Equation (9) is the same as calculating a Feynman amplitude in the frame of generalized Dyck paths; that is, one sums overall generalized Dyck paths of length 2N and to each whole path is then attributed a weight which is given by the

Boltzmann factor.

where [[beta].sub.E]([omega]) = 1/[T.sub.E]([omega]) and exp[- [[beta].sub.E]([omega])[omega]] is the effective Boltzmann factor. From the Hawking temperature [T.sub.H] = ((d- 1)/4[pi])([r.sub.h]/[R.sup.2]) and (8), we can introduce the effective mass [M.sub.E] = [[2(d-1)].sup.d-1] [M.sup.d]/[[2(d-1)M+[omega]].sup.d-1] and, then, we Taylor expand [M.sub.E] in [omega]/M.

and the leading term gives the thermal Boltzmann factors for the emanating radiation.

THE BOLTZMANN FACTOR: VAPOR PRESSURE AND A MECHANICAL ANALOG.

How does this binding energy compare to the thermal energy in the Boltzmann factor? At room temperature [k.sub.B]T [congruent to] 0.025 eV, so we note that [l.sub.v] [much greater than] [k.sub.B]T, which shows via the Boltzmann factor that the fraction of molecules escaping the liquid (i.e., the fraction of molecules having energy equal to or greater than [l.sub.v] is small.

In the case of a gas of particles, [beta] is the

Boltzmann factor associated to the temperature, H is the Hamiltonian (a function describing the energy) of the i-th microstate, [N.sub.ij] is the number of particles of the j-th species in the i-th microstates and [[mu].sub.ij] is the chemical potential describing the energy related to exchanges of particles.

According to Arrhenius equation R decrease is determined by the

Boltzmann factor [[exp.sup.(-Ea/fcBT)]] and becomes significant as [E.sub.a] exceeds [k.sub.B]T.

The rate of SCG is thermally activated as governed by the

Boltzmann factor.

However, if the energy is higher, the

Boltzmann factor (i.e., [e.sup.-[DELTA]Ui[right arrow]i+1/kT]) will be calculated and compared with a random number, generated by the computer, in the range of 0 and 1 to determine whether the new configuration could be accepted or not.

The rate of this process is proportional to the 12 K phonon density which is strongly suppressed by the

Boltzmann factor [e.sup.-E*/kT] and is negligibly small when the temperature of the helium bath is less than 500 mK.

For each of the ring polymers, we evaluate the corresponding average potentials [[bar.U].sub.2] and [bar.U] and accumulate their

Boltzmann factors to calculate the effective potentials for the given configuration according to Eqs.