Non equilibrium processes in condensedmatter physics and biological physics

Hans FogedbyDepartment of Physics and Astronomy

University of Aarhus

Work supported by The Danish Natural Science Research Council

First talk at NORDITA, February 2, 2006

Growing bacterial colony and a molecular motor model

2

Aarhus

3

Copenhagen

4

NORDITA

5

In these talks I will choose units such thatPlancks constant h is zero

Atomic forces just make the gue hang togetherHow it behaves is largely classical physics

Well, it is not the whole story butApologies to Planck!

Non equilibrium processes in condensedmatter physics and biological physics

Bacterial colony

Simulation

Phase diagram Bio Motor

Ag deposition on Pt Myosin

Animation

Kinesin

7

Outline of talk

What is statistical physics ? Aspects of equilibrium physics Non equilibrium physics Theoretical methods Example: A growing bacterial colony Example: A biological motor Summary

What is statistical physics ?

Statistical physics enjoys a very special position among the subfields of physics. Its principles hold whenever there are many (more than three?) particles or subunits interacting with each other to make a "complex" system. Its subject vary from microsopic and mesoscopic scales to macroscopic and cosmic scales. Its working tools range from rigorous mathematics to numerical simulations. Its applications extend to almost all branches of natural sciences and, perhaps, economy and sociology.

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Aspects of equilibrium physicsThe legacy of Maxwell, Boltzmann, and Gibbs

James Clerk Maxwell (1831-1879)Theory of electromagnetismKinetic theory of gasesMaxwell distribution

Ludwig Boltzmann (1844 -1906)Statistical interpretation of the 2nd lawEntropy, Ensemble, Boltzmann distributionBoltzmann equation, H - theorem

Josiah Willard Gibbs (1839 -1903)Vector analysisThermodynamics, Gibbs phase ruleFoundation of statistical mechanics

Boltzmann Entropy (fundamental)

Entropy S is given by

W is the number of microstates (thermodynamic weigth)2nd law of thermodynamics

T is the temperature, W is the workS grows due to irreversible processes(the direction of time)

S maximum in thermal equilibrium

logS k W=

dE TdS dW= +

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Gibbs method (practical)

Heat bath at temperature TSystem in microstate n with energy E(n) Probability P(n) given by Boltzmann factor

Partition function Z is given by

Free energy F, 2nd law of thermodynamics

Microscopic foundation ofthermodynamics

F minimum in thermal equilibrium

exp[ ( ) / ]n

Z E n kT=

dWSdTdF +=

ZkTF log=23 -1

( ) exp[ ( ) / ]1.38 10 JK

P n E n kTk

=

=

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The challenge of non equilibrium physics

Driven systems out of equilibrium:

Fluid flow in porous media

Deposition sputtering corrosion

Chemical reactions

Random walk interface growth

Turbulence convection advection

Bacterial growth molecular motors

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Theoretical methodsThe issues in non equilibrium

Ensemble not known in advance

No Boltzmann - Gibbs scheme

No partition function defined

Ensemble defined via dynamics

Equation of motion for probability distribution P(q,t)

Equation of motion for fluctuating variable q(t)

Computer algorithm

Master equation:Equation of motion for distribution

Fokker Planck equation:Equation of motion for distribution P(q,t)

Langevin equation:Equation of motion for fluctuating variable q(t)driven by noise

Numerical simulation:Computer generated dynamics call of random generator

( )i j i j i j ij

dP w P w Pdt

=

( )2

2

12

dP d P d FPdt dq dq

=

( ), ( ) (0) ( )dq F t t tdt

= + < >=

( )iP t

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Master equation for two-levelsystem

1 2

11 2

2

2 1 1

1 2 2

''Detailed balance'' in case of heat bath and energies of system 1 and 2

exp( ( ) / )

Ratesexp( / )exp( / )

E EP E E kTP

w E kTw E kT

=

1 1 2 2

12 1 2 1 2 1

21 2 1 2 1 2

1 2 1

2 1 2

Master equations( ), ( )

Steady state

P P q P P qdP w P w PdtdP w P w Pdt

P wP w

= =

=

=

=

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Langevin equation for random walk

2

Diffusion, many random walkersDensity ( , )Diffusion equation

( , ) ( , )

diffusion coefficient

n r t

dn r t D n r tdt

D

=

Fluctuating variable ( )Fluctuating noise ( )Langevin equation

( ) ( )

Solution

( ) ' ( ')t

x tt

dx t tdt

x t dt t

=

=

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Langevin equation for randomwalk in force field

constant dampinga is potential theis )(

)()0()(nscorrelatio Noise

)(

equation Langevin

>=