Connectivity master equation dynamics

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

It is widely appreciated that chemical and biochemical reactions in the condensed phase are stochastic. It has been more than 60 years since Delbriick studied a stochastic chemical reaction system in terms of the chemical master equation. Kramers theory, which connects the rate of a chemical reaction with the molecular structures and energies of the reactants, is established as a central component of theoretical chemistry [77], Yet study of the dynamics of chemical and biochemical reaction systems, in terms of either deterministic differential equations or the stochastic CME, is not the exclusive domain of chemists. Recent developments in the simulation of reaction systems are the work of many sorts of scientists, ranging from control engineers to microbiologists, all interested in the dynamic behavior of biochemical reaction systems [199, 210],... 

The time scales of the structural transitions in (NaCl)35Cl mean that it is impossible to use conventional molecular dynamics to investigate the interfunnel dynamics. Instead we use the master equation method outlined in Section III.D. To reduce the computational expense and numerical difficulties we recursively removed from our sample those minima that are only connected to one other minimum—these dead-end minima do not contribute directly to the probability flow between different regions of the PES. The resulting pruned sample had 1624 minima and 2639 transition states. RRKM theory in the harmonic approximation was used to model the individual rate constants,. ... 

Once an adequate sample of minima and transition states has been found, we begin the dynamical analysis. Connectivity between minima and transition states has already been determined by the triples calculation (i.e., downhill searches). The free energy of each stationary point is calculated (using the vibrational frequencies), and from that the transition rates may be calculated. Then we can construct a Cv vs. T plot, determine equilibrium probability distributions, solve the Master equation, constmct the rate disconnectivity graph, and perform a full pathway analysis. 

The theoretical method developed here provides a rigorous approach to the description of the internal dynamics of flexible aliphatic tails. The treatment is able to link the master equations used in connection with the RIS approximation to the multivariate Fokker Planck or diffusive equations, avoiding loosely defined phenomenological parameters. 

In Section 2 the formalism of the Master equation, our main tool in the microscopic approach developed in this chapter, is laid down. This formalism, which constitutes a convenient intermediate between purely microscopic and macroscopic theories, accounts for microscopic dynamics through the fluctuations of the macrovariables. We review the main assumptions at the basis of this description, the formal properties of its solutions, and some results established in the early literature on this subject in connection with bifurcations leading to steady-state solutions. We subsequently focus on dynamical bifurcation phenomena and discuss, successively, thermodynamic fluctuations near Hopf bifurcation (Section 3) and in the regime of deterministic chaos (Section 4). A summary and suggestions for further study are given in the final Section 5. 

But these models are not very realistic for actual flexible polymer chains, and one cannot expect to understand polymer dynamics without turning to models which take into account the molecular nature of polymers. In polymers, each bond is subjected to particular anisotropic constraints due to neighboring bonds. Rouse (21) proposed to model the chain by a sequence of beads separated by springs. The random forces exerted by the viscous environment are localized on the beads. In spite of its crudeness, this early model contains the two essential features of pol3rmer dynamics, i.e. the connectivity and the flexibility. It leads to a master equation for the orientation probability ... 

The temperature dependency of the apparent energy of activation of the bleeching process as exhibited by the curvature in the Arrhenius plot (Fig. 3) is typically found for, e. g., dynamic mechanical relaxation processes (17), which leads to the connection whith the free volume theory. The latter processes are best described by the WLF-equation (18), log a =Cj(T-Tg)/(C +T-Tg), i. e., a master plot is obtained when plotting the logarithm of... 

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