# Ensembles apply to trajectories, not frames

It is common to initially think of ensembles as applying on a frame-by-frame basis. However, ensembles are only valid on trajectories, not individual frames. To explain this, let’s consider a couple simple common misconceptions.

## Combining volume ensembles doesn’t combine volumes

It might seem like `Ensemble(volume_A) & Ensemble(volume_B)`

should be the
same as `Ensemble(volume_A & volume_B)`

. But this is absolutely *not* the
case. We can see this easily by using `PartOutXEnsemble`

as the example
ensemble. `PartOutXEnsemble(volume)`

creates an ensemble in which at
least one frame must be outside of `volume`

. So the difference between
`and`

-ing together the two ensembles vs. `and`

-ing together the two
volumes can be described like this:

`PartOutXEnsemble(volume_A) & PartOutXEnsemble(volume_B)`

: there is at least one frame in the trajectory outside of`volume_A`

, and at least one frame outside of`volume_B`

. These two frames do not need to be the same frame.`PartOutXEnsemble(volume_A & volume_B)`

: there is at least one frame which is outside of`volume_A & volume_B`

, which is the intersection of`volume_A`

and`volume_B`

.

Note that the second case does NOT mean that a single frame is
simultaneously outside of both `volume_A`

and `volume_B`

: it is outside
the intersection, not the union. If what you want is a ensemble of
trajectories which contain at least one frame that is simultaneously outside
of `volume_A`

and outside of `volume_B`

, you can write that as
`PartOutXEnsemble(volume_A | volume_B)`

.

The above trajectory is in `PartOutXEnsemble(volume_A & volume_B)`

but not
in `PartOutXEnsemble(volume_A) & PartOutXEnsemble(volume_B)`

. To be in
that ensemble, it would need to have a frame in the red area, or the outside
both volumes in the white area. The volume given by `volume_A & volume_B`

is the purple area, and some frames are outside of that.

## Complementary frames do not generate the logical inverse ensemble

Another case where your intuition can lead you astray is when thinking about
complementary and inverse ensembles. For example, since
`InXEnsemble(volume)`

consists of frames inside of `volume`

and
`OutXEnsemble(volume)`

consists of frames outside of `volume`

, you might
mistakenly think that `InXEnsemble(volume) | OutXEnsemble(volume)`

allows
all trajectories.

However, if you think about the whole trajectory, you’ll see this is not the
case. `InXEnsemble`

requires that *all* frames be in the given volume;
`OutXEnsemble`

requires that *all* frames be outside the given volume. A
trajectory tested with `InXEnsemble(volume) | OutXEnsemble(volume)`

must
satisfy one of the two ensembles: either all frames inside or all frames
outside. It can *not* include a transition across edge of the volume.

`OutXEnsemble`

is complementary to `InXEnsemble`

in the sense that
`OutXEnsemble(volume) == InXEnsemble(~volume)`

, but
`OutXEnsemble(volume) != ~InXEnsemble(volume)`

. This again comes back to
the statement in the previous section that combining volume ensembles does
not combine volumes, because ```
InXEnsemble(volume) | InXEnsemble(~volume) !=
InXEnsemble(volume | ~volume)
```

.

To find the actual logical inverse of an ensemble, we should return to its
set-theoretic definition. For `InXEnsemble`

, this is:

where \(t\) is the time (frame number), \(x\) is the order parameter
function which defines the volume, and \(V_x\) is the extent of the
volume. To take the logical `not`

of that, we apply the standard rules
that \(\forall\) becomes \(\exists\) and \(\in\) becomes
\(\notin\), giving us:

This is, of course, the definition for a `PartOutXEnsemble`

. And if we
think about this in words, it makes perfect sense: if the ensemble requires
that *all* frames be in some volume, then the set of all trajectories which
do not satisfy that ensemble would have *at least one* frame outside that
volume.