Ind. Eng. Chem. Res. 1999, 38, 845-850
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The Beauty of Self-Adjoint Symmetry† Doraiswami Ramkrishna* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
Rutherford Aris Department of Chemical Engineering & Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
We discuss the aesthetic attributes of self-adjoint symmetry in the analysis of natural systems and find them to be in accord with the physical traits. The elegance of its symmetric structure, besides being beautiful, is a boon to efficient computation. 1. Introduction
2. Symmetry
Concerning the beauty of mathematics there is no great difference of opinion, though there is a great range in the degree to which this opinion is informed. A proper regard for it is bred in the matematician’s bone. To those who use mathematics only as a tool, its existence is acknowledged, but it is the sound of a distant trumpet that may have little rallying power. To the educated nonmathematician it is a cultural axiom. To those who think that mathematicsssomething they say they were “never any good at”sis balancing their checkbooks, it is a notion incomprehensible, if not risible, to be dismissed by an “if you say so” (or, in Minnesota, by a “whadever”). It is informed user of mathematics with whom we shall be concerned and want to discuss whether the canons of aesthetics in the engineering applications of mathematics are merely those of mathematics itself or whether they have their own peculiar glories. The canons by which we judge the aesthetic value of anything are formally the same; in Aquinas’ language integritas, consonantia, and claritas.1 These can be translated as integrity, consonance (in the sense of coherence), and clarity (in the sense of radiance or brightness).2 By integrity we understand that the thing holds together and can be appreciated as a whole. In consonance we have the idea that the work should have just proportions and so that it is seen, not just as a whole, but as being also appropriate in its methods and emphases. Clarity is not only a matter of style and presentation, but of the intrinsic luminance of the underlying principles. In large measure, such principles in chemical engineering science are mathematical in nature. This does not diminish the roles of chemistry and physics in chemical engineering; they have their several luminosities, but it is with the mathematics that we are concerned here. That the aesthetic motivation in science is vital to the free play of scientific ideas has long been recognized and nowhere better expressed than in Chandrasekhar’s later reflections. Speaking of E. A. Milne, he writes, “...his obvious enjoyment in the flow of his ideas and in the course and texture of his arguments, transports the reader to an equal measure of enjoyment”.3
One of the most pervasive elements of aesthetic judgment is the consideration of symmetry. It stands sometimes as the ideal to be aspired to, sometimes as the canon by which the success of deliberate departure must be measured. The widespread and wonderful ramifications of symmetry have received an excellent exposition by Ian Stewart and Martin Golubitsky in their book, Fearful Symmetry, and we shall not attempt to trace them all here, but merely take a short path to the symmetry of self-adjointness.4 At the lowest level, mathematical manipulation, if functions like cultivated speech and rewards the user with its power of self-correction. None but a mathematician could have written You boil it in sawdust: you salt it in glue: You condense it with locusts and tape: Still keeping one principal object in view To preserve its symmetrical shape.5 Who has not enjoyed the pleasure of being introduced to the conic in homogeneous plane coordinates and recognized at once the suitability of both form and substance in the equation ax2 + by2 + cz2 + 2fyz + 2gzx + 2hxy ) 0? The choice of the triplets of letters for coordinates and coefficients and the cyclic order of the factors are small and arbitrary matters but bespeak the mathematician’s instinct. When matrix notation is used and the conic is written:
* Author for correspondence. † This paper is dedicated to Roy Jackson, whose papers have always been known for their mathematical elegance.
xT(AT - A)x* ) 0
[ ][ ]
a h g x [x y z] h b f y ) 0 g f c z the corresponding matrix is symmetric, being identical with its transpose. Also, a symmetric matrix has real eigenvalues. For, if AT is the transpose of A and λ, an eigenvalue satisfying Ax ) λx is complex, its complex conjugate λ*, must also be an eigenvalue. Let the corresponding eigenvectors be x and x*. Now,
xTx* ) x*Tx ) |x|2 + |y|2 + |z|2 is nonzero and because A is symmetric
(λ - λ*)xTx* ) xTATx* - xTAx* ) so that λ is real.
10.1021/ie980252x CCC: $18.00 © 1999 American Chemical Society Published on Web 01/28/1999
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Not all is lost, however, if A is unsymmetric, for some unsymmetric matrices are symmetrizable. This term is used if a symmetric, strictly positive definite matrix S with the property ATS ) SA can be found. Such a matrix A has all the virtues of one that is symmetric, which suggests the existence of a deeper, inherent concept of symmetry, i.e., that of self-adjointness. Of specific concern to us in this article is the elegance of the symmetry of self-adjointness. In that the beauty of quantum mechanics is attributed to the elegance of hermitian operators, the beauty of self-adjointness is well established but is often treated without reference to the many other fields of application it has served. We dwell on its application to engineering in general, although the examples cited are often from chemical engineering.
vector by λ. For a self-adjoint operator it follows from above that such an eigenvalue must be real and that the eigenvectors corresponding to two distinct eigenvalues must be orthogonal. A result of far-reaching consequence is that all the eigenvectors of the self-adjoint operator together form a basis set, which implies that every vector can be expanded in terms of this set with coefficients of expansion that are readily calculated by using the orthogonality property of the eigenvectors. Some details have been suppressed in favor of brevity but it is fair to summarize by saying that a fine dissection of the Hilbert space into orthogonal subspaces is made possible with the eigenvectors of T and that any vector has readily identifiable pieces, one in each subspace, that can be added to recover the vector.
3. Linear Operators
5. Partitioned Operators
A matrix is more than just a notational device. It represents a linear operator in a finite dimensional vector space transforming each vector into another by preserving linear combinations. Thus, for arbitrary choices of R, β, x, and y we must have (A(Rx + βy) ) RAx + βAy. An eigenvalue λ of the operator A and the corresponding eigenvector x satisfy the relation Ax ) λx, which identifies the vector x as one whose transformation by A does not alter its “direction”. Although this is only a matter of definition that begs the question of existence, it is a significant consequence of symmetry (or more generally self-adjointness) that it guarantees the existence of eigenvalues and eigenvectors of an operator. Before we probe further into this gift of symmetry, we look more carefully at the concept of symmetry itself. The symmetry of a matrix A is rather self-evident, but that of the more abstract linear operator that simply preserves linear combinations is less obvious. We shall henceforth refer to the symmetry of a general linear operator as self adjointness.
Our focus on the real, symmetric matrix as a selfadjoint operator is rooted in the natural role of real numbers in numerous applications. There are other applications, however, that involve complex numbers (such as in quantum mechanics) in which the symmetric matrix must be replaced by the hermitian matrix, which has the property that the ijth element is the complex conjugate of jith element. It is a relatively simple matter to show that the hermitian matrix is self-adjoint, but we are not so much interested here in hermitian matrices themselves as we are in the abstraction underlying their self-adjointness. For this we must return to our discussion on the meaning of self-adjointness. Where the operator T does not satisfy the property of self-adjointness, an adjoint operator T* can be associated with it characterized by the relation that for any two vectors x and y in the Hilbert space the inner product of Tx and y is the same as that of x and T*y. For a complex valued matrix A the adjoint operator can then be shown to be the transpose of its complex conjugate. It is often necessary to consider partitioned operators, which represent a matrix array of operators that can act on a space composed of ordered replicas of a particular Hilbert space or, more generally, ordered sets of different Hilbert spaces. A typical vector in such a space would be an ordered array of vectors from the individual Hilbert spaces that would be transformed by the partitioned operator to another one of its kind. The inner product on such a “composite” space would sum the inner products of corresponding elemental pairs with positive weights that could be negotiated in seeking the self-adjointness of the partitioned operator. It is easy to show that the partitioned operator is self-adjoint in the composite space if it is a matrix array with diagonal terms that are self-adjoint in the composite space if it is a matrix array with diagonal terms that are selfadjoint in their respective Hilbert spaces and the offdiagonal operators are adjoints of each other. Although we have made the discussion somewhat abstract here, there is a surprising number of concrete applications that belong to this category. Natural systems can be described in terms of a vector in an appropriate Hilbert space. The choice of the system vector is generally dictated by physical laws governing the evolution of the system through space and time, which yield a relation between vectors, some known and others that result from transformation (frequently linear) of the (unknown) system vector. Our
4. What Is Self-Adjointness? Self-adjointness is a property of symmetry assigned to operators in Hilbert spaces. The vectors of a Hilbert space are characterized by the existence of an “inner product” that transforms ordered pairs of vectors into the field of numbers (generally complex) associated with the vector space. Thus every pair of vectors is transformed into a complex number subject to certain rules that can be satisfied in a countless variety of ways by suitable definitions of inner products. Indeed, the choice of a proper inner product is profoundly important, for in it lies the essence of the symmetry we seek. Regardless of the choice, the inner product leads to the “norm” or “magnitude” of a vector, and the angle (consequently orthogonality) between vectors. A linear operator T that preserves linear combinations of vectors under transformation is said to be selfadjoint if for any pair of vectors x and y the inner product between Tx and y is the same as that between x and Ty. Because reversing the order of the pair of vectors x and y in the inner product obtains the complex conjugate of the first, the inner product between Tx and x becomes real when T is self-adjoint. As pointed out earlier, a number λ qualifies for an “eigenvalue” of the operator T if one or more (linearly independent) nonzero vectors (called eigenvectors) can be found in the space on which the action of T is the same as multiplying the
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focus is on problems in which the transformations are either self-adjoint or, more importantly, can be rendered self-adjoint by negotiating the parent Hilbert space of the system vector and its inner product. It should now be evident that this exercise is in quest of a hidden symmetry, the unfolding of which produces a picturesquely prismatic partitioning of the Hilbert space in which the skeletal structure of the unknown vector is revealed. For this view to emerge, however, it is crucial that the vector be traced to its most natural habitat, one that not only relies on mathematical symmetry but, as we shall see, resonates with nature. The harmony between the intrinsic elegance of mathematics and nature’s ways is the subject of this paper.
constraint on mole fractions. Furthermore, any mole fraction vector x has the curious property that (x, xˆ ) ) 1. Of even more general import is the symmetry that the matrix D of transport coefficients for multicomponent systems acquires from irreversible thermodynamics through its being expressible as the product of two symmetric, positive definite matrices L and G. The positive definite symmetry of G follows from it being a Hessian matrix associated with the free energy at or near equilibrium, whereas that of L arises from the principle of microscopic reversibility.7 The matrix D is not symmetric because
6. Symmetrizable Operators
which is not necessarily the product LG; the positive definite, symmetrix matrix L-1 leads to the relation DTL-1 ) GLL-1 ) G ) L-1LG ) L-1D so that we conclude from our prior deliberations that LG (i.e., D) is symmetrized by the inner product weighted by the matrix L-1. The principle of microscopic reversibility is thus the origin of the self-adjoint symmetry of D! The consequence of such a symmetry is an unwavering approach of systems to equilibrium from being very near it. Other examples of such symmetrization have been discussed at length elsewhere.8
It should be apparent that the symmetric matrix is the plainest of the self-adjoint family. On the other hand, the symmetrizable matrix A mentioned in section 2 is considerably more subtle because its self-adjointness is rooted in the inner product defined by weighting one of the pair with the positive definite, symmetric matrix (or more generally a self-adjoint operator) S. We may represent this inner product (x,y) t (Sx)Ty where on the right-hand side we have the regular inner product in Euclidean space between the vectors Sx and y. The self-adjointness of A emerges from the following elementary matrix manipulations
(Ax,y) ) (SAx)Ty ) xTATSy ) xTSAy ) (x, Ay) where we have used the property ATS ) SA that had been imposed on the symmetrizing matrix S. A delightful example of the foregoing symmetrization lies in the classic treatment of first-order reaction systems by Wei and Prater.6 Such a system consists of n chemical species, each of which undergoes transformation reversibly with linear kinetics to every other species on the surface of a catalyst. The matrix K of rate constants (kij for the transformation of Aj to Ai) is generally unsymmetric, but a principle arising from statistical mechanics, that of microscopic reversibility,7 stipulates that if the system is globally at equilibrium, each of the reactions must be individually at equilibrium, which translates to the relation kijxˆ j ) kjixˆ i where xˆ j is the mole fraction of Aj in the reaction mixture at equilibrium. It is then readily seen that the diagonal matrix X with its ijth element given by (1/xˆ j)δij which derives its property of positive definiteness from the positivity of mole fractions, satisfies the relation KTX ) XK. K is thus rendered self-adjoint under the inner product weighted with the diagonal matrix of reciprocal mole fractions because of microscopic reversibility! In symbols, the inner product is written as n
(x,y) )
1
∑ ˆ j)1 x
j
x∑ n
xjyj
||x|| )
j)1
1 xˆ j
xj
2
(1)
Besides the self-adjointness of K, the foregoing inner product is also a source of some further harmony, for the equilibrium mole fraction vector xˆ , which is the eigenvector corresponding to the zero eigenvalue, stands automatically “normalized” since the requirement of its norm as defined in eq 1 being unity is indeed the usual
DT ) (LG)T ) GL
7. Diffusion of a Gas in a Liquid We consider a closed system under isothermal conditions containing a nonvolatile liquid in contact with a finite volume of gas that is sparingly soluble in the liquid. On the dissolution of the gas in the liquid, the pressure continually depletes to an equilibrium value characterized by the gas solubility under the prevailing pressure and temperature. Diffusion could occur free from the effects of natural convection because concentration gradients for a sparingly soluble gas will be small. The formulation of the problem is straightforward in that diffusion in the liquid is described by the unsteady-state diffusion equation with the boundary condition as follows. At the top surface, the concentration is at the solubility that could be regarded as being proportional to the prevailing pressure, while the concentration gradient at the bottom surface vanishes because of its impenetrability. The pressure varies at a rate determined by the rate at which the gas molecules enter the liquid, i.e., the flux at the top surface. The fine details are available elsewhere;9 here we merely outline the problem directly in dimensionless form. The dimensionless concentration is chosen to be the ratio of the local, prevailing concentration in the liquid to that if it were uniformly saturated at the initial pressure of the gas. The dimensionless gas concentration v(x, t) satisfies
∂v(x,t) ∂2v(x,t) , t > 0, 0 < x < 1 ) ∂t ∂x2
(2)
with boundary conditions
|
∂v(x,t) ∂t
x)1
) 0,
v(0,t) ) v1(t)
(3)
the first of which represents impenetrability of the container’s bottom to the solute, whereas the second relates to equilibrium at the gas-liquid interface. The
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decrease in pressure as a result of diffusion into the liquid is described by the equation
|
∂v(x,t) ∂x
R
) x)0
∂v(0,t) ∂t
(4)
If initially there were no dissolved gas in the liquid we have the initial condition
v(x,0) ) 0, 0 < x < 1, v1(0) )1
[ ]
v(x,t) v(t) t v1(t)
[ | ]
-
dv(t) ) Lv(t), dt
∫01 f(x) g(x) dx + rf1g1, f)
[ ] f(x) f1
g)
[ ] g(x) g1
(6)
One senses that the (positive) weight r must somehow measure the preference of the solute between the liquid and the gas space. The negotiation for symmetry results in a weight precisely equal to the constant equilibrium molar ratio of the solute in the gas space to that in the given volume of liquid that represents the capacity of the gas space for the solute relative to that of the liquid and is given by R-1. Thus for f and g constrained to satisfy eq 3 we have
〈Lf,g〉 ) 〈f,Lg〉 provided r ) 1/R in eq 6. What must this be but a resonance in the physics to the notion of symmetry in the mathematics? 8. Convective Diffusion in Fully Developed Flow We consider the entry of a fluid in fully developed flow (through a conduit of constant cross-section shown Figure 1) into a zone marked by imposition of some peripheral condition at the tube wall, which results in the diffusion of energy. Diffusion occurs both radially and axially; the flow is in the x-direction but its velocity is only a function of y. The differential equation in dimensionless form is given by
∂2u ∂u 1 ∂ ∂u y + 2 - V(y) ) 0, y ∂y ∂y ∂x ∂x
( )
0 < y < 1, - ∞ < x < ∞ (7)
with boundary conditions
The boundary conditions in eq 3 restrict the domain of this system vector to a linear subspace. That this is a natural description of the system is reflected in the coupled mass balance equations for the gas phase as well as the liquid phase as a result of diffusion; these equations, viz., 2 and 4, imply that the rate of change of the system vector is given by a “matrix” differential operator acting on the system vector to produce another vector of the same kind, viz., a continuous function over the liquid domain and a number meant for the gasliquid interface. The operator, denoted L, is shown below in terms of which the boundary value problem represented by eqs 2-5 can be succinctly stated
d2 0 dx2 Lt ; d -R 0 dx x)0
〈f,g〉 t
(5)
The first issue is the Hilbert space to which the system vector belongs. The natural choice of course is the concentration field, v(x,t) in the liquid that controls the diffusion process. However, if the liquid initially contained no dissolved gas, the concentration field would vanish identically as in eq 5. No diffusion process can be initiated from such an initial state without having to bring to bear the presence of the solute gas at the liquid surface. The concentration at a single point, however, would seem not to be of consequence because the effect of an initial condition usually manifests itself through its inner product with each of the eigenfunctions of the diffusion operator; in this case all the inner products will vanish, producing no diffusion into the liquid! Here one begins to sense the special role of the solute concentration at the liquid surface (or the pressure in the gas space) in initiating the diffusion process. (As the solute diffuses into the interior of the liquid, fresh replenishment from the gas phase may be presumed to maintain the concentration at the solubility.) This special role of the surface concentration earns it specific mention over and above the stipulation of the entire concentration profile in the liquid domain. The system vector is thus a function defined over the liquid domain representing the (dimensionless) concentration profile there and a number that represents the (dimensionless) concentration at the gas-liquid interface or the pressure in the gas space. The dimensionless pressure is defined by dividing the pressure by its initial value. We write for the system vector
-
The inner product between any two vectors f and g in this space superposes that between the functions and a contribution from the product of the numbers with a positive weight that must be negotiated for self-adjointness
v(0) )
[] 0 1
The foregoing formulation of the Hilbert space, spurred primarily by physical considerations, is essential to accomodate the second issue, which is that of symmetry.
|u| < ∞
u(x,1) )
{
0, - ∞ < x < 0 1, 0 e x e ∞
(8)
The bidirectional convective diffusion operator in eq 7 is essentially nonself-adjoint because it is unresponsive to any form by symmetrization by manipulating the inner product in the Hilbert space of functions defined over the flow region. This situation manifests an interesting alternative form of symmetry. Regarding the axial coordinate as one along which the transverse temperature profile evolves, the problem is to seek the temperature at any x and y. Suppose one were to consider along with the temperature profile u(x,y) the total axial flow of energy v(x,y) (by diffusion and convection) over the partial crosssection of the tube extending up to radius y:
v(x,y) )
[
∫0y -
]
∂u(x,η) + V(η) u(x, η) η dη ∂x
(9)
This definition of v(x,y) implies that its radial derivative is simply the total energy flux weighted with the local circumference at y, a relation that can be rearranged
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Figure 1.
to obtain an expression for the axial derivative of u(x,y), i.e.,
∂u(x,y) 1 ∂v(x,y) ) V(y) u(x,y) ∂x y ∂y
(10)
The energy transport process can now be described by differential equations for the axial evolution of u(x,y) and v(x,y) written to display the axial derivatives on the left-hand side. Whereas the equation for v(x,y) has just been outlined above, that for v(x,y) is obtained by simply recognizing that its change in the axial direction is brought about by radial diffusion at the periphery. Thus
∂u(x,y) ∂v(x,y) )y ∂x ∂y
(11)
Alternatively, we may view this formulation as mathematically decomposing the second-pair partial differential equation into a pair of first-order partial differential equations in the two dependent variables u(x,y) and v(x,y). The procedure is readily verified by eliminating the function v(x,y) by calculating its mixed second derivative from eqs 10 and 11 and recovering eq 7 in the process. Equations 10 and 11 are conveniently cast in the following “matrix” form
[ ] [
][ ]
V(y) ∂ u(x,y) -y-1(∂/∂y) u(x,y) ) v(x,y) ∂x v(x,y) y(∂/∂y) 0
(12)
The Hilbert space at hand is a collection of ordered pairs of functions of the radial coordinate. Each consists of functions square integrable over the unit interval, the first with weight y and the second with weight y-1. The transport process produces an operator on this Hilbert space that is concerned with the reciprocal effects of the temperature profile and the axial energy flow profile. What is most pertinent to our discussion is that this operator is symmetrizable! In other words, the reciprocal relation between the temperature profile and the axial energy admits an inner product in the Hilbert space with respect to which the operator is self-adjoint. Although the precise derivation of this symmetry is mired somewhat in algebraic details,10 it can be broadly understood from the “hermitian” character of the 2 × 2 “matrix” differential operator, as its off-diagonal coefficients y(∂/∂y) and -y-1(∂/∂y) are adjoint images of each other. It is thus in the category of self-adjoint partitioned operators mentioned in section 7. The inner product between any two real-valued elements u t [v1(y),u2(y)] and v t [v1(y), v2(y)] in the Hilbert space is chosen to be
〈u,v〉 t
∫01[ηu1(η) v1(η) + η-1u2(η) ν2(η)] dη
(13)
differential operator owes its ultimate symmetry to the homogeneous boundary conditions derived from eq 8, viz.,
u1(1) ) 0
u2(0) ) 0
(14)
the latter arising from the application of boundedness of u (and hence its derivative) at y ) 0 to eq 11. Letting the matrix differential expression in eq 12 be L, it is now a simple matter to use the inner product of eq 13 to arrive at
〈Lu,v〉 - 〈u,Lv〉 ) -u2(1) v1(1) + v2(1) u1(1) + u2(0) v1(0) - v2(0) u1(0) (15) The application of the boundary conditions of eq 14 to both u and v makes the right-hand side of eq 15 vanish so that one has the self-adjointness
〈Lu,v〉 ) 〈u,Lv〉 in which the boldface on L is to signify its restriction to elements in the Hilbert space that satisfy eq 14. Whereas its consequent symmetry is “a thing of the beauty”sour etheral themes, its gift of analytical solutions to numerous boundary value problems complies also with the earthly demands of practicality. Thus the Graetz problem with axial diffusion10 and several other boundary value problems of interest to engineers8,9 can be solved in this manner.
9. Concluding Remarks It is our hope that this discussion has aroused the aesthetic instincts of the reader and will have shown that concept and the consequences of self-adjointness also play to the practical demands of engineering by providing a structure of computation. The algebraic elementssinner products, eigenvalues and vectors, orthogonal expansions, etc.sseem to bond with the physical in a way that has often been thought remarkable, or even unreasonable. It is this congruence of beauty that makes it worthwhile to study the mathematics and to design heat exchangers. As Horace (Quintus H. Flaccus, not Greeley!) said, “You get full marks for mixing the useful with the beautiful and delighting at the same time as you instruct.”11 Literature Cited (1) Aquinas, T. Summa Theologica i, q.5, a.4, ad 1.
which renders the matrix differential operator on the right-hand side of eq 12 “formally” self-adjoint. The
(2) Hough, G. An Essay on Criticism; Norton: New York, 1966, Sec. 16.
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(3) Chandrasekhar, S. Truth and Beauty: Aesthetics and Motivations in Science; University of Chicago Press: Chicago, 1987. (4) Stewart, I.; Golubitsky, M. Fearful Symmetry: Is God a Geometer? Blackwell: Oxford, 1992. (5) Dodgson, C. L. (Lewis Carroll). The Hunting of the Snark. Fit the fifth, Stanza 24. In The Complete Works of Lewis Carroll. Modern Library: New York (Undated). (6) Wei, J. T.; Prater, C. D. The Structure and Analysis of Complex Reaction Systems. Adv. Catal. 1962, 13, Ch. 5. (7) Onsager, L. Reciprocal relations in Irreversible Processes. Phys. Rev. 1931, 37, 405-426. (8) Ramkrishna, D.; Amundson, N. R. Linear Operator Methods in Chemical Engineering with Applications to Transport and Chemical Reaction Systems.
(9) Ramkrishna, D. Advances in Transport Processes, Vol. III Majumdar, A. S., Mashelkar, R. A., Eds. Wiley Easter: New Delhi, 1983. (10) Papoutsakis, E.; Ramkrishna, D.; Lim, H. C. The Extended Graetz Problem with Dirichlet Wall Boundary Conditions. (11) Horace. Ars Poetica 343. See Brink, C. O. Horace on Poetry. Cambridge University, Press: Cambridge, 1963.
Received for review April 20, 1998 Revised manuscript received July 23, 1998 Accepted July 29, 1998 IE980252X
