Chemical potential average

Fluctuations of observables from their average values, unless the observables are constants of motion, are especially important, since they are related to the response fiinctions of the system. For example, the constant volume specific heat of a fluid is a response function related to the fluctuations in the energy of a system at constant N, V and T, where A is the number of particles in a volume V at temperature T. Similarly, fluctuations in the number density (p = N/V) of an open system at constant p, V and T, where p is the chemical potential, are related to the isothemial compressibility iCp which is another response fiinction. Temperature-dependent fluctuations characterize the dynamic equilibrium of themiodynamic systems, in contrast to the equilibrium of purely mechanical bodies in which fluctuations are absent. 

The chemical potential of particles belonging to species a and (3 is measured by using the classical test particle method (as proposed by Fischer and Heinbuch [166]) in parts II and IV of the system i.e., we calculate the average value of (e) = Qxp[—U/kT]), where U denotes the potential energy of the inserted particles. 

Abbreviations N is the total number of particles Pdim is the average density of dimers in each of the parts I and III is the average density of monomers in each of the parts II and IV p is the average density at the middle of parts II and IV this value has been used to calculate the excess chemical potential from Eq. (148). All remaining symbols are explained in the text. 

FIG. 21 Dependence of the average density on the configurational chemical potential. The solid line denotes the grand canonical Monte Carlo data, the long dashed fine corresponds to the osmotic Monte Carlo results for ZL = 40, and the dotted line for ZL = 80. (From Ref. 172.)... 

Figure 5.18. Schematic representation of the density of states N(E) in the conduction band and of the definitions of work function d>, chemical potential of electrons p, electrochemical potential of electrons or Fermi level p, surface potential x> Galvani (or inner) potential

Probe methods like particle insertion and test particle methods (29-32) are quite useful for computing chemical potentials of constituent particles in systems with low densities. Test particles are randomly inserted the average Boltzmann factor of the insertion energy yields the free energy. For dense systems these methods work poorly because of the poor statistics obtained. 

The temperature, pore width and average pore densities were the same as those used by Snook and van Megen In their Monte Carlo simulations, which were performed for a constant chemical potential (12.). Periodic boundary conditions were used In the y and z directions. The periodic length was chosen to be twice r. Newton s equations of motion were solved using the predictor-corrector method developed by Beeman (14). The local fluid density was computed form... 

With applications to protein solution thermodynamics in mind, we now present an alternative derivation of the potential distribution theorem. Consider a macroscopic solution consisting of the solute of interest and the solvent. We describe a macroscopic subsystem of this solution based on the grand canonical ensemble of statistical thermodynamics, accordingly specified by a temperature, a volume, and chemical potentials for all solution species including the solute of interest, which is identified with a subscript index 1. The average number of solute molecules in this subsystem is... 

Note that the additional factor within the average, the n j (1 — bj), would be zero for any solvent configuration in which a solvent molecule is found in the inner shell. Thus, this expression involves a potential distribution average under the constraint that no binding in the inner shell is permitted. We can formally write the full expression for the excess chemical potential as... 

The overlap distribution method is more reliable than the direct averaging or single-direction FEP/NEW calculation. For example, it is capable of producing a good estimate of the chemical potential of dense fluids where the latter methods fail... 

The thermodynamic chemical potential is then obtained by averaging the Boltzmann factor of this conditional result using the isolated solute distribution function Sa Sn). Notice that the fluctuation contribution necessarily lowers the calculated free energy. 

We now use a trick to partition this exact expression for the chemical potential into classical and quantum correction parts [29]. To do this we multiply and divide inside the logarithm of the excess term by the classical average... 

Once the QFH formula for the excess chemical potential is linearized in (11.37), the logarithmic expression can be expanded to first order and all or part of the classical-average term can be integrated by parts to yield the Wigner-Kirkwood correction to the free energy. Then if (11.40) is reorganized, computation of the chemical potential can be viewed as a classical average with a modified interaction potential of the same form as (11.3). 

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