Perturbation theory, thermodynamic

To proceed we employ a perturbational approadi that enables us to calculate (the absolute value of) the grand-potential density. The key idea in any perturbation theory is to somehow link properties of the system of interest to those of a reference system whcjse properties can readily be calculated. One then needs to perturb the reference system in a controllable manner so that one switches continuously from the known reference system to the system of interest assuming that one can obtain system properties at any instant of the applied perturbation. 

Let us apply this general philo.sophy to the grand potential for the system depicted schematically in Fig. 5.12. Clearly, Eq. (1.66) represents the (exact differential of the) grand potential for the situation shown in Fig. 5.12. Because the fluid substrate potential in Eq. (5.83) depends on the (vector) position r of a fluid molecule rather than a subset of Cartesian coordinates, it is immediately clear- that we are dealing with a fluid that is inhomogeneous in all three spatial directions. According to our discussion in Section 1.6.1, 

However, we also notice from Eq. (5.83) that in the absence of the switching function, the remaining fluid substrate potential would depend only on the x-coordinate of a fluid molecule. In this case, fluid properties would be translationally invariant in the x- and y-directions. Hence, in this case, the exact differential of the grand potential would be given by Eq. (1.63), where this higher symmetry of the confined fluid has alread been exploited. As a consequence we obtain a closed expression for the grand potential [see Eq. (1.65)1 in terms of the tran.sverse stress T as we show in Section 1.6.1. 

The discussion at the beginning of this section therefore suggests taking the fluid confined between undecorated stirfaces as a reference system. For the reference system, we may derive a molecular expression for Tyy = r = r, where 

Following the general philosophy of perturbation theory, we replace the fluid substrate potential for the system of inter st by 

The most general approach to the statistical-mechanical problems considered in this chapter is to evaluate the appropriate partition functions for the systems of interest. Given the quantum-mechanical energies, Ek, of the states of the entire system, the partition function has the form 

In simulations, an alternative to perturbation theory is umbrella sampling. It is used to connect the two configurations of interest (e.g., a protein plus a bound ligand in solution versus a protein in solution and a free ligand in solution see Fig. 10(b)) by an appropriate configurational coordinate. Since the calculations correspond exactly to those used with a reaction coordinate in activated dynamics (Chapt. IV.E) we do not repeat the description of the umbrella sampling method.,70a 

The central idea of thermodynamic perturbation theory is that the potential energy function can be partitioned in a convenient way i.e., one can write 

To introduce the formulation, we consider the exact connection between the unperturbed and perturbed systems. We focus on the Helmholtz free energy, A, which is the quantity of interest at constant N, T, and V, where N is the number of particles, T is the temperature, and V is the volume of the system the alternative case (constant N, T, and P), which leads to the Gibbs free energy, can be treated similarly. The Helmholtz free energy for the potential energy function V(r A) can be written in terms of the partition function Zxas 

Equation 58 can be interpreted as the Boltzmann factor for the perturbation averaged over the unperturbed (reference) system this is indicated in the expression on the left-hand side of the equation by the angular brackets with the subscript 0. Introducing Eq. 57 into Eq. 53, we obtain 

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Larsen B, Rasaiah J C and Stell G 1977 Thermodynamic perturbation theory for multipolar and ionic fluids Mol. Phys. 33 987... 

Wertheim M S 1987 Thermodynamic perturbation theory of polymerization J Chem. Phys. 87 7323... 

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... 

Finally, the associative term is computed by using generalizing thermodynamic perturbation theory. One then obtains [38]... 

Tobias, D. J. Brooks III, C. L., Calculation of free energy surfaces using the methods of thermodynamic perturbation theory, Chem. Phys. Lett. 1987,142, 472-476... 

Thermodynamic perturbation theory represents a powerful tool for evaluating free energy differences in complex molecular assemblies. Like any method, however, FEP has limitations of its own, and particular care should be taken not only when carrying out this type of statistical simulations, but also when interpreting their results. We summarize in a number of guidelines the important concepts and features of FEP calculations developed in this chapter ... 

This takes the conventional form of standard thermodynamic perturbation theory, but with the decisive feature that interactions with only one molecule need be manipulated. Here (.., )r indicates averaging for the case that the solution contains a distinguished molecule which interacts with the rest of the system on the basis of the function AUa, i.e., the subscript r identifies an average for the reference system. Notice that a normalization factor for the intramolecular distribution cancels between the numerator and denominator of (9.22). 

Free energy calculations rely on a well-known thermodynamic perturbation theory [6, 21, 22], which is recalled in Chap. 2. We consider a molecular system, described by the potential energy function U(rN), which depends on the coordinates of the N atoms rN = (n, r2,..., r/v). The system could be a biomolecule in solution, for example. We limit ourselves to a classical mechanical description, for simplicity. Practical calculations always consider differences between two or more similar systems, such as a protein complexed with two different ligands. Therefore, we consider a change in the system, such that the potential energy function becomes ... 

The exp-6 model is not well suited to molecules with large dipole moments. To account for this, Ree9 used a temperature-dependent well depth e(T) in the exp-6 potential to model polar fluids and fluid phase separations. Fried and Howard have developed an effective cluster model for HF.33 The effective cluster model is valid for temperatures lower than the variable well-depth model, but it employs two more adjustable parameters than does the latter. Jones et al.34 have applied thermodynamic perturbation theory to... 

Thermodynamic Perturbation Theory for Weakly Interacting Superparamagnets... 

Feg). Subsequently, thermodynamic properties of spins weakly coupled by the dipolar interaction are calculated. Dipolar interaction is, due to its long range and reduced symmetry, difficult to treat analytically most previous work on dipolar interaction is therefore numerical [10-13]. Here thermodynamic perturbation theory will be used to treat weak dipolar interaction analytically. Finally, the dynamical properties of magnetic nanoparticles are reviewed with focus on how relaxation time and superparamegnetic blocking are affected by weak dipolar interaction. For notational simplicity, it will be assumed throughout this section that the parameters characterizing different nanoparticles are identical (e.g., volume and anisotropy). 

We will consider dipolar interaction in zero field so that the total Hamiltonian is given by the sum of the anisotropy and dipolar energies = E -TEi. By restricting the calculation of thermal equilibrium properties to the case 1. we can use thermodynamical perturbation theory [27,28] to expand the Boltzmann distribution in powers of This leads to an expression of the form [23]... 

All results obtained below with the thermodynamic perturbation theory are limited to the case of axially symmetric anisotropy potentials (see the Appendix, Section A.2), and all explicit calculations are done assuming uniaxial anisotropy (see the Appendix, Section B). 

Figure 3.2. Equilibrium linear susceptibility in reduced units X = x Hi[/m) versus temperature for three different ellipsoidal systems with equation x ja +y lb + jc < I, resulting in a system of N dipoles arranged on a simple cubic lattice. The points shown are the projection of the spins to the xz plane. The probing field is applied along the anisotropy axes, which are parallel to the z axis. The thick lines indicate the equilibrium susceptibility of the corresponding noninteracting system (which does not depend on the shape of the system and is the same in the three panels) thin lines show the susceptibility including the corrections due to the dipolar interaction obtained by thermodynamic perturbation theory [Eq. (3.22)] the symbols represent the susceptibility obtained with a Monte Carlo method. The dipolar interaction strength is itj = d/ 2o = 0.02.

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

Because of the long-range and reduced symmetry of the dipole-dipole interaction, analytical methods such as the thermodynamic perturbation theory presented in Section II.B.l. will be applicable only for weak interaction. Numerical simulation techniques are therefore indispensable for the study of interacting nanoparticle systems, beyond the weak coupling regime. 

Thermodynamic perturbation theory is used to expand the Boltzmann distribution in the dipolar interaction, keeping it exact in the magnetic anisotropy (see Section II.B.l). A convenient way of performing the expansion in powers of is to introduce the Mayer functions fj defined by 1 +fj = exp( cOy), which permits us to write the exponential in the Boltzmann factor as... 

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