Partition functions description

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Several methods have been developed for the quantitative description of such systems. The partition function of the polymer is computed with the help of statistical thermodynamics which finally permits the computation of the degree of conversion 0. In the simplest case, it corresponds to the linear Ising model according to which only the nearest segments interact cooperatively149. The second possibility is to start from already known equilibrium relations and thus to compute the relevant degree of conversion 0. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

The theories of hydration we have developed herein are built upon the potential distribution theorem viewed as a local partition function. We also show how the quasi-chemical approximations can be used to evaluate this local partition function. Our approach suggests that effective descriptions of hydration are derived by defining a proximal... 

The mesoscopic description is introduced by defining functions 4> (q) and 4>B(q) that have the meaning of averaged over some mesoscopic volume values of the microscopic concentration operators. The conditional partition function, Z(4>t) (y =A,B), is the partition function for the system subject to the constraint that the microscopic operators 4>T(q) are fixed at some prescribed... 

Subject areas These are broad areas of usage or function that help partition the description of the system behavior so that one area can be modeled and specified somewhat separately from the others. They need not correspond to separately implemented components that would be an internal view. 

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

The most accurate theories of reaction rates come from statistical mechanics. These theories allow one to write the partition function for molecules and thus to formulate a quantitative description of rates. Rate expressions for many homogeneous elementary reaction steps come from these calculations, which use quantum mechanics to calculate the energy levels of molecules and potential energy surfaces over which molecules travel in the transition between reactants and products. These theories give... 

So far, the effects of the chain ends were neglected in our stochastic model for the restricted chain. Therefore, n must be much larger than the number of steps needed to form the largest excluded polygon. The partition function, which incorporates the chain-end effects and which could be also employed for exact statistical description of short non-self-intersecting chains can be obtained as follows Assume, as before, that we eliminate only lowest-order polygons of t steps. Therefore, the first t — 1 steps in the chain are described as a sequence of independent events. Eq (9), then, will be replaced by... 

As seen from our discussion in Chapter 3, which dealt with onedimensional problems, in many relevant cases one actually does not need the knowledge of the behavior of the system in real time to find the rate constant. As a matter of fact, the rate constant is expressible solely in terms of the equilibrium partition function imaginary-time path integrals. This approximation is closely related to the key assumptions of TST, and it is not always valid, as mentioned in Section 2.3. The general real-time description of a particle coupled to a heat bath is the Feynman-Vernon... 

It is often impossible to obtain the quantized energies of a complicated system and therefore the partition function. Fortunately, a classical mechanical description will often suffice. Classical statistical mechanics is valid at sufficiently high temperatures. The classical treatment can be derived as a limiting case of the quantum version for cases where energy differences become small compared with ksT. 

In addition to being a function of T, the partition function is also a function of V, on which the quantum description of matter tells us that the molecular energy levels, , depend. Because, for single-component systems, all intensive state variables can be written as functions of two state variables, we can think of q(T, V) as a state function of the system. The partition function can be used as one of the independent variables to describe a single-component system, and with one other state function, such as T, it will completely define the system. All other properties of the system (in particular, the thermodynamic functions U, H, S, A, and G) can then be obtained from q and one other state function. 

From a statistical thermodynamic standpoint, the description of the folding/unfolding equilibrium in proteins requires the specification of the system partition function, Q defined as the sum of the statistical weights of all the possible states of the molecule (see Freire and Biltonen, 1978a) ... 

According to the hierarchical approach, the number of states that need to be considered in the partition function is 2"cu, where ncu is the number of cooperative folding units. In order to develop a complete description of a system composed of ncu interacting cooperative folding units it is necessary to evaluate the intrinsic energetics of each... 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

Lattice models play a central role in the description of polymer solutions as well as adsorbed polymer layers. All of the adsorption models reviewed so far assume a one-to-one correspondence between lattice random-walks and polymer configurations. In particular, the general scheme was to postulate the train-loop or train-loop—tail architecture, formulate the partition function, and then calculate the equilibrium statistics, e.g., bound fraction, average loop... 

It is interesting that Eq. (4.98), p. 97, offers a discrete-state partition function for the description of the inner-sphere contribution to the thermodynamics. But the discrete coordinate is an occupation number for a precisely defined configurational region, and parameters required for this discrete-state partition function are obtained by molecular-level calculations. Therefore, molecular realism isn t the first casualty of these theories, although strong approximations are typically accumulated after the formulation of quasi-chemical theories. 

The minimization of the canonical transition state partition function as in Eq. (2.13) is generally termed canonical variational RRKM theory. This approach provides an upper bound to the more proper E/J resolved minimization, but is still commonly employed since it simplifies both the numerical evaluation and the overall physical description. It typically provides a rate coefficient that is only 10 to 20% greater than the E/J resolved result of Eq. (2.11). 

To calculate the static thermodynamic and molecular ordering properties of a system of molecules, the configurational partition function Qc of the system must be derived. Qc does not contain the kinetic energy, intramolecular and intermolecular vibrations, and very small rotations about molecular bonds. Qc does contain terms which deal with significant changes in the shapes of the molecules due to rotations about semiflexible bonds (such as about carbon-carbon bonds in n-alkyl [i.e., (-CH2-)X] sections) in a molecule. For mathematical tractability in deriving Qc, the description of the molecules in continuum space is mapped onto a... 

The equilibrium properties of an adsorbed layer can be examined based on the chemical or electrochemical potentials of the constituents of this layer and the equilibrium equations derived in the section above. This is the simplest approach, although problems might appear in the description of the adsorbed layer properties during a surface phase transition [18]. Alternatively, the chemical potentials may be used for the determination of the grand ensemble partition function of the adsorbed layer, which in turn is used for the derivation of the equilibrium equations. This approach is mathematically more complex, but it leads to a better description of an adsorbed layer when it undergoes a phase transformation [18]. The present analysis for simplicity is restricted to the first approach. 

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