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 next step in the calculation is to choose pw to) to be the equilibrium density matrix, pwe- One of the differences between quantum and quantum-classical response theories appears at this stage. In quantum mechanics, the quantum canonical equilibrium density is pj = exp(—/ .ff) which, when... 

Consider a system in the canonical ensemhle. Assume that this system is made up of two subsystems, I and II, of different densities at equilibrium. Consider for example a liquid phase at equiUbrium with its vapor. Each of the subsystems is in the fxPr ensemble. 

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)... 

The usual context for linear response theory is that the system is prepared in the infinite past, —> -x, to be in equilibrium witii Hamiltonian H and then is turned on. This means that pit ) is given by the canonical density matrix... 

To conclude, the introduction of species-selective membranes into the simulation box results in the osmotic equilibrium between a part of the system containing the products of association and a part in which only a one-component Lennard-Jones fluid is present. The density of the fluid in the nonreactive part of the system is lower than in the reactive part, at osmotic equilibrium. This makes the calculations of the chemical potential efficient. The quahty of the results is similar to those from the grand canonical Monte Carlo simulation. The method is neither restricted to dimerization nor to spherically symmetric associative interactions. Even in the presence of higher-order complexes in large amounts, the proposed approach remains successful. 

The canonical ensemble was developed as the appropriate description of a system in thermal equilibrium with its surroundings by free exchange of energy. Following the discussion of classical systems the density operator of the canonical ensemble is introduced axiomatically as... 

The YBG equation is a two point boundary value problem requiring the equilibrium liquid and vapor densities which in the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T. If we accept the applicability of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the interfacial contribution to the free energy, through the condition of mechanical stability, and consequently, the droplet size dependence of the surface tension must be obtained. 

Now that the entanglement of the XY Hamiltonian with impurities has been calculated at Y = 0, we can consider the case where the system is at thermal equilibrium at temperature T. The density matrix for the XY model at thermal equilibrium is given by the canonical ensemble p = jZ, where = l/k T, and Z = Tr is the partition function. The thermal density matrix is diag-... 

Remark. We assumed that Y(t) is a Markov process. Usually, however, one is interested in materials in which a memory effect is present, because that provides more information about the microscopic magnetic moments and their interaction. In that case the above results are still formally correct, but the following qualification must be borne in mind. It is still true that p y0) is the distribution of Y at the time t0, at which the small field B is switched off. However, it is no longer true that this p(y0) uniquely specifies a subensemble and thereby the future of Y(t). It is now essential to know that the system has aged in the presence of B + AB, so that its density in phase space is canonical, not only with respect to Y, but also with respect to all other quantities that determine the future. Hence the formulas cannot be applied to time-dependent fields B(t) unless the variation is so slow that the system is able to maintain at all times the equilibrium distribution corresponding to the instantaneous B(t). 

It is assumed that the system is in a state of thermodynamic equilibrium at temperature T prior to the application of the forces Fj. j5( —oo) must consequently be the canonical density matrix, p0,... 

The next step is the choice of the basis set of these operators. The choice of the basis set depends on the ensemble. Kadanoff and Swift have considered a grand canonical ensemble with the temperature, chemical potential, and the velocity as the equilibrium parameters. The states that describe a situation where these equilibrium parameters vary slowly are the local equilibrium states. Linear combinations of the densities of the conserved operators acting on the equilibrium state ( ), where L ) =0) provide the desired local... 

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-... 

Let us consider a system in equilibrium, described in the absence of external perturbations by a time-independent Hamiltonian Ho. We will be concerned with equilibrium average values which we will denote as (...), where the symbol (...) stands for Trp0... with p0 = e H"/ Vre the canonical density operator. Since we intend to discuss linear response functions and symmetrized equilibrium correlation functions genetically denoted as Xba(, 0 and CBA t,t ), we shall assume that the observables of interest A and B do not commute with Ho (were it the case, the response function %BA(t, t ) would indeed be zero). This hypothesis implies in particular that A and B are centered A) =0,... 

The transport of a sub-critical Lennard-Jones fluid in a cylindrical mesopore is investigated here, using a combination of equilibrium and non-equilibrium as well as dual control volume grand canonical molecular dynamics methods. It is shown that all three techniques yield the same value of the transport coefficient for diffusely reflecting pore walls, even in the presence of viscous transport. It is also demonstrated that the classical Knudsen mechanism is not manifested, and that a combination of viscous flow and momentum exchange at the pore wall governs the transport over a wide range of densities. 

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