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## Microcanonical averages and trajectory averages

In the preceding sections and were defined in terms of the microcanonical average over initial conditions,

 (2.27) (2.28)

The assumption was made that the change in over the correlation time is insignificant. This means that the frozen' Hamiltonian (at fixed ) can be used, so the distribution is unchanging in time . All resulting averages are constant in time (or, if they involve multiple times, they are functions of time differences only), and the choice of initial time is arbitrary.

However, these averages need not be taken using an average over phase space. By ergodicity [79], they are equal to time averages over a single trajectory. Namely, the conservative force (2.3) can be written

 (2.29)

seen to be the mean force due to motion of a single particle in the system. Similarly, the auto-correlation is written
 (2.30)

There is now an issue of convergence: the number of independent samples of any quantity along a trajectory is , and the fractional error of the estimate of the above quantities converges slowly as . Therefore very long trajectories are required to get good estimates. However, this is often easier than performing the dimensional integral over the energy shell which would be required for the explicit evaluation of the phase-space average, especially since the integrand in (2.30) already involves propagation forward in time. This technique of evaluation of a multi-dimensional integral using a random sample of points taken from the distribution function is called Monte Carlo integration [161]. However, in practice, rather than compute and take the Fourier transform, is most efficiently estimated directly from the Fourier transform of . This approach is discussed in detail in Appendix B.

In an identical fashion to that shown in Fig. 2.2a, Eq.(2.30) describes as the projection of the function onto the axis. However only a single trajectory is involved, so is noisy, and an average over the time axis is required. In this figure, one can imagine the box' of allowed , values now bounded by . The average converges in the limit .

An instructive convergence issue arises if we extend this single-trajectory estimate to the noise intensity . Naive use of (2.14) and (2.30) would give

 (2.31)

This does not converge because the integral involves an infinite number of similarly-sized contributions (the integrand is similar in size for all , ). Averaging over any finite cannot remove this divergence. Another option is to look back to Eq.(2.8) which is responsible for the appearance of in the spreading rate. So I could define
 (2.32)

which is the same integral (2.31) with different limits. This is simply proportional to the energy variance after time for the single trajectory involved, divided by . The term inside the square brackets is simply a random walk (on timescales ), giving a Gaussian distribution whose variance grows linearly. Therefore this estimate of will not converge, rather it will wander for all , taking values with a distribution whose mean is the correct . In effect this reproduces exactly the stochastic energy spreading whose variance is desired! However, it is more convergent' than (2.31). A convergent estimate can only be created by limiting the integration further, to give
 (2.33)

which will converge once . In essence the convergence arises because the limits do not allow the number of independent -squared-sized patches' of the integrand to grow as fast as , where is defined as above. The nearly-convergent case (2.32) corresponds to exactly growth.

The above considerations will not relate directly to the numerical method of finding (which will be via ). However they serve to warn and provide intuition about convergence when a single trajectory is used for estimation.

Next: Review of the linear Up: Review of classical dissipation Previous: Irreversible growth of average
Alex Barnett 2001-10-03