1. Stochastic Thermodynamics in Mesoscopic Chemical Oscillation Systems
Zhonghuai Hou (侯中怀)
XiaMen
Department of Chemical Physics
Hefei National Lab for Physical Science at Microscale
University of Science & Technology of China (USTC)
First of all, thanks very much for the organizers to give me this oppurtuinity to talk about my recent work here. Here is the title of my talk today, “stochastic thermodynamics in mesoscopic chemical oscillation systems”.

2. 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
Our main research interest covers a few topics, …. In a word, we focus on nonequilibrium, nonlinear and complex problems. This talk is about “thermodynamics of small systems’.
Nonequilibrium + Nonlinearity + Complexity

3. Irreversibility Paradox?
Microscopic
Reversibility
Macroscopic
Time Arrow
t-t, p-p
the 2nd law
How the second law emerges as the system size grows?
Well, the question of how reversible microscopic equations of motion can lead to irreversible macroscopic behavior, such as the second law, has been one of the central issues in statistical mechanics for over one hundred years. A related problem is how …. In my opinion, to answer this question, one of the keys is to understand thermodynamics of small systems. This one, itself, is also of particular importance to study the properties of nano-systems, such as…
Key: Thermodynamics of Small Systems !
Molecular Motors 2~100nm
Solid Clusters ~10nm
Quantum Dots ~100nm
Subcellular
reactions…

4. Small Systems? Fluctuations begin to dominate
Heat, Work: Stochastic Variables
Distribution is more important
Polymer Stretching
Protocol：X(t)
So, what’s different for small systems? First of all,as system dimensions decrease, fluctuation away from equilibrium begins to dominate its behavior. Actually, fluctuations can lead to significant deviations from the system’s average behavior, therefore, small systems may not be well described by classical thermodynamics. For a small system, heat, work and entropy production are stochastic variables. For example, consider a polymer stretching experiment under a given protocol that control the length of the polymer. Even if we repeat the protocol exactly each time, the heat and work during the experiment differs from time to time. So the distribution of these thermodynamic variables are more important than their average values.
Heat
Work
Physics Today, 58, 43, July 2005

5. Fluctuation Theorem Nonequilibrium Steady States 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)
In this field, maybe the most important progress in recent years is the so-called fluctuation theorems, tells that the fluctuations must satisfy some strict relations. For instance, in a NE state, distribution of the entropy production \sigma obeys a so-called detailed FT. According to this equality,
Adv. In Phys. 51, 1529(2002); Annu. Rev. Phys. Chem. 59, 603(2008); ……

6. Stochastic Thermodynamics (ST)
Stochastic process(Single path based)
A Random Trajectory
Trajectory Entropy
Total Entropy Change
Originally, FT was established in many-particle systems. A few years ago, it was extended to general stochastic processes and single trajectory level. For a random trajectory, we can define a trajectory entropy. From the balance equation of the entropy, a total entropy change and medium entropy change can be well defined, where the medium entropy change is related to the exchanged heat of the system with bath.
It can be proved that the \DS_tot satisfies detailed FT and integrated form of FT. And the mean value of DS_tot is non-negative which is the 2nd law.
Exchange heat
Fluctuation Theorems
Second Law
Prof. Udo Seifert

7. 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)
Such a scheme has found many applications, for instance, in driven colloidal particles, optimal two-state system, as well as chemical reaction networks.

8. Chemical Oscillation Systems
Our Work
Stochastic
Thermodynamics
Chemical Oscillation Systems
Our work, we try to apply ST to chemical oscillation systems.

9. Chemical Oscillation Self-Organization far from Equilibrium
Important: signaling, catalysis
Nanosystems: Fluctuation matters
Synthetic Gene Oscillator
CO+O2 Rate Oscillation
Chemical oscillation is a kind of self-organization behavior far from equilibrium. It is widely found in cell-signaling process and catalysis reactions. There two examples show experimental observations of oscillation in synthetic gene regulatory network and oxidation of CO on nanoparticle surfaces. WE see that for such small systems, the oscillations are stochastic.

10. Modeling of Chemical Oscillations
N Species, M reaction channels, well-stirred in V
Reaction j:
Rate:
Macro- Kinetics: Deterministic, Cont.
For a macroscopic system, the dynamics of a oscillation system is described by nonlinear differential equations. Usually with the change of a control parameter, the system may go through a Hopf bifurcation from a steady state to a oscillation state. Hopf bifurcation can be viewed as a nonequilibrium phase transition.
Hopf bifurcation
Nonequilibrium Phase Transition (NPT)

11. Modeling of Chemical Oscillations
Mesoscopic Level: Stochastic, Discrete
Master Equation
Kinetic Monte Carlo Simulation (KMC)
Gillespie’s algorithm
Exactly
Approximately
For small systems, however, the reactions are stochastic, discrete events. The molecular numbers are stochastic variables, whose probability distribution function satisfies the master equation. We can use Kinetic Monte Carlo method to simulate the reactions. If the system size is not too small, the master equation is equivalent to a chemical Langevin equation. We can use the KMC method or Langevin equations to generate stochastic trajectories in the molecular number or concentration state space.
Internal Noise
Deterministic kinetic equation

12. Our concern… How ST applies ? Fluctuation Theorem ? Second law?
N, V
(Small)
How ST applies ?
Fluctuation Theorem ?
Second law?
Role of
Bifurcation? ……
Small
Far From
Equilibrium
Stochastic
Process
So for chemical oscillations in small system, we want to know whether FT is valid or not, and what’s the role of the Bifurcation, any new features related to the oscillation properties ?

13. The Brusselator Concentration： Stochastic Oscillation
Molecular number：
State Space Random Walk
We consider a conceptual model, the Brusselator.
We pay our attention to the region near the Hopf bifurcation.

14. Dynamic Irreversibility
Path and Entropy
Master Equation：
Random Path：Gillespie Algorithm
Entropy：
Dynamic Irreversibility
In this case, we can generate a random path by Kinetic Monte Carlo simulation according to the Gillespie method. The probability distribution function satisfies master equation. We can show that the total entropy change along such a trajectory is related to the probability of the forward path over that of the backward path. Therefore, DS_tot is a measure of so-called dynamic irreversibility.

15. Entropy Change Along Limit Cycle
Stochastic Oscillation: Closed Orbit (Limit Cycle)
We can then calculate the entropy production along a limit cycle, which is a closed orbit in the molecular number state space. The distribution of DS_tot is not Gaussian, and not sensitive to the bifurcation. We can see that 2nd law violation fluctuations have non-zero probability, but the 2nd law is true on an ensemble level.
Distribution not sensitive to Hopf Bifurcation (HB)
2nd-Law Violation Events happens( )
Second Law:

16. Fluctuation Theorem Holds
It can be validated the detailed form of FT exactly holds. But the HB does not show any effects here.

17. NPT: Scaling Change Abruptly
Role of HB？
Above HB
Entropy Production
Below HB
So, what’s the role of HB? We find that before the bifurcation, the entropy production is not dependent on the system size. However, in the deterministic oscillatory region, P increase linearly with system size.
Scaling changes abruptly when the bifurcation happens.
 Universal for Oscillation Systems?
Entropy production and fluctuation theorem along a stochastic limit cycle
T Xiao, Z. Hou, H. Xin. J. Chem. Phys. 129, (Sep 2008)

18. General Meso-Oscillation Systems
Chemical Langevin Euqations(CLE):
Fokker-Planck Equations(FPE):
Large size scaling, we can use CLE.

19. Path Integral … Trajectory Entropy Entropy Change Along Path: Entropy
Total
System
Medium
We can get the expression for the E.P., which depends on the details of the trajectory.
Entropy
Production

20. Stochastic Normal Form Theory
Centre Manifold: Oscillatory Motion
Stable Manifold: Decay Much faster
To go further, we use the stochastic normal form theory developed by ourselves about 2 years ago. According to our theory, the dynamics of the system near HB is dominated by the 2-dimensional oscillatory motion on the centre manifold. The amplitude and phase angle of the oscillation obey a simplified stochastic normal form equation. All other N-2 modes are stable and decay much faster than the oscillatory mode.
T Xiao, Z. Hou, H. Xin. ChemPhysChem 7, 1520(2006); New J. Phys. 9, 407(2007)

21. Slow Oscillatory Mode Dominants
Analytical Result
Slow Oscillatory Mode Dominants
Using this theory, we can obtain the analytical expression for the entropy production. Since the first term depends on the system size V, while the second term does not, it concludes that the slow oscillatory mode dominates the entropy production. In addition, we show the E.P. is proportional to the product of the oscillation frequency and amplitude square.

22. Scaling Relations: Universal
Normal form theory tells:
Scaling relation
Our theory also gives the expression for the oscillation amplitude near the HB. Substitute this into the formula for EP, we can obtain the final scaling relations.

23. General Picture Universal FT Holds
The general picture is, FT holds no matter the HB happens or not, but the EP shows distinct scaling behavior with the system size. Our theory also demonstrates that the scaling of EP with V is an universal phenomenon in meso-oscillation systems.
Stochastic Thermodynamics in mesoscopic chemical oscillation systems
T Xiao, Z. Hou, H. Xin. J. Phys. Chem. B 113, 9316(2009)

24. 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

25. Thank you Acknowledgements Detail work: Dr. Tiejun Xiao (肖铁军)
Support: National science foundation of China