Boltzmann probability

This result comes from the idea of a variational rate theory for a diffusive dynamics. If the dynamics of the reactive system is overdamped and the effective friction is spatially isotropic, the time required to pass from the reactant to the product state is expected to be proportional to the integral over the path of the inverse Boltzmann probability. 

Consider the free energy difference, AA ( ), obtained from sampling the system with the non-Boltzmann probability density p C), given by ... 

The angle bracket denotes that the configurational integral is taken over the initial state. The conformational sampling indicated by Equation 4 is generated according to the Boltzmann probability associated with the initial state potential. As discussed in Section 2.1, convergence of conformational... 

What we do instead is to adopt the famous Metropolis algorithm, in which the generation of configurations is biased towards those that make the most significant contribution to the integral. The method generates states with a probability proportional to exp(- (Z)/kBT) (equal to their Boltzmann probability) and then counts each of them equally. 

At this deeper level, Boltzmann probabilities are geometry ... 

Let us now see how the metric geometrical properties of Boltzmann probabilities are inherited by the final thermodynamic properties. From (13.70a, b), we can write for a and its first and second differentials... 

There are numerous approaches to modeling the jump rate.1 Below, three progressively more realistic models are presented. All three approaches produce the same basic result—the jump rate is a product of the vibration frequency in the initial stable site and a Boltzmann probability of a sufficient energy fluctuation for the jump. 

Diagonal matrix elements of the general expression, that is, Boltzmann probabilities. 

Figure 1.1 Illustration of the Boltzmann probability law of Equations (1.6) and (1.7). The state probability distribution is plotted at two different temperatures for a system with ten possible microstates with energy ranging from 10 21 to 10-20 joules. At the lower temperature (T = 273 K), the lower-energy states are significantly more probable than the higher-energy states. At the higher temperature (T = 1000 K), the energy distribution becomes more uniform than at the lower temperature.

AB, corresponding to the left and right sides of Equation (10.14). If the probability of complex existing (the relative amount of time the system is found in the AB state) is p, then the Boltzmann probability law says that the energy difference between the two possible states is —ksT In As we shall see, the probability p is a function of V, the volume of the system. We define Ka = which will be shown to be related to (but not equal to) the equilibrium constant for the reaction. 

As is apparent in the figure, a minimum in energy occurs where the molecules A and B are separated by tq. Conformations in or near this energy correspond well to the AB complex in Equation (10.14). Values of r that are sufficiently greater than ro correspond to the A + B uncomplexed state. To make this definition precise, we note that the probability of the distance between A and B, RAB, is directly related to U(r) following the Boltzmann probability law ... 

Gill (1972) was the first to suggest that charge transport in polymers occurred by polaron hopping. The application of polaron theory to transport in polymers was first described by Sahvun (1984). Schein et al. (1990), and Schein (1992). The models described by Sahvun and Schein and coworkers lead to a mobility that is a product of a Boltzmann probability of energy coincidence and the probability a carrier will hop to an adjacent site by thermal activation once... 

According to the normal equations of classical statistical mechanics, which are used to express velocities and momenta distributed in three dimensions, the probability that the location of a hole is between x and x + dx, y and y + dy, z and z + dz that its translational momenta lie between and + dp, Py and Py + dpy, p and p + dp that its breathing momentum is between p and p + dp/, and, finally, that its radius is between r and r+ dr, is proportional to the Boltzmann probability factor,... 

Glucose is a system in which the conformational average of multiple low energy isomers cannot be ignored. That is, the free energy of solvation is a Boltzmann probability-weighted average over conformations ... 

Another way of statingthis is that the probability Pi of a particular configuration Ni is proportional to its Boltzmann probability divided by the Boltzmann probability of all the other configurations or states ... 

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