Spatially uniform force

The thermodynamic force (affinity) X is a pivotal concept in thermo dynamics of nonequilibrium processes because of its relationship to the concept of driving force of a particular irreversible process. Evidently, thermodynamic forces arise in spatially inhomogeneous systems with, for example, temperature, concentration, or pressure inhomogeneity. In spatially uniform homogeneous systems, such forces arise either in the presence of chemically reactive components that have not reached thermodynamic equiHbrium via respective chemical transformations or at the thermodynamic possibility of some phase transformations. 

The simplest example of an oscillation in a bulk mode is the classical Helmholtz resonator. Consider a gas-filled chamber of volume V, into which is inserted a tube of length I and of cross-sectional area that is open to the atmosphere. Assume that the atmospheric pressure remains constant at the outer end of the tube at a value p, and consider slowly varying processes for which the pressure p at the inner end of the tube is equal to the spatially uniform pressure within the chamber. With friction neglected, a force balance on the gas in the tube is — p) = (a O dv/dt, where p is the average gas density in the tube and v is the average outward velocity of gas in the tube. Conservation of mass for the gas of density p in the chamber is t" dp/dt —... 

The elastic term takes into account the elastic restoring force, tending to establish a spatially uniform LC ordering. It is expressed as... 

The force fields used by scientists for simulations have been developed with distinct traditions, each appropriate for its own use. In computational chemistry, interest has been in bulk properties of solutions and proteins, in the thermodynamic limit in which boundary conditions do not appear explicitly and where equilibrium (i.e., zero flux of all species) is present. The thermodynamic limit of computational chemistry implies a spatial uniformity of bulk properties that can be analyzed with periodic boundary conditions if the period is longer than the spatial inhomogeneities of the bulk solution. Contrarily, in computational electronics, the interest has focused on electron devices, which exchange charge with their environment through geometrically and electrically complex boundaries and where internal dielectric discontinuities exist. Simulations are usually performed by varying the applied bias in order to reproduce transient nonequilibrium conditions and to obtain a record of the response of the simulated devices. 

This is the elastic energy per unit length along the crest of the buckle. The resultant force has the spatially uniform value which is given in terms of... 

We have repeatedly used the term hydrodynamic, and we now give it a more precise definition. By a hydrodynamic process we mean one for which the local thermodynamic variables, temperature, chemical potential (or density), and velocity, are determined by the past history of their boundary values. The normal solution to the Boltzmann equation, as well as its generalization obtained in the previous Sections, then clearly corresponds to a hydrodynamic process. The significance of the term hydro-dynamic may be clarified by the consideration of some processes of non-hydrodynamic type. A process of relaxation in momentum space in a spatially uniform gas is clearly non-hydrod5mamic, since the local thermod5mamic variables are not at all pertinent to its description. Another example is provided by processes in a Knudsen gas. Here there is an essential dependence on the particular form of the boundary forces. An insensitivity to the nature of the boundary forces is implied in the definition of a hydrod5mamic process, for which it is immaterial whether a thermal reservoir is constructed of, say, copper or aluminum, and... 

Thus, you can see that the propagative compression results in a final pressure are 2.3 times that produced by spatially uniform compression. There is the added effect of shock with propagative compression that can enhance forces exerted on internal vessel components. 

In nonequilibrium thermodynamics, the quantities Xy(r, r) are viewed as thermodynamic driving forces that give rise to the flows described by the current densities Jj(r, f). For systems in equilibrium, the thermodynamic parameters AtBAy(r, i) are spatially uniform. Then the driving forces [Xy(r, i) vanish. Consequently, the current densities Jj(r, /) vanish for systems in equilibrium. 

The particle motion is driven by the action of the electric field on the double layer. Although the total electric force on the particle and double layer is zero - because the particle charge is balanced by the diffuse layer charge - a motion is still generated, for the field pushes the particle in one direction and it pulls the double layer ions, and the surrounding fluid, in the opposite direction. As described above, this motion is spatially uniform, except close to the electrodes, and so there is no ESA except from the electrodes. 

The value o+l <0.4 found for H2 shows that even in the lowest state the molecules are rotating freely, the intermolecular forces producing only small perturbations from uniform rotation. Indeed, the estimated (3vq<135° corresponds to Fo <28 k, which is small compared with the energy difference 164 k of the rotational states j = 0 and j= 1, giving the frequency with which the molecule in either state reverses its orientation. The perturbation treatment shows that with this value of Fo the eigenfunctions and energy levels in all states closely approximate those for the free spatial rotator.9... 

Research has previously shown that bacteria are not uniformly distributed in soil, reflecting soil structure and available nutrients (Richaume et al., 1993). The distribution of microorganisms throughout the soil can also be considered from the applied ecological perspective of patch dynamics, where patch formation is a reflection of intrinsic and extrinsic forces (Rao et al., 1986). The same authors also showed spatial variability in the degradation of pesticides applied to a soil system. 

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