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editing(sed): Describing jump dynamics
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This explains the presence of jump dynamics and why they are important.
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# Understanding the breakdown of Stokes--Einstein--Debye {#sec:sed}

@Sengupta2013

## Jump Dynamics {#sec:jump_dynamics}

@Nair2019
@Karmakar2014 growing length scales
@Zasetsky2010
@Laage2006

Our current understanding of the breakdown of the Stokes--Einstein--Debye relations,
Current work in describing the Stokes--Einstein--Debye relations
centers around finding quantities which satisfy
some proportionality criteria. [@Sengupta2013;@Kawasaki2019]
These studies are looking to find the quantity which adheres
to the Stokes--Einstein--Debye relations,
rather than the tangential question of why does it break down?
A foundational assumption of the Stokes--Einstein--Debye relations
is that the particles undergo Brownian motion,
that is, movement occurs as a sequence of small independent jumps.
What if the issue with dynamic heterogeneities
and non-Gaussian behaviour,
is rather than observing many jumps,
over the timescale of the relaxation
only a small number of jumps are observed.
The single particle relaxation times
provide a method to investigate this idea.

This is depicted in @fig:rotational_jumps
which has both the traditional $\tau_1/\tau_2$ quantity
in addition to the ratio of the molecular relaxation quantities $\tau_{T2}/\tau_{T4}$.
These both display motion of small jumps,
which becomes large jumps as the temperature drops.
that is, movement occurs as a random sequence of small independent jumps.
It is possible that dynamic heterogeneities result
from only observing a small number of jumps
resulting in far more correlated motion.
The single particle relaxation provides
tools which we can use to investigate this hypothesis.

## Jump Dynamics and Dynamic Heterogeneities {#sec:jump_dynamics}

The presence of jump dynamics in rotational relaxations (see @sec:intro_jump_dynamics)
has been observed in many experiments, [@Zasetsky2010;@Laage2006]
with the quantity $\tau_1/\tau_2$ being a measure of
how much large angular jumps contribute to relaxation,
from $4$ being no contribution, to $1$ being all jump dynamics.
This same quantity can be calculated for the molecular relaxations,
giving $\tau_{T2}/\tau_{T4}$.
These measures of rotational jump dynamics are depicted in @fig:rotational_jumps,
where both quantities describe the rotational relaxation by large angular jumps
at low temperatures.
Plotting the change in displacement and orientation over time,
@fig:molecule_trajectory_fast shows a fast molecule
which exhibits jump dynamics in both rotational motion as described above,
and also translational motion.
This motion can be described by
an occasional large jump to a new local environment
resulting in a change in orientation and position,
which is followed by a long period in that new environment.
The displacements between each configuration
is the type of irreversible relaxation we are measuring
in the last passage time $\tau_L$.
A notable observation about the positions in @fig:molecule_trajectory_fast
is that each jump appears to be governed by
a relaxation measured by the last passage time.

![jump dynamics](../Projects/Dynamics/figures/rotational_jumps.svg){width=80% #fig:rotational_jumps}

This describes angular jumps of the rotations,
however what does this actually look like
and does it also occur with translations?
@Fig:molecule_trajectory shows the translational and rotational displacement
of two molecules at each time point.
For a fast molecule (@fig:molecule_trajectory_fast)
there is some time spent in the initial cage,
before a large jump including both angular and translational displacement
puts it within a new cage.
This is in contrast to the slow molecule (@fig:molecule_trajectory_slow)
which spends all the time within the initial cage,
however the size of that cage is significantly larger.

:::{class=subfigures id=fig:molecule_trajectory}

![Fast Particle](../Projects/Dynamics/figures/molecule_trajectory_fast.svg){width=48% #fig:molecule_trajectory_fast}
![Slow Particle](../Projects/Dynamics/figures/molecule_trajectory_slow.svg){width=48% #fig:molecule_trajectory_slow}

The distinct positions of the fast particle are visible as clustered regions,
with time spent in each one before a large rearrangement
which is comprised of both a large translational and rotational motion.
The slow particle has not moved from it's initial state.

:::

![A trajectory of a fast particle with each point marking
the translational and orientational displacement
at a point in time.
The progression of time is described by the colour of the point.
The particle spends long periods of time in one configuration
before quickly moving to the next,
which typically requires both a translational and rotational motion.
](../Projects/Dynamics/figures/molecule_trajectory_fast.svg){width=48% #fig:molecule_trajectory_fast}

With the motion of particles so heavily influenced by Jump dynamics
we can investigate the role they play in dynamic heterogeneities.
In @sec:dynamic_heterogeneities we found that
the presence of dynamic heterogeneities
is a short time phenomenon,
decaying to the expected Gaussian distribution over longer times.
Are the dynamic heterogeneities present
because we are only observing a single jump
rather than the walk of Brownian motion.
To test this hypothesis we can use the distribution of last passage times $\tau_L$.
In taking the relaxation time of each particle within a simulation,
we have the distribution of relaxation times that take place.
So making the assumption that once relaxation has taken place,
the particle is completely independent of it's initial configuration
it will then need to undergo another relaxation event to move further.
This can be modelled by drawing consecutive relaxation times for a particle
we have sampled the distribution of relaxation times that can take place.
This allows us to model the motion like we have observed in @fig:molecule_trajectory_fast.
We can make the assumption
that all relaxations are drawn from the same distribution,
that is, each jump between local environments is equivalent.
This also makes the assumption that once the particle
has undergone relaxation it is completely independent
of it's previous configuration.
This sequence of jumps can be modelled
by drawing consecutive relaxation times
from our experimentally determined distribution of relaxation times.
This modelling of consecutive relaxations is shown in @fig:jump_heterogeneities
for the last passage time $\tau_L$,
which was chosen as a timescale for which
the relaxation is considered irreversible.
When we consider long enough length scales (or timescales),
the heterogeneities within the dynamics disappear.
Where the long relaxation times
dominate the averaging over a single time period,
they are rare occurrences,
so over multiple relaxation times
their effect averages out.
When we consider motion over enough jumps,
the heterogeneities within the dynamics disappear,
the long relaxation times dominate the averaging
over a single time period,
however they are rare occurrences
so their effect is lower over multiple relaxation times.
The reduction in the dynamic heterogeneities
was also present when plotting the molecular heterogeneities
in @fig:molecular_heterogeneities.
As the length scale increases from $\tau_F$ to $\tau_L$ to $\tau_D$,
the dynamic heterogeneities decrease,
and the same is true of the rotational heterogeneities,
from $\tau_{T4}$ to $\tau_{T3}$ and to $\tau_{T2}$.

![Heterogeneous dynamics taking progressively more relaxation
times](../placeholder_figure.png){width=80% #fig:jump_heterogeneities}

So here we do observe an increasing length scale
as the liquid is supercooled;
the length over which the observed dynamics
can be considered Brownian.
This can be an important consideration,
when comparing the different relaxations.
The ratio $\tau_1/\tau_2$ shows that
the rotational relaxation is dominated
by jump dynamics in the supercooled liquid,
while the diffusion constant
is a much longer timescale phenomenon.
Is the different regions of dynamics
responsible for the observed decoupling
of rotational and translational motion?

In the picture of the decoupling of diffusion and rotation @fig:trans_rot_otp
there are quantities on two different length scales,
the diffusion measured over a long distance,
Expand All @@ -101,6 +105,20 @@ with fast particles dominating the diffusion
while slow particles dominate the structural relaxation
and rotational relaxation.

### Increasing Length Scale

There are a range of studies which predict
an increasing length scale of the dynamic heterogeneities. [@Ediger2000]
There are experimental results which both observe this length scale,
while others reject it.
Here we postulate that the length scale is that of
the size of the jumps the particles take
which is alternatively the length over which
the dynamics can be described as Brownian.
There are already observations of this length scale increasing
upon cooling,
in the form of the ratio $\tau_1/\tau_2$ increasing
as the temperature decreases.
This also answers a question posed by @Ahn2013,
"Are rare, long waiting times between rearrangement events
responsible for the slowdown of dynamics at the glass transition?"
Expand All @@ -122,7 +140,6 @@ which was acknowledged by Einstein [@Einstein1907;@Bian2016]
that the inertia of a particle is neglected.
This means that the time resolution of observations becomes important
for the motions of particles. [@Pusey2011;@Li2013]

The result that the changing length scale of a measurement
has an effect on the resulting Stokes--Einstein--Debye relation
has also been observed in models of water.
Expand Down Expand Up @@ -246,7 +263,7 @@ in the coupling of rotational and translational motion.
happen to have similar timescales with no connection between them,
3. correlation, where the regions of large rotational motions
are also the regions of large translational motions, and
4. coupling, where rotations are required for translations to occur.
4. Coupling, where rotations are required for translations to occur.

As we go from 1 to 4,
the requirements for establishing these become more stringent.
Expand Down

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