Fractal nature of the phase Fractal nature of the phase space and energy landscape space and energy landscape

topologytopologyGerardo G. Naumis

Instituto de Física, UNAM. México D.F., Mexico.

XVIII Meeting on Complex Fluids, San Luis Potosí, México.

• IntroductionIntroduction Relaxation and flexibility in polymers, proteins, colloids Relaxation and flexibility in polymers, proteins, colloids

and fluids. and fluids.

Energy landscape formalismEnergy landscape formalism

• Topography of the phase space and energy Topography of the phase space and energy landscape.landscape.

• A modified Monte-Carlo method to test the A modified Monte-Carlo method to test the topology and topography.topology and topography.

• Applications to the most simple fluidApplications to the most simple fluid

• ConclusionsConclusions

In many systems, we are interested in following the temporal evolution In many systems, we are interested in following the temporal evolution

Failure in folding: Alzheimer's disease, cystic fibrosis, BSE (Mad Cow disease), an inherited form of emphysema, and even many cancers are believed to result from protein misfolding.

WHY IS PROTEIN FOLDING SO DIFFICULT TO UNDERSTAND?It's amazing that not only do proteins self-assemble -- fold -- but they do so amazingly quickly: some as fast as a millionth of a second. It takes about a day to simulate a nanosecond (1/1,000,000,000 of a second). Unfortunately, proteins fold on the tens of microsecond timescale (10,000 nanoseconds). Thus, it would take 10,000 CPU days to simulate folding -- i.e. it would take 30 CPU years!

But is also important to understand the “mechanical flexibility”:But is also important to understand the “mechanical flexibility”:

Example: experimental results in coloids from the group of David Weitz

High volume fraction supercooled fluid: volume fraction 0.56

Dynamics and rheology in dense colloids, glasses, jamming in granular media, etc.

Highlighted particles are slow by over a timestep of 3600 seconds. At this timestep, the largest slow cluster percolates.

One timestep (18 seconds) later, the percolating supercooled fluid sample has broken up.

The third sample is a glass at volume fraction 0.56. The highlighted particles are particles which are slow over an entire experiment, a timestep of 39,000 seconds. At this timestep, the largest slow cluster percolates. Over the experimentally accessible timescales, this percolating cluster never breaks up!

Glass: volume fraction 0.60

There are many approaches to solve these problems, but in fact the Hamiltonian contains all this information…

Energy landscapes and rigidityEnergy landscapes and rigidity

2

1 21

( , ,..., )2

Ni

Ni

PH V r r r T V E

m

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1 2 1 2( , ,..., , , ,..., )N Np p p r r r������������������������������������������������������������������������������������

The mechanical state of the system is represented as a point in phase space:

1( )V r E

The allowed part of the phase space is determined by the “energy landscape”

Basin

ln ( , , )S k E V N

intvibF F F

vib jumps

( )E Statistics of landscape

Decoy tree (protein:villin) 

•Saddle points•Distribution of energy basins•Size of each basin

Some predictions were made about the range of the potential-roughness using catastophy theory. Short range: rough landspaceScience, Vol 293, Issue 5537, 2067-2070 , 14 September 2001

FRACTALS!

FRACTALES: AUTOSIMILARIDAD

Rigidity TheoryRigidity Theory

With N hindges, how many bars do I need to make the system rigid?

Flexible Isostatic Rigid

4x (2 freedom degrees)-(# constraints)=# flexible movements

1 0 -1

f =(3N-constraints)/3N Fraction of “floopy modes since:

# flexible movements=# of normal modes of vibration with zero frequency

f is a function of <r>. In the Maxwell approximation: f=2-5<r>/6.

2 2)(2

(2

)V l

l

Maxwell Counting (1860)

<r>

f

2.0 2.2 2.4

1/33 ( (2 3)) / 3

2rr

rf N x r N

526

f r

f=0, <r>=2.4

Flexible Rigid

2 22 21 2( , )1 2 1 22 2

m mV Q Q Q Q

Interpretation in terms of energy landscapesInterpretation in terms of energy landscapes

Ak AkBkAk AkBk

1q 2q

1Q2Q

1 3( ,..., ) 0NE V Q Q K

221

1 2 1( , )2

mV Q Q Q

Floppy modes provide channels in the landscape: a lot of entropy!!

Channels are not flat; there is a small curvature along the floopy coordinate, since floppy modes are not at non-zero frequency.

1Q2Q

Energy landscapes and rigidityEnergy landscapes and rigidity

There are many approaches to solve these problems, but the Hamiltonian in fact contains all this information…

A A B B C Cr x r x r x r

1

(1 )2 4 3

x y x ySe Ge As

r x y x y

00

0

expDT

T T

0 exp( / )cC TS Adams-Gibbs equation:

( )

K

Tp

c

T

c TS dT

T

0 0exp( / )DT T

Tatsumisago et. al.,Phys. Rev. Lett. 64, 1549 (1990).

Intermediate phase (P. Boolchand et. al., J. of Optoelectronics and Advanced Materials Vol. 3, 703 (2001)).

Boolchand et. al., J. of Non-Cryst. Solids 293, 348 (2001).

<r>

Self-organization: Thorpe, et. al., J. Non-Cryst. Solids 266, 859 (2000).Barré et. al., Phys. Rev. Lett. 94, 208701(2005)

Rigidity of proteins and glasses

To read more: G. Naumis, “Energy landscape and rigidity”, Phys. Rev. E71, 026114 (2005).

Results of Mike Thorpe, Arizona State University

2/ 2 / 2,x L

Topology of the phase space: the role of constraints…

Hard-disks

Restrictions:

1/ 2 / 2,x L

1 2x x

Boundary: Box, 1 2x x

( ) / 2HR Z NFor N particles:

1 1

( ) ( )N N

l ll l

F P r C P m r

����������������������������

Center of mass minimization observed in colloids!!! A. Van Blaaderen, Science 301, 471 (2003).

MBM L

22 2 L1 2 /L

,

pR mL ,

A simple example that explains the method…

mL 58 1

2 2MB /M L

2 ( )( ) ( )

2k

t k

DN zp p

DN

( )( ) 1 ( )

2B

kt k

k L

zp p

DN

( ) ( ) ( )k k kz z z

1( ) ( ).

( )B

k kk LB

z zM

Probabilidad de caer en un sitio de frontera:

Sumando sobre toda la frontera:

Definimos una coordinación y probabilidad promedio en los sitios de frontera:

1( ) ( )

( )B

k kk LB

p pM

( ) ( ) ( )k k kp p p

21 ( )

( ) ( )2 2

B

k kk L

z pDN DN

2( ) ( ( ) / 2 )( ) ( )

(1 ( ( ) / 2 ))

R

B Bk

p DNM p

z DN

( ) ( ) ( ) / ( )B B BM p M M

2( ) ( ( ) / 2 )ln / ln( / )

(1 ( ( ) / 2 ))

R

fk

p DNd

z DN

The difference in dimensions between the phase space and the boundary is:

Application for the case of simple fluids: a hard disk system

N=100 disks in a box, with hard core repulsion

CONCLUSIONSCONCLUSIONS• The topology of the phase space and the topography of

the energy landscape are important to understand several thermodynamical and relaxation phenomena.

• This explains diverse features of simulations in associative fluids.

• A method to obtain the fractal dimension using the Monte-Carlo rejection ratio was proposed.

• The application of this method to a simple fluid shows the fractal nature of the phase space and that freezing occurs when the surface scales as the volume in phase space.

• To read more: G.G. Naumis, Phys. Rev. E71, 056132 (2005).