Macro-state

Figure 2. Schematic representation of the four conceptually different paths (the heavy lines) one may utilize to attack the phase-coexistence problem. Each figure depicts a configuration space spanned by two macroscopic properties (such as energy, density. ..) the contours link macrostates of equal probability, for some given conditions c (such as temperature, pressure. ..). The two mountaintops locate the equilibrium macro states associated with the two competing phases, under these conditions. They are separated by a probability ravine (free-energy barrier). In case (a) the path comprises two disjoint sections confined to each of the two phases and terminating in appropriate reference macrostates. In (b) the path skirts the ravine. In (c) it passes through the ravine. In (d) it leaps the ravine.

Observed Macro-state Micro-state Quantum infor-... 

Jammer, when he refers to researches in modern physics, presumably means the philosophical difficulties created by quantum physics. Quantum theory was first introduced to explain a number of experimental laws concerning phenomena of thermal radiation and spectroscopy which are inexplicable in terms of classical radiation theory. Eventually it was modified and expanded into its present state. The standard interpretation of the experimental evidence for the quantum theory concludes that in certain circumstances some of the postulated elements such as electrons behave as particles, and in other circumstances they behave as waves. The details of the theory are unimportant to us except in respect of the Heisenburg uncertainty relations . One of these is the well known formula Ap Aq > hl4ir where p and q are the instantaneous co-ordinates of momentum and position of the particle, Ap and Aqi are the interval errors in the measurements of p and q, and h is the Universal Planck s constant. The interpretation of this formula is, therefore, that if one of these co-ordinates is measured with great precision, it is not possible to obtain simultaneously an arbitrarily precise value for the other co-ordinate. The equations of quantum theory cannot, therefore, establish a unique correspondence between precise positions and momenta at one time and at another time nevertheless the theory does enable a probability with which a particle has a specified momentum when it has a given position. Thus quantum theory is said to be not deterministic (i.e, not able to be precisely determined) in its structure but inherently statistical. Nagel [25] points out that this theory refers to micro-states and not macro-states. Thus although quantum... 

A macro-state of availabflity is assigned to the generic -th component it can take two mutnally exclusive values ... 

The macro-state ON of the component /can be further specified by a vector... 

The macro-state OFF has a munber of possible sub-states that specify the condition of the component while not working (e.g., under corrective maintenance, under preventive maintenance, waiting for the availability of the repair team, imder test, etc). A repair rate / , is associated to the generic component /, whose macro-state is OFF -, the value is assumed to be dependent directly on the type of the maintenance action, on the values of the IFs and on the degradation state of the component. 

The repair rates, /x, of those components that at the current time t enter in an OFF macro-state, i.e., iit) = OFF . This vector is used by the MC module to sample the duration T ofthe maintenance... 

At each transition the failure rates are updated to values that depend on the degradation states reached by the components. A weight is accumulated in the counters associated to the discrete bins in which the macro-states of the components are OFF", equal to the fraction of time in which the components are unavailable in the bin. After performing all the Monte Carlo histories, the content of each counter divided by the time interval Dt and by the number of histories gives an estimate of the mean unavailability of the component in that counter time. A similar accumulation is done for the counters devoted to the recording of the availability state of the system. 

In particular, given the relationship between the degradation states and the failure rates, the Porward Issue is put into practice by assessing the vector ofthe degradation state, D, for those components whose macro-state is ON . This assessment is performed by fuzzy logic models (one for each degradation mechanism of each component) built on FRBs which link the IPs to the vectors D (Baraldi et al. 2009). 

We shall consider the system of all objects which can he affected as a consequence of some initiative disastrous event as a whole along with all its macro states. The system of objects can be represented as an oriented network. The objects are nodes and the direction of possible events propagation (transitions) represents oriented edges. The disaster starts by a strong initial event on one of the objects, and it spreads with some probability to the next neighbouring object at a random time. In [6] is shown that despite the strong dependency of successive events, after some arrangements... 

Generally, both parent and offspring events can be considered as macro states of the largest model. The... 

Now consider the most complex one-way model, in which a hranching is allowed. It means that SEC consists of one parent event represented by a parent macro state, one final state and at least two macro states, each of which can be a parent event of single subsequent SET. From this point of view, such SEC contains more than two absorbing states. In the above example, beside the control room blaze another two events caused by the defect in cable-junction box can be considered a damage to the equipment joined to the junction box and power supply interruption for all the object. 

We denote by i the number of non-operational components, / = 0,1,— k. The Markov process governing this system is denoted by X(t),t > 0, and the space of macro-states, representing the number of non-operational components, is 0,1,...,k. When the system occupies the macro-state i, there are n-i operational components. A failure of one unit implies a transition between the macro-states / —> i-l-1,/ = 0,1,..k. A failure of the system occurs in the transition — k->- — k-Fl. 

We will assume that the lifetimes of the identical components follow phase-type distributions PH(a,T) of order m. So, when the system occupies the macrostate i, the phase of the operational component / = 1,2,will be denoted by h, h = 1,..., w. Then, we can construct the exponential states when the system is up, they are grouped in macro-states given by,... 

Now, we construct the infinitesimal generator of the system, denoted by S. The expression of this matrix in terms of the transition among macro-states is. 

From the forward Kolmogorov equations for matrices and considering that the initial macro-state is 0, that is, X(0) = 0, aU the components are operational, we have,... 

Matrix S is the matrix of a quasi-birth process (QB process) with an absorbent macro-state. 

The lifetime of the system is the time up to the absorption by the macro-state n — k +. This is a phase-type distribution with representation 08,S) being p the initial vector of probabilities, p= (a .. S"K., a, 0) S the parameter matrix, and S the absorption matrix ... 

The rate of occurrence of failures for a unit at time t is defined as the mean number of failures per unit time at time t. The rate of occurrence of failures of the components is denoted by vi(t). A failure of a component occurs when there is a transition from an operational macro-state i to the macro-state i-1. When the system occupies macro-state i, there are n — i operational components, and only one of these can fail, while the others do not change. This is governed by the matrix T T . .. .. T . On the other... 

In the present Section the calculations of the paper are illustrated. Let a 2-out-of-3 system be. The system operates when at least two components are operational, so the system fails when one component fails. The macro-states are / = 0,1,2, indicating the number of non-operational components. The unit time are denoted by u.t. 

The probability vector of staying in the macro-states given in (3) is for this particular case... 

Almost 7 0% of the time the system occupies macro-state 2 (the system is not operational) after a time t -10. 

In contrast to this microscopic description of many-particle systems, thermodynamics aims at a macroscopic description of such systems. Typical macroscopic variables are for example the volume of the system, its total mass, the number of moles of its chemical components and its total energy. In any case, the number of thermodynamic or macroscopic variables is much less than the number of the microscopic degrees of freedom. Hence, the transition from a microscopic to a macroscopic description involves a drastic reduction of the information about the system. This means that any particular macro-state must be realized by a very large number of micro-states among which the system rapidly fluctuates. In the course of a macroscopic measurement of the system, however, such microscopic fluctuations will not be observed. 

The important point of (3.1) is the fact that it not merely defines a quantity U but actually represents an autonomous statement by saying that the left-hand side is independent of the history of the particular process A—>B but depends only on the initial and the terminal macro-states A and B of the process. Such variables which are uniquely related to macro-states and independent of the processes between the macro-states are called state variables. Thus, the first law of thermodynamics can also be expressed by the statement that the internal energy U is a state variable. 

Another way of formulating the first law is to replace the finite processes between macro-states A and B in (3.1) by infinitesimal or differential processes such that... 

After having postulated the existence of equilibrium states, we shall try to formulate a criterion which distinguishes an equilibrium state from nonequilibrium macro-states. Recalling the intuitive description of the evolution towards the equilibrium state at the beginning of this section, one could be inclined to characterize equilibrium as the most probable state of the system that is compatible with... 

The latter quantity must first be defined more closely, so that it may be applied to examples other than the one which has been discussed. Q is the total number of distinguishable micro-states, or complexions , which are confined within a given macro-state of a system, this macro-state being characterized by fixed values of the energy, volume and numbers of particles of specified kinds. Thus... 

The term (macro)state of a system A, both equilibrium and non-equilibrium, means a class of equivalence on the set of all its possible microstates - microscopic arrangements of all its constituents (particles, matter units) within all (by us distinguished, defined) parts, cells, of its whole, by them occupiable, volume V, or, better said, of a given (state) space of, 4. ... 

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