Ensemble constraints

Environments with stable energetic stresses are frequently divided into nearly decoupled spatial or compositional subsystems. Tliis is true of quasi-stable energetic redox couples at hydrothermal vents and of tlie weakly coupled 6000 K spectrum of solar visible light and 300 K terrestrial tliemial black body [55]. Tlie separate components may constitute internally near-equilibrium subsystems, defined individually by simple ensemble constraints. 

Determine the probability density, p, or the partition function as a function of the Hamiltonian and the ensemble constraints. This is a taxing task, requiring elements of probability and combinatorial theory. 

It can be quickly verified that although the results in Eq. 4.61 and Eq. 4.82 are different by a factor hE/E, the derived thermodynamic properties of an ideal gas remain the same. This means that thermodynamic properties do not depend on the exact number of microscopic states, or more generally the actual value of the partition function. Instead, they depend on the functional dependence of the partition function on NVE properties and how it changes with the ensemble constraints. 

Again, aU that is needed to derive macroscopic thermodynamics from microscopic relations is the functional dependence of the partition function on ensemble constraints. 

The equivalence of ensembles becomes apparent at the thermodynamic limit. Regardless of the ensemble constraints, a system at equilibrium will attain one observable value for each thermodynamic state variable. 

Constraints. The ensemble constraints, whether NVE,NVT,yxPT, etc., must be defined. 

It is customary in statistical mechanics to obtain the average properties of members of an ensemble, an essentially infinite set of systems subject to the same constraints. Of course each of the systems contains the... 

Since other sets of constraints can be used, there are other ensembles and other partition functions, but these tliree are the most important. 

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... 

An explicit example of an equilibrium ensemble is the microcanonical ensemble, which describes closed systems with adiabatic walls. Such systems have constraints of fixed N, V and E < W< E + E. E is very small compared to E, and corresponds to the assumed very weak interaction of the isolated system with the surroundings. E has to be chosen such that it is larger than (Si )... 

Consider two systems in thennal contact as discussed above. Let the system II (with volume and particles N ) correspond to a reservoir R which is much larger than the system I (with volume F and particles N) of interest. In order to find the canonical ensemble distribution one needs to obtain the probability that the system I is in a specific microstate v which has an energy E, . When the system is in this microstate, the reservoir will have the energy E = Ej.- E due to the constraint that the total energy of the isolated composite system H-II is fixed and denoted by Ej, but the reservoir can be in any one of the R( r possible states that the mechanics within the reservoir dictates. Given that the microstate of the system of... 

Notice that those distribution functions that satisfy Eq. (4-179) still constitute a convex set, so that optimization of the E,R curve is still straightforward by numerical methods. It is to be observed that the choice of an F(x) satisfying a constraint such as Eq. (4-179) defines an ensemble of codes the individual codes in the ensemble will not necessarily satisfy the constraint. This is unimportant practically since each digit of each code word is chosen independently over the ensemble thus it is most unlikely that the average power of a code will differ drastically from the average power of the ensemble. It is possible to combine the central limit theorem and the techniques used in the last two paragraphs of Section 4.7 to show that a code exists for which each code word satisfies... 

As is well known, we can consider the ensemble of many molecules of water either at equilibrium conditions or not. To start with, we shall describe our result within the equilibrium constraint, even if we realize that temperature gradients, velocity gradients, density, and concentration gradients are characterizations nearly essential to describe anything which is in the liquid state. The traditional approaches to equilibrium statistics are Monte Carlo< and molecular dynamics. Some of the results are discussed in the following (The details can be found in the references cited). 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

Several techniques are available to calculate dJf /dQ. Ciccotti and coworkers [17, 19-27] have developed a technique, called blue-moon ensemble method or the method of constraints, in which a simulation is performed with fixed at some value. This can be realized by applying an external force, the constraint force, which prevents from changing. From the statistics of this constraint force it is possible to... 

Before we derive the appropriate expressions to calculate cL4/d from constrained simulations, we note an important difference between sampling in constrained and unconstrained simulations. There are two ways to gather statistics at (x) = . In unconstrained simulations, the positions are sampled according to exp —iiU while the momenta are sampled according to exp —j3K. If a constraint force is applied to keep fixed the positions are sampled according to A( (x) — x) exp —iiU. The momenta, however, are sampled according to a more complex statistical ensemble. Recall that... 

The Fukui function is primarily associated with the response of the density function of a system to a change in number of electrons (N) under the constraint of a constant external potential [v(r)]. To probe the more global reactivity, indicators in the grand canonical ensemble are often obtained by replacing derivatives with respect to N, by derivatives with respect to the chemical potential /x. As a consequence, in the grand canonical ensemble, the local softness sir) replaces the Fukui function/(r). Both quantities are thus mutually related and can be written as follows ... 

Figure 26. The proposed workflow of structural kinetic modeling Rather than constructing a single kinetic model, an ensemble of possible models is evaluated, such that the ensemble is consistent with available biological information and additional constraints of interest. The analysis is based upon a (thermodynamically consistent) metabolic state, characterized by a vector S° and the associated flux v° v(S°). Since based only on the an evaluation of the eigenvalues of the Jacobian matrix are evaluated, the approach is (computationally) applicable to large scale system. Redrawn and adapted from Ref. 296.

In the enumeration of chirality elements of flexible molecules all arrangements are taken into account which are permitted by the given constraints under the observation conditions. Here, one must always assume a rigid skeletal model and freely rotating ligandsF That arrangement for which the lowest number of chirality elements is found equal zero determines the number of chirality elements for the whole ensemble. 

In addition to the study of atomic motion during chemical reactions, the molecular dynamics technique has been widely used to study the classical statistical mechanics of well-defined systems. Within this application considerable progress has been made in introducing constraints into the equations of motion so that a variety of ensembles may be studied. For example, classical equations of motion generate constant energy trajectories. By adding additional terms to the forces which arise from properties of the system such as the pressure and temperature, other constants of motion have been introduced. 

---