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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
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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
>=