Partition function infinite level system

Example Partition Function of a System with an Infinite Number of Levels... 

Hence, in the light of our both accounts of causality, the molecular dynamics model represents causal processes or chains of events. But is the derivation of a molecule s structure by a molecular dynamics simulation a causal explanation Here the answer is no. The molecular dynamics model alone is not used to explain a causal story elucidating the time evolution of the molecule s conformations. It is used to find the equilibrium conformation situation that comes about a theoretically infinite time interval. The calculation of a molecule s trajectory is only the first step in deriving any observable structural property of this molecule. After a molecular dynamics search we have to screen its trajectory for the energetic minima. We apply the Boltzmann distribution principle to infer the most probable conformation of this molecule.17 It is not a causal principle at work here. This principle is derived from thermodynamics, and hence is statistical. For example, to derive the expression for the Boltzmann distribution, one crucial step is to determine the number of possible realizations there are for each specific distribution of items over a number of energy levels. There is no existing explanation for something like the molecular partition function for a system in thermodynamic equilibrium solely by means of causal processes or causal stories based on considerations on closest possible worlds. 

We will now do the same for a system with an infinite number of equidistant energy levels, separated by Ae. For such a system the partition function becomes the following mathematical series with a well-known sum ... 

The above two examples illustrate that the value of the partition function is an indicator for how many of the energy levels are occupied at a particular temperature. At T = 0, where the system is in the ground state, the partition function has the value q = 1. In the limit of infinite temperature, entropy demands that all states are equally occupied and the partition function becomes equal to the total number of energy levels. 

Equation 17.54 is a useful conclusion. The (translational) partition function, originally defined as an infinite sum of negative exponentials of the energy levels, is equal to an expression in terms of the mass of the gas particles, the absolute temperature, the system volume, and some fundamental universal constants. This expression lets us calculate explicit values for q, which can then be used to determine values for energy, entropy, heat capacity, and so on. These calculated values—determined from a statistical rather than a phenomenological perspective—can then be compared to experimental values. We will thus get the first chance to see how well a statistical approach to thermodynamics compares with experiment. 

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