Equilibrium ensemble

An explicit example of an equilibrium ensemble is the microcanonical ensemble, which describes closed systems with adiabatic walls. Such systems have constraints of fixed N, V and E < W< E + E. E is very small compared to E, and corresponds to the assumed very weak interaction of the isolated system with the surroundings. E has to be chosen such that it is larger than (Si )... 

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

The ensemble density p g(p d ) of a mixing system does not approach its equilibrium limit in die pointwise sense. It is only in a coarse-grained sense that the average of p g(p,. d ) over a region i in. S approaches a limit to the equilibrium ensemble density as t —> oo for each fixed i . 

For practical calculations, the microcanonical ensemble is not as useful as other ensembles corresponding to more connnonly occurring experimental situations. Such equilibrium ensembles are considered next. 

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. 

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

Entropy and Equilibrium Ensembles.—If one can form an algebraic function of a linear operator L by means of a series of powers of L, then the eigenvalues of the operator so formed are the same algebraic function of the eigenvalues of L. Thus let us consider the operator IP, i.e., the statistical matrix, whose eigenvalues axe w ... 

Bloch s Equation for the Density Matrix.—Bloch s equation will be derived here both for the equilibrium ensemble density matrix of Eq. (8-204), and for the equilibrium grand ensemble density matrix of Eq. (8-219). 

The average ( ) in (5.6) is over many repetitions of the transformation initiated from an equilibrium ensemble of configurations at A = 0, with the same prescribed... 

Create an equilibrium ensemble of starting configurations. To create N initial conformations representative of the equilibrium ensemble for Hamiltonian - // (z. A = 0), one can, for instance, save conformations at regular intervals during a long equilibrium simulation. In some cases, accelerated sampling procedures may be necessary. 

The equilibrium ensemble of the configurations of meso-tartaric acid 10 a—c has umin = 0. From the definition of umm follows immediately that for all achiral molecules min = 0 holds. This applies also to those cases where asymmetric C-atoms are present but cancel each other because of overall molecular symmetry. 

The calculation of the transmission coefficient for adiabatic electron transfer modeled by the classical Hamiltonian Hajis based on a similar procedure developed for simulations of general chemical reactions in solution. The basic idea is to start the dynamic trajectory from an equilibrium ensemble constrained to the transition state. By following each trajectory until its fate is determined (reactive or nonreactive), it is possible to determine k. A large number of trajectories are needed to sample the ensemble and to provide an accurate value of k. More details... 

D. Electron spin dynamics in the equilibrium ensemble Spin-dynamics models Outer-sphere relaxation... 

The Redfield matrix elements are defined in full analogy with the case where the conventional electron spin relaxation processes in a non-equilibrium ensemble are considered, but the rates are in general different from... 

Keywords error analysis principal component block averaging convergence sampling quality equilibrium ensemble correlation time ergodicity... 

The fundamental perspective of this review is that simulation results are not absolute, but rather are intrinsically accompanied by statistical uncertainty [4-8]. Although this view is not novel, it is at odds with informal statements that a simulation is "converged." Beyond quantification of uncertainty for specific observables, we also advocate quantification of overall sampling quality in terms of the "effective sample size" [8] of an equilibrium ensemble [9,10]. 

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