Generalized ensembles

Such an ensemble generalized ground-state energy functional, E = E[N, v] = E[p[N. i l- represents the thermodynamic potential of the N, v -representation, with the corresponding generalized Hellmann-Feynman expression for its differential (see equations (17), (22) and (27)) ... 

The above derivation leads to the identification of the canonical ensemble density distribution. More generally, consider a system with volume V andA particles of type A, particles of type B, etc., such that N = Nj + Ag +. . ., and let the system be in themial equilibrium with a much larger heat reservoir at temperature T. Then if fis tlie system Hamiltonian, the canonical distribution is (quantum mechanically)... 

When a system is not in equilibrium, the mathematical description of fluctuations about some time-dependent ensemble average can become much more complicated than in the equilibrium case. However, starting with the pioneering work of Einstein on Brownian motion in 1905, considerable progress has been made in understanding time-dependent fluctuation phenomena in fluids. Modem treatments of this topic may be found in the texts by Keizer [21] and by van Kampen [22]. Nevertheless, the non-equilibrium theory is not yet at the same level of rigour or development as the equilibrium theory. Here we will discuss the theory of Brownian motion since it illustrates a number of important issues that appear in more general theories. 

There are tliree steps in the calculation first, solve the frill nonlinear set of hydrodynamic equations in the steady state, where the time derivatives of all quantities are zero second, linearize about the steady-state solutions third, postulate a non-equilibrium ensemble through a generalized fluctuation dissipation relation. 

An important point for all these studies is the possible variability of the single molecule or single particle studies. It is not possible, a priori, to exclude bad particles from the averaging procedure. It is clear, however, that high structural resolution can only be obtained from a very homogeneous ensemble. Various classification and analysis schemes are used to extract such homogeneous data, even from sets of mixed states [69]. In general, a typical resolution of the order of 1-3 mn is obtained today. 

It is very convenient to be able to choose a nonstandard ensemble for a simulation. Generally, it is more straightforward to do this in MC, but MD teclmiques for various ensembles have been developed. We consider MC implementations first. 

It is possible to devise extended-system mediods [79, 82] and constrained-system methods [88] to simulate the constant-A/ r ensemble using MD. The general methodology is similar to that employed for constant-... 

Panagiotopoulos A Z, Quirke N, Stapleton M and Tildesley D J 1988 Phase equilibria by simulation in the Gibbs ensemble. Alternative derivation, generalization and applioation to mixture and membrane equilibria Mol. Phys. 63 527-45... 

Master equation methods are not tire only option for calculating tire kinetics of energy transfer and analytic approaches in general have certain drawbacks in not reflecting, for example, certain statistical aspects of coupled systems. Alternative approaches to tire calculation of energy migration dynamics in molecular ensembles are Monte Carlo calculations [18,19 and 20] and probability matrix iteration [21, 22], amongst otliers. 

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is... 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

Since the averaging operator is not normalized and in general (1), 1 for g 7 1, it is necessary to compute Zq to determine the average. To avoid this difficulty, we employ a different generalization of the canonical ensemble average... 

Consider the generalized distribution Pq(r ) to be generated in the Gibbs-Boltzmann canonical ensemble (9 = 1) by an effective potential W,(r /3) which is defined... 

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. 

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