Microcanonical averages and trajectory averages

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

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

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

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

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) |

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.