Variable-length ensemble

The fixed-length ensembles are best suited for studying the critical exponents V and 2A4 - 7, while the variable-length ensembles are best suited for studying the connective constant fj, and the critical exponents agi g (fixed-endpoint) or 7 (free-endpoint). Physically, the free-endpoint ensembles correspond to linear polymers, while the fixed-endpoint ensembles with x = 1 correspond to ring polymers,... 

Experimental dependences of resistivity p(T) obtained by the procedure [5], are presented in Fig. 1. As can be seen from Fig. 1, the value of resistivity and the character of its temperature dependence essentially differ for different samples. Sample I is the ensemble of the disordered carbon particles, sample II consists of the particles of disordered carbon with an insignificant amount of CNT (see Table 1). It was stated earlier the conductivity of such materials is described by the hopping model with the variable length of jump (Eq. (1)), that is confirmed experimentally (Fig. 2). Sample HI consists of the particles of disordered carbon up to 60 nm in size and CNT s ropes 2.4 mn in diameter, i.e., this sample is non uniform, and its conductivity can be described within the proposed model taking into account the first and the terms (Eq. (4)). 

Figure 3. Illustration of the transitions between states in a Expanded Ensemble simulation of the chemical potential. The tagged chain of variable length (anisotropic coupling) is depicted in black (solid circles).

Variable-length, fixed-endpoint ensemble (variable N, fixed x)... 

Incomplete enumeration is a quasi-static algorithm that generates a batch of SAWs from the variable-iV, variable-x ensemble (eq. (2.18)) with p = 0. The idea is to take a standard algorithm for systematically enumerating all self-avoiding walks up to some length Nmix, and modify it so that it pursues each branch of the SAW tree only with some probability / < 1 ... 

Apart from trapping, there also exist situations where, as far as ensemble average ( ) is concerned, the mean square displacement does not exist. This corresponds to a jump length distribution X(x) emerging from an Levy stable density for independent identically distributed random variables of the symmetric jump length x, whose second moment diverges. The characteristic function of this Levy stable density is [14,33,34]... 

Of course, completely symmetric mixtures, where the chain lengths Na, Nb of both species A, B are strictly equal, Na = Nb, in practical reality are rather the exception than the rule. Although asymmetric binary fluids can be studied in a fully grand-canonical ensemble (where both chemical potentials, /J.A, /J-B, and the volume, V, of the system, and the temperature T are the independent control variables see, e.g., Mognetri et for a study of short alkanes dissolved in carbon... 

Thermodynamics of irreversible processes delivers a complete description of nonequilibrium states by extending the system of coordinates with a set of hidden variables. Following this route an extended version of the law of mass action can be formulated which holds true for stationary states even in open systems. It identifies stationary nonequilibrium states of cell ensembles as optimised patterns with universal features The reduced version of the cell size distributions of bacteria or yeast belong to the p = 3 class, while the length distribution of the intracellular proteins is described asp>= 1 type. 

Fixed-length, free-endpoint ensemble (fixed N, variable x)... 

---