We start with the general potential case (with constant
isotropic mass).
The deformation field
must be defined everywhere in space
(note that it does not change in time).
The deformation parameter controls the hamiltonian

(D.4) |

where is the (time-dependent) particle position, and is the force on the particle. Therefore is a function of the particle position alone. The constant is chosen such that the time-average of is zero; this is necessary for the asymptotic value to be zero, to allow finite moments of . We will not restrict ourselves to phase-space-volume-preserving deformations (

The integral of (D.5) gives
the
general expression

We treat the time-integral terms in (D.6) in increasing powers
of , (writing with Einstein summation notation):

and so on. What constraints arise on the coefficients for there to be no diffusive growth terms? The first result shows that since the integrand is an exact time derivative, there is no diffusive growth for any . In general, any boundary-type terms (those coming from exact time derivatives, shown with square brackets) cannot cause diffusion since all functions of are stationary and bounded (since the energy-surface is bounded in phase-space); so these terms can be ignored. The time-integral term in (D.8) will vanish if is any antisymmetric tensor. Any symmetric component in will give a quadratic form in the momentum, and therefore a fluctuating function whose time-integral grows diffusively. The one important exception to this is when this quadratic form is proportional (up to an exact time derivative) to , the Hamiltonian. is a constant of the motion, so gives uniform linear (non-diffusive) growth of the time-integral. This will correspond to the special nature of dilation in the hard-walled case discussed in Sec. D.3.

We can show that the higher terms (D.9), etc cause diffusive growth, unless either all the coefficients , etc are zero, or unless the coefficients fall into the `trivially special' subspace corresponding to at all accessible . (This latter case imposes linear conditions on the coefficients; we exclude the resulting deformation-less subspace). For any given -order term, for instance (D.9) with , the coefficient tensor must be totally symmetric with respect to interchange of its last indices (nonsymmetric parts cancel in the summation because of this symmetry in the factors). Note that this symmetry forbids the type of argument that allowed an antisymmetric to give no diffusive growth for . Manipulation by parts conserves the symmetry of the terms, the power of (namely ), and the number of time-derivatives, or `dots' (namely 2). (It also generates ignorable boundary-type terms). There are only two cases:

- Both the dots are on the same , in which case the integrand is a (non-zero) function of position alone (since is a function of position alone).
- The dots are on different 's, giving a quadratic form in multiplied by a (non-zero) function of .

The only other way to prevent diffusive growth is by
making the integrand a multiple of (plus exact time-
derivatives).
Case 2) cannot give the required term so it is no help.
(The only term can come from an isotropic tensor).
It is unknown whether case 1) can give a term proportional
to
, which could add to an existing term to give .
It seems this latter occurrence could only happen for particular forms
of
, for instance the scaling potentials given below
^{D.1}.
If it happens, it would give a `dilation-type' special deformation
corresponding to conservation of .

In conclusion this gives the form of a `special' deformation
(in a general potential) as

(D.10) |