Macroscopic system states

Rate processes occurring both close to equilibrium and far from equilibrium pose challenges for theory. For close-to-equilibrium rates, the microscopic details of the reactive event in a dense many-body system are crucial to an understanding of the rate, and the task of theory is to formulate tractable models for realistic situations. For far-from-equilibrium, noise-induced rate processes, complex macroscopic system states may be involved in the transition process, and the underlying deterministic system may itself lead to chaotic dy-... 

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... 

Thermodynamics is a deductive science built on the foundation of two fundamental laws that circumscribe the behavior of macroscopic systems the first law of thermodynamics affirms the principle of energy conservation the second law states the principle of entropy increase. In-depth treatments of thermodynamics may be found in References 1—7. 

We present and discuss results for MD modeling of fluid systems. We restrict our discussion to systems which are in a macroscopically steady state, thus eliminating the added complexity of any temporal behavior. We start with a simple fluid system where the hydrodynamic equations are exactly solvable. We conclude with fluid systems for which the hydrodynamic equations are nonlinear. Solutions for these equations can be obtained only through numerical methods. 

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-... 

Now — L is the Landau-Ginzburg free energy, where m2 = a(T — Tc) near the critical temperature, is a macroscopic many-particle wave function, introduced by Bardeen-Cooper-Schrieffer, according to which an attractive force between electrons is mediated by bosonic electron pairs. At low temperature these fall into the same quantum state (Bose-Einstein condensation), and because of this, a many-particle wave function (f> may be used to describe the macroscopic system. At T > Tc, m2 > 0 and the minimum free energy is at = 0. However, when T [Pg.173]

When the adsorbent molecides are not independent, we can no longer use the relation (D.2) for the GPF of the system. In this case, we must start from the GPF of the macroscopic system from which we can derive the general form of the BI for any concentration of the adsorbent molecule. The derivation is possible through the McMillan-Mayer theory of solution, but it is long and tedious, even for first-order deviations from an ideal solution. The reason is that, in the general case, the first-order deviations would depend on many second-virial coefficients [the analogue of the quantity B2(T) in Eq. (D.9)]. For each pair of occupancy states, say i and j, there will be a pair potential [/pp(R, i,j), and the corresponding second-virial coefficient... 

The rigor and power of equilibrium thermodynamics is purchased at the price of precise operational definitions. In this section, we wish to carefully define four of the most important thermodynamic terms system, property, macroscopic, and state. Although each term has an everyday meaning, it is important to understand the more rigorous and precise aspects of their usage in the thermodynamic context. 

Definition Two macroscopic systems having all the same numerical values of the independent intensive properties are said to be in the same state (regarded as identical for thermodynamic purposes). 

The irreversibility inherent in the equations of evolution of the state variables of a macroscopic system, and the maintenance of a critical distance from equilibrium, are two essential ingredients for this behavior. The former confers the property of asymptotic stability, thanks to which certain modes of behavior can be reached and maintained against perturbations. And the latter allows the system to reveal the potentialities hidden in the nonlinearity of its kinetics, by undergoing a series of symmetry breaking transitions across bifurcation points. 

Until now we assumed that we have the maximum information on the many-particle system. Now we will consider a large many-body system in the so-called thermodynamic limit (N- °o, V—> >, n = NIV finite) that means a macroscopic system. Because of the (unavoidable) interaction of the macroscopic many-particle system with the environment, the information of the microstate is not available, and the quantum-mechanical description is to be replaced by the quantum-statistical description. Thus, the state is characterized by the density operator p with the normalization... 

So far we have ignored bound states, or composite particles, which may form as a result of the interaction due to an attractive part of the potential. Of course, the behavior of macroscopic systems such as thermodynamic, transport, and optical properties, is essentially influenced by the existence of bound states. A particular problem of special interest in connection with these bound states is the ionization phenomenon, or more general, the problem of chemical reactions. 

In Fig. 3 the sheet is macroscopically stretched as indicated by the arrows. As a consequence, transversal contraction takes place perpendicular to the stretching direction. Wrinkles will appear in the macroscopically stretched state and - provided the system is linearly elastic and no plastic deformations occur - disappear upon relaxation [22],... 

A system that is made up of a homogeneous mass of a substance at equilibrium can be described as being in a certain thermodynamic state that is characterized by certain properties. If forces of various types act on the system or more of the substance is added, the system is changed to a different state. It is remarkable that only a small number of properties have to be specified to completely characterize the equilibrium state of a macroscopic system. For a system containing a single substance, three properties suffice, if they are properly chosen. For example, the... 

When other kinds of work are involved, it is necessary to specify more variables, but the point is that when a small number of properties are specified, all the other properties of the system are fixed. This is in contrast with the very large number of properties that have to be specified to describe the microscopic state of a macroscopic system. In classical physics the complete description of a mole of an ideal gas would require the specification of 3NA components in the three directions of spatial coordinates and 3NA components of velocities of molecules, where NA is the Avogadro constant. 

The term density matrix has already occurred in Section 2.2. There the density matrix (p) generally used in simulations was consequently called average density matrix. This matrix gives the average state of the spin system, and the spectrum of the whole macroscopic system is determined from this quantity. 

In this example we take the most microscopic viewpoint of macroscopic systems. We regard them as being composed of np 1023 number of particles. Gibbs (1902) has realized that the state variable of classical mechanics, namely... 

We note that the classical equilibrium entropy (i.e., the eta-function evaluated at equilibrium states) acquires in the context of the Microcanonical Ensemble an interesting physical interpretation. The entropy becomes a logarithm of the volume of the phase space that is available to macroscopic systems having the fixed volume, fixed number of particles and fixed energy. If there is only one microscopic state that corresponds to a given macroscopic state, we can put the available phase space volume equal to one and the entropy becomes thus zero. The one-to-one relation between microscopic and macroscopic thermodynamic equilibrium states is thus realized only at zero temperature. 

The most fundamental experimental observation on which equilibrium thermodynamics is based on is the observation that all externally unforced macroscopic systems (with some exceptions, namely glasses, that we shall mention later in Section 4) can be prepared in such a way that their behavior shows some universal features. Subsequent investigation of these features leads then to the formulation of equilibrium thermodynamics. The preparation process consists of letting the macroscopic systems evolve sufficiently long time without external influences. The states reached when the preparation process is completed are called equilibrium states. The approach to equilibrium states is thus a primary experience the behavior at equilibrium states is the secondary experience. An investigation of the secondary experience leads to equilibrium thermodynamics. We may expect that an investigation of the primary experience (i.e., an... 

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