Dynamic Decay and Superadiabatic Forces in the van Hove Dynamics of Bulk Hard Sphere Fluids

1 great introduction to the dynamics of the van Hove function

2 connection to earlier studies of the van Hove function within DDFT

3 connection between the behaviour of hard spheres and a Lennard Jones fluid

1 Fig. 3 is difficult to read

In this manuscript the authors study the dynamics of the van Hove function
of a hard-sphere fluid. This problem has been studied before with various
theoretical approaches, but the present manuscript makes use of some ideas
in Power Functional Theory (PFT) which is a variational approach to dynamical
systems, out of equilibrium. This additional use of PFT is a clear extension
to earlier studies that employed Dynamical Density Functional Theory (DDFT).
The manuscript gives a detailed and good to read introduction to the DDFT/PFT
approach to the dynamics of the van Hove function and discusses the application
to hard spheres in detail. Moreover, there is an additional insight by
comparing some details of the van Hove functions of hard spheres and that of
a Lennard-Jones fluid.

There are a few minor points that I have picked up while reading (with great
joy, if I may add) through the manuscript.

The authors mention that a shortcoming of DDFT is that it does not pick up
all the details of the out of equilibrium dynamics of a system. To this I
fully agree. A bit later the author mention that even an equilibrium system
shows some dynamics and give the example of the dynamical decay of the van
Hove function. These two statements are somewhat separated in the manuscript,
but if I put them next to each other it seems that either I have missed some
details, by taking these statements out of the context of the manuscript, or
the authors should clarify this point. With the discussion of the force
densities in Fig. 6 it seems that DDFT is missing some contributions even
if the system considered is in equilibrium.

Along a similar line I would like the authors to clarify their discussion
of the long time diffusion in Sec. 2.6. Is it correct that if one were to
apply standard DDFT using the ansatz in Eq.(48) one would obtain Eq.(57)
with C_drag = 0? Is this super-adiabatic contribution to the modified
Einstein relation the answer, or part of the answer, to my previous question?
It would be helpful if the authors could mention this at the end of Sec.
2.6.

One final point is that I find Fig.3 rather difficult to read. The combination
of color and Morse code challenging. Maybe it would help to name the lines
of the plot at t/tau=2 from top to bottom or the other way around, so that the
results are easy to identify.

Once these minor points have taken into account, I can highly recommend the
publication of this manuscript in Sci Post.

1 clarify the role of DDFT and the equilibrium dynamics of the van Hove function

2 clarify the role C_drag in 'standard' DDFT

3 clarify Fig. 3

1 - complete exposition of a theory for the van Hove functions in the language
of dynamic density functional and power functional theory

2 - full determination by simulation of all contributing forces identified
in this framework ("exact" reference results)

3 - power functional expressions for these forces are not specific to the van Hove
setting but are general for nonequilibrium Brownian systems

4 - link between drag force and long-time self diffusion

1 - Fits for almost all parameters of the power functional are on the
van Hove data themselves (not from a "general" theory or different systems)

This ms reports an "anatomical study" of the self and distinct van Hove
function for hard spheres at a particular density (rho* = 0.73). Some
comparison to LJ data from previous work is made.

The van Hove functions are viewed as time-dependent density profiles of
the self particle Gs and of the distinct particles Gd. Their time derivative
is a divergence of a current, i.e. proportional to a force density (in BD).
The force density is split into adiabatic and superadiabatic terms according
to the framework of PFT as developed by the group. All terms can be determined
in simulations, thus "exact" results are available to compare with theoretical
approximations.

The theoretical analysis focuses on an in-depth analysis of DDFT results and
their extension using PFT. DDFT gives the exact adiabatic dynamics if the exact
equilibrium functional for the two-species mixture of self and distinct particles
is known. The paper analyses the best currently available functionals for this
mixture and finds deficiencies.

Going beyond DDFT, three types of superadiabatic
forces are taken into account (drag, viscoelastic, structural). These general
types had been identified before and they show distinct signatures in different
settings. They are derived from a certain approximation for the power functional.
A highlight is the derived link of long-time self diffusion constant to the drag
force (viscoelastic and structural forces do not contribute), and the simple
relation for the long-time self diffusion constant to the overall strength of the
drag force.

The types of superadiabatic forces are general and a small velocity expansion
for the corresponding power functional terms has been developed by the group.
However, these terms contain prefactors and memory kernels which were mainly
fitted to the van Hove simulation results themselves, only the viscoelastic term
could be used from an analysis of a different (sheared) system. Overall, the fits
provide a very good medium- to long-time description of Gs and Gd.

I strongly recommend publication of this work. It gives a self-contained theoretical
framework for one of the basic correlation functions in liqiuds which is accessible
both theoretically and experimentally.

1 - I do not fully understand 3.3 and Fig. 5: Do you mean that you calculate

V_ad,a(r) = mu - dF / drho_a(r) (a=s,d)

for different versions of F with rho_a(r) inputted from simulations (for a
particular time)? And then you simulate the equilibrium profile with
this V_ad,a (which should be equal to the rho_a input)?

Perhaps one could write this equation.

Fig 5 is rather small, and the input Gs and Gd curves are hardly distinguishable.
Perhaps use symbols for those?

2 - The quenched functional appears to me rather good in describing the adiabatic
forces but you state it is not reliable. If you use the quenched functional + the
superadiabatic fits, how does it compare to the simulated Gs and Gd?

3 - If you fit the drag amplitude only to the largest times in your simulations,
will DL from (57) get closer to the simulation result? For large times velocities are small and the small velocity expansion should be good.

4 - Can you comment on the relation of the venerable Vineyard approximation for Gd to your framework?

5 - Typo: Medi*n*a-Noyola on p. 2