Factor Boltzmann

We have started considering the behavior of those systems consisting of an enormous number of particles, for some special cases. In particular, we have obtained the one-dimensional Boltzmann distribution. It is easy to generalize this approach to a system in three-dimensional space. Let dN be the number of particles near a point x, y, z in a volume dV -dxdydz. Keeping in mind eq. (3.2.4), it is possible to write 

This expression describes the relationship of molecules of ideal gas concentration and their coordinates in an external three-dimensional force field. It can be seen that the probability for the gas particles to be in volume dV around a point x, y, z is proportional to the exponential multiplier 

This defines its importance in chemical kinetics. The Boltzmann factor determines the exponential (temperature) part of Arrhenius formula for the ratios of the majority of chemical reactions. 

This factor is common to the solution of many problems in various areas of physics and chemistry. 

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The idea may be illustrated by considering first a method for increasing the acceptance rate of moves (but at the expense of trying, and discarding, several other possible moves). Having picked an atom to move, calculate the new trial interaction energy for a range of trial positions t = 1.. . k. Pick the actual attempted move from this set, with a probability proportional to the Boltzmann factor. This biases the move selection. 

From the potential energy, calculate the Boltzmann factor, exp(—iC(r )/cBT). 

Add the Boltzmann factor to the accumulated sum of Boltzmann factors and the potent energy contribution to its accumulated sum and return to step 1. 

The density of states increases rapidly with energy but the Boltzmann factor decrease exponentially, meaning that Pcanon(T, E) is bell-shaped, with values that can vary by mar orders of magnitude as the energy changes. In the multicanonical method the simulatic... 

The effect of temperature in Monte Carlo simulations is primarily to modulate the strength of intermolecular interactions, since temperature enters the simulation only through the Boltzmann factor exp(-AE/kT), where AErepresents a difference in potential... 

In this expression, cos 0 is the average value of cos 0 the weighting factor used to evaluate the average is given by the Boltzmann factor exp(-V /RT), where R is the gas constant in the units of and T is in degrees Kelvin. Note that the correction factor introduced by these considerations reduces to unity if... 

A Boltzmann factor in which the energy of crystallization appears in a negative exponent. According to Eq. (4.11), this energy increases-hence the exponential decreases-with increasing r. 

The statistical problem. The relative probability associated with the placement of the two coils such that d = °° is unity The solution is so dilute that we can place the polymer molecules anywhere. It is at smaller d s that the placement probability drops off because of a generally unfavorable AG associated with the overlap. We assume that the decrease in probability is described by the Boltzmann factor and write... 

Now we evaluate the probability of a fluctuation 6p in terms of a Boltzmann factor ... 

If the spectmm is observed in absorption, as it usually is, and at normal temperatures the intensities of the transitions decrease rapidly as v" increases, since the population of the uth vibrational level is related to Nq by the Boltzmann factor... 

There is considerable literature on material imperfections and their relation to the failure process. Typically, these theories are material dependent flaws are idealized as penny-shaped cracks, spherical pores, or other regular geometries, and their distribution in size, orientation, and spatial extent is specified. The tensile stress at which fracture initiates at a flaw depends on material properties and geometry of the flaw, and scales with the size of the flaw (Carroll and Holt, 1972a, b Curran et al., 1977 Davison et al., 1977). In thermally activated fracture processes, one or more specific mechanisms are considered, and the fracture activation rate at a specified tensile-stress level follows from the stress dependence of the Boltzmann factor (Zlatin and Ioffe, 1973). 

The sticking coefficient at zero coverage, Sq T), contains the dynamic information about the energy transfer from the adsorbing particle to the sohd which gives rise to its temperature dependence, for instance, an exponential Boltzmann factor for activated adsorption. 

This means that particle configurations where at least two particles overlap, i.e., have a distance r smaller than the diameter cr, are forbidden. They are forbidden because the Boltzmann factor contains a term, exp(—oo) 0, that leads to a vanishing statistical weight. Hence we have an ensemble of... 

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