# Thermodynamic average

To obtain thermodynamic averages over a canonical ensemble, which is characterized by the macroscopic variables (N, V, T), it is necessary to know the probability of finding the system at each and every point (= state) in phase space. This probability distribution, p(r, p), is given by the Boltzmann distribution function. [c.41]

Thermodynamic average. An average over all points in phase space at a single time. [c.42]

It is hoped that the point that is being dynamically followed will eventually cover all of phase space and that the dynamic average will converge to the desired thermodynamic average. A key concept that ties the two averaging strategies together is the ergodic hypothesis. This hypothesis states that for an infinitely long trajectory the thermodynamic ensemble average and the dynamic average become equivalent to each other, [c.42]

Also included in this table are some average execution times for the thermodynamic subroutines measured for the CDC 6400 of the Computer Center, University of California, Berkeley. [c.352]

Born-Haber cycle A thermodynamic cycle derived by application of Hess s law. Commonly used to calculate lattice energies of ionic solids and average bond energies of covalent compounds. E.g. NaCl [c.64]

To promote auto-ignition, especially under marginal operating conditions — cold starting, for example— a high compression ratio is necessary, generally between 15 and 22 according to the type of technology, e.g., direct or prechamber. This distinction along with other purely thermodynamic considerations such as average specific heat of the gases present in the cylinder, explain the generally high efficiency of the diesel engine. [c.212]

If there are no reactions, the conservation of the total quantity of each species dictates that the time dependence of is given by minus the divergence of the flux ps vs), where (vs) is the drift velocity of the species s. The latter is proportional to the average force acting locally on species s, which is the thermodynamic force, equal to minus the gradient of the thermodynamic potential. In the local coupling approximation the mobility appears as a proportionality constant M. For spontaneous processes near equilibrium it is important that a noise term T] t) is retained [146]. Thus dynamic equations of the form [c.26]

In recent years a variety of powerful new molecular dynamics and Monte Carlo methods have been developed to address the intrinsic and extrinsic multiple time scale respectively. Accurate numerical integrators are required for questions involving real dynamical problems such as transport and energy relaxation. Thus in Sec. 2 we discuss accurate numerical integrators for intrinsic multiple time scale problem. On the other hand, in the simulation of biomolecular systems, one is often interested in computing equilibrium averages and thermodynamic quantities. For this purpose, the exact time dependence is not required, since all that is needed is the correct and efficient sampling of the thermally accessible configurations of the system, a problem made difficult by the extrinsic multiple time scales connected with the omnipresent energy barriers in systems with rough energy landscapes. A variety of techniques, such as stochastic dynamics, Monte Carlo, Hybrid Monte Carlo, J-Walking etc can be used, some of which are discussed in Sec. 3. First new methods for generating accurate dynamical trajectories are described and then methods based on inaccurate dynamics for sampling state space in systems with rough energy landscapes are treated. [c.298]

If a sufficiently large number of iterations have been performed, the ensemble average of any given property should not change signihcantly with additional iterations. However, there will be fluctuations in any given property computable as a root-mean-square deviation from the ensemble average. These fluctuations can be related to thermodynamic derivatives. For example, fluctuations in energy can be used to compute a heat capacity for the fluid. Alter- [c.304]

The Monte Carlo method samples phase space by generating random configurations from a Boltzmann distribution at a given temperature. Averages computed from a properly equilibrated Monte Carlo simulation correspond to thermodynamic ensemble averages. Thus, the Monte Carlo method can be used to find average energies and equilibrium structural properties of complex interacting systems. [c.19]

After initial heating and equilibration, the trajectory may be stable for thousands of time points. During this phase of a simulation, you can collect data. Snapshots and CSV files (see Collecting Averages from Simulations on page 85) store conformational and numeric data that you can later use in thermodynamic calculations. [c.75]

Monte Carlo simulations are commonly used to compute the average thermodynamic properties of a molecule or a system of molecules, and have been employed extensively in the study of the structure and equilibrium properties of liquids and solutions. Monte Carlo methods have also been used to conduct conformational searches under non-equilibrium conditions. [c.95]

A thermodynamically stable system conserves energy. Thus, by monitoring the potential energy one can confirm that a stable (and productive) phase of the simulation has begun. Absence of systematic drift in computed averages is often used as a check on the stability of a Monte Carlo trajectory. Fluctuations in the energy [c.98]

Chemistry students are generally more familiar with the absorption of light by molecules than with more classical optical phenomena such as refraction and scattering. In Chap. 10 we attempt to remedy this situation by detailed development of those aspects of light scattering that are especially pertinent to the study of polymers. The discussion of this topic brings together certain basic ideas from electromagnetism, optics, and solution thermodynamics. Even though it is explicitly through an expression for osmotic pressure that the molecular weight enters the light-scattering equations, these two techniques give different averages for M. This makes these two methods complementary rather than redundant, and, in combination, they provide information concerning the width of the molecular weight distribution. [c.496]

As noted above, all of the colligative properties are very similar in their thermodynamics if not their experimental behavior. This similarity also extends to an application like molecular weight determination and the kind of average obtained for nonhomogeneous samples. All of these statements are also true of osmotic pressure. In the remainder of this section we describe osmotic pressure experiments in general and examine the thermodynamic origin of this behavior. [c.544]

The effect of interparticle forces on the entrainment of Group A powders has been studied (12). When a bed is fluidized at a superficial gas velocity equal to the terminal velocity of the average particle, it takes many hours to entrain the bed because of the interparticle forces hoi ding the particles within. As the gas velocity reaches the terminal velocity of the largest 1% of the particles, it can stiH take nearly 20 minutes to empty the bed. These thermodynamic interparticle forces combine with hydrodynamic drag reduction forces so that Group A particles behave as larger clusters of particles. These clusters vary in size from the single particle at low soHds concentration to as high as 25 mm for a dense bed (13). [c.73]

Thermochemistry. Thermodynamic considerations ate of utmost importance in fluorinations. Table 1 is based on JANAF data (25) for CH, which indicate an average carbon-hydrogen bond strength of 410.0 kj/mol (98 kcal/mol) based on the atomization energy of CH. [c.274]

Gas Diffusion. For very long-Hved foams, film mpture is negligible and drainage slows to a stop as hydrostatic equiUbrium is attained. Nevertheless, the foam is still not in thermodynamic equiUbrium and continues to evolve with time. This occurs through an entirely different, though very general, means gas diffusion. Smaller bubbles have a greater interfacial curvature and hence, by Laplace s law, have a higher internal pressure than larger bubbles. This results in a diffusive flux of gas from smaller to larger bubbles. Thus with time small bubbles shrink while large bubbles grow. This process is known as coarsening, or ripening, and results in the net increase in the average bubble size over time. It is ultimately driven by surface tension and serves to decrease the total interfacial surface area with time. This process has many similarities to the phase separation of binary Hquids and metal alloys (22—24). [c.430]

Temperature. Temperature is the measurement of the average kinetic energy, resulting from heat agitation, of the molecules of a body. The most widely used scale is the Celsius scale for which the freezing and boiling poiats of water are used as defining poiats. The ice poiat is the temperature at which macroscopic ice crystals are ia equiUbtium with pure Hquid water under air that is saturated with moisture at standard atmospheric pressure (101.325 kPa). One degree on the Celsius scale is 1.0% of the range between the melting and boiling poiats of water. The unit of thermodynamic temperature is the Kelvin, defined as 1/273.16 of the thermodynamic temperature of the triple point of water. The relation of the Kelvin and Celsius scales is defined by the International Temperature Scale of 1990 (22). The international temperature scale between —259 and 962°C is based on a number of defining fixed points, use of a standard platinum resistance thermometer, and the following formula for the resistance, R, as a function of temperature, /, above 0°C, [c.20]

Scattering. Use of the classical techniques of small-angle scattering of light and x-rays to study polymer blends has been reviewed (77). Scattering techniques provide a sensitive method for determining whether blends are homogeneous or heterogeneous and to foUow the kinetics of phase-separation processes (78). Small-angle neutron scattering (sans) and reflection have emerged as powerful tools for investigating many aspects of polymer blends (76,79—91). The sans technique can be used to obtain thermodynamic interaction energies for miscible blends however, one of its more unusual aspects is the abiUty to determine conformational information about the components. For example, the radius of gyration of a styrene—acrylonitrile (AN) copolymer containing 19% AN by weight as a function of its weight average molecular weight when blended with a deuterated poly(methyl methacrylate) has been determined (79,83). The chain is slightly more expanded in the blend than the unperturbed chain dissolved in an ideal or 9 solvent, as would be expected for a solvent which is somewhat better than an ideal one. Thus, it is confirmed that the copolymer is moleculady dissolved in PMMA in the same sense as polymers are in low molecular weight solvents this requires mixing at the segmental level for the blend (79,80,83). The sans technique has become an important tool in elucidating the phase behavior and quantifying the interactions in polyolefin blends (88—91). A significant issue in the use of sans for these purposes is that one of the samples generally needs to be deuterated to produce scattering contrast. This can pose some synthesis challenges. Of more serious concern is the fact that deuterated polymers appear to have measurably different interactions with other molecules than their hydrogenous counterparts (86—90). [c.411]

Because there is no single material called RDF and because the composition and therefore fuel properties depend on the composition of the starting MSW and the methods of processing, it is impossible to give what might be an average set of fuel properties. Table 4 gives the results of a typical RDF fuel analysis from a waste-to-energy plant in Maine (26). A number of analyses have examined how the presence of ash as well as the high moisture content of RDF affect its quaUty as a fuel. The thermodynamic balance for possible drying of the fuel has also been examined (27). It is unlikely that any RDF production process will be able to afford drying the material. [c.545]

Charge carriers in a semiconductor are always in random thermal motion with an average thermal speed, given by the equipartion relation of classical thermodynamics as m v /2 = 3KT/2. As a result of this random thermal motion, carriers diffuse from regions of higher concentration. Applying an electric field superposes a drift of carriers on this random thermal motion. Carriers are accelerated by the electric field but lose momentum to collisions with impurities or phonons, ie, quantized lattice vibrations. This results in a drift speed, which is proportional to the electric field = p E where E is the electric field in volts per cm and is the electron s mobility in units of cm /Vs. [c.346]

The CFTI method is highly efficient, has improved convergence properties and enables new ways of exploring energy landscapes of flexible molecules. The efficiency is due to the fact that calculation of the free energy gradient with respect to an arbitrary number of coordinates may be performed at essentially the same cost as a standard one-dimensional TI simulation under the same conditions [2]. This is because the most expensive terms to evaluate, dU/d k, may be expressed in terms of simple algebraic transformations of the Cartesian gradient dU/dqj, which is known at each step of a simulation [2, 8]. A single simulation yields derivatives of free energy with respect to all conformational degrees of interest, yielding a complete local characterization of conformational space, not just the derivative along a one-dimensional reaction path [2, 8]. This enables the determination of stability of structures with respect to perturbations, location of minima on the free energy surface, and finding minimum free energy paths connecting different states. The accelerated convergence may be achieved by selecting all soft degrees of freedom as the fixed coordinates. In the case of peptides these would be the backbone 4>, Ip, and some of the sidechain dihedrals [8, 9, 10]. The sampling of the restricted conformational space of remaining hard degrees of freedom and solvent is very fast - simulations of 20-50 ps were sufficient to obtain precise gradient values in the studied cases. Simulations of similar length are sometimes used in the standard approach to free energy profiles, where only the reaction coordinate is constrained. However in these methods, because of the size of the available conformational space, the convergence of thermodynamic averages is often assumed rather than actually achieved. [c.166]

The DPDPE simulations are the first example of use of CFTI to accelerate convergence of thermodynamic averages. This approach has two important advantages. First, because all soft degrees of freedom of the solute are fixed, the simulations do not suffer from conformational sampling problems common in other fre< energy simulation protocols, and all averages converge [c.173]

The force acting on an atom is therefore the negative energy gradient. Molecular mechanics methods are often used because they are fast and can be applied to a large number of molecules with many atoms. The reason for this advantage is the so-called Born-Oppenheimer approximation, which separates the movement of the electrons from the much slower movement of the nuclei and makes it possible to describe the energy of a molecule or molecular system as a function of the atomic coordinates. We will see this in more detail in Section 7.2.3. This simplification does not necessarily mean that the results of a molecular mechanics calculation are less reliable than, for example, high-level quantum mechanical data. Force field parameters are often adjusted to correctly reproduce experiments, which normally represent thermodynamic averages and not a particular geometry or conformation of the molecule tmder consideration. A second important point to mention is the transferability of the parameter set, which is generated using a limited set of experimental data and small test molecules. Nevertheless, these parameters can be applied to a much greater number of problems and larger molecular systems. [c.338]

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. [c.63]

It is, in fact, possible to consider at least three different contact angles for a given system Let us call them 6 , Oat, and app. The first, 6 , is the microscopic angle between the liquid and the ridge of solid it is the angle determined by the balance of surface stresses taking into account local deformations. The second, 0ih> is the thermodynamic or intrinsic angle obtained in the derivation of Eq. X-18 and in general is the angle obtained in a free energy minimization for the overall system. Finally, for a variety of reasons outlined in the preceding sections, one will measure experimentally an apparent angle 0app> which is some kind of average such as that in Eq. X-27. [c.372]

This paper presents the theoretical background and some practical applications of a new conformational free energy simulation approach, aimed at correcting the above shortcomings. The new method, called Conformational Free energy Thermodynamic Integration (CFTI), is based on the observation that it is possible to calculate the conformational free energy gradient with respect to an arbitrary number of conformational coordinates from a single simulation with all coordinates in the set kept fixed [2, 8]. The availability of the conformational gradient makes possible novel techniques of multidimensional conformational free energy surface exploration, including locating free energy minima by free energy optimization and analysis of structural stability based on second derivatives of the free energy. Additionally, by performing simulations with all "soft degrees of freedom of the system kept fixed, free energy averages converge very quickly, effectively overcoming the conformational sampling problem. [c.164]

A thermodynamically stable system conserves energy. Thus, by monitoring the potential energy one can confirm that a stable (and prodnciive) phase ol the simnlaiion has begun. Absence of systematic drift in computed averages is often used as a check on th e stability of a Mon te Carlo trajectory. Fluctuation s in the energy [c.98]

I his approach to the calculation of free energy differences. Equation (11.6), is gener attributed to Zwanzig [Zwanzig 1954]. To perform a thermodynamic perturbation calculal we must first define and and then run a simulation at the state X, forming ensemble average of exp[—(jfy — x)/ bT] as we proceed. Analogously, we could ru simulation at the state Y and obtain the ensemble average of exp[-(jfx - JifY)/kBT]- T1 if X corresponds to ethanol and Y to ethane thiol, the free energy difference could obtained from a simulation of ethanol in a periodic box of water as follows. For e configuration we calculate the value of the energy for every instantaneous conformat of ethanol in which the oxygen atom is temporarily assigned the potential energy p meters of sulphur. Alternatively, we could simulate ethane thiol and for each configurat calculate the energy of the system in which the sulphur is mutated into oxygen. [c.581]

Equation 2 gives the relation between stresses and accelerations obtained from momentum balances. To proceed further requires use of the constitutive equations which codify the material properties through additional relations between the stresses (t, etc) and the rates of strain (du/dx, etc). The constitutive equation for a given fluid is found empirically or theoretically by use of some theory of material properties. The simplest model is one in which the various stresses are expressed as linear combinations of the rates of strain. When the fluid is homogeneous and isotropic, this relation leads to the Navier-Stokes equations. Fluids that obey these equations are by definition Newtonian. The conditions of homogeneity and isotropy ensure that only two material constants, the shear viscosity, ]1, and the dilational viscosity, X, are needed to describe the fluid. By defining pressure as the negative of the average of the three normal stresses, CJ, one finds that X = 2/3 fi, hence only a single material viscosity ]1 is required. As defined, the pressure is usually identified with the thermodynamic pressure for purposes such as determining physical properties. Equations 4 and 5 show the constitutive equation so derived, and equation 6 shows the form taken by the equation of motion in the X-direction when these are inserted into the force balance. [c.88]

Copper Phthalocyanine Blue. CPC blue exists in several polymorphic modifications, two of which, the red-shade blue alpha and green-shade blue beta form, are of great commercial significance. Beta is the thermodynamically more stable phase and is the product resulting from manufacture by the two basic processes using either phthalonittile or phthaUc anhydride as starting materials, either in the presence of a solvent or by a dry baking process. The alpha form is usually obtained by conversion from the beta form and has to be stabili2ed to prevent phase reconversion. The pronounced tendency of the alpha form to flocculate in fluid media is suppressed by special surface treatments and/or the introduction of a small amount of aromatically bound chlorine to form soHd solutions between CPC and its chlorinated derivatives. A so-called semichloroCPC containing an average of about 0.75 chlorine atoms per molecule provides a stable pigment even upon incorporation in high temperature plastics. [c.30]

See pages that mention the term

**Thermodynamic average**:

**[c.174] [c.187] [c.202] [c.312] [c.97] [c.320] [c.97] [c.41] [c.344] [c.230] [c.582] [c.375] [c.40] [c.75] [c.358] [c.428] [c.584]**

Computational biochemistry and biophysics (2001) -- [ c.41 , c.42 ]