EEFFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES
CARLOS VALERO VALDESDEPARTAMENTO DE MATEMATICASUNIVERSIDAD DE GUANAJUATOGUANAJUATO, GTO, MEXICO
Abstract.
The purpose of this paper is to provide equations to model theevolution of effective diffusion over a Riemannian fiber bundle (under the hy-pothesis of infinite diffusion rate along compact fibers). These equations areobtained by projecting the diffusion equation onto the base manifold of thefiber bundle. The projection (or dimensional reduction) is achieved by inte-grating the diffusion equation along the fibers of the bundle. This work gen-eralizes an put into a general framework previous work on effective diffusionover channels and the interfaces between curved surfaces. Introduction
Understanding spatially constrained diffusion is of fundamental importance invarious sciences, such as biology, chemistry and nano-technology. However, solvingthe diffusion equation in arbitrarily constrained geometries is a very difficult task.One way to tackle it consists in reducing the degrees of freedom of the problemby considering only the main direction(s) of transport. For example, the study ofdiffusion on thin channels can be carried out by reducing several spatial degrees offreedom to a single one by means of a projection method. More concretely, considera process in a channel modeled by a density function P that obeys the diffusionequation ∂P∂t ( x, t ) = D ∆ P ( x, t ) , subject to the restriction that there is no density flow along the channel’s wall(s).We can construct an effective density function by letting(1.1) ρ ( u, t ) = lim h (cid:55)→ Total concentration of P in R ( h ) h , where the region R ( h ) is the section of the channel within two transversal crosssections that are an arc distance of h apart over a base curve that “follows” thechannel’s geometry (see Figure 1.1), and the variable u is the arc-length parame-ter on this curve. It turns out that this effective density function ρ obeys in anapproximate manner an equation of the form (known as a generalized Fick-Jacobsequation)(1.2) ∂ρ∂t ( u, t ) = ∂∂u (cid:18) σ ( u ) D ( u ) ∂∂u (cid:18) ρ ( u, t ) σ ( u ) (cid:19)(cid:19) , Date : 26 January 2015.Partially supported by CONACyT grant 135106. a r X i v : . [ m a t h - ph ] F e b FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 2
Figure 1.1.
Region between to transversal cross section of a channel.where σ is given by(1.3) σ ( u ) = lim h (cid:55)→ Area ( R ( h )) h . The function D is known as the effective diffusion coefficient and it encapsulatesthe effect of the channel’s geometry on the diffusion process along the base curve.Much work has been done (see [1, 8, 3, 4, 12, 6, 5, 7, 10, 11]) to find explicitformulas for D in terms of geometrical quantities associated to the channel, so thatthe Fick-Jacobs equation models the evolution of ρ as closely as possible . We candistinguish two cases.(1) Infinite transverse diffusion rate.
In this case it is assumed that the densityfunction P stabilizes instantly in the transversal directions of the channel.In mathematical terms this means that P is constant along these transver-sal directions. This assumption results in an effective diffusion coefficientthat depends on the curvature function of the base curve and -th ordergeometrical quantities of the cross sections, such as width or area.(2) Finite transverse diffusion rate.
In this case the finite time of transversalstabilization of P is taken into account. This is characterized mathemati-cally in that the resulting formulas for D involve the curvature function ofthe base curve, and tangential and curvature information of the channel’swall(s).The selection of the base curve is very important in the dimension-reduction tech-nique described above. This is demonstrated by the fact (see [11, 10]) that if forchannels of constant width the base curve is chosen properly, then the formulas for D coincide for the finite and infinite transversal diffusion rate cases. FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 3
Motivated by the above discussion, we know describe the main purpose of thepaper. We develop a very general theory, in the infinite transversal diffusion ratecase, for projecting the diffusion equation in a space of dimension n to a base spaceof dimension m with m < n . We do this in the context of the theory of fibrebundles . Such objects have a total space E , a base space M and a projection map π : E → M . In the case discussed above, the total space E is the channel, M isthe base curve, and π sends points on the transversal cross sections (fibers) to theirbase point in the base curve. The process of passing from the density P to thereduced density ρ is a particular case of a very well known construction: that ofintegrating a differentiable form along the fibers of the bundle. Using these toolswe are able to give global and coordinate-free proofs of all our results. In thisgeneral setting, the effective diffusion D becomes an endomorphism of the tangentbundle of M , i.e for every x in M we have that D ( x ) is a linear map in the tangentspace T x M of M . We will compute the effective diffusion using local frames insteadof local coordinates, which results in a simpler and more geometric way of doingcalculations. The paper is organized as follows • In section 2, we show how to reduce the continuity equation in the fiberbundle E to a reduced continuity equation in its base space M . We provethat if Fick’s law holds on E (for a constant diffusion coefficient D ) and ifwe have infinite diffusion rate in fiber direction, then the reduced continuityequation becomes a diffusion equation in M (see Proposition 3). This lastequation involves an effective diffusion coefficient D , which is a bundleendomorphism of the tangent space of M . • In section 3 we compute the effective diffusion for channels of constantwidth over arbitrary curves on the plane. We have obtained this resultpreviously in [11], and our re-derivation of the formula serves as a test caseof our general theory. • In section 4 we compute the effective diffusion endomorphism D correspond-ing to the interface of two equidistant surfaces in 3-dimensional space. Wedo this by showing that the principal directions of the base surface areeigenvectors of D , and then computing the eigenvalues of D in these direc-tions. Using this result, we show that the calculation of D made by Ogawain [9] for an elliptical cylinder, is just an approximation to ours obtainedby only considering the two first terms in a series expansion of the functionarctanh (the inverse of the hyperbolic tangent). • In section 5 we compute the effective diffusion on the surface (not theinterior) of a circular channel over an arbitrary curve in R . The mainpoint of the calculation is to illustrate how our techniques still apply tofibre bundles whose fibers are manifolds without boundary (in this case,circles). • In the Appendix we make a brief review of the geometrical concepts neededfor the construction of the effective diffusion endomorphism.2.
The effective diffusion equation
Let E be fiber bundle over an m -dimensional manifold M , having compact fibers(with or without boundary) of dimension k , and projection map π : E → M . Wewill assume that M and E are orientable, and with Riemannian metrics <, > E and <, > M . FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 4
The continuity equation.
The continuity equation on the fibre bundle E is givenby(2.1) ∂P∂t + div ( J ) = 0 , where the density function P is a time dependent function on E , and the densityflow J is a time dependent vector field in E . The divergence of J is given by(2.2) div ( J ) = ( ∗ ( d ( ∗ J (cid:91) ))) (cid:93) , where d is the differential operator acting on differentiable forms in E , ∗ is theHodge star operator, and the (cid:91) -operator converts vector fields to 1-forms (see theAppendix). Integrating the continuity equation along the fibers.
Since we are assumingthat the fibers of E are compact, by integrating along them we can define theoperator π ∗ sending l -forms in E to ( l − k )-forms in M . For any l − form ω on E we define π ∗ ( ω ) x ( X , . . . , X l − k ) = ˆ π − ( x ) β ω where β ω ( Y , . . . , Y k ) = ω ( ˜ X , . . . , ˜ X l − k , Y , . . . , Y k ) and ˜ X i is a lift of X i , i.e Dπ ( ˜ X i ) = X i . Observe that ˜ X i is defined on the wholefiber π − ( x ) . By applying ∗ to 2.1 and using formula 2.2 we get(2.3) ∂∂t ( ∗ P ) + d ( ∗ J (cid:91) ) = 0 . If we apply π ∗ and then ∗ to the above equation we obtain(2.4) ∂ρ∂t + ∗ ( π ∗ ( d ( ∗ J (cid:91) ))) = 0 , where(2.5) ρ = ∗ ( π ∗ ( ∗ P )) = π ∗ ( P µ E ) µ M , and µ E and µ M be the metric volume forms in E and M . Remark.
Formula 2.5 is a generalization of formula 1.1.
Neumann Boundary conditions.
We will need adequate boundary conditionon J in order to write equation 2.4 as a continuity equation in M . If the fibers of E are manifolds without boundary then π ∗ commutes with d (see [2, pg. 62]), i.efor any differential form ω we have that(2.6) d ( π ∗ ω ) = π ∗ ( dω ) When the fibers of E are manifolds with boundary we will assume that ω vanisheson all the vectors perpendicular to the boundary ∂E of E . This last condition isequivalent to the assuming that there is no density flow across ∂E , i.e J is parallelto the boundary of E . This last condition ensures that formula 2.6 still holds. FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 5
Integrating the density flow along the fibers.
We will now use formula 2.6 toshow that we can write 2.4 as a continuity equation in M . We need to find a timedependent vector field j in M such thatdiv ( j ) = ∗ ( π ∗ ( d ( ∗ J (cid:91) ))) . Using formula 2.6, and the definition of the divergence of j , we can write the aboveformula as ∗ ( d ( ∗ j (cid:91) )) = ∗ ( d ( π ∗ ( ∗ J (cid:91) ))) . This last equation is satisfied if we let(2.7) j = ( − m − ( ∗ ( π ∗ ( ∗ J (cid:91) ))) (cid:93) , where the (cid:93) -operator converts 1-forms to vector fields (see Appendix). The effective continuity equation.
We will refer to the time dependent function ρ given by formula 2.5 as the effective density function, and to the time dependentvector field j given by 2.7 as the effective density flow . We proved above that theseobjects satisfy the equation(2.8) ∂ρ∂t + div ( j ) = 0 , which we will refer to as the effective continuity equation . The diffusion equation.
Fick’s law establishes that J ( x, t ) = − D ( x ) ∇ P ( x, t ) , where for x in E we have that D ( x ) is a linear operator from T x E to T x E , i.e anendomorphism of T E x . The simplest choice of D is to let it be scalar multiplicationby a constant D . By using more general D ’s we can model the inhomogeneity andanisotropy of E . Assuming Fick’s law, the continuity equation in E becomes the diffusion equation (2.9) ∂P∂t ( x, t ) = div ( D ( x ) ∇ P ( x, t )) , In our work we will always assume that the endomorphism D on T E is multiplica-tion by a positive scalar D , in which case the diffusion equation becomes ∂P∂t ( x, t ) = D ∆ P ( x, t ) , where the laplacian operator ∆ applied to P is ∆ P = div ( ∇ P ) . Problem.
If Fick’s law holds in E for D = D , does the effective density ρ obeysa diffusion equation in M ?There is an important case when we can answer the above question positively(see Proposition 3 in the next paragraph). FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 6
An effective diffusion equation for infinite fiber diffusion rate.
If the fibersof E are “small enough” compared to the “size” of M , then it is to be expected thatthe density P will stabilize faster along the fibers than along M . If we assumethat this stabilization occurs infinitely fast, then P must be constant along thefibers of E . We borrow the nomenclature from the physics literature, and refer tothis situation by saying that there is an infinite fiber diffusion rate . Under thisassumption, can write P = π ∗ Q = Q ◦ π for a time dependent function Q in M . Using 2.5 and the projection formula (see[2, pg. 63]) , we obtain ρ = π ∗ ( π ∗ ( Q ) µ E ) µ M = Qπ ∗ ( µ E ) µ M , which allows us to obtain Q in terms of ρ as(2.10) Q = ρσ , where(2.11) σ = π ∗ ( µ E ) µ M . Observe that if R is a region in M then(2.12) ˆ R σµ M = ˆ π − ( R ) µ E = vol E ( π − ( R )) . Remark.
Formula 2.11 is a generalization of that given by 1.3.If Fick’s law holds in E for D = D , then we have J (cid:91) = − D d ( π ∗ Q ) = − D π ∗ ( dQ ) , where π ∗ is the pull-back of forms under π . Hence, we can write the effective flow2.7 as(2.13) j = − ( − m − D ( ∗ ( π ∗ ( ∗ ( π ∗ ( dQ ))))) (cid:93) . If we define D by(2.14) D = ( − m − (cid:18) D σ (cid:19) ( (cid:93) ◦ ∗ ◦ π ∗ ◦ ∗ ◦ π ∗ ◦ (cid:91) ) , where ◦ stands for composition of operators, then by combining formulas 2.10 and2.13 we obtain(2.15) j = − σ D ( ∇ Q ) = − σ D (cid:16) ∇ (cid:16) ρσ (cid:17)(cid:17) . The above formulas now lead to the following result.
Proposition 1.
For the case of infinite fiber diffusion rate, the effective densityfunction ρ satisfies the equation (2.16) ∂ρ∂t ( x, t ) = div (cid:18) σ ( x ) D ( x ) (cid:18) ∇ (cid:18) ρ ( x, t ) σ ( x ) (cid:19)(cid:19)(cid:19) , where D : T X → T X is the vector bundle morphism defined in formula 2.14.
FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 7
Proof.
This follows from the effective continuity equation 2.8 and formula 2.15. Asdefined by formula 2.14, D sends vector fields in M to vector fields in M . To provethat for any x in M it is in fact linear map D ( x ) : T x M → T x M , we show that D ( f X ) = f D ( X ) for any smooth function f : M → R and vector field X in M .We have that ( ∗ ◦ π ∗ ◦ (cid:91) )( f X ) = ( π ∗ f )( π ∗ ( X (cid:91) )) . Using this and the projection formula (see [2, pg. 63]), we obtain that π ∗ ( ∗ ◦ π ∗ ◦ (cid:91) )( f X ) = f π ∗ ( π ∗ ( X (cid:91) )) , and hence D ( f X ) = f D ( X ) . (cid:3) Definition 2.
We will refer to the endomorphism D of T M , given by formula 2.14,as the effective diffusion endomorphism.Remark.
Equation 2.16 is a generalization of 1.2.
Proposition 3.
For the case of infinite fiber diffusion rate, there exists a metricin M such that the effective density function ρ satisfies the diffusion equation (2.17) ∂ρ∂t ( x, t ) = div ( D ( x ) ∇ ρ ( x, t )) , where D is the effective diffusion endomorphism. Furthermore, this choice of metricis such that for any region R in M we have thatvol M ( R ) = vol E ( π − ( R )) . Proof.
Let <, > E and <, > M be any metrics in E and M . Define a new metric <, > (cid:48) M in M by the formula < X, Y > (cid:48) M = σ /m < X, Y > M , for σ defined in 2.11. Then, since σ /m = (cid:18) π ∗ ( µ E ) µ M (cid:19) /m we have that σ (cid:48) M = π ∗ ( µ E ) µ (cid:48) M = π ∗ ( µ E )( σ /m ) m/ µ M = 1 . Equation 2.17 then follows directly from Proposition 1, and the last part of theProposition follows from equation 2.12. (cid:3) Constant-width channels on the plane
In this section we compute the effective diffusion function of a channel of constantwidth w over a curve C on the plane. Such a channel can be represented as the set E = { x + vN ( x ) | x ∈ C and − w/ ≤ v ≤ w/ } , where N is a unit normal field to C . For a reasonable curves (e.g compact) andsmall w , the space E is a fibre bundle with projection map π : E → M given by π ( p ) = x where p = x + vN ( x ) . FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 8
Figure 3.1.
Lifting tangent vector T to vector (1 − κv ) T along afiber π − ( x ) Let T be a unit tangent field to C that makes the frame T, N positively oriented.To perform integration over the fibers of E we need to compute the lift of T to E .If we define T ( p ) = (1 − κ ( x ) v ) T ( x ) N ( p ) = N ( x ) then T is such a lift of T (see Figure 3.1), and Dπ ( N ) = 0 . Let T ∗ be the dual fieldto T , and T ∗ , N ∗ be the dual frame to T , N . The matrix of <, > E in the frame T , N is given by g = (cid:18) (1 − κv )
00 1 (cid:19) . Computing σ . The volume element in E is µ E = (1 − κv ) T ∗ ∧ N ∗ , and hence π ∗ ( µ E ) = (cid:32) ˆ w/ − w/ (1 − κv ) dv (cid:33) T ∗ = wT ∗ . Since the volume form in C is T ∗ , we conclude that σ = wT ∗ T ∗ = w. Computing D . Observe that π ∗ ( T ∗ ) = T ∗ , and by formula 7.1 in the Appendix we have that ∗ ( T ∗ ) = g det( g ) / N ∗ = (1 − κv ) − N ∗ . FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 9
Hence π ∗ ( ∗ ( π ∗ ( T ∗ ))) = ˆ w/ − w/ (1 − κv ) − dv = 1 κ log (cid:18) κw/ − κw/ (cid:19) Using formula 2.14 we obtain D = D κw log (cid:18) κw/ − κw/ (cid:19) = (cid:18) D κw (cid:19) arctanh ( κw/ . (3.1)We obtained this formula in [11] by different methods.4. The interface between two equidistant surfaces in 3-d space
Let S be an orientable surface in R and N a unit normal field to this surface.For small w > and a “reasonable surface” the space(4.1) E = { x + vN ( x ) | x ∈ S and − w/ ≤ v ≤ w/ } . is a fibre bundle over S with projection map π : E → M given by π ( p ) = x, where p = x + vN ( x ) . The surface S has principal directions fields T and T , with corresponding principalcurvatures κ and κ . If we define T ( p ) = (1 − κ ( x ) v ) T , T ( p ) = (1 − κ ( x ) v ) T , N ( p ) = N ( x ) , then T and T are lifts of T and T , and Dπ ( N ) = 0 . The metric in <, > E isrepresented in the T , T , N frame by g = (1 − κ v ) − κ v )
00 0 1 . We will let T ∗ , T ∗ , N ∗ be the dual frame to T , T , N . Computing σ . The volume form in E is µ E = (1 − κ v )(1 − κ v ) T ∗ ∧ T ∗ ∧ N ∗ , and hence π ∗ ( µ E ) = (cid:32) ˆ w/ − w/ (1 − κ v )(1 − κ v ) dv (cid:33) T ∗ ∧ T ∗ , = w (1 + κ κ w / T ∗ ∧ T ∗ . Since the volume form in M is T ∗ ∧ T ∗ , we conclude from formula 2.11 that σ = w (1 + κ κ w / . FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 10
Computing D . Using formula 7.1 in the Appendix we obtain that ∗ ( T ∗ ) = g det( g ) / T ∗ ∧ N ∗ = (cid:18) − κ v − κ v (cid:19) T ∗ ∧ N ∗ , ∗ ( T ∗ ) = − g det( g ) / T ∗ ∧ N ∗ = − (cid:18) − κ v − κ v (cid:19) T ∗ ∧ N ∗ . From these formulas and the identities ∗ T ∗ = T ∗ , ∗ T ∗ = − T ∗ , we obtain ∗ ( π ∗ ( ∗ ( T ∗ ))) = − (cid:32) ˆ w/ − w/ (cid:18) − κ v − κ v (cid:19) dv (cid:33) T ∗ , ∗ ( π ∗ ( ∗ ( T ∗ ))) = − (cid:32) ˆ w/ − w/ (cid:18) − κ v − κ v (cid:19) dv (cid:33) T ∗ . By evaluating the above integrals, using formulas π ∗ ( T ∗ ) = T ∗ , π ∗ ( T ∗ ) = T ∗ , and the fact that the matrix of the metric <, > M is the identity, formula 2.14 yields D ( T ) = D T and D ( T ) = D T . where D = D ( wκ κ + 2( κ − κ ) arctanh ( κ w/ κ w (1 + κ κ w / , (4.2) D = D ( wκ κ − κ − κ ) arctanh ( κ w/ κ w (1 + κ κ w / . (4.3)We have just proved the following result. Proposition 4.
Let S be a surface in R with principal direction fields T and T , and corresponding principal curvatures κ and κ . For the bundle E given by4.1, we have that the fields T and T are eigenvectors of D with correspondingeigenvalues D and D given 4.2 and 4.3.Remark. At an umbilical point x of S (i.e where κ ( x ) = κ ( x )) the above propo-sition is still valid for any pair of orthonormal vectors T , T in T x S .It is important to observe that if we want to write the effective diffusion equation2.17 in coordinates, we need to express the principal direction fields T and T interms of the corresponding coordinate fields. The reason for this is that the diver-gence operator that enters into the effective diffusion equation needs coordinatesfor its computation. Remark.
We can express eigenvalues of D in terms of the gaussian and mean cur-vatures K = κ κ and H = 12 ( κ + κ ) by using the identities κ = H + (cid:112) H − K and κ = H − (cid:112) H − K. We now discuss some applications of Proposition 4 to specific families of surfaces.
FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 11
Spheres.
In this case we have that the principal curvatures κ and κ satisfy κ = κ = 1 /r, where r is the radius of the sphere. From formulas 4.2 and 4.3, we obtain that theeigenvalues of D are(4.4) D = D = D r r + w = D w r . Recovering the one dimensional case.
Let C be any curve on the plane withcurvature function κ , and let S = C × R ⊂ R . The principal fields of S are theunit tangent T to the curve and (0 , , , with corresponding eigenvalues κ and .Using formulas 4.2 and 4.3 we obtain D = (cid:18) D κw (cid:19) arctanh ( κw/ , D = D . Hence, the eigenvalue D of D coincides with the case of curves in the plane dis-cussed in section 3 (see Formula 3.1).We will now show how to write equation 2.16 in local coordinates. Consider thecoordinates ( s, z ) in S where s is the arc-length parameter of the curve C and z isstandard z -coordinate in R . The coordinate fields ∂∂s and ∂∂z are the principal direction fields of S , and D (cid:18) ∂∂s (cid:19) = (cid:18) D arctanh ( κw/ κw (cid:19) ∂∂s , D (cid:18) ∂∂z (cid:19) = D ∂∂z . Since the metric matrix is the identity, we have that ∇ ρ = ∂ρ∂s ∂∂s + ∂ρ∂z ∂∂z . We conclude that in this case formula 2.16 becomes(4.5) ∂ρ∂t = (cid:18) D arctanh ( κw/ κw (cid:19) ∂ ρ∂s + ∂ ρ∂z . If we use the series expansionarctanh ( x ) = x + x x . . . , we obtain D arctanh ( κw/ κw = 1 + w κ
12 + w κ
180 + · · ·
If we use only the first two terms of the above series in equation 4.5 we obtain ∂ρ∂t = (cid:18) w κ (cid:19) ∂ ρ∂s + ∂ ρ∂z . This last formula is that obtained by Ogawa in [9].
FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 12
Figure 4.1.
Eigenvalues D and D of the effective diffusion D for D = 1 , on a torus with inner radius r = 1 , outer radius R = 2 and width w = 1 / . The torus.
For a torus with inner radius r and outer radius R we have that κ = − /r and κ = − cos( θ ) R + r cos( θ ) , where θ is the variable parametrizing the parallels of the torus. In Figure 4.1 weshow the graphs of D and D obtained by using the above values of κ and κ forspecific values of r, R and w .5. Effective diffusion in the surface of a tube
Let C be a curve in three dimensional space, and let T, N, B be the correspondingSerret-Frenet frame. We will let E be the set of points of the form (for a constantradius r ) p = x + r cos( θ ) N + r sin( θ ) B where x ∈ C. We want to construct a lift of T to E . To do this, consider x = x ( s ) , T = T ( s ) , N = N ( s ) and B = B ( s ) as functions of the arc length parameter s of C . From theformula π ( p ( s )) = x ( s ) we obtain Dπ (cid:18) dpds (cid:19) = T and Dπ (cid:18) dpdθ (cid:19) = 0 , where (using the Serret-Frenet formulas) dpds = (1 − κr cos( θ )) T + τ r ( − sin( θ ) N + cos( θ ) B ) ,dpdθ = − r sin( θ ) N + r cos( θ ) B, FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 13 for κ and τ the curvature and torsion of C . Hence, if we define T = (1 − κr cos( θ )) T, N = − r sin( θ ) N + r cos( θ ) B, then T is lift of T and Dπ ( N ) = 0 . The metric matrix in the T , N frame is g = (cid:18) (1 − κr cos( θ )) r (cid:19) . Computing σ . The volume form in E is µ E = r (1 − κr cos( θ )) T ∗ ∧ H ∗ , so that π ∗ ( µ E ) = (cid:18) ˆ π r (1 − rκ cos( θ )) dθ (cid:19) T ∗ = 2 πrT ∗ and hence σ = 2 πrT ∗ T ∗ = 2 πr. Computing D . We have that π ∗ ( T ∗ ) = T ∗ , and ∗T ∗ = g det( g ) / H ∗ = r (1 − κr cos( θ )) − H ∗ , so that π ∗ ( ∗T ∗ ) = ˆ π (cid:18) r − κr cos( θ ) (cid:19) dθ = 2 πr rκ (cid:32)(cid:114) − rκ − (cid:33) T ∗ . Hence D = D (1 + rκ ) (cid:32)(cid:114) − rκ − (cid:33) . = D (cid:114) − r κ Conclusions
We have shown that the diffusion equation on the total space of a fiber bundlecan be projected onto a diffusion equation on its base space, under the hypothesisof infinite diffusion rate along the fibers. We provided a general formula for theeffective diffusion endomorphism of the reduced diffusion equation, that we laterapplied to obtain explicit formulas for diverse fiber bundles. Of particular interestwas the computation of the effective diffusion endomorphism associated to the in-terface of two equidistant surfaces in 3-dimensional space, in terms of the principalcurvatures of the base surface.
FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 14 Appendix
The sharp and flat operators.
We can see a 1-form α as a vector field α (cid:93) definedby < α (cid:93) , X > = α ( X ) , where <, > is the metric of the space under consideration. Similarly, a vector field X can be seen as a 1-form X (cid:91) defined by X (cid:91) ( Y ) = < X, Y > . For a local frame X , . . . , X n the metric can be expressed as a symmetric matrix g with coefficients g ij = < X i , X j >, and we will write g ij = ( g − ) ij . Let X , . . . , X n be the 1-forms forming the dual frame to X , . . . , X n , so that X i ( X j ) = δ ij . For a vector field X = n (cid:88) i =1 a i X i we have that X (cid:91) = n (cid:88) i =1 a i X i where a i = n (cid:88) j =1 g ij a j . For a 1-form α = n (cid:88) i =1 α i X i we have that α (cid:93) = n (cid:88) i =1 α i X i where α i = n (cid:88) j =1 g ij α j . Hodge star operator.
The Hodge star operator ∗ maps l -forms to ( n − l ) forms,where n is the dimension of the space under consideration, and it is defined so thatfor l -forms ω and η we have that ω ∧ ( ∗ η ) = < ω, η > µ, where µ is the volume form of the metric. For monomials ω = α ∧ . . . ∧ α l and η = β ∧ . . . β l , where the α i ’s and the β j ’s are 1-forms, we have that < ω, η > = det( < α i , β j > ) where < α i , β j > = < α (cid:93)i , β (cid:93)j > . The ∗ -operator satisfies the duality relation, where for an l -form ω we have that ∗ ∗ ω = ( − l ( n − l ) ω. If for a local frame of vector fields X , . . . , X n we have the metric matrix g ij = (cid:104) X i , X j (cid:105) , then the coefficients g ij of g − are g ij = < X i , X j > . FFECTIVE DIFFUSION ON RIEMANNIAN FIBER BUNDLES 15
The metric volume form is given by µ = det( g ) / X ∧ . . . ∧ X n . Using the above formulas we obtain ∗ ( X ∧ . . . ∧ X n ) = det( g ) − / , and(7.1) ∗ X i = det( g ) / n (cid:88) j =1 g ij ι j ( X ∧ . . . ∧ X n ) , where ι j ( X ∧ . . . ∧ X n ) = ( − j +1 X ∧ . . . ∧ X i − ∧ ˆ X j ∧ X i +1 ∧ . . . ∧ X n , and the hat in ˆ X j indicates that that term has been removed as a factor in theabove wedge product. References [1] Anatoly E. Antipov, Alexander V. Barzykin, Alexander M. Berezhkovskii, Yurii A.Makhnovskii, Vladimir Yu. Zitserman, and Sergei M. Aldoshin. Effective diffusion coeffi-cient of a brownian particle in a periodically expanded conical tube.
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