Probability density, canonical equilibrium

Such a method has recently been developed by Miller. et. al. (28). It uses short lengths of classical trajectory, calculated on an upside-down potential energy surface, to obtain a nonlocal correction to the classical (canonical) equilibrium probability density Peq(p, ) at each point then uses this corrected density to evaluate the rate constant via eq. 4. The method appears to handle the anharmonic tunneling in the reactions H+HH and D+HH fairly well (28), and can... 

The grand canonical ensemble describes a system of constant volume, but capable of exchanging both energy and particles with its environment. Simulations of open systems under these conditions are particularly useful in the study of adsorption equilibria, surface segregation effects, and nanoscopically confined fluids and polymers. Under these conditions, the temperature and the chemical potentials jti,- of the freely exchanged species are specified, while the system energy and composition are variable. This ensemble is also called the jx VT ensemble. In the case of a one-component system it is described by the equilibrium probability density... 

The MC technique is a stochastic simulation method designed to generate a long sequence, or Markov chain of configurations that asymptotically sample the probability density of an equilibrium ensemble of statistical mechanics [105, 116]. For example, a MC simulation in the canonical (NVT) ensemble, carried out under the macroscopic constraints of a prescribed number of molecules N, total volume V and temperature T, samples configurations rp with probability proportional to, with, k being the Boltzmann constant and T the... 

Classical mechanical formulas must agree with those obtained by taking the limit of quantum mechanical formulas as masses and energies become large (the correspondence limit). This limit does not affect the formula representing the equilibrium canonical probability density, so it must therefore be the same function of the energy as that of quantum statistical mechanics. For a one-component monatomic gas or liquid of N molecules without electronic excitation but with intermolecular forces, the classical energy (classical Hamiltonian function Jf) is expressed in terms of momentum components and coordinates ... 

A-dimensional phase space). A probability density function/(r, /A) characterizes the equilibrium state of the system, so that/,p )dr dp is the probability to find the system in the neighborhood dr dp = dr i,.dpN of the corresponding phase point. In a canonical ensemble of equilibrium systems characterized by a temperature T the function/ r, p ) is given by... 

It is a well-established fact that, within the context of the canonical ensemble, the equilibrium probability for finding expectation values is determined by the equilibrium density operator... 

If the oscillator is weakly coupled to the bath, in canonical thermal equilibrium the probability of finding the oscillator in the nth state is of course P q = e / En/Zq, where ft = I/kT and the oscillator s canonical partition function is Zq = e In addition, the oscillator s off-diagonal (in this energy representation) density matrix elements are zero. The average oscillator energy (in thermal equilibrium) is Eeq = n13nPnq-... 

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