Microcanonical partition functions

If Q (E) is differentiable in the ordinary sense the partition function of a generalized ensemble with m intensive parameters is the m-fold Laplace transform of the microcanonical partition function e, ... 

The treatments of Flory,93 Gibbs and Di Marzio,91 and Milchev94 differ in the way they calculate the second factor ftnter- This microcanonical partition function describes the number of ways in which the K chains can be put on the lattice,... 

When treating polyatomics it is convenient to define an average molecular partition function, In = (lnQ)/N, for an assembly of N molecules. In the dilute vapor (ideal gas) this introduces no difficulty. There is no intermolecular interaction and In = (In Q)/N = ln(q) exactly (q is the microcanonical partition function). In the condensed phase, however, the Q s are no longer strictly factorable. Be that as it may, continuing, and assuming In = (In Q)/N, we are led to an approximate result which is superficially the same as Equation 5.10,... 

The set of independent variables (N, V, T) defines the canonical partition function. Systems defined by the number of molecules, the total energy, and the volume (N, E, V) lead to the microcanonical partition function, and systems specified by N, the temperature, and a pressure, namely (N, p, T), lead to the isothermal-isobaric partition function, denoted A. 

Micro-canonical ensemble fiCE (each system has constant N, V, and U the walls between systems are rigid, impermeable, and adiabatic each system keeps its number of particles, volume, and energy, and it trades nothing with neighboring systems). The relevant partition function is the microcanonical partition function Cl ( N, V, U) ... 

This system is illustrated in Figure 12.2. For a given microstate i and corresponding energy Ei of the subsystem, the reservoir can obtain Qr(E0 — Ei) microstates, where Qr is the microcanonical partition function of the heat reservoir. For each state i obtained by the subsystem, the total number of states available to the composite system is enumerated by 2r, the partition function for the reservoir. According to our standard assumption that the probability of a state is proportional to the number of microstates available to the system ... 

For a system of non-interacting monatomic particles (an ideal gas) the microcanonical partition function is proportional to VN. Based on Q, VN, we can derive the state equation known as the ideal gas law ... 

In order to evaluate the canonical partition function Q for a gas, we shall consider the system to be composed of an aggregate of essentially independent particles (molecules). As we shall see later, a crystal may be considered to a good approximation as an aggregate of independent harmonic oscillators. Each of these has its own microcanonical partition function ... 

For a one-component ideal gas, the microcanonical partition function for an individual molecule is q Therefore, under all ordinary conditions, we may write for a gas... 

We will now develop expressiorts for the microcanonical partition functions q and q to substitute into Eqs. (10) and (13). 

The interpretation of this result is that to obtain the canonical energy of the system, one computes the thermodynamic derivative with respect to P of the extended-system microcanonical partition function and adds back k T/2 for the thermostat kinetic energy and a constant E, which merely changes the energy scale. Similarly, the heat capacity at constant volume is given by... 

Finally, the microcanonical partition function, based on the kinetic energy constraint and the compressibility, can be written using the approach outlined in Sect. 3 as ... 

In the present context pe E,, N ) is the microcanonical partition function—a smn over the im-normalized probabilities. This function is in turn directly related to the system entropy... 

The invariant measure of the whole level set Eb is defined as the microcanonical partition function... 

The high epoxidation selectivity of cyclooctene was investigated computationally. Four different chiral conformations and two enantiomeric forms of each conformation were identified. Ring inversion further increased the degeneracy, yielding a total of 16 conformers. An evaluation of microcanonical partition functions quantified the entropy contributions and therefore the equilibrium composition at different temperatures could be calculated. The results suggested that the high epoxidation selectivity for cyclooctene is related to a poor (Tc-ati c-c orbital overlap in the predominant conformation. ... 

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