**Scientific Report**

One of the goals of
the workshop, i.e. to bring together researchers of different fields working on

scale transitions, was well
achieved. The research disciplines represented by the speakers and

participants were mathematics,
physics, bio- and chemical engineering, theoretical and applied

mechanics as well as
computational and material sciences. The workshop lasted one full week

with 50 participants from
within and outside the

One of the key
challenges in scale transitions is to retain the relevant physical parameters
by

either coarse-graining or
homogenizing the fine scale variations. Different methodological

approaches and terminologies are
used in various fields, and it was obvious throughout the

workshop that the respective
communities are often not aware of the work done in neighboring

areas. Among the originally
scheduled issues were: nonlinear behaviour, defects,
discreteness,

interfaces and surfaces,
non-affinity, and time-dependent behavior.

The program was
scheduled through a daily focus on a different theme:

1. spatial
scale transitions,

2. temporal
scale transitions,

3. discrete to
continuous transitions,

4. scale
transitions with space-time interactions, and

5. complex
fluids, soft matter or granular matter.

Within these topics,
special sessions were organized for (i) flash-poster
presentations, followed

by a poster session;
(ii) definition of the goal and presentation of open questions beforehand;
(iii)

an interactive session
in small groups on the definition of homogenization versus coarse-graining;

(iv) an interactive session in small groups on particular
difficulties characterizing soft-matter and

the importance of
handling inhomogeneities in scale-transitions, and
(v) the concluding session,

revealing the “Lessons
Learned”.

Scale transitions in
space and time are typically presented in a space-time diagram, as shown

below for the particular
case of metals:

Among the Lessons
Learned are the results of the interactive workshop 1:

W1-A: The terminology
used in various fields and their characteristic differences evolved into the

summary in the Table below.

Homogenization (mostly
in space) leads to the same type of eqs. (possibly with a reduced set of

dofs [degrees of freedom]). In contrast, CG
[=coarse-graining] generally applies to timedependent

phenomena and is based on
ensemble averaging and can lead to different equations,

reduced dofs.
It relies on the correlation between fine scale fluctuations, which should NOT
be

averaged out but, instead,
lead to irreversibility and emergent behavior. Other terms and methods

were discussed –
Hierarchical Multiscale Modeling (HMM), up-scaling,
micro-macro transitions –,

but they can be
classified in one of these two main groups.

W1-B Q2: The second
main question was: “What is the methodological complexity that

discriminates upscaling
in time from upscaling in space?” Discussions among
participants

indicated that length-scale
jumps are easier to identify; thinking in terms of length-scales is more

intuitive for the majority of researchers
working on solids. Multi-scale in time invokes the

correlation of fine, rapidly
varying phenomena; the community working on small oscillatory and

wave phenomena is likely
to be more familiar in this respect. Furthermore, the role of

discreteness for time-scaling vs.
discrete phenomena in space was discussed. Discreteness

seems to bring a coupling
between space and time. A general problem is the infinite propagation

speed in some models, which
assume quasi-static deformations. Discretization is an

approximation that makes
propagation-speeds finite. The participants seemed to indicate that the

history-dependent behaviour
is the main complexity to handle in time-scale transitions in solids.

However, within
GENERIC (see below), basic ingredients to do so seem to be well in place if
time

scale separation holds.

W2-A: Among the
results from Workshop 2 are the different interpretations of soft-matter
(versus

solids) and what makes soft
materials special. The following issues can be special for soft-matter:

fluid- and solid-like
behavior is important at the same time; different time- and length-scales are

involved; the systems are
highly heterogeneous/ disordered; energy/kBT is around unity; they

display long time tails;
small changes can lead to big effects.

W2-B: The role of inhomogeneities for scale transitions needed first a
definition of non-affinity vs.

inhomogeneity (what comes first and leads to the other?).
Then the size/scale of inhomogeneities

is linked the scale-jump
and possibly determines its upper-limit. The question what one can do

when the length-scale of inhomogeneities changes (sometimes rapidly) during a
process

remained unclear. In any case
one has to be careful with smoothing out inhomogeneities,
since

they make the system
behavior change with their presence and state. The length scale set by

inhomogeneities can lead to new state-variables and
parameters like porosity, a structure tensor

or, e.g.,
pair-correlation functions.

The outcome of all
discussions during the lectures and the various discussion sessions were

summarized in the Lessons
Learned session, which can be summarized as follows:

· Mathematical tools exist that justify the
limit passage from micro to macro, for

homogenization of energy-driven
systems. A "smart" Γ-limit (‘smart’ indicating the
"right"

variables were chosen) but
there is no general recipe (except for special/trivial cases). There

seems to be, however, a
(partially unexplored) connection between GENERIC (see below)

and the mathematical
tools at hand.

· GENERIC received extensive attention in the programme and was profiled as a very powerful

methodology, enabling systematic
scale transitions relying on coarse graining and correlation

of fluctuations. The
scale separation in time is thereby essential (Hütter).
Understanding the

link between the handling
of the entropy terms versus some proposed stochastic methods

(with
particular probability distribution functions) remains open but seems feasible.

· The notion of history was often mentioned as a
really complicating factor in time scale

transitions, but this does not
seem to be justified completely. Handling history dependent

behaviour can be done by incorporating the related
micro-information into a state variable

and thereby avoiding (as
for example in Maxwell fluids) the memory kernel. The choice of the

"right" state variable is essential and requires a good
physical insight in the system studied.

· There is no essential contradiction between
kinetics (evolution) controlled solution methods

vs. equilibrium-driven
methods. The equilibrium-driven methods usually only provide a

necessary condition and hence a
lower-bound for the real evolution path. The kinetics

controlled methods are more rich
and detailed and incorporate the physics of the evolution.

Phase field models are
well developed methods to resolve several energy driven-kineticslimited

microstructure evolution processes.
They are nowadays mature and reach out now to

coupling to other scales and
more complex mechanical behavior (kinetics in mechanics and

materials science do not refer
to inertia!).

· Dissipation is inherently coupled to fine
scale fluctuations. The precise definition of dissipative

events can be
scale-dependent, e.g. where systems may be either considered as open or

closed. The issue of a fluctuation-dissipation
theorem and related emergence of entropy was

not worked out to the
end.

· Biophysical systems are characterized by
complexity. To handle this, use is made of an

efficient scale separation map.
This map classifies each of the processes at different spatial

and temporal scales.
Model for each of the processes being available, most emphasis is

given to the coupling
between different processes (= resolving multiple processes at different

scales). This is not a true
scale transition (= resolving a single process across the scales),

and the methods used
therefore more focus on solving the materiomics loop.

· Peridynamics was presented as a
promising method, based on solving integral equations

rather than PDEs. The method seems to show great promise for problems
that present

discontinuities, but there are still
many challenges ahead before its potential can be fully

exploited. It was applied to
non-local elasticity, but how to go beyond (finding the kernel for

nonlinear material behaviour) is an open issue. The numerical implementation
and solution

seems very convenient,
relying on a discrete (MD-like) solution strategy. Problems to be

resolved are the proper
handling of various types of boundary conditions and interface

conditions (resolving
heterogeneities). At this stage, its relation to large scale transitions and

to SPH (smooth particle
hydrodynamics) was not yet fully clear.

· GENERIC (General Equation for the
Non-Equilibrium Reversible-Irreversible Coupling): For

this promising method,
most important aspect is the choice of internal variables, for which a

healthy insight and intuition
is required. For a scale transition in time, time-scale separation is

required.

· Spelling checkers: Some mathematical tools
exist now that allows for a proof of the limit

passage. In a primitive form,
GENERIC can be used as a spelling checker, but this

methodology seems more powerful
than just that.

· An inherent space-time coupling seems to exist
upon studying the *same *process at different

scales. This does not
exclude the existence of fast-large and slow-small processes (as

exemplified for bio-systems). A
true scale transition (for the same process) perpendicular to

the diagonal in the
x-t-diagram does not seem trivial (if it exists at all). More importantly, a

space-time coupling may abruptly
change due to the occurrence of instabilities (e.g.

microstructure evolution, strain
bursts, dislocation avalanches, etc., where a rapid evolution

occurs at a larger length
scale). Such instabilities may seriously compromise the scale

separation assumed up front and
hence may render several methods inapplicable (e.g.

homogenization for localization of
deformation). This implies that, even when one succeeds

to develop a macro-model
based on a micro-model, the occurrence of an instability may call

for the explicit
incorporation of the finer scale that controls the physics of the instability.

Many of the original
questions that were posed by organizers and participants (see sheets on the

website) were discussed and
partially answered. In summary, a better focus and improved insight

was gained, but also many
*new *questions were raised and challenges for further work and future

collaborations could be identified.

Some pending issues
discussed that were unresolved at this stage:

1.
Does time-scale separation hold in metal plasticity?

2.
How to extend and handle interfacial problems and
heterogeneities in GENERIC (seems

3.
feasible though)?

4.
Does it make sense at all to treat scale transition in time
separate from those in space?

5.
What are the conceptual differences for different CG methods, in
particular since some CG

6.
methods only resolve a small iterative step,
which is insufficient.

We thank all
participants for their contributions, input and active participation. Financial
support is

acknowledged from the

organizers strongly acknowledge
the hospitality at the Lorentz-Center and the efficient and

competent help of the local
staff.

Scientific
coordinators:

K. Bertoldi
K. Bertoldi (

H. Steeb H. Steeb (Enschede/Bochum, Netherlands/Germany)

S. Luding S. Luding (Enschede, Netherlands)

M. Geers M. Geers (Eindhoven, Netherlands)

E. van der Giessen E. van der Giessen
(Groningen, Netherlands)