Reciprocal theorem

The usual emphasis on equilibrium thermodynamics is somewhat inappropriate in view of the fact that all chemical and biological processes are rate-dependent and far from equilibrium. The theory of non-equilibrium or irreversible processes is based on Onsager s reciprocity theorem. Formulation of the theory requires the introduction of concepts and parameters related to dynamically variable systems. In particular, parameters that describe a mechanism that drives the process and another parameter that follows the response of the systems. The driving parameter will be referred to as an affinity and the response as a flux. Such quantities may be defined on the premise that all action ceases once equilibrium is established. 

The kinetic constants of the system enter into the phenomenological L-coefficients, which are parameters of state. According to the reciprocity theorem of Onsager, the cross-coefficients L+r and Lr+ are identical. Now the definition of the efficiency 17 emerges directly from the dissipation function... 

The first and second terms on the right are self-energies of the central ion and the ionic atmosphere, and the third contains the interaction energies of the central ion with its ionic cloud and vice versa. According to Green s reciprocal theorem, these energies are equal and are given by... 

Both Newton s equation of motion for a classical system and Schrodinger s equation for a quantum system are unchanged by time reversal, i.e., when the sign of the time is changed. Due to this symmetry under time reversal, the transition probability for a forward and the reverse reaction is the same, and consequently a definite relationship exists between the cross-sections for forward and reverse reactions. This relationship, based on the reversibility of the equations of motion, is known as the principle of microscopic reversibility, sometimes also referred to as the reciprocity theorem. The statistical relationship between rate constants for forward and reverse reactions at equilibrium is known as the principle of detailed balance, and we will show that this principle is a consequence of microscopic reversibility. These relations are very useful for obtaining information about reverse reactions once the forward rate constants or cross-sections are known. Let us begin with a discussion of microscopic reversibility. 

As already noted, the reciprocal theorem does not include specifically conditions where sliding between particles takes place. Geometrical alterations between granular material affect reciprocity of pressure and displacement. 

Reciprocity theorem gives dA dF = dA2dFdAi dAi (b) Diffuse view factor between surfaces dAi and A2. 

There is the reciprocity theorem for the view factor For any object, sum to unity for all view factors View factor to itself is zero for plane surfaces View factor to itself is zero for convex surfaces. 

As a consequence of Lorentz reciprocal theorem (see Happel and Brenner, 1965) the grand resistance matrix R(xJV, e ) possesses many internal symmetries, greatly reducing the number of its independent elements. Another important feature of R is that it depends only on the instantaneous configuration ( N, eN) of the particulate phase. 

The coil sensitivity is characterized by means of the reciprocity theorem by the ratio where is the virtual field induced by a coil carrying... 

Comparison of these two expressions yields the Reciprocal Theorem ... 

On account of (1.3.10) the left-hand side vanishes, and the right-hand side may be rewritten, such that one obtains the Reciprocity Theorem... 

Formulae (10.33) and (10.34) have a clear physical interpretation. Let us introduce a unit conductance Sq = 1 S. According to the reciprocity theorem (see Chapter 8), the integral term in equation (10.33) can be treated as the complex conjugate of the electric field E (divided by the unit conductance Eo)>... 

Theorem 27 (The reciprocity theorem) The wavefield at a point r" generated by a point source located at a point r is equal to the wavefield at a point r generated by a point source located at a point r" ... 

The reciprocity theorem shows that for any wavefield we can switch between the receiver and source positions without changing the values of the observed field. This result plays an extremely important role in wavefield imaging and inversion, especially in wavefield migration, which will be discussed in Chapter 15. 

Note that we can show, by methods analogous to those used in deriving the reciprocity theorem 27 (formula (14.24)) for a scalar wavefield, that Gj satisfies the reciprocity relationship ... 

We can also use the 1-D Green s theorem, equation (15.51), to prove the reciprocity theorem in one dimension, similar to the general reciprocity theorem 27 ... 

Taking into account the reciprocity theorem (14.24), we can write the last formula in the form... 

In addition to the theoretical advantages offered by the principle of minimum potential energy, our elastic analyses will also be aided by the reciprocal theorem. This theorem is a special example of a more general class of reciprocal theorems and considers two elastic states (u ), or(i)) and where each... 

The proof of the reciprocal theorem is based upon two fundamental realizations. First, we note the relation... 

Based on this latter result, the reciprocal theorem is immediate. 

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