State ergodic

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

The ergodic hypothesis essentially states that no matter where a system is started, it is possible to get to any other point in phase space. For U and A this leads to the following expressions. 

Another problem with microcononical-based CA simulations, and one which was not entirely circumvented by Hermann, is the lack of ergodicity. Since microcanoriical ensemble averages require summations over a constant energy surface in phase space, correct results are assured only if the trajectory of the evolution is ergodic i.e. only if it covers the whole energy surface. Unfortunately, for low temperatures (T << Tc), microcanonical-based rules such as Q2R tend to induce states in which only the only spins that can flip their values are those that are located within small... 

With a computer program that evaluates r as a function of time for a given U(r) we can use the ergodic hypothesis (which states that the time average over a long time is equal to the configuration average) and write... 

Central to many developments in this book is the concept of ergodicity. Let us consider a physical system consisting of N particles. Its time evolution can be described as a path, or trajectory, in phase space. If the system was initially in the state... 

In a perfect crystal at 0 K all atoms are ordered in a regular uniform way and the translational symmetry is therefore perfect. The entropy is thus zero. In order to become perfectly crystalline at absolute zero, the system in question must be able to explore its entire phase space the system must be in internal thermodynamic equilibrium. Thus the third law of thermodynamics does not apply to substances that are not in internal thermodynamic equilibrium, such as glasses and glassy crystals. Such non-ergodic states do have a finite entropy at the absolute zero, called zero-point entropy or residual entropy at 0 K. 

If the n-steps transition probability elements are defined as the probability to reach the configuration j in n steps beginning from the configuration i and Ilj, = n (qjMarkov chain is ergodic (the ergodicity condition states that if i and j are two possible configurations with 0 and Ilj 0, for some finite n, pij(nl 0 ) and aperiodic (the chain of configurations do not form a sequence of events that repeats itself), the limits... 

Eq. (14), which was originally postulated by Zimmerman and Brittin (1957), assumes fast exchange between all hydration states (i) and neglects the complexities of cross-relaxation and proton exchange. Equation (15) is consistent with the Ergodic theorem of statistical thermodynamics, which states that at equilibrium, a time-averaged property of an individual water molecule, as it diffuses between different states in a system, is equal to a... 

In principle, the transition from state A to B needs to be unbiased and ergodic... 

The transition probabilities W% C C) cannot be arbitrary but must guarantee that the equilibrium state P C) is a stationary solution of the master equation (5). The simplest way to impose such a condition is to model the microscopic dynamics as ergodic and reversible for a fixed value of X ... 

Zero-one law for steady states of weakly ergodic reaction networks... 

Let us take a weakly ergodic network iV and apply the algorithms of auxiliary systems construction and cycles gluing. As a result we obtain an auxiliary dynamic system with one fixed point (there may be only one minimal sink). In the algorithm of steady-state reconstruction (Section 4.3) we always operate with one cycle (and with small auxiliary cycles inside that one, as in a simple example in Section 2.9). In a cycle with limitation almost all concentration is accumulated at the start of the limiting step (13), (14). Hence, in the whole network almost all concentration will be accumulated in one component. The dominant system for a weekly ergodic network is an acyclic network with minimal element. The minimal element is such a component Amin that there exists an oriented path in the dominant system from any element to Amin- Almost all concentration in the steady state of the network iV will be concentrated in the component Amin-... 

Of course, similar results for perturbations of zero eigenvalue are valid for more general ergodic chemical reaction network with positive steady state, and not only for simple cycles, but for cycles we get simple explicit estimates, and this is enough for our goals. 

---