 [A(n, N) + B(n, N)], where the functions A and B are of order 1 and they approach definite limiting functions of p as N -> 00. We generally take A = A(p) and B = B(p) although a more complex (n, N)-dependence may be found in some applications, e.g. Ref. 8. Note that in order to have 0 ~ p ~ 1, we generally assume A(O) = 0 and B(l) = 0, where the argument is p from now on. In most applications the functions A(p) and B(p) are in fact simple polynomials of low order in p. Furthermore, typically B(O) = 0 so that the state p = 0 (n = 0) is an absorbing state of the stochastic dynamics. This property is most easily visualized in the epidemiological interpretation: no infection can go on if there are no carriers left to spread it. The advantage of the N-dependence chosen for the time steps (3) and the general single-step setup of the transition probabilities (4)-(6) is that one can (5) (6)

3 develop a systematic fluctuation expansion l - 2 Kampen. 1 One puts Relaxation Time Scales of Critical Stochastic Models 1775 termed the n-expansion by van n = Np(t) + VNx (7) where the appropriate power,.;n, in the fluctuation term is determined selfconsistently. One can then set up a systematic expansion in powers of 1/.;N such that the leading terms in N yield the Eq. (1) dt = B(p) - A(p). (8) In the next-to-leading order the fluctuation variable is approximately treated as continuous, in the range -00 < x < 00, and its distribution P is Gaussian, of the form e-[xlw(t)]2 P (x, t) ex wet) (9) (for a more mathematically precise formulation see [1-2]). The actual calculations beyond the lowest orders are extremely cumbersome. Details of the n-expansion will not be reviewed here; see Refs o~--~ ~~ p Fig. 1. The "flow diagrant" encountered, e.g., in the directed percolation-type transition, as a function of some connectivity-determining paranteteil which controls the shape of the curve F(p). As mentioned, the functions A, B, F, see (1) an(i (8), are usually quite simple in most applications. As an example, Fig. 1 schematically depicts variation in the location of the stable fixed point from p* > 0 to p* = O. Since the fixed points, i.e.,

4 1776 V. Privman fj B. D. Lubachevsky the zeroes of F(;», determine the long-time asymptotic behavior of the macroscopic system (I), it is instructive to linearize near p*, dp '(_ *) dt ~ -A P - P (10) where A > 0 for stable fixed points. The system dynamics is then characterized by the exponential relaxation time scale,..., A -1. Critical fixed points such as the border-line case shown in Fig. I, correspond to A = O. The relaxation is then no longer exponential, fluctuation effects are expected to be more profound and interesting, and approximation schemes such as the O-expansion are generally less reliable. Our discussion from now on will be focused on the critical fixed point case, and we first describe some published 10 and new results for the simplest, quadratic fixed point at p = 0, F(p) = _p2 (ll) The conventional wisdom in the theories of strongly fluctuating systems, e.g., near critical points, is that the "universality class" or the type of fluctuations is largely determined by the "bulk" Ginsburg-Landau form expansion which near a critical point at zero density is essentially the series of F(p) in powers of p. Surprisingly, however, in stochastic models considered here the situation is more complex in that a microscopic paramater 10 may control the fluctuation features. Thus, let us consider the transition rate functions A(p) = (1- y)p+ yp2 B(p) = (1- y)p(l- p) (12) (13) see (4)-(5). Obviously, this choice of the transition probabilities satisfies all the general conditions mentioned earlier and also yields (11) for all values of the "microscopic" parameter y, where 0 ~ y ~ 1. For y = 0 the rates are those of the directed mean-field percolation model at its critical point. s For y = 1 the n -+ n + 1 transition probability vanishes and the resulting model can be treated much more easily by analytical methods. By (ll) the average density vanishes asymptotically for large times as '" lit. A remarkable property of the discrete evolution for finite N is that the density actually reaches the absorbing state at n = 0 (p = 0) in a finite time for each discrete process realization. The distribution of these first-passage times received much attention in the theory of Markov Processes. It is usually approximately exponential and can be characterized by the first-moment average, T. This time scale can be exponentially large with N for persistent percolation state (i.e., for p* > 0).6-8 However, for critical points one generally expects a power-law divergence of T(N) as N -+ (Xl. Another time scale that also measures the onset of non negligible fluctuations, i.e., deviations from the average evolution given by (I), can be determined as the time T for which the two terms in (7) become comparable, Np(T)~..fNW(T). (14)

5 Relaxation Time Scales of Critical Stochastic Models 1777 Finally, the spectrum of the transition probability operator (matrix) defined by the rates (4)-(6), which is a tridiagonal stochastic matrix, can be used to define relaxation time scales e via spectral gaps; see Refs. 8, for details. Results for the model defined by (12)-(13) were as follows. Numerical studies for percolations yielded e(y = 0, N) IX.../N, while results of Ref. 10 included the exact value e(y = 1, N) = N, as well as analytical calculation of r(y < 1, N) IX IN/(I- y) and r(y = 1, N) IX N. All these results were for fixed y in the limit N The use of the scaling description near y = 1 was proposed inlo: the limits y -+ 1 and N can be taken while keeping the product (1- y)n fixed of order 1, reminiscent of scaling combinations used in scaling theories of phase transitions. Then the shape of the crossover between the different N-dependences near y = 1 can be studied. Numerical studies lo concentrated on the time scale T(y, N) and the leading scaling function 9 defined via the scaling relation T(y, N) ~ Ng [(1- y)n]. (15) Note that the fixed-y limiting behavior is incorporated in the scaling description as usual via the large-argument form of the scaling function, 9 (g ~ 1) ~ 1/ Vi, Corrections to this leading order scaling form were also conjectured in Ref. 10 based on numerical studies. However, we are going to present evidence that unlike the leading order behavior, the N -dependence of the corrections may actually be different for various time scales such as T, r, (. To this end we point out that generally analytical expressions for T to the extent they are available [1,7]' are rather complex to analyze. However, at y = 1 in the present model the average time spent by the system at each given value n > 0 is statistically independent of the prior or following evolution and is given simply by ~t/prob(n -+ n - 1). When averaged over the initial n values no = 0, 1,..., N, the time scale T is then 1 N no N ~t 1 N N n T(y = 1, N) = N + 1 E E ~ = N(N + 1) E A(y = 1, n, N)' (16) no=l n=l n=l For the choice of weights (12)-(13) the result is simple, N ~ N n 11'2 N ( 1 ) T(y = 1, N) = N + 1 ~ n 2 = 6" N - N + 1 In N - (1 + IE) + 0 N (17) where IE = is the Euler's constant. Numerical Monte Carlo estimate of the leading coefficient 9 (0) = 11'2/6 := was only accurate to the first two digits, i.e., := 1.6; see Ref. 10. The form of the logarithmic correction, not present for the time scale (y = 1, N) = N, could not be easily guessed based on numerical simulation results. Quite generally, the convergence of numerical estimates of the first-passage times for critical stochastic processes described here was

6 1778 V. Privman fi B. D. Lubache-usky at best sufficient in probing only the leading-order behavior despite the investment of a substantial computer effort1g-ll and use of advanced algorithmsy,15 The exponential nature of the first-passage time distribution makes it difficult to measure by numerical simulations. In order to evaluate the importance of the fluctuation effects vs. the effects of the discretization, N < 00, on the deviations from the smooth average behavior p:= pet) and generally, the possibility of considering the N-dependence as smooth, nearly continuous for N ~ 1, the following model was considered. ll The rates (12)-(13) were replaced by A = [(1- y)p+ yp2]sin 2 (~), B = [(1- y)p(1- p)]sin 2 (~), (18) (19) for p > 0 and A = B = 0 for p = O. To the extent that one can use the standard n-expansion arguments, the N == 00 evolution equation should be dp _ 2. 2 (1) dt = -p SID ~. (20) However, this evolution law has infinitely many critical fixed points, at p. = (k1r)-l, k = 1, 2,..., accumulating at p = O. For finite N, the p > 0 fixed points are not exact because 1 is not commensurate with the period of the sine function. The only absorbing state is thus at p = O. Numerical studies of T(y, N) up to N = 700 as well as some analytical results for (,(y = 1, N) obtained in Ref. 11 indicated that the N-dependence is no longer smooth for the multiple-fixed-point model. In fact, the relaxation time scales vary by orders of magnitude for consecutive N values. There is, however, the overall trend of increase, '" N, of the time scales as N -> 00, according to T:= N 0(y, N), (21) where the function 0 (y, N) fluctuates widely and is difficult to quantify. There was no qualitative y-dependence in the form of these fluctuations, for all 0 ~ y ~ l. As an illustration, we evaluated numerically the sum in (16) for y = 1, N T (y = 1, N) 1 L N n N = N + 1 n=l n 2 sin 2 (N/n) (22) for N = 1,..., Figure 2 depicts the values thus obtained on the logarithmic scale, illustrating the overall trend", N and the irregularity of residual variation, 0; see (21). These results indicate that "noisy" features in discrete

7 I Brief Reviews Relaxation Time Scales of Critical Stochastic Models z ~ t: 10 C 5 o N Fig. 2. The average (first-moment) passage time T(y = 1, N), to the absorbing state at n = 0, for the model defined by (18)-(19), for N = 1,.,., These data were obtained by numerical evaluation of the sum (22) and plotted as In(T IN) vs. N. stochastic dynamics need not necessarily be dominated by statistical effects; Irregular behavior may result as well from the near-fixed-point conditions which lead to long waiting times, sensitive to N, and consequently, nonsmooth N-dependence. In summary, this survey presented recent new studies of effects of discrete density and time on single-variable stochastic dynamics models. Applications of new scaling ideas and numerical simulations in studies of general, universal properties were emphasized. Acknowledgment One of the authors (V.P.) wishes to acknowledge the award of a Guest Research Fellowship at Oxford from The Royal Society and partial support by the Science and Engineering Research Council (UK). References 1. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, 1981). 2. S. N. Ethier and T. G. Kurtz, Markov Processes. Characterization and Convergence (Wiley, 1986). 3. J. Gani, in Stochastic Models in Biology, ed. M. Kimura, G. KalIianpur, and T. Hida (Springer-Verlag, 1987), Lecture Notes in Biomathematics Vol. 70, p. 176.

8 1780 V. Privman fj B. D. Lubachevsky 4. W. Liu, in Mathematical Approaches to Problems in Resource Management and Epidemiology, ed. C. Castillo-Chavez, S. A. Levin, and C. A. Shoemaker (Springer-Verlag, 1989), Lecture Notes in Biomathematics Vol. 81, p F. Bauer, in Mathematical Approaches to Problems in Resource Management and Epidemiology, ed. C. Castillo-Chavez, S. A. Levin, and C. A. Shoemaker (Springer-Verlag, 1989), Lecture Notes in Biomathematics Vol. 81, p B. Derrida, in Fundamental Problems in Statistical Mechanics VII, ed. H. van Beijeren (Elsevier, 1990), p O. Golinelli and B. Derrida, J. Physique 50, 1587 (1989). 8. V. Privman and 1. S. Schulman, J. Stat. Phys. 64, 207 (1991). 9. B. Gaveau and L. S. Schulman, J. Phys. A24, L475 (1991). 10. V. Privman, N. M. ~vakic, and S. S. Manna, Phys. Rev. Lett. 66, 3317 (1991). 11. R. Bidaux, B. D. Lubachevsky, and V. Privman, Phys. Rev. A, in print (1991). 12. H. K. Janssen, B. Schaub, and B. Schmittmann, J. Phys. A21, L427 (1988). 13. J. O. Vigfusson, J. Stat. Phys. 27, 339 (1982). 14. C. R. Doering and M. A. Burschka, Phys. Rev. Lett. 64, 245 (1990). 15. B. D. Lubachevsky, J. Compo Phys. 94, 255 (1991).