# Statistical Mechanical Averages

Statistical Mechanical Averages [c.312]

Statistical Mechanical Averages [c.312]

Statistical mechanical averages in a molecular dynamics run are obtained by simply averaging an energetic or structural value over time steps. Thus if the values x , i are being computed in a trajectory, the statistical mechanical average is just [c.312]

Statistical mechanical averaging is a fundamental aspect of molecular dynamics. While there may be many reasons for wanting to generate a molecular dynamics trajectory, using it to obtain statistical mechanics averages may be the most common. In the micro-canonical ensemble of constant volume and number of particles, the appropriate equilibrium macroscopic average of some variable x(r) depending on the configuration r of the particles is given by. [c.319]

A molecular dynamics (MD) calculation collects statistical information as it progresses. So, for example, if the calculated position vector of atom A at times ti,t2, -, tn is rA(f2), .., t-A(tn), then the statistical-mechanical average is [c.64]

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by [c.582]

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 [c.374]

According to statistical mechanics, for the canonical ensemble one may calculate (U), the average energy of all the members of the ensemble, while for the grand canonical ensemble one can calculate two averages, (N) and (U). Of crucial importance, however, is the probability of observing significant variations (fluctuations) from these averages in any particular member of the ensemble. Fortunately, statistical mechanics yields an answer to these questions. [c.376]

If one denotes the averages over a canonical distribution by (.. . ), then the relation A = U-T S and U = (W) leads to the statistical mechanical coimection to the themiodynamic free energy A [c.398]

Statistical mechanics and kinetic theory, as we have seen, are typically concerned with the average behaviour of an ensemble of similarly prepared systems. One usually hopes, and occasionally can demonstrate, that the variations of these properties from one system to another in the ensemble, or that the variation with time of the properties of any [c.687]

Colloidal particles can be seen as large, model atoms . In what follows we assume that particles with a typical radius <3 = lOO nm are studied, about lO times as large as atoms. Usually, the solvent is considered to be a homogeneous medium, characterized by bulk properties such as the density p and dielectric constant t. A full statistical mechanical description of the system would involve all colloid and solvent degrees of freedom, which tend to be intractable. Instead, the potential of mean force, V, is used, in which the interactions between colloidal particles are averaged over [c.2667]

Equilibrium average properties are calculated using a statistical weighting of the probability Pq(r) of Eq. (3) raised to the power of q as required by the generalized statistical mechanics. The so-called q-expectation value is written [c.199]

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is [c.199]

The average of the step function, using the action for a Boltzmann weight can be pursued by standard statistical mechanics. It may require more elaborate sampling techniques such as the Umbrella sampling [20]. [c.277]

There is an energy of interaction between solute and solvent. Because of this, the solute properties dependent on energy, such as geometry, vibrational frequencies, total energy, and electronic spectrum, depend on the solvent. The presence of a solvent, particularly a polar solvent, can also stabilize charge separation within the molecule. This not only changes the energy, but also results in a shift in the electron density and associated properties. In reality, this is the result of the quantum mechanical interaction between solute and solvent, which must be averaged over all possible arrangements of solvent molecules according to the principles of statistical mechanics. [c.206]

Subsequent to equilibration, averages over the trajectory can be accumulated to describe statistical mechanical properties. For example, to calculate an average bond length, the bond should first be selected, prior to collecting molecular dynamics data or playing back snapshots, and made a named selection with the Select/Name Selection menu item. Then, the named selection should be placed in the Average only or Avg. graph column of the Molecular Dynamics Averages dialog box invoked by the Averages button of the Molecular Dynamics Options dialog box. A molecular dynamics simulation will then average the bond length. The average may be viewed after the sampling by re-opening the [c.316]

In addition to being able to plot simple instantaneous values of a quantity x along a trajectory and reporting the average,

Molecular Thermodynamics, the Kinetic Theory of Gases, and Statistical Thermodynamics. Molecular thermodynamics attempts to explain observed physical properties via individual molecular properties (117). It uses classical thermodynamics, as well as concepts from statistical thermodynamics and chemical physics (118). The molecular view of thermodynamics can be traced back to 1738, when Bernoulli proposed that a gas consisted of individual molecules in constant motion, colliding with each other and with their containment vessels. During the nineteenth century. Maxwell and Boltzmann developed this into the kinetic theory of gases (119,120,121), which appHes the laws of mechanics to microsystems. From this theory expressions for the pressure of a gas, its internal energy, and its specific heat capacity can be derived (122). Statistical thermodynamics, or equihbrium statistical mechanics, provides the mathematical glue between the kinetic theory and macrosystems, ie, systems containing large numbers of molecules. It takes advantage of the fact that molecules ate very numerous and that average properties can be estimated through probabiUty and statistical analysis (122). [c.248]

An intriguing approach to the problem of statistical fragmentation is offered by the maximum entropy principle of statistical mechanics as recast in the more general terms of information theory (Jaynes, 1979 Weaire and Rivier, 1984). In the fracture of a body, physical forces which might bring about regularity in the fragment size or structure are assumed to be absent. Thus, fragmentation is totally random and the only constraints are that the process is space filling and that certain observable features such as the average fragment size or total surface area are satisfied. Such constraints serve as undetermined multipliers in a variational expression of the maximum entropy [c.311]

Equation (6.2) shows that the driving force increases almost linearly with decreasing temperature and we might well expect the growth speed to do the same. The decrease in growth rate below 24°C is therefore quite unexpected but it can be accounted for perfectly well in terms of the movements of molecules at the solid-liquid interface. We begin by looking at solid and liquid salol in equilibrium at T, . Then AG = 0 from eqn. (6.2). In other words, if a molecule is taken from the liquid and added to the solid then the change in Gibbs free energy, AG, is zero (see Fig. 6.4). Flowever, in order to move from positions in the liquid to positions in the solid, each molecule must first free itself from the attractions of the neighbouring liquid molecules specifically, it must be capable of overcoming the energy barrier q in Fig. 6.4. Due to thermal agitation the molecules vibrate, oscillating about their mean positions with a frequency v (typically about 10 s ). The average thermal energy of each molecule is 3/cT where k is Boltzmann s constant. But as the molecules vibrate they collide and energy is continually transferred from one molecule to another. Thus, at any instant, there is a certain probability that a particular molecule has more or less than the average energy 3/cT, . Statistical mechanics then shows that the probability, p, that a molecule will, at any instant, have an energy q is [c.59]

The average solvent structure caused by granularity, packing, and hydrogen bonding gives rise to important effects that are ignored by continuum electrostatic approaches. Statistical mechanical theories based on distribution functions and integral equations are sophisticated approaches that can provide a rigorous framework for incorporating such effects into a description of solvation [3,76]. A complete review of integral equations would be beyond the scope of this chapter therefore we provide only a brief overview of this vast field. [c.144]

About 1902, J. W. Gibbs (1839-1903) introduced statistical mechanics with which he demonstrated how average values of the properties of a system could be predicted from an analysis of the most probable values of these properties found from a large number of identical systems (called an ensemble). Again, in the statistical mechanical interpretation of thermodynamics, the key parameter is identified with a temperature, which can be directly linked to the thermodynamic temperature, with the temperature of Maxwell s distribution, and with the perfect gas law. [c.3]

The partition function Z is given in the large-P limit, Z = limp co Zp, and expectation values of an observable are given as averages of corresponding estimators with the canonical measure in Eq. (19). The variables and R ( ) can be used as classical variables and classical Monte Carlo simulation techniques can be applied for the computation of averages. Note that if we formally put P = 1 in Eq. (19) we recover classical statistical mechanics, of course. [c.93]

The rotational isomeric state (RIS) model assumes that conformational angles can take only certain values. It can be used to generate trial conformations, for which energies can be computed using molecular mechanics. This assumption is physically reasonable while allowing statistical averages to be computed easily. This model is used to derive simple analytic equations that predict polymer properties based on a few values, such as the preferred angle [c.308]

Statistical mechanical averages in a molecular dynamics run are obtained by simply averaging an energetic or structural value over time steps. Thus if the values (xj, i are being computed in a trajectory. the statistical mechanical average is jiist [c.312]

At this point it is important to make some clarifying remarks (1) clearly one caimot regard dr in the above expression, strictly, as a mathematical differential. It caimot be infinitesimally small, since dr much be large enough to contain some particles of the gas. We suppose instead that dr is large enough to contain some particles of the gas but small compared with any important physical lengtii in the problem under consideration, such as a mean free path, or the length scale over which a physical quantity, such as a temperature, might vary. (2) The distribution fiinction / (r,v,t) typically does not describe the exact state of the gas in the sense that it tells us exactly how many particles are in the designated regions at the given time t. To obtain and use such an exact distribution fiinction one would need to follow the motion of the individual particles in the gas, that is, solve the mechanical equations for the system, and then do the proper countmg. Since this is clearly impossible for even a small number of particles in the container, we have to suppose that / is an ensemble average of the microscopic distribution fiinctions for a very large number of identically prepared systems. This, of course, implies that kinetic theory is a branch of the more general area of statistical mechanics. As a result of these two remarks, we should regard any distribution fiinction we use as an ensemble average rather than an exact expression for our particular system, and we should be carefiil when examining the variation of the distribution with space and time, to make sure that we are not too concerned with variations on spatial scales that are of the order or less than the size of a molecule, or on time scales that are of the order of the duration of a collision of a particle with a wall or of two or more particles with each other. [c.666]

It is important to realize that MC simulation does not provide a way of calculating the statistical mechanical partition function instead, it is a method of sampling configurations from a given statistical ensemble and hence of calculating ensemble averages. A complete sum over states would be impossibly time consuming for systems consisting of more than a few atoms. Applying the trapezoidal rule, for instance, to the configurational part of 2 vT Mihails discretizing each atomic coordinate on a fine grid then the dimensionality of the integral is extremely high, since there are 3N such coordinates, so the total number of grid points is astronomically high. The MC integration method is sometimes used to estimate multidimensional mtegrals by randomly sampling points. This is not feasible here, since a very small proportion of all pomts would be sampled in a reasonable time, and very few, if any, of these would have a large enough Boltzmaim factor to contribute significantly to the partition fimction. MC simulation differs from such methods by sampling points in a noiumifomi way, chosen to favour the important contributions. [c.2256]

Subsequent to equilibration, averages over the trajectory can be accumulated to describe statistical mechanical properties, for example, to calculate an average bond length, the bond should first be selected, prior to collecting molecular dyriam ics data or playing back snapshots, and made a named selection with the Select/Nam e-Selection menu item. Then, the named selection shoii Id be placed in the. Average on ly or, Avg. graph column of the Molecular Dynam ics. Averages dialog box invoked by the Averages button of th e Molecular Dyii am ics Option s dialog box.. A m o I ecu la r dyn am ics si rn ulation will then average th e bond length. I he average may be viewed after the sampling by re-opening the [c.316]

Due to the noncrystalline, nonequilibrium nature of polymers, a statistical mechanical description is rigorously most correct. Thus, simply hnding a minimum-energy conformation and computing properties is not generally suf-hcient. It is usually necessary to compute ensemble averages, even of molecular properties. The additional work needed on the part of both the researcher to set up the simulation and the computer to run the simulation must be considered. When possible, it is advisable to use group additivity or analytic estimation methods. [c.309]

Statistical mechanics states that the macroscopic values of certain quantities, like the energy, can be obtained by ensemble averaging over a very large number of possible states of the microscopic system. In many realms of chemistry, these statistical averages are what computational chemistry requires for a direct comparison with experiment. Afundamental principle of statistical mechanics, the Ergodic Hypothesis, states that it is possible to replace an ensemble average by a time average over the trajectory of the microscopic system. Molecular dynamics thus allows you to compute a time average over a trajectory that, in principle, represents a macroscopic average value. These time averages are fundamental to the use of molecular dynamics. [c.311]

The atoms in a solid vibrate, or oscillate, about their mean positions, with a frequency v (typically about lO s" ). The crystal lattice defines these mean positions. At a temperature T, the average energy (kinetic plus potential) of a vibrating atom is 3kT where k is Boltzmann s constant (1.38 X lO Jatom" K" ). But this is only the average energy. As atoms (or molecules) vibrate, they collide, and energy is continually transferred from one to another. Although the average energy is 3kT, at any instant, there is a certain probability that an atom has more or less than this. A very small fraction of the atoms have, at a given instant, much more - enough, in fact, to jump to a neighbouring atom site. It can be shown from statistical mechanical theory that the probability, p, that an atom will have, at any instant, an energy q is [c.181]

This expression has a formal character and has to be complemented with a prescription for its evaluation. A priori, we can vary the values of the fields independently at each point in space and then we deal with uncountably many degrees of freedom in the system, in contrast with the usual statistical thermodynamics as seen above. Another difference with the standard statistical mechanics is that the effective Hamiltonian has to be created from the basic phenomena that we want to investigate. However, a description in terms of fields seems quite natural since the average of fields gives us the actual distributions of particles at the interface, which are precisely the quantities that we want to calculate. In a field-theoretical approach we are closer to the problem under consideration than in the standard approach and then we may expect that a simple Hamiltonian is sufficient to retain the main features of the charged interface. A priori, we have no insurance that it [c.806]

Ab-initio studies of alloy phase stability are usually based on the Ising-type Hamiltonian, whose parameters, often called the effective cluster interactions (ECI), serve as an input in determination of the equilibrium properties of the system via the methods of statistical mechanics. An alternative formulation, which completely avoids determination of ECIs by calculating the electronic structure and solving the statistical part of the problem in one step, is employed in the concentration-wave method. It is based on the grandcanonical potential fl configurationally averaged within the coherent potential approximation (CPA) and on a mean-field solution of the statistical part of the problem including usually also the Onsager cavity field. [c.39]

Let P a a ) be the probability of transition from state a to state a. In general, the set of transition probabilities will define a system that is not describ-able by an equilibrium statistical mechanics. Instead, it might give rise to limit cycles or even chaotic behavior. Fortunately, there exists a simple condition called detailed balance such that, if satisfied, guarantees that the evolution will lead to the desired thermal equilibrium. Detailed balance requires that the average number of transitions from a to a equal the number of transitions from a to a [c.328]

Another drawback to using Shannon information as a measure of complexity is the fact that it is based on an ensemble of all possible states of a system and therefore cannot describe the information content of a single state. Shannon information thus resembles traditional statistical mechanics - which describes the average or aggregate behavior of, say, a gas, rather than the motion of its constituent molecules - more so than it docs a complexity theory that must address the complexity of individual objects. [c.616]

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. [c.1180]

See pages that mention the term

**Statistical Mechanical Averages**:

**[c.319] [c.376] [c.1006] [c.230] [c.311] [c.318] [c.318] [c.301] [c.299] [c.1033] [c.2482]**

See chapters in:

** Hyperchem computation chemistry
-> Statistical Mechanical Averages
**