1 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

2

3 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

4 The way to comprehensibility "The most incomprehensible thing about the world is that it is at all comprehensible." (Albert Einstein) A complicated phenomenon A complicated model with “hidden truth” inside The “logically transparent” model Experiment Computational experiment Theory

5 The travel plan Gaussian mixtures for unstable systems Phenomenon of molecular individualism Successful bimodal approximations Shock waves Spinodal decomposition Polymer molecule in flow: essentially non-Gaussian behavior of simplest models Neurons, uncorrelated particles, and multimodal approximation for molecular individualism Conclusion and outlook

6 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)

7 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. Small perturbation of normal distribution P(x)=Ae -(x-M, C (x-M))/2 (1+  (x))

8 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.

9 Cascade of peaks dissociation Second result: the peak parallelogram The peak cube First result: the peak dumbbell

10 The similarity and the difference between Ellipsoid and Parallelepiped

11 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 R n. Suppose it is a sufficiently big set of some of vertices of an unknown parallelepiped with unknown dimension, m  n,  S  2 m. Please find the edges of this parallelepiped. What is the complexity of this problem?

12 Several forms of molecules in a flow

13 Polymer stretching in flow A schematic diagram of the polymer deformation (S.Chu, 1998).

14 The Fokker-Planck equation (FPE)  t  (x,t)=  x {  *(x) D [  x -F ex (x,t)][  (x,t)/  *(x)]}. x=(x 1,x 2,…x n ) is a conformation vector;  (x,t) is a distribution function; D is a diffusion matrix; U(x) is an energy (/kT) ; F ex (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

15 Kinetics of gases The Boltzmann equation (BE)  t f(x,v,t)+(v,  x f(x,v,t))=Q(f,f) The Maxwell distribution (Maxwellian): f M n,u,T (v)=n(m/2  kT) 3/2 exp(-m(v-u) 2 /2kT) Local Maxwellian is f M n(x),u(x),T(x) (v). If f(x,v)=f M n(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?

16 Tamm-Mott-Smith approximation for shock waves (1950s): f is a linear combination of two Maxwellians ( f TMS = af hot + bf cold ) n Variation of the velocity distribution in the shock front at M=8,19 (Zharkovski at al., 1997)

17 The projection problem:  t a ( x,t ) = ?  t b ( x,t ) = ? Coordinate functionals F 1,2 [f(v)]. Their time derivatives should persist (BE  t F 1,2 =TMS  t F 1,2 ): BE  t F 1,2 [f(x,v,t)]=  (  F 1,2 [f]/  f){-(v,  x f(x,v,t))+Q(f,f)}dv; TMS  t F 1,2 [f TMS ]=  t (a(x,t))  (  F 1,2 [f]/  f)f hot (v)dv+  t (b(x,t))  (  F 1,2 [f]/  f)f cold (v)dv. There exists unique choice of F 1,2 [f(v)] without violation of the Second Law: F 1 =n=  fdv - the concentration; F 2 =s=  f(lnf-1)dv - the entropy density. Proposed by M. Lampis (1977). Uniqueness was proved by A. Gorban & I. Karlin (1990).

18 TMS gas dynamics n The gas consists of two ideal equilibrium components (Maxwellians); n Each component can transform into another (quasichemical process); n The basis of coordinate functionals is the pair: the concentration n and the entropy density s.

19 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 X B =n B /(n A +n B )

20 Kinetic description of spinodal decomposition n Ginzburg-Landau free energy G=  [(g(u(x))+ 1 / 2 K(  u(x)) 2 ]dx, where u(x)=X B (x)-  X B  ; n Infinite-dimensional Fokker-Planck equation for distribution of fields u(x); n Perturbation theory expansions, or direct simulation, or…? n Model reduction: we do not need the whole distribution of fields u(x), but how to construct the appropriate variables?

21 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(x 1 ),u(x 2 )), depends on u 1,u 2, and r=  x 1 -x 2 . In LBM theory  2 (u(x 1 ),u(x 2 ))=(1+  (r)u 1 u 2 )  1 (u 1 )  1 (u 2 ), The highest correlation functions S n (r)=  u n-1 (x 1 )u(x 2 ) , In LBM theory S n (r)=  u n   u 2  (r). The main variables:  1 (u),  (r), and  1 (u)=Aexp(-(u-a) 2 /2  1 2 )+Bexp(-(u-b) 2 /2  2 2 ). LMB project FPE on  (r) and this  1 (u).  1 (u) for two moments of time

22 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), M 2. FENE-P model: f=[1- M 2 /b] -1. Gaussian manifold  (x)=(1/2  M 2 ) 1/2 exp(-x 2 /2M 2 ) is invariant with respect to mean field models  is the elongation rate

23 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

24 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  (M 2 )>0 is the domain of exponential instability.

25 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

26 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.

27 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 =5  10 2, b=10 3, and b =5  The extension Q is made dimensionless with the equilibrium end-to-end distance Q 0. P.Ilg, I. Karlin et al., 2002

28 The steps of molecular individualism n Black dots are vertices of the Gaussian parallelepiped. Quasi- stable polymeric conformations are associated with each vertex. n Zero, one, three, and four-dimensional polyhedrons are drawn. n Each new dimension of the polyhedron adds as soon as the corresponding bifurcation occurs.

29 Neurons and particles for FPE The approximation for distribution function Quasiequilibrium (MaxEnt) representation Dual representation If, then the distribution function  is the Gaussian Parallelepiped

30 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

31 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):

32 Conclusion n The highest form of the art of anzatz is to represent a complicated system as a mixture of ideal subsystems. n Gaussian polyhedral mixtures give us a technical mean for description of complex kinetic systems with instabilities as simple mixtures of ideal stable systems. n Molecular individualism is a good problem for development of the methods of Gaussian polyhedral mixtures. n Presentation of particles (neurons) gives us a new technique for solution of multidimensional problems as well, as a new way to construct phenomenology.

33 We work between complexity and simplicity and try to find one in the other n "I think the next century will be the century of complexity". Stephen Hawking n But... “Nature has a Simplicity, and therefore a great Beauty”. Richard Feynman

34 Thank you for your attention. Authors n Alexander Gorban, n Iliya Karlin n ETH Zurich, Switzerland, n Institute of Computational Modeling Russian Academy of Sciences

35

36 Painted by Anna GORBAN