Boltzman factor

Let us consider an V-component fluid in a volume V, at temperature T, and at chemical potentials /r = mi, > Mv - The fluid is in contact with an impermeable solid surface. We assume that the fluid particles interact between themselves via the pair potential denoted by u pir), and interact with the confining surface via the potential (a,f3= 1,2,. ..,V). The potential v ir) contains a hard-wall term to ensure that the solid surface is impermeable. For the sake of convenience, the hard-wall term is assumed to extend into the bulk of the solid [46,47], such that the Boltzman factor (r), and the local density Pa r) are cutoff at a certain distance z = z, ... 

As before (see Chapter 3), an equilibrium constant K represents the chemical effects on free energy. Multiplying K by the Boltzman factor,... 

Further improvement of the centroid method came with the introduction of centroid dynamics.Here the fundamental idea is to construct a centroid Hamiltonian in the full phase space of the system and the bath. The Boltzman factor is then the one obtained from this centroid Hamiltonian while seal time dynamics is obtained by running classical trajectories. This method has been applied to realistic systems " and recently derived from first principles.244 The main advantage of the centroid methodology is that thermodynamic quantum effects can be computed numerically exactly as it is not too difficult to converge numerically the computation of the centroid potential. 

The average is taken over all possible configurations that the system of interest can assume (i.e. weighted by the Boltzman factor). Now, the displacement of a particle from its initial position, r, is where... 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

Due to the Boltzman factor, the population of this level is only 0.04 at 7 30 K. Thus, in a good approximation, only the ground doublet contributes to the magnetic moment mt A T) of the Er subsystem at lower temperatures and the temperature dependence of mEl follows the Brillouin function Bi/2(x)=tanh(x).-... 

In a static magnetic field, each unpaired electron-spin of a free radical precesses about the axis parallel or antiparallel to the magnetic field (z-direction). The quantum state of the electron spin is expressed by a or p corresponding to these precessions. The population of the P spin state is larger than that of the a state by a Boltzman factor under the thermal equilibrium condition. so that the spin system has a total magnetization along the z-axis. The x-and y-components of the magnetic moments of electron spins are cancelled out under the thermal equilibrium condition because of the incoherence of the precession motion. 

Since the intermolecular potential energy of a configuration of hard spheres is either zero or infinite, the Boltzman factor, exp(-pt/Af), is either one or zero and the configurational partition function is independent of temperature. Thus, the full behavior of this model is described by a single isotherm. 

In Eq. (3-51), and in a few equations which will follow the vibrational frequencies are not expressed as v in wavenumbers, but as co in Hz. In Eq. (3-51), A is the scattering cross-section of the scattering nucleus (since the H atom has a very large cross section, namely Ah = 82.5 barn, Ac = 5.5 barn, molecules containing hydrogen are very suitable for neutron experiments), kf and ko are the moduli of the wave-vectors of the scattered and incident neutron respectively, e " the Debye-Waller factor or temperature factor, K = kf - ko, M the mass of the unit cell, g(vibrational states, and P = h/2kBT (where ks and T are the Boltzman factor and the temperature respectively). 

It has been proposed that in the magnetization dynamics of singledomain particles there is a characteristic crossover temperature T below which the escape of the magnetization from the metastable states is dominated by quantum barrier transitions, rather than by thermal over barrier activation. Above T the escape rate is given by the rate of the thermal transitions, determined by the Boltzman factor, = 1/exp(—t/Z/cT), where U is the barrier separating two metastable states. In a thermally activated regime it should vanish when the temperature approaches zero. 

The Boltzman factor e is of the order of 20 for a phosphatidylinositol membrane under the conditions of these experiments. The values of Kr and Kr indicate that the conditions for the reaction in solution and at the interface are rather different. In solution, the association takes place in a homogeneous medium on the other hand, the complex formation at the interface is a heterogeneous reaction in which the ion comes from the aqueous phase and combines with a carrier molecule that is bound to the membrane. The detailed mechanism of this heterogeneous reaction is not clear, however, and therefore we cannot explain why the reaction at the membrane is so much slower than in solution. It is possible that the carrier molecule at the interface is stabilized in a conformation that is less favorable for complex formation. 

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