Statistical thermodynamics ensemble

Let A = yc, , 1 < / < Nc, be conformations generated for C using a computational method. Because the global free energy minimum conformation is expected to statistically dominate the thermodynamic ensemble, the predicted binding activity for C is determined by (C)=min F y. ) = F(yf ). 

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... 

One of the most powerful tools molecular simulation affords is that of measuring distribution functions and sampling probabilities. That is, we can easily measure the frequencies with which various macroscopic states of a system are visited at a given set of conditions - e.g., composition, temperature, density. We may, for example, be interested in the distribution of densities sampled by a liquid at fixed pressure or that of the end-to-end distance explored by a long polymer chain. Such investigations are concerned with fluctuations in the thermodynamic ensemble of interest, and are fundamentally connected with the underlying statistical-mechanical properties of a system. 

Occasionally alternative expressions of the PDT (9.5) have been proposed [23-25], These alternatives arise from consideration of statistical thermodynamic manipulations associated with a particular ensemble, and the distinguishing features of those alternative formulae are relics of the particular ensemble considered. On the other hand, relics specific to an ensemble are not evident in the PDT formula (9.5). These alternative formulae should give the same result in the thermodynamic limit. 

The correction displayed is negligible relative to 1, in the macroscopic limit. The independence in the thermodynamic limit of the PDT on a choice of simulation ensemble used for statistical evaluation is a difference from the partition functions encountered in Gibbsian statistical thermodynamics. 

The molecular modelling of systems consisting of many molecules is the field of statistical mechanics, sometimes called statistical thermodynamics [28,29], Basically, the idea is to go from a molecular model to partition functions, and then, from these, to predict thermodynamic observables and dynamic and structural quantities. As in classical thermodynamics, in statistical mechanics it is essential to define which state variables are fixed and which quantities are allowed to fluctuate, i.e. it is essential to specify the macroscopic boundary conditions. In the present context, there are a few types of molecular systems of interest, which are linked to so-called ensembles. 

Monte Carlo Methods. Although several statistical mechanical ensembles may be studied using MC methods (2,12,14), the canonical ensemble has been the most frequently used ensemble for studies of interfacial systems. In the canonical ensemble, the number of molecules (N), cell volume (V) and temperature (T) are fixed. Hence, the canonical ensemble is denoted by the symbols NVT. The choice of ensemble determines which thermodynamic properties can be computed. 

The maximum entropy method (MEM) is an information-theory-based technique that was first developed in the field of radioastronomy to enhance the information obtained from noisy data (Gull and Daniell 1978). The theory is based on the same equations that are the foundation of statistical thermodynamics. Both the statistical entropy and the information entropy deal with the most probable distribution. In the case of statistical thermodynamics, this is the distribution of the particles over position and momentum space ( phase space ), while in the case of information theory, the distribution of numerical quantities over the ensemble of pixels is considered. 

We take a rather simple approach to the treatment of statistical thermodynamics and the derivations that follow. A rigorous discussion requires introduction of the canonical ensemble and is beyond the scope of our immediate interest. The details can be found, for example, in textbooks on statistical mechanics [269]. The approach taken here yields all of the needed results, and is a compact introduction to the theory. 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

Thermodynamics is based on the atomistic view, that is, that matter consists of elementary particles such as atoms and molecules that cannot be divided into smaller units. The three different states of matter are the result of the simultaneous interaction of a very large number, usually N = Na =6.02x 1023, of elementary particles. Thus, the macroscopic behavior of an ensemble of particles can be mathematically described as a state function that can be related to the individual behavior on a molecular scale, leading to the scientiLcally rigorous framework of statistical thermodynamics (Gcpel and Wiemhcfer, 2000). 

Consider an ensemble S of N objects, belonging to m classes (surface segment types in COSMOSPACE) of identical objects (surface segments in COSMOSPACE). Let N, be the number of objects of class i. Then x, =Ni/N is the relative concentration. Let there be N sites that can be occupied by the N objects. Each two of these sites form pairs. Hence, all objects occupying the N sites are paired. The interaction energy of the pairs will be described by a symmetric matrix, Ey, where i and j denote two different classes of objects. Let Z be the partition sum of the ensemble. From statistical thermodynamics, the chemical potential of objects of class i is given by... 

To illuminate the binding situation we can retreat to the statistical thermodynamics point of view. Experimentally accessible state functions (AG, AH, AS, ACp etc.) relate to the differences of average states in the ensembles of all constituents of the system including counterions, buffers, solvents etc. before... 

Statistical thermodynamics gives us the recipes to perform this average. The most appropriate Gibbsian ensemble for our problem is the canonical one (namely the isochoric-isothermal ensemble N,V,T). We remark, in passing, that other ensembles such as the grand canonical one have to be selected for other solvation problems). To determine the partition function necessary to compute the thermodynamic properties of the system, and in particular the solvation energy of M which we are now interested in, of a computer simulation is necessary [1],... 

Statistical thermodynamics has defined, in addition to the particle partition function z, the canonical ensemble partition function Zas follows ... 

In addition to these complications in interpreting H/D exchange data, it must be bom in mind that hydrogen exchange provides a static measure of protein flexibility proteins in solution exist as an ensemble of different conformations. The population of each conformation is determined by its Gibbs free energy according to the standard statistical thermodynamic relationship... 

Nonequilibrium statistical thermodynamics is comprised of different methods allowing the solution of nonequilibrium problems, like the Liouville equation (which is used mostly for gases) and some approaches employing ensembles. However, the complexity of systems with random fluxes causes serious difficulties when methods of nonequilibrium statistical thermodynamics are applied. 

The equipartition theorem, which describes the correlation structure of the variables of a Hamiltonian system in the NVT ensemble, is a central component of the held of statistical mechanics. Although the intent of this chapter is to introduce aspects of statistical thermodynamics essential for the remainder of this book -and not to be a complete text on statistical mechanics - the equipartition theorem provides an interpretation of the intrinsic variable T that is useful in guiding our intuition about temperature in chemical reaction systems. 

Surface tension is one of the most basic thermodynamic properties of the system, and its calculation has been used as a standard test for the accuracy of the intermolecular potential used in the simulation. It is defined as the derivative of the system s free energy with respect to the area of the interface[30]. It can be expressed using several different statistical mechanical ensemble averages[30], and thus we can use the molecular dynamics simulations to directly compute it. An example for such an expression is ... 

In real situations surface and volume changes are often made with systems that are at equilibrium with their environment, characterized by a set of chemical potentials p, rather than keeping In ] fixed, as in [2.2.7 and 8j. In other words, area changes in open systems are considered. In statistical thermodynamics the conversion from closed to open implies the transition from the canonical to the grand canonical ensemble. The characteristic function of the latter is nothing other than the sum of the bulk and surface mechemical work terms (see [1.3.3.12] and [I.A6.23D which are the quantities of interest ... 

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