Transcript The Kinetic Basis of Molecular Individualism and the

The Kinetic Basis of Molecular Individualism
and the Difference Between Ellipsoid and
Parallelepiped
Alexander Gorban
ETH Zurich, Switzerland,
and Institute of Computational
Modeling Russian Academy of
Sciences
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The Kinetic Basis of Molecular Individualism
and the Difference Between Ellipsoid and
Parallelepiped
Alexander Gorban
ETH Zurich, Switzerland,
and Institute of Computational
Modeling Russian Academy of
Sciences
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The way to comprehensibility
"The most incomprehensible thing about the world is
that it is at all comprehensible." (Albert Einstein)
A complicated
phenomenon
Experiment
A complicated
model with
“hidden truth”
inside
The “logically
transparent”
model
Computational
experiment
Theory
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The travel plan
Gaussian
mixtures for
unstable systems
Phenomenon
of molecular
individualism
Successful bimodal approximations
Shock
waves
Spinodal
decomposition
Conclusion
and
outlook
Polymer molecule in flow: essentially
non-Gaussian behavior of simplest models
Neurons, uncorrelated particles, and
multimodal approximation for molecular individualism
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Normal distribution
Everyone was assuming
normality; the theorists
because the empiricists
had found it to be true,
and the empiricists
because the theorists had
demonstrated that it must
be the case. (H.Poincaré
attributed to Lippmann)
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Multidimensional normal distribution
-(x-M, C (x-M))/2
P(x)=Ae
M - mean vector,
C- inverse of covariance matrix,
A - a constant for unit normalisation,
( , ) - usual scalar product.
The data form an ellipsoidal cloud.
Small perturbation of normal distribution
P(x)=Ae-(x-M, C (x-M))/2(1+(x))
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Typical multimodal distribution
for systems with instabilities
Normal or “almost normal”
distributions are typical for stable
systems.
For systems with m-dimensional
instabilities the typical distribution is
m-dimensional parallelepiped with
normal or “almost normal” peaks in
the vertices.
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Cascade of peaks
dissociation
First result:
the peak dumbbell
The peak cube
Directions of
Instability
Second result: the peak parallelogram
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The similarity and
the difference between
Ellipsoid and Parallelepiped
In n-dimensional
space
Ellipsoid
(normal)
k principal
Difference
(complexity of components
phenomenon) (kn)
Similarity
kn
(complexity of parameters
description)
Parallelepiped
(multimodal)
m
2 distinct
states
(mn)
mn+kn
parameters
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What is the complexity of
a parallelepiped?
The way from edges to vertices is easy.
But is it easy to go back, from vertexes to edges?
The problem:
Let us have a finite set S in Rn. Suppose it is a
sufficiently big set of some of vertices of an
unknown parallelepiped with unknown dimension,
mn, S2m.
Please find the edges of this parallelepiped.
What is the complexity of this problem?
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Several forms of molecules in a flow
Dumbbell
Kinked
Half-dumbbell
Folded
At the highest strain rates, distinct conformational
shapes with differing dynamics were observed.
(S.Chu at al., 1997)
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Polymer stretching in flow
A schematic
diagram of
the polymer
deformation
(S.Chu, 1998).
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The Fokker-Planck equation (FPE)
t(x,t)=x{*(x) D [x-Fex(x,t)][(x,t)/ *(x)]}.
x=(x1,x2,…xn) is a conformation vector;
(x,t) is a distribution function;
D is a diffusion matrix;
U(x) is an energy (/kT);
Fex(x,t) is an external force (/kT).
The equilibrium distribution is: *(x)=exp-U(x).
The hidden truth about molecular
individualism is inside the FPE
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Kinetics of gases
The Boltzmann equation (BE)
tf(x,v,t)+(v,xf(x,v,t))=Q(f,f)
The Maxwell distribution (Maxwellian):
fMn,u,T(v)=n(m/2kT)3/2exp(-m(v-u)2/2kT)
Local Maxwellian is fMn(x),u(x),T(x)(v).
If f(x,v)=fMn(x),u(x),T(x)(v) (1 + small function),
then there are many tools for solution of BE
(Chapman-Enskog series, Grad method, etc.).
But what to do, if f has not such form?
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Tamm-Mott-Smith approximation for shock
waves (1950s): f is a linear combination of
two Maxwellians (fTMS=afhot+bfcold)
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Variation of the velocity distribution in the shock front at
M=8,19 (Zharkovski at al., 1997)
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The projection problem:
ta(x,t)=? tb(x,t)=?
Coordinate functionals F1,2[f(v)].
Their time derivatives should persist (BE tF1,2=TMS tF1,2):
BE tF1,2[f(x,v,t)]=(F1,2[f]/f){-(v,xf(x,v,t))+Q(f,f)}dv;
TMS tF1,2[fTMS]=
t(a(x,t)) ( F1,2[f]/f)fhot(v)dv+ t(b(x,t)) (F1,2[f]/f)fcold(v)dv.
There exists unique choice of F1,2[f(v)]
without violation of the Second Law:
F1=n=fdv - the concentration;
F2 =s=f(lnf-1)dv - the entropy density.
Proposed by M. Lampis (1977).
Uniqueness was proved by A. Gorban & I. Karlin (1990).
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TMS gas dynamics
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The gas consists of two ideal equilibrium
components (Maxwellians);
Each component can transform into another
(quasichemical process);
The basis of coordinate functionals is the pair:
the concentration n and the entropy density s.
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Spinodal decomposition and the free energy
If a homogeneous mixture of A •and B •is rapidly cooled,
then a sudden phase separation onto A and B can set in.
Any small fluctuation of composition grows, if
XB=nB/(nA+nB)
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Kinetic description of
spinodal decomposition
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Ginzburg-Landau free energy
G=[(g(u(x))+1/2K(u(x))2]dx,
where u(x)=XB(x)-XB;
Infinite-dimensional Fokker-Planck equation
for distribution of fields u(x);
Perturbation theory expansions, or direct
simulation, or…?
Model reduction: we do not need the whole
distribution of fields u(x), but how to construct
the appropriate variables?
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Langer- Bar-on- Miller (LBM) theory
of spinodal decomposition (1975).
Variables
1(u) - distribution of volume on the values of u.
The pair distribution function, 2(u(x1),u(x2)),
depends on u1,u2 , and r=x1-x2 .
In LBM theory
2(u(x1),u(x2))=(1+(r)u1u2) 1(u1) 1(u2),
The highest correlation functions
Sn(r)=un-1(x1)u(x2),
1(u) for two
In LBM theory Sn(r)= un  u2 (r).
moments of time
The main variables: 1(u), (r),
and 1(u)=Aexp(-(u-a)2/212)+Bexp(-(u-b)2/222).
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LMB project FPE on (r) and this 1(u).
Mean field model for polymer molecule
in elongation flow
Potential U(x) is
quadratic, but
with the spring
constant
dependent on
second moment
(variance), M2.
FENE-P model:
f=[1- M2 /b]-1.
where
 is the elongation rate
Gaussian manifold (x)=(1/2M2)1/2exp(-x2/2M2)
is invariant with respect to mean field models
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Gaussian manifold may be non-stable with
respect to mean field models
Deviations of moments dynamics from the Gaussian
solution in elongation flow
FENE-P model.
Upper part: Reduced
second moment.
Lower part: Reduced
deviation of fourth moment
from Gaussian solution
for different elongation
rates
I. Karlin, P. Ilg, 2000
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Two-peak approximation, FENE-P model in
elongation flow
Phase trajectories
for two-peak
approximation.
The vertical axis
corresponds to the
Gaussian manifold.
The triangle with
(M2)>0 is the
domain of
exponential
instability.
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Two-peak approximation, FPE, FENE model
in elongation flow
a) A stable
equilibrium on
the vertical
axis, one
Gaussian
stable peak;
b) A stable
two-peak
configuration.
Fokker-Planck
equation.
is the effective potential well
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Dynamic coil-stretch transition is not a
stretching of ellipsoid of data, but it’s
dissociation and shifting
Distributions of molecular
stretching for coiled
(one-peak distribution)
and stretched (two-peak
distribution) molecules.
The distribution of
distances between fixed
points on a molecule
becomes non-monotone.
The dynamic coil-stretch transition exists both for FENE and
FENE-P models for constant diffusion coefficient. It is the
first step in the cascade of molecular individualism.
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Radial distribution function for polymer
extension in the flow (it is non-Gaussian!)
FENE-P model,
The Reynolds number
(Taylor Scale) Re=160,
the Deborah number De =10,
b is the dimensionless finiteextensibility parameter.
b varies from top to bottom as
b =5102, b=103, and b =5103.
The extension Q is made
dimensionless with the equilibrium
end-to-end distance Q0.
P.Ilg, I. Karlin et al., 2002
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The steps of molecular individualism
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Black dots are vertices
of the Gaussian
parallelepiped. Quasistable polymeric
conformations are
associated with each
vertex.
Zero, one, three, and
four-dimensional
polyhedrons are drawn.
Each new dimension of
the polyhedron adds as
soon as the
corresponding
bifurcation occurs.
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Neurons and particles for FPE
The approximation for distribution function
Quasiequilibrium (MaxEnt) representation
Dual representation
If
, then the distribution
function  is the Gaussian Parallelepiped
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Geometry of Anzatz
Defect of
invariance is the
difference
between the
initial vector field
and it’s projection
on the tangent
space of the
anzatz manifold
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Equations for particles
The initial kinetic equation:
Equations of motion (P - projectors):
The orthogonal projections P (J) can be computed by adaptive
minimization of a quadratic form (T is a tangent space to
anzatz manifold):
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Conclusion
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The highest form of the art of anzatz is to represent a
complicated system as a mixture of ideal subsystems.
Gaussian polyhedral mixtures give us a technical
mean for description of complex kinetic systems with
instabilities as simple mixtures of ideal stable systems.
Molecular individualism is a good problem for
development of the methods of Gaussian polyhedral
mixtures.
Presentation of particles (neurons) gives us a new
technique for solution of multidimensional problems as
well, as a new way to construct phenomenology.
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We work between complexity and
simplicity and try to find one in
the other
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"I think the next century will be the
century of complexity".
Stephen Hawking
But ... “Nature has a Simplicity, and
therefore a great Beauty”.
Richard Feynman
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Thank you for your attention.
Authors
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Alexander Gorban,
Iliya Karlin
ETH Zurich,
Switzerland,
Institute of
Computational
Modeling Russian
Academy of
Sciences
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Painted by
Anna GORBAN
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