Macroscopic variable

Generally speaking, intermolecular forces act over a short range. Were this not the case, the specific energy of a portion of matter would depend on its size quantities such as molar enthalpies of formation would be extensive variables On the other hand, the cumulative effects of these forces between macroscopic bodies extend over a rather long range and the discussion of such situations constitutes the chief subject of this chapter. 

To define the thennodynamic state of a system one must specify fhe values of a minimum number of variables, enough to reproduce the system with all its macroscopic properties. If special forces (surface effecls, external fields—electric, magnetic, gravitational, etc) are absent, or if the bulk properties are insensitive to these forces, e.g. the weak terrestrial magnetic field, it ordinarily suffices—for a one-component system—to specify fliree variables, e.g. fhe femperature T, the pressure p and the number of moles n, or an equivalent set. For example, if the volume of a surface layer is negligible in comparison with the total volume, surface effects usually contribute negligibly to bulk thennodynamic properties. 

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... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

T is the free energy fiinctional, for which one can use equation (A3.3.52). The summation above corresponds to both the sum over the semi-macroscopic variables and an integration over the spatial variableThe mobility matrix consists of a synnnetric dissipative part and an antisyimnetric non-dissipative part. The syimnetric part corresponds to a set of generalized Onsager coefficients. 

Relationships from thennodynamics provide other views of pressure as a macroscopic state variable. Pressure, temperature, volume and/or composition often are the controllable independent variables used to constrain equilibrium states of chemical or physical systems. For fluids that do not support shears, the pressure, P, at any point in the system is the same in all directions and, when gravity or other accelerations can be neglected, is constant tliroughout the system. That is, the equilibrium state of the system is subject to a hydrostatic pressure. The fiindamental differential equations of thennodynamics ... 

The NAs such as DNA usually used in the experiments consist of 10" -1 o nucleotides. Thus, they should be considered as macrosystems. Moreover, in experiments with wet NA samples macroscopic quantities are measured, so averaging should also be performed over all nucleic acid molecules in the sample. These facts justify the usage of the macroscopic equations like (3) in our case and require the probabilities of finding macromolecular units in the certain conformational state as variables of the model. 

The chemical composition of vermiculite can be quite variable (145). The megascopic varieties are generally trioctahedral, and the clay-si2e varieties contain both dioctahedral and trioctahedral varieties (144). Smectite minerals do not commonly occur as macroscopic single crystals. 

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. 

Kinetic theories of adsorption, desorption, surface diffusion, and surface reactions can be grouped into three categories. (/) At the macroscopic level one proceeds to write down kinetic equations for macroscopic variables, in particular rate equations for the (local) coverage or for partial coverages. This can be done in a heuristic manner, much akin to procedures in gas-phase kinetics or, in a rigorous approach, using the framework of nonequihbrium thermodynamics. Such an approach can be used as long as... 

This review is structured as follows. In the next section we present the theory for adsorbates that remain in quasi-equilibrium throughout the desorption process, in which case a few macroscopic variables, namely the partial coverages 0, and their rate equations are needed. We introduce the lattice gas model and discuss results ranging from non-interacting adsorbates to systems with multiple interactions, treated essentially exactly with the transfer matrix method, in Sec. II. Examples of the accuracy possible in the modehng of experimental data using this theory, from our own work, are presented for such diverse systems as multilayers of alkali metals on metals, competitive desorption of tellurium from tungsten, and dissociative... 

If it cannot be guaranteed that the adsorbate remains in local equilibrium during its time evolution, then a set of macroscopic variables is not sufficient and an approach based on nonequihbrium statistical mechanics involving time-dependent distribution functions must be invoked. The kinetic lattice gas model is an example of such a theory [56]. It is derived from a Markovian master equation, but is not totally microscopic in that it is based on a phenomenological Hamiltonian. We demonstrate this approach... 

For a macroscopic variable A Sj), the time evolution of the expectation value, Eq. (11), is obtained by the master equation explicitly as... 

The sometimes contradictory results from different workers in relation to the elements mentioned above extends to other elements . Some of these differences probably arise from variations in test methods, differences in the amounts of alloying additions made, variations in the amounts of other elements in the steel and the differing structural conditions of the latter. Moreover, the tests were mostly conducted at the free corrosion potential, and that can introduce further variability between apparently similar experiments. In an attempt to overcome some of these difficulties, slow strain-rate tests were conducted on some 45 annealed steels at various controlled potentials in three very different cracking environments since, if macroscopic... 

Two properties, in particular, make Feynman s approach superior to Benioff s (1) it is time independent, and (2) interactions between all logical variables are strictly local. It is also interesting to note that in Feynman s approach, quantum uncertainty (in the computation) resides not in the correctness of the final answer, but, effectively, in the time it takes for the computation to be completed. Peres [peres85] points out that quantum computers may be susceptible to a new kind of error since, in order to actually obtain the result of a computation, there must at some point be a macroscopic measurement of the quantum mechanical system to convert the data stored in the wave function into useful information, any imperfection in the measurement process would lead to an imperfect data readout. Peres overcomes this difficulty by constructing an error-correcting variant of Feynman s model. He also estimates the minimum amount of entropy that must be dissipated at a given noise level and tolerated error rate. 

The differential lengths and velocities considered must be small compared with the macroscopic distances and velocity intervals over which there are significant changes in the gross properties of the gas. On the other hand, they must be sufficiently large so that there are a large number of particles contained in the differential space-velocity volume this allows f(r,v,t) to be a continuous function of its variables.2... 

If the interval r is large compared with the time for a collision to be completed (but small compared with macroscopic times), then the arguments of the distribution functions are those appropriate to the positions and velocities before and after a binary collision. The integration over r2 may be replaced by one over the relative distance variable r2 — rx as noted in Section 1.7, collisions taking place during the time r occur in the volume g rbdbde, where g is the relative velocity, and (6,e) are the relative collision coordinates. Incomplete collisions, or motions involving periodic orbits take place in a volume independent of r when Avx(r) and Av2(r) refer to motion for which a collision does not take place (or to the force-field free portion of the... 

Chapter 10, the last chapter in this volume, presents the principles and applications of statistical thermodynamics. This chapter, which relates the macroscopic thermodynamic variables to molecular properties, serves as a capstone to the discussion of thermodynamics presented in this volume. It is a most satisfying exercise to calculate the thermodynamic properties of relatively simple gaseous systems where the calculation is often more accurate than the experimental measurement. Useful results can also be obtained for simple atomic solids from the Debye theory. While computer calculations are rapidly approaching the level of sophistication necessary to perform computations of... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

The system states (dependent variables) are the pressure, p, and the superficial (Darcy) velocity, v. The density, p, and viscosity, p, are fluid properties, and g is the acceleration of gravity. The porosity, < )(z), and permeability, fc(z), represent the macroscopic properties of the media. Both are spatially dependent and are represented as continuous functions of position z, as explicitly noted. While the per-... 

Terminal velocity, linear thermodynamics intermediate regimes and maximum flux, 25-27 regression theorem, 18-20 Test particle density, multiparticle collision dynamics, macroscopic laws and transport coefficients, 100-104 Thermodynamic variables heat flow, 58-60... 

Despite the importance of initiators, synthesis conditions, and diluents on the properties of a gel, composition is, of course, the most important variable. When growing polymeric chains are first initiated, they tend to grow independently. As the reaction proceeds, different chains become connected through cross-links. At a critical conversion threshold, called the gel point or the sol-gel transition, enough growing chains become interconnected to form a macroscopic network. In other words, the solution gels. The reaction is typically far... 

---