Boltzmann probability function

The Boltzmann probability function P can be written either in a discrete energy representation or in a continuous phase space formulation. 

Other ensembles may be used as well, and the perturbation equations have been derived for the isothermal, isobaric ensemble (constant T,P,N). An additional important relation is the Boltzmann probability function... 

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. 

In its more advanced aspects, kinetic theory is based upon a description of the gas in terms of the probability of a particle having certain values of coordinates and velocity, at a given time. Particle interactions are developed by the ordinary laws of mechanics, and the results of these are averaged over the probability distribution. The probability distribution function that is used for a given macroscopic physical situation is determined by means of an equation, the Boltzmann transport equation, which describes the space, velocity, and time changes of the distribution function in terms of collisions between particles. This equation is usually solved to give the distribution function in terms of certain macroscopic functions thus, the macroscopic conditions imposed upon the gas are taken into account in the probability function description of the microscopic situation. 

In contrast, the probability of activated flow is given by the Boltzmann distribution function in Eqn. 7-29 ... 

The search for the form of W of vulcanized rubbers was initiated by polymer physicists. In 1934, Guth and Mark2 and Kuhn3) considered an idealized single chain which consists of a number of links jointed linearly and freely, and derived the probability P that the end-to-end distance of the chain assumes a given value. The resulting probability function of Gaussian type was then substituted into the Boltzmann equation for entropy s, which reads,... 

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. 

This equation (Wilson, 1964) was developed on the basis of a liquid molecular model which assumes that interactions between molecules of two different components depend on the volume fraction of each component in the vicinity of a molecule of a given component. The volume fractions are determined from the probability of finding a molecule of one component or another in the vicinity of a molecule of the given component. The probabilities, and hence the volume fractions, are expressed through Boltzmann distribution functions of energy in terms of binary interaction parameters. The result for volume fractions is... 

The partition function can be considered as an average excited state number-operator, since it is the probability-weighted sum of energy states, each counted with a factor of 1. It may also be viewed as the normalization factor for the Boltzmann probability distribution. 

All possible contacts have been taken into account, and flie normalized Boltzmann probability distributions, as well as the Shannon information entropies, have been calculated for each pair of the molecules (i.e., AE, BE, CE, DE) as functions of temperature T within flie range from 100 K to 400 K. The results (Fig. 15.9) show that flie molecule A (wifli flie simplest structure) does not recognize molecule E at any temperature T under study. The ignorance (Shannon s entropy) of A with respect to E is of the order of 5 bits. Transforming molecule A into B improves the... 

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