5.2. Time evolution

The dynamical invariants derived in section 4 and the extensions considered in section 5.1 have <ne thing in common-they are associated with the asymptotic, smooth, is»-, ariant measure for the return mapping. They are properties of T°. But how long does it take for these equilibrium probability distributions to be attained and how rapidly is the equilibrium value of the metric entropv built up Answers to these questions might enable us to get a little closer to a rigorous characterization o; gravitational entropy along the lines discussed in the introduction and nearer to tracing the evolution o' the space-time dynamics by the change in its metric entropy.

In 1812 Gauss, writing to Laplace [55], asked for an estimate of the rate at which the partial quotients of the cfe of an arbitrary irrational number approach theiir equilibrium distribution. The first quantitative answer to this question was supplied by Kuzmin [56] in 1928 but his results were subsequently strengthened by Levy [57], Sziisz [58] and Babenko [61].

where [y] = ([yi], [y2],...) then a„(x) iis the nth coordinate of the F-expansion of x and we can expand t in the nested sequential form,



Another way to obtain the equilibrium distribution (4.27) is to assume that such a distribution a exists and /x„ -* /x„ as n « so p.*. must satisfy

Referring back to equations (4.22>-(4.24) we see that the probability, f.in(x). ot a Mixmaster cycle being coded by x after n cycles (iterations) is linked to such a probability after (n 1) cycles by the functional recursion relation





and clearly /Mx) *(1 + x)"1 satisfies this equation if we note that the right-hand side simplifies on using the algebraic relation (и + .г+ l)~l (n + x)~l«(n + x)"1 -(w + x +1)"*.


with 0 e (0,1). This result was tightened first, by Livy [57] who showed the remainder was C(0H) with (3 « 0.68, and then by Sziisz [58] who improved the estimate of 0 to,


Now Kuzmin [56] showed that (5.11) has a solution of a form which generalizes (4.27), namely

Further refinements have recently been made by Wirsing [60] and Babenko [61].

The importance of these results is that they show a Mixmaster model with arbitrary initial data will evolve with exponential rapidity toward a state wherein its characteristics are statistically chaotic and independent of the initial data. This convergence rate is faster than the rate of round-off error generation in machine computations and so one imagines that numerical analyses will be stable.t Also, by reference to equations (2.32) and (4.30) we see that after n iterations the value of the metric entropy will also be within <9(0.48)" of its asymptotic value ?r2/6(ln If. We can translate this into rough temporal terms by recalling that the time coordinate r in which the Mixmaster oscillations are played out is logarithmically related to the synchronous coordinate time, t, so by (4.7) we have r = In r' (since abc - t in the Kasner phases (3.15)). Now n iterations of the Mixmaster return mapping correspond to an evolution time r„ where [31,35]



and r0 is the starting time corresponding to n = 0. Therefore, n - ln(ln /) and the equilibrium metric entropy is approached as ~e" ~ In t, for t-* 0.

The rate at which other Mixmaster parameters converge can be evaluated by calculating the consequences of results like (5.13) for the ergodic theorem [59]. If we take into account the deviation of fin from fix in (4.28) then one obtains, for



and of course (4.28) is obtained in the limit as n

Just as we used the ergodic theorem to calculate the geometric and arithmetic mean cycle lengths of the Mixmaster sc the relation (5.16) can be used to calculate the rate at which these means are approached. In the case of the arithmetic mean cycle length, the formula (4.37) generalizes tc-


+ For an interesting discussion of such problems in stochastic systems see Benettin et al. [62].