Nonergodic behavior

As mentioned in the introduction, blinking NCs exhibit a nonergodic behavior. In particular the ensemble average intensity (I) is not equal to the time average 7. Of course in the ergodic phase—that is, when both the mean on and off times are finite—we have (I) = 7, in the limit of long measurement time. More generally... 

Figure 12. PDF of C (T t ) for different r - t /T and a = 0.8. Diamonds are numerical simulations. Curves are analytical results without fitting For r = 0, Eq. (32) is used (full line) for r = 0.01, 0.1 and 0.5 Eq (35) is used (dashed) and for r = 0.9 and 0.99, Eq. (38) is used (full lines). If compared with the cases a = 0.3 and 0.5, the distribution function exhibits a weaker nonergodic behavior, namely for r = 0 the distribution function peaks on the ensemble average value of 1/2.

In systems with two degrees of freedom, two-dimensional tori separate the equi-energy surface into two disjoint parts. Thus, orbits on one part of the equi-energy surface cannot go into the other. Therefore, the existence of two-dimensional tori in systems of two degrees of freedom results in nonergodic behavior of the system. 

Static susceptibility and other Nonergodic behavior at Ergodic behavior at all... 

Even if a system is formally ergodic, its behavior during computer simulations may resemble those of nonergodic systems. This means that the system does not properly explore phase space, and, therefore, the calculated statistical averages might... 

Third, what is the dynamical origin of Maxwell s demon As is well known since the work of Maxwell, Szilard, and Brillouin, nonequilibrium conditions are necessary for systems to do information processing. Therefore, in studying biochemical reactions, we are interested in how nonequilibiium conditions are maintained at the molecular level. From the viewpoint of dynamics, in particular, the following problem stands out as crucial Does any intrinsic mechanism of dynamics exist which helps to maintain nonequilibrium conditions in reaction processes In other words, are there any reactions in which nonergodicity plays an essential role for systems to exhibit functional behavior ... 

In the third part, those contributions are collected which discuss nonergodic and nonstationary behavior in systems with many degrees of freedom, and seek new possibilities to describe complex reactions, including even the evolution of living cells. 

IV. Anomalous Behavior of Variance of Nonergodic Adiabatic Invariant... 

In Section II we will review thermodynamics and the fluctuation-dissipation theorem for excess heat production based on the Boltzmann equilibrium distribution. We will also mention the nonequilibrium work relation by Jarzynski. In Section III, we will extend the fluctuation-dissipation theorem for the superstatisitcal equilibrium distribution. The fluctuation-dissipation theorem can be written as a superposition of correlation functions with different temperatures. When the decay constant of a correlation function depends on temperature, we can expect various behaviors in the excess heat. In Section IV, we will consider the case of the microcanonical equilibrium distribution. We will numerically show the breaking of nonergodic adiabatic invariant in the mixed phase space. In the last section, we will conclude and comment. 

IV. ANOMALOUS BEHAVIOR OF VARIANCE OF NONERGODIC ADIABATIC INVARIANT... 

Our DLS data exhibit the behavior anticipated from the macroscopicaUy observed transition from viscous, fluid-like behavior for all samples with ( ) < 0.06 to elastic, solid-like behavior for the samples with ( ) > 0.09. The autocorrelation functions fully decay for all fluid-like samples (< ) < 0.06), while they are characterized by a nondecaying component at long times for the two solid-like samples ( ) > 0.09). This clearly indicates that our system undergoes an ergodic to nonergodic transition, that is, a fluid to solid transition as the volume fraction increases beyond a critical value <1), where 0.06 <, < 0.09. 

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