Stochastic behavior

Probabilistic rules can be introduced into the CA in several ways. In "the speed of light," a rule that contains a random element leads to the propagation of a wavefront. When updating each cell, we could make a random choice between several different rules to introduce stochastic behavior, but we could also determine the future state of the cell by reference not to the... 

Determine the key chemical parameters that govern non-stochastic behavior. 

G. Casati, B. Chirkov, J. Ford, and F.M. Izrailev, in G. Casati and J. Ford, (Eds.), Stochastic Behavior in Classical and Quantum Hamiltonian Systems of Lecture Notes in Physics, Vol. 93, Springer, Berlin, 1979. 

Dendrimer 15 was the object of a very interesting theoretical work by Amatore et al.55 discussing the possibility to observe a stochastic behavior of electrochemical events. In the limiting case of only one dendrimer 15 adsorbed on the electrode surface, the current measured in a chronoamperometric experiment would show a random and discontinuous succession of single electron transfer events, while for an array of 7800 dendrimers, corresponding to complete coverage of an electrode of 500 nm in radius, the stochastic nature of the phenomenon is no longer clearly discernable. 

If one chooses Pi(Vi, 0) = fi ) a non-stationary Markov process is defined, called the Wiener process or Wiener-Levy process. ) It is usually considered for f >0 alone and was originally invented for describing the stochastic behavior of the position of a Brownian particle (see VIII.3). The probability density for t > 0 is according to (2.2)... 

The reader will have no difficulty in verifying that the two consistency conditions in 2 are obeyed. This process was originally constructed to describe the stochastic behavior of the velocity of a Brownian particle (see VIII.4). Clearly it has zero average and the autocorrelation function is simply... 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

Conclusion. In classical statistical mechanics the evolution of a many-body system is described as a stochastic process. It reduces to a Markov process if one assumes coarse-graining of the phase space (and the repeated randomness assumption). Quantum mechanics gives rise to an additional fine-graining. However, these grains are so much smaller that they do not affect the classical derivation of the stochastic behavior. These statements have not been proved mathematically, but it is better to say something that is true although not proved, than to prove something that is not true. 

XVII. Stochastic behavior of quantum systems Subject Index... 

Because of the unavailability of a method for solving the classical many-body problem directly, the harmonic approximation was sometimes stretched, or stochastic behavior assumed too early, in an effort to... 

If all possible combinations were equally probably, we would observe stochastic behavior like primary nucleation, so that crystal growth kinetics would be virtually unpredictable. However, a few molecular paths for crystal growth are highly preferred over others, these paths combine in an ensemble to provide the macroscopic observations of crystal growth described in the next section. 

The Lorenz and Rossler models are deterministic models and their strange attractors are therefore called deterministic chaos to emphasize the fact that this is not a random or stochastic behavior. 

We have demonstrated the stochastic formation of (S)- and (,R)-5-pyrimidyl alkanol 12 from pyrimidine-5-carbaldehyde 11 and i-V Zn without the intervention of a chiral auxiliary. Even in the reactions performed in toluene alone, stochastic behavior of the formation of (S)- and (A)-12 was observed in the presence of achiral silica gel. We believe that the approximate stochastic behavior in the formation of alkanols fulfils one of the conditions necessary for chiral symmetry breaking by spontaneous absolute asymmetric synthesis. 

Fluctuations are inherent to any experimental chemical system. Even if these fluctuations are infinitesimally small, they are sufficient to drive the system away from an unstable state. The optically active state is characterized by two equivalent options starting from an unstable racemic situation, the system can evolve into either an R configuration or into an S one. However, each individual experiment remains unpredictable as to which of the optically active states the system will move towards. For a large number of experiments an equal and random distribution between R and S dominance is expected if the initial conditions do not involve any preferences. Due to this unpredictability of the chiral configuration, the phenomenon of mirror-symmetry breaking introduces another element of stochastic behavior into chemical reactions different from that occurring in clock reactions [38,39]. 

Enzyme-catalyzed reactions involve multi-molecular enzyme-substrate association. Therefore, even when the overall reaction is unimolecular, the enzyme mechanism is generally non-linear. If a system has more than one copy of the enzyme and a small number of the reactant molecules, then one needs the CME framework to represent the stochastic behavior of the system. Note that in cellular regulatory networks, the substrates themselves may be proteins that are present in small numbers of copies. Recall from Section 5.1, for example, that the mitogen-activated protein (MAP) is the substrate of MAP kinase, and the MAPK is the substrate of MAPKK. 

The stochastic behavior of biochemical reactions has been observed in single living cells as well as in in vitro experiments. Sunney Xie and colleagues have observed a stochastic, randomly timed bursting behavior in the production of proteins, i.e., translation of an mRNA [26, 217]. The observed number of the protein molecules produced in each burst, np, follows a geometric distribution,... 

Casati, G., Chirikov, B.V., Izraelev, F.M. and Ford, J. (1979). Stochastic behavior of a quantum pendulum under a periodic perturbation, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, eds. G. Casati and J. Ford (Springer, New York). 

---