# Ensemble average

It is a strong condition. For a system of particles, the infinitely long time must be much longer than O (e" ), whereas the usual observation time window is O (1). (When one writes y = O (x) and z = o (x), it implies that lim v/z = finite 0 and lim z/x = 0.) However, if by reasonable fiinctions one means large variables 0(N), then their values are nearly the same everywhere in the region of motion and the trajectory need not be truly ergodic for the time average to be equal to the ensemble average. Ergodicity of a trajectory is a difficult mathematical problem in mechanics. [c.387]

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if [c.387]

Next, let X. be either/>. or q., ( = ,...,/). Consider the ensemble average (x. d Wdx-) [c.390]

Consider an ensemble composed of constituents (such as molecules) per unit volume. The (complex) density operator for this system is developed perturbatively in orders of the applied field, and at. sth order is given by The (complex). sth order contribution to the ensemble averaged polarization is given by the trace over the eigenstate basis of the constituents of the product of the dipole operator, N and = Tr A pp [c.1182]

If individual scattering particles are far enough apart and their spatial distribution is such that the relative phases of their contributions to a scattered wave are random, the intensity distribution in the diffraction pattern will be the sum of contributions from all particles [5]. If the particles are identical (monodisperse) but have random orientations, or if they differ in size and shape (polydisperse), the resulting pattern will reflect an ensemble average over the sample. In either case there will be a spreading of the incident beam, so-called small-angle scattering. Flow small tire angles are depends on the wavelength of the radiation and the size of the particles long wavelengths give larger angles, but they also tend to be more strongly absorbed by the sample, so that there is a trade-off between resolution and intensity. [c.1364]

How long should we run This depends on die system and the physical properties of interest. Suppose that we are interested in a variable X, defined such that its ensemble average X= (x) = 0. (Here and tliroughout we use script letters for instantaneous dynamical variables, i.e., fiinctions of coordinates and momenta, to distinguish them from averages and thennodynamic quantities.) A characteristic time, i, may be defined, over which the correlations (/(O)x(t)) decay towards zero. The simulation mn time should be significantly longer than t. The time scales of properties of interest will vary from one system to another they may not be predictable in advance, and this will have a bearing on the length of simulation required. [c.2242]

The ensemble average here includes an unweighted average over inserted particle coordinates. In practice this means randomly inserting a test particle, many times, and averaging the Boltzmaim factor of the associated energy change. More details of free energy calculations will be given later. [c.2248]

MC simulation is a method of concentrating the sampled points in the important regions, namely the regions with high Boltzmaim factor e P a random walk is devised, moving from one point to the next, with a biasing probability chosen to generate the desired distribution. Unfortunately, a consequence of this approach is that it is no longer possible to estimate the partition fiinction itself, merely ratios of sums over states, that is, ensemble averages. Suppose that we have succeeded in selecting states y with probability proportional to (T) = exp -p7f(f)]. Then, if we have conducted observations or steps in the process, the ensemble average becomes an average over steps [c.2256]

In the calculation of ensemble averages, we correct for the weighting as follows [c.2258]

A consideration of the transition probabilities allows us to prove that microscopic reversibility holds, and that canonical ensemble averages are generated. This approach has greatly extended the range of simulations that can be perfonned. An early example was the preferential sampling of molecules near solutes [77], but more recently, as we shall see, polymer simulations have been greatly accelerated by tiiis method. [c.2259]

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. [c.2555]

The second problem is related to tire limitations in generating really long duration trajectories tliat can sample all tire relevant confonnational spaces of proteins. To observe reversible folding of even a moderate sized protein requires simulations tliat span tire millisecond time scale. More importantly, making comparisons witli experiments involves generating many (greater tlian perhaps 100) folding trajectories so that a reliable ensemble average is obtained. Thus, we need to make progress on botli fronts (force fields and enhanced sampling techniques in long duration simulations) before straightforward all-atom simulations become routine. [c.2645]

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]

Ensemble Averages and the q-Expectation Value [c.201]

Since the averaging operator is not normalized and in general (1), 1 for g 7 1, it is necessary to compute Zq to determine the average. To avoid this difficulty, we employ a different generalization of the canonical ensemble average [c.201]

The initial energy - E XoA t), VoA(t)) - is a function of the coordinates and the velocities. In principle, the use of momenta (instead of velocities) is more precise, however, we are using only Cartesian coordinates, making the two interchangeable. We need to sample many paths to compute ensemble averages. Perhaps the most direct observable that can be computed (and measured experimentally) is the state conditional probability - P A B,t) defined below [c.275]

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by [c.132]

In the condition of mixing, equation (A2.2.24), if the fiinction g is replaced by p, then the mtegral on die left-hand side is the expectation value of/ and at long times approaches the equilibrium value, which is the microcanonical ensemble average (J), given by the right-hand side. The condition of mixing is a sufiBcient condition for this result. The condition of mixing for equilibrium systems also has die implication that every equilibrium time-dependent correlation fimctioii, such as (f p, q)g, q))), approaches a limit of the uncorrelated product (/) (g) as t —>. [c.388]

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]

When a system is not in equilibrium, the mathematical description of fluctuations about some time-dependent ensemble average can become much more complicated than in the equilibrium case. However, starting with the pioneering work of Einstein on Brownian motion in 1905, considerable progress has been made in understanding time-dependent fluctuation phenomena in fluids. Modem treatments of this topic may be found in the texts by Keizer [21] and by van Kampen [22]. Nevertheless, the non-equilibrium theory is not yet at the same level of rigour or development as the equilibrium theory. Here we will discuss the theory of Brownian motion since it illustrates a number of important issues that appear in more general theories. [c.687]

It is worth discussing the fact that a free energy can be directly relevant to the rate of a dynamical process such as a chemical reaction. After all, a free energy function generally arises from an ensemble average over configurations. On the other hand, most condensed phase chemical rate constants are indeed thennally averaged quantities, so this fact may not be so surprising after all, although it should be quantified in a rigorous fashion. Interestingly, the free energy curve for a condensed phase chemical reaction (cf figure A3.8.1) can be viewed, in effect, as a natural consequence of Onsager s Imear regression hypothesis as it is applied to condensed phase chemical reactions, along with some additional analysis and simplifications [7]. [c.884]

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]

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in temis of an ensemble average over the microscopic nonlmear polarizability tensors of the individual constituents. [c.1189]

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average [c.1290]

In principle, these fomuilae may be used to convert results obtained at one state point into averages appropriate to a neighbouring state point. For any canonical ensemble average [c.2247]

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]

The ensemble average in the Widom fomuila, ((exp -p is sometimes loosely referred to as the [c.2263]

At a continuous phase transition, a correlation length (see section b3.3.2.1) diverges and an order parameter, typically the ensemble average of the corresponding dynamical variable, becomes macroscopically large. The divergence heralding the transition is describable in tenns of universal exponent relations. Effects of finite size close to continuous phase transitions are well studied [24, 132]. By contrast, a first-order phase transition is abrupt, as one phase becomes tliennodynamically more stable than another there are no transition precursors. In the thennodynamic limit, there is a step-function discontinuity in most properties, including thennodynamic derivatives of the free energy. Again it is possible to describe the effects of finite size [132. 133]. [c.2266]

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. [c.2556]

[c.3043]

Free Dynamics In simulations one usually represents a single protein molecule and one or a few ligand molecules. In principle, one might then obtain an estimate of the binding constant by monitoring the state of the protein during a long simulation in which ligand were observed to bind and unbind many times, and determining the fraction of time, xpL during which a ligand molecule was bound, and then, by equating time-average with ensemble-average properties, write Ka = xpl/ xpCl), with xp = I - xpp. However, the association and dis.sociation rates will nearly always be too slow to make [c.133]

See pages that mention the term

**Ensemble average**:

**[c.266] [c.387] [c.401] [c.832] [c.848] [c.885] [c.1503] [c.2246] [c.2262] [c.2483] [c.2485] [c.2497] [c.3040] [c.3040] [c.40] [c.150] [c.152] [c.159] [c.368]**

Computational biochemistry and biophysics (2001) -- [ c.41 , c.270 ]

Modelling molecular structures (2000) -- [ c.60 ]