Tiejun Xiao

Effects of internal noise in mesoscopic chemical systems near Hopf bifurcation

The effects of internal noise in mesoscopic chemical oscillation systems have been studied analytically, in the parameter region close to the deterministic Hopf bifurcation. Starting from chemical Langevin equations, stochastic normal form equations are obtained, governing the evolution of the radius and phase of the stochastic oscillation. By stochastic averaging, the normal form equation can be solved analytically. Stationary distributions of the radius and auto-correlation functions of the phase variable are obtained. It is shown that internal noise can induce oscillation; even no deterministic oscillation exists. The radius of the noise-induced oscillation (NIO) becomes larger when the internal noise increases, but the correlation time becomes shorter. The trade-off between the strength and regularity of the NIO leads to a clear maximum in its signal-to-noise ratio when the internal noise changes, demonstrating the occurrence of internal noise coherent resonance. Since the intensity of the internal noise is inversely proportional to the system size, the phenomenon also indicates the existence of an optimal system size. These theoretical results are applied to a circadian clock system and excellent agreement with the numerical results is obtained.

Coherence resonance induced by colored noise near Hopf bifurcation.

Effects of colored noise near supercritical Hopf bifurcation, especially noise induced oscillation (NIO) and coherence resonance (CR), have been studied analytically in the Brusselator model, using the stochastic normal form method. Two types of colored noise are considered: one is the standard Gaussian colored noise generated by the Ornstein-Uhlenbeck (OU) process and the other is the so-called power-limited (PL) process. Depending on the noise intensity and noise type, it is found that the autocorrelation time, most probable radius and signal to noise ratio of the NIO may show nontrivial dependencies on the noise correlation time tau(c). Interestingly, for OU-type noise with intensity above a threshold, SNR is a bell-shaped function of tau(c), indicating enhancement of CR by noise correlation; and for PL-type noise, SNR may show double maxima when tau(c) is changed, demonstrating a new kind of multiresonance phenomenon. These theoretical predictions are well reproduced by numerical simulations.

Stochastic thermodynamics in mesoscopic chemical oscillation systems.

Stochastic thermodynamics in mesoscopic chemical oscillation systems is discussed on the basis of chemical Langevin equation for the state variables with particular attention paid to a parameter region close to the deterministic Hopf bifurcation. The Langevin dynamics defines stochastic trajectories in the state space and therefore trajectory dependent entropy and entropy production according to the schemes proposed by Udo Seifert (Phys. Rev. Lett. 2005, 95, 040602). The total entropy change along a stochastic trajectory obeys the fluctuation theorems. By using the stochastic normal form theory, we derive explicit theoretical expressions for the mean entropy production in the stationary state. The resulting entropy production in the large system volume V limit can scale linearly or independent with V when the control parameter is above or below the Hopf bifurcation while it is of V(1/2) at the bifurcation. We verify the above relations by direct simulation with a stochastic circadian clock model.

Entropy production in a mesoscopic chemical reaction system with oscillatory and excitable dynamics.

Stochastic thermodynamics of chemical reaction systems has recently gained much attention. In the present paper, we consider such an issue for a system with both oscillatory and excitable dynamics, using catalytic oxidation of carbon monoxide on the surface of platinum crystal as an example. Starting from the chemical Langevin equations, we are able to calculate the stochastic entropy production P along a random trajectory in the concentration state space. Particular attention is paid to the dependence of the time-averaged entropy production P on the system size N in a parameter region close to the deterministic Hopf bifurcation (HB). In the large system size (weak noise) limit, we find that P ∼ N(β) with β = 0 or 1, when the system is below or above the HB, respectively. In the small system size (strong noise) limit, P always increases linearly with N regardless of the bifurcation parameter. More interestingly, P could even reach a maximum for some intermediate system size in a parameter region where the corresponding deterministic system shows steady state or small amplitude oscillation. The maximum value of P decreases as the system parameter approaches the so-called CANARD point where the maximum disappears. This phenomenon could be qualitatively understood by partitioning the total entropy production into the contributions of spikes and of small amplitude oscillations.

Entropy production and fluctuation theorem along a stochastic limit cycle

Entropy production along a trajectory in the stochastic irreversible Brusselator model of chemical oscillating reactions is discussed. Particular attention is paid to a parameter region near the deterministic supercritical Hopf bifurcation. In the stationary state, detailed fluctuation theorem holds due to the reversibility in the state space, which is verified by direct simulations via Gillespies algorithm [J. Comput. Phys. 22, 403 (1976); J. Phys. Chem. 81, 2340 (1977)]. In addition, we have considered how the entropy production along a noisy limit cycle depends on the system size. Interestingly, in the large system size limit, the entropy production approaches a constant value when the control parameter stays at the deterministic steady state region, while it increases linearly in the deterministic oscillatory region. Such simulation results can be well understood by a stochastic normal form analysis.

Stochastic thermodynamics for delayed Langevin systems

We discuss stochastic thermodynamics (ST) for delayed Langevin systems in this paper. By using the general principles of ST, the first-law-like energy balance and trajectory-dependent entropy s(t) can be well defined in a way that is similar to that in a system without delay. Because the presence of time delay brings an additional entropy flux into the system, the conventional second law � �s tot� 0 no longer holds true, wheres tot denotes the total entropy change along a stochastic path and �·� stands for the average over the path ensemble. With the help of a Fokker-Planck description, we introduce a delay-averaged trajectory-dependent dissipation functional η(χ(t))whichinvolvestheworkdonebyadelay-averagedforce ¯ F(x,t)alongthepath χ(t)andequalsthemedium entropy changes m(x(t)) in the absence of delay. We show that the total dissipation functional R = �s + η, wheres denotes the system entropy change along a path, obeysR� 0, which could be viewed as the second law in the delayed system. In addition, the integral fluctuation theoreme −R �= 1 also holds true. We apply these concepts to a linear Langevin system with time delay and periodic external force. Numerical results demonstrate that the total entropy change � �s totcould indeed be negative when the delay feedback is positive. By using an inversing-mapping approach, we are able to obtain the delay-averaged force ¯ F(x,t) from the stationary distribution and then calculate the functional R as well as its distribution. The second lawR� 0 and the fluctuation theorem are successfully validated.

Internal noise enhanced oscillation in a delayed circadian pacemaker

The effect of internal noise in a delayed circadian oscillator is studied by using both chemical Langevin equations and stochastic normal form theory. It is found that internal noise can induce circadian oscillation even if the delay time τ is below the deterministic Hopf bifurcation τ(h). We use signal-to-noise ratio (SNR) to quantitatively characterize the performance of such noise induced oscillations and a threshold value of SNR is introduced to define the so-called effective oscillation. Interestingly, the τ-range for effective stochastic oscillation, denoted as Δτ(EO), shows a bell-shaped dependence on the intensity of internal noise which is inversely proportional to the system size. We have also investigated how the rates of synthesis and degradation of the clock protein influence the SNR and thus Δτ(EO). The decay rate K(d) could significantly affect Δτ(EO), while varying the gene expression rate K(e) has no obvious effect if K(e) is not too small. Stochastic normal form analysis and numerical simulations are in good consistency with each other. This work provides us comprehensive understandings of how internal noise and time delay work cooperatively to influence the dynamics of circadian oscillations.