Stochastic Thermodynamics in Mesoscopic Chemical Oscillation Systems
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Transcript of Stochastic Thermodynamics in Mesoscopic Chemical Oscillation Systems
Stochastic Thermodynamics in Mesoscopic Chemical Oscillation Systems
Zhonghuai Hou (侯中怀 )2009.12 XiaMen
Email: [email protected] 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 )