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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 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

A complicatedmodel with

“hidden truth” inside

The “logicallytransparent”

model

ExperimentComputational

experimentTheory

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The travel plan

Gaussian mixtures for

unstable systems

Phenomenon of molecular individualism

Successful bimodal approximationsShockwaves

Spinodaldecomposition

Polymer molecule in flow: essentially non-Gaussian behavior of simplest models

Neurons, uncorrelated particles, and multimodal approximation for molecular individualism

Conclusion and

outlook

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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

M - mean vector,

C- inverse of covariance matrix,

A - a constant for unit normalisation,

( , ) - usual scalar product.

The data form an ellipsoidal cloud.

P(x)=Ae-(x-M, C (x-M))/2

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 instabilitiesNormal 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

Second result: the peak parallelogram

The peak cube

Directions ofInstability

First result: the peak dumbbell

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The similarity and the difference between Ellipsoid and Parallelepiped

In n-dimensionalspace

Ellipsoid(normal)

Parallelepiped(multimodal)

Difference(complexity ofphenomenon)

k principalcomponents(kn)

2m distinctstates(mn)

Similarity(complexity ofdescription)

knparameters

mn+knparameters

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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, mn, S2m.

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 conformationalshapes 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 gasesThe 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/2kT)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) 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

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 energyIf 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

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),

In LBM theory Sn(r)= un u2 (r).The main variables: 1(u), (r), and 1(u)=Aexp(-(u-a)2/21

2)+Bexp(-(u-b)2/222).

LMB project FPE on (r) and this 1(u).

1(u) for two

moments of time

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Mean field model for polymer molecule in elongation flow

where

Potential U(x) is quadratic, but with the spring constant dependent on second moment

(variance), M2.

FENE-P model:

f=[1- M2 /b]-1.

Gaussian manifold (x)=(1/2M2)1/2exp(-x2/2M2) is invariant with respect to mean field models

is the elongation rate

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Gaussian manifold may be non-stable with respect to mean field modelsDeviations 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 finite-extensibility parameter.

b varies from top to bottom as b =5102, b=103, and b =5103.

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 Black dots are vertices

of the Gaussian parallelepiped. Quasi-stable 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 FPEThe 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

Equations of motion (P - projectors):

The initial kinetic equation:

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

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

"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

Alexander Gorban, Iliya Karlin ETH Zurich,

Switzerland, Institute of

Computational Modeling Russian Academy of Sciences

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Painted by Anna GORBAN