Stochastic Thermodynamics in Mesoscopic Chemical Oscillation Systems

Zhonghuai Hou (侯中怀 )2009.12 XiaMen

Email: hzhlj@ustc.edu.cnDepartment of Chemical PhysicsHefei National Lab for Physical Science at MicroscaleUniversity of Science & Technology of China (USTC)

Our Research Interests

Nonlinear Dynamics in Mesoscopic Chemical Systems

Dynamics of/on Complex Networks

Nonequilibrium Thermodynamics of Small Systems (Fluctuation Theorem)

Mesoscopic Modeling of Complex Systems

Nonequilibrium + Nonlinearity + Complexity

Irreversibility Paradox

Microscopic Reversibility

MacroscopicTime Arrow?

t-t, p-p the 2nd law

How the second law emerges as the system size grows?

How the second law emerges as the system size grows?

Key: Thermodynamics of Small Systems !

Key: Thermodynamics of Small Systems !

Quantum Dots 2~100nm

Molecular Motors 2~100nm

Solid Clusters 1~10nm

Subcellular reactions…

Small Systems?

Fluctuations begin to dominate Heat, Work: Stochastic Variables Distribution is more important

Protocol ： X(t)

Physics Today, 58, 43, July 2005

Polymer Stretching

Heat Work

Fluctuation TheoremNonequilibrium Steady States

( )lim ln

( )tB

tt

Pk

t P

/ ,S Q T

/S t

Adv. In Phys. 51, 1529(2002); Annu. Rev. Phys. Chem. 59, 603(2008); ……

Second Law: Must have P(-)>0 Second law violation ‘events’ P(+)/P(-) grows exponentially with size and time For large system and long time, the 2nd Law holds

overwhelmingly For small system and short time, 2nd Law violating

fluctuations is possible (Molecular motor)

0

Stochastic Thermodynamics (ST)

0 1 2 1j

j j n ru u u u u u u

A Random Trajectory

Trajectory Entropy ln ;s p u

tot ms s s Total Entropy Change

R t u u

Fluctuation Theorems

Stochastic process(Single path based)

Second Law

, 1tot tots stot totp s p s e e

0tots Prof. Udo Seifert Prof. Udo Seifert

Exchange heat Exchange heat

Many Applications…… Probing molecular free energy landscapes by periodic loading

PRL(2004) Entropy production along a stochastic trajectory and an

integral fluctuation theorem , PRL (2005) Experimental test of the fluctuation theorem for a driven two-

level system with time-dependent rates, PRL (2005) Thermodynamics of a colloidal particle in a time-dependent

non-harmonic potential, PRL(2006) Measurement of stochastic entropy production, PRL(2006) Optimal Finite-Time Processes In Stochastic

Thermodynamics, PRL(2007) Stochastic thermodynamics of chemical reaction networks,

JCP(2007) Role of external flow and frame invariance in stochastic

thermodynamics, PRL(2008) Recent Review: EPJB(2008)

Our Work

Stochastic Thermodynamics

Stochastic Thermodynamics

Chemical Oscillation Systems

Chemical Oscillation

Self-Organization far from Equilibrium Important: signaling, catalysis Nanosystems: Fluctuation matters

Synthetic Gene Oscillator CO+O2 Rate Oscillation

Modeling of Chemical Oscillations

Macro- Kinetics: Deterministic, Cont.

N Species, M reaction channels, well-stirred in VReaction j:

j X X v Rate:

( ) jW VX

1

( ( ))( ( ) )

Mji

ij ij

W td X t VF

dt V

XX

Oscillation

Co

nce

ntr

atio

n

Control parameter

Hopf Bifurcation

Stale focusHopf bifurcation

Nonequilibrium Phase Transition (NPT)

Modeling of Chemical Oscillations

Mesoscopic Level: Stochastic, Discrete

1

;; ;

M

j j j jj

P tW P t W P t

t

X

X ν X ν X XMaster Equation

Kinetic Monte Carlo Simulation (KMC)Gillespie’s algorithm

Exactly

( , )j

Approximately 1 2

1 1

1 ( )

M Mj ji

ij ij jj j

W WXdt

dt V V VV

X X

Chemi cal Langevi n Equati on (CLE)

V Deterministic kinetic equation

Internal Noise

Our concern…

• Small• Far From Equilibrium• Stochastic Process

• Small• Far From Equilibrium• Stochastic Process

How

ST a

pplie

s ?

Fluctuation Theorem ?

Second law?

Role of Bifurcation?

……

The Brusselator

(X+1,Y-1)(X,Y-1)

(X-1,Y)

(X-1,Y-1)

(X+1,Y)(X,Y)

(X+1,Y+1)(X,Y+1)

Y

X

(X-1,Y+1)

(a)

Molecular number：State Space Random Walk

1.4 1.6 1.8 2.0 2.2 2.4 2.60.4

0.8

1.2

1.6

2.0

2.4

2.8

Con

cent

ratio

n X

1

Control parameter B

V=1E4

Stochastic OscillationA=1, B=1.95

Concentration：Stochastic Oscillation

Path and Entropy

0 1 2 1j

j j n ru u u u u u u

Random Path ： Gillespie Algorithm

Entropy： ln ;s p u

R t u u

Master Equation：

0;0ln

;n

ps

p t

u

u

1;ln

;

j j

m jj j

Ws

W

u r

u r

0 0| ;0ln

| ;tot R

n n

p ps

p p t

u u u

u u uDynamic

Irreversibility

,t X t Y tu

Entropy Change Along Limit CycleStochastic Oscillation: Closed Orbit (Limit Cycle)

tot ms s 0 nu u

Distribution not sensitive to Hopf Bifurcation (HB) 2nd-Law Violation Events happens( ) Second Law:

0ms 0ms

Fluctuation Theorem Holds

msm mp s p s e

2 3 4 51

2

3

4 b=1.9 b=2.1

lnP

lnV

Above HB1P V

NPT: Scaling Change Abruptly

1lim tott

P st

Entropy Production

Below HB

0P V

Universal for Oscillation Systems?

Entropy production and fluctuation theorem along a stochastic limit cycle T Xiao, Z. Hou, H. Xin. J. Chem. Phys. 129, 114508(Sep 2008)

Role of HB？

1.4 1.6 1.8 2.0 2.2 2.4 2.60.4

0.8

1.2

1.6

2.0

2.4

2.8

Con

cent

ratio

n X

1

Control parameter B

V=1E4

Stochastic OscillationA=1, B=1.95

General Meso-Oscillation SystemsChemical Langevin Euqations(CLE):

Fokker-Planck Equations(FPE):

; 1( ) ; ;

2i iji ji j

p tf p t G p t

t x V x

xx x x x

1 1

1( )

M Mj j

jx v w v w tV

x x

1

( ) ,M

iif v w

x x 1

,M

i jijG v v w

x x1

2kj

k k jj

Gf f

V x

1Γ G 2H = Γf 1,...,T

Nf ff

1.4 1.6 1.8 2.0 2.2 2.4 2.60.4

0.8

1.2

1.6

2.0

2.4

2.8

Con

cent

ratio

n X

1

Control parameter B

V=1E4

Stochastic OscillationA=1, B=1.95

0

0

|ln

|

t im ii

t

p ts V dt H s

p t

x

xx

Path Integral …

Trajectory Entropy ln ,s p x

Entropy Change Along Path:

System

Entropy Production

lim mi ii st

SP V H

t

x

0 0 1ln ln ts p p x x

Medium

Total

0 0 0

1

|ln

|tot mt t

p t ps s s

p t p

x x

x x

Stochastic Normal Form Theory

3

20

2r r

i

drr C r t

dt Vr V

dC r t

dt r V

2 1j j j j ju u r t

V

Centre Manifold: Oscillatory Motion

Stable Manifold: Decay Much faster

T Xiao, Z. Hou, H. Xin. ChemPhysChem 7, 1520(2006); New J. Phys. 9, 407(2007)

Analytical Result

212 21 , 2

2 k jkj kjj ks

k j

P V L L r L D

T T TL T J Γ T

3 ,..., Ndiag

TJ Λ

1 1 T D T G T

Slow Oscillatory Mode Dominants

Scaling Relations: Universal

2 2 2( 2 / ) / ( 2 )m r rr C V C

20 0,

lnlnlim lim 1/ 2 0,

ln ln1 0.

m

V V

VrP

V V

Normal form theory tells:

Scaling relation

General Picture

0 50 100

0.00

0.02

0.04

0.06

P(

s m)

sm

Below Onset Above

2 3 4 5 6

1

2

3

4

5

Below Onset Above

log

PlogV

Stochastic Thermodynamics in mesoscopic chemical oscillation systemsT Xiao, Z. Hou, H. Xin. J. Phys. Chem. B 113, 9316(2009)

FT HoldsFT Holds

UniversalUniversal

Concluding Remarks ST applies to mesoscopic oscillation

systems with trajectory reversibility

Oscillatory motion(circular flux) leads to the dynamic irreversibility

FT holds for the total entropy change along a stochastic limit cycle

The scaling of E.P. with V changes abruptly at the HB (NPT), which can be explained by the stochastic normal form theory

Acknowledgements

Support: National science foundation of China

Thank you

Detail work: Dr. Tiejun Xiao (肖铁军 )