Ensemble fixed distance

For pairs of like chromophores at a fixed distance and with random and uncorrelated static orientations, the decay of emission anisotropy of the indirectly excited chromophore varies with time, tending to zero (Berberan-Santos and Valeur, 1991) in contradiction to earlier works where it was reported to be 4% of that of the directly excited chromophore. Therefore, because the probability that emission arises from the directly excited chromophore is 1/2, the decay of emission anisotropy of the latter levels off at r0/2. This can be generalized to an ensemble of n chromophores (with random and uncorrelated static orientations) the decay of emission anisotropy of the directly excited chromophore levels off at r0/n. 

One may also write down the equation for the force in this fixed distance ensemble... 

It has been argued that the unzipping transition for the quenched averaged RANI case is second order [41]. However for real DNA, it is not the quenched averages that matter. There is strong ensemble dependence and sample to sample variation. This has been exploited to identify point mutants by a comparison of the unzipping force in a fixed distance ensemble [42]. An experimental determination of the unzipping phase boundary for a real DNA has been reported in Ref. [43]... 

With the probability distribution of Eq. (3), the entropy in a fixed distance ensemble can be written as... 

For example, consider a local Monte Carlo scheme for a Lennard-Jones liquid in the NVT (constant particle number, volume, and temperature) ensemble We choose one particle at random, and move it a fixed distance away from the previous position in an arbitrary direction. Eor the reverse move (from j to ij 0j = at the two prefactors cancel out. Pi oc exp[— 3 (i)] according to the canonical Boltzmann distribution. Equation (1.13) indicates that the move is always accepted if the energy of the system is lowered by the displacement. If the energy increases, the move is accepted with probability exp (— 3AE), that is, we draw a random number between 0 and 1 and accept the move if the random number is smaller than exp (—PA ). 

Another example of phase transitions in two-dimensional systems with purely repulsive interaction is a system of hard discs (of diameter d) with particles of type A and particles of type B in volume V and interaction potential U U ri2) = oo for < 4,51 and zero otherwise, is the distance of two particles, j l, A, B] are their species and = d B = d, AB = d A- A/2). The total number of particles N = N A- Nb and the total volume V is fixed and thus the average density p = p d = Nd /V. Due to the additional repulsion between A and B type particles one can expect a phase separation into an -rich and a 5-rich fluid phase for large values of A > Ac. In a Gibbs ensemble Monte Carlo (GEMC) [192] simulation a system is simulated in two boxes with periodic boundary conditions, particles can be exchanged between the boxes and the volume of both boxes can... 

With the total number of monomers and the volume of the system fixed, a number of statistical averages can be sampled in the course of canonical ensemble averaging, like the mean squared end-to-end distance Re), gyration radius R g), bond length (/ ), and mean chain length (L). 

One of the most powerful tools molecular simulation affords is that of measuring distribution functions and sampling probabilities. That is, we can easily measure the frequencies with which various macroscopic states of a system are visited at a given set of conditions - e.g., composition, temperature, density. We may, for example, be interested in the distribution of densities sampled by a liquid at fixed pressure or that of the end-to-end distance explored by a long polymer chain. Such investigations are concerned with fluctuations in the thermodynamic ensemble of interest, and are fundamentally connected with the underlying statistical-mechanical properties of a system. 

The use of the periodic boundary conditions in the two directions perpendicular to the interface normal (X and Y) implies that the system has infinite extent in these directions. To make the computational cost reasonable, one must truncate the number of interactions that each molecule experiences. The simplest possible technique is to include, for each molecule i, the interaction with all the other molecules that are within a sphere of radius which is smaller than half the shortest box axis. One selects, from among the infinite possible images of each molecule, the one that is the closest to the molecule i under consideration. This is called the minimum image convention, and more details about its implementation can be found elsewhere [2]. To arrive at the correct bulk properties, any ensemble average calculated by this technique must be corrected for the contribution of the interactions beyond the cutoff distance. The fixed analytical corrections are calculated by assuming some simple form of the statistical mechanics distribution function for distances greater then R. ... 

In the simplest case, the R mode is characterized by a low frequency and is not dynamically coupled to the fluctuations of the solvent. The system is assumed to maintain an equilibrium distribution along the R coordinate. In this case, ve can exclude the R mode from the dynamical description and consider an equilibrium ensemble of PCET systems with fixed proton donor-acceptor distances. The electrons and transferring proton are assumed to be adiabatic with respect to the R coordinate and solvent coordinates within the reactant and product states. Thus, the reaction is described in terms of nonadiabatic transitions between two sets of intersecting free energy surfaces ( R, and ej, Zp, corresponding to... 

The deviation of the adsorption isotherm at a uniform electrode surface from the linear behavior (Henri isotherm) is related to the interaction between the adsorbed species. A substantiated derivation of its form can only be made on the basis of an analysis of the statistical-mechanical properties of the whole ensemble of adsorbed ions, which, in turn, requires the knowledge of the interaction potential between the ions, U, as a function of their distance, R, along the surface. This quantity is defined as a difference between the energies of the system, when these two ions are fixed at distance R or are very far from each other. 

The (N, P, T) ensemble will sometimes have advantages over the N, V, T). Evidently in the latter the values of V and T necessary to give a certain pressure are not known in advance, and the result can be far from the conditions of interest. If one wants to compare results at a common pressure, or to compare them with experimental results at fixed pressure, it may often be sensible to fix the pressure and use the (N, P, T) ensemble. The equation of state, in the form (V(N, P, T)), is measured rather more directly in the (N, P, T) ensemble and may sometimes be more precise. This possible advantage can certainly be realized for hard-core particles, where the (N, V, T) pressure determination requires an often dubious extrapolation of g2 to the contact distance of the hard cores. For other thermodynamic quantities, such as the energy, the (N, P, T) method seems to be marginally less economical. 

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