Laplacian transport towards irregular interfaces: the mathematics
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1 Laplacian transport towards irregular interfaces: the mathematics Denis S GREBENKOV Laboratoire de Physique de la Matière Condensée CNRS Ecole Polytechnique, France denisgrebenkov@polytechniqueedu Web: Denis S Grebenkov, CNRS Ecole Polytechnique p1/35
2 Plan Laplacian transport phenomena Theoretical approach Numerical and experimental study Conclusion, questions and perspectives Denis S Grebenkov, CNRS Ecole Polytechnique p2/35
3 Oxygen diffusion towards and through alveolar membranes Denis S Grebenkov, CNRS Ecole Polytechnique p3/35
4 Oxygen diffusion towards and through alveolar membranes Dichotomic tree Volume 4 5 liters Area m 2 Human lung 23 generations : convection 15 diffusion 8 Denis S Grebenkov, CNRS Ecole Polytechnique p3/35
5 Oxygen diffusion towards and through alveolar membranes Laplacian transport C = 0 in the volume C = C 0 at the entrance J = D C Fick s law J n = WC on the membrane Alveoli Alveolated duct Airflow C=C 0 2 C = 0 n C C = 1 Λ Λ = D/W characteristic length of the problem D diffusion coefficient W membrane permeability Denis S Grebenkov, CNRS Ecole Polytechnique p3/35
6 Oxygen diffusion towards and through alveolar membranes Laplacian transport Airflow C=C 0 2 C = 0 C = 0 C = C 0 C n = C Λ in the volume at the entrance on the membrane Alveoli Alveolated duct n C C = 1 Λ Λ = D/W characteristic length of the problem D diffusion coefficient W membrane permeability Denis S Grebenkov, CNRS Ecole Polytechnique p3/35
7 Heterogeneous catalysis Denis S Grebenkov, CNRS Ecole Polytechnique p4/35
8 Heterogeneous catalysis Chemical reaction C = C 0 C n = C Λ A A on the source on the surface Denis S Grebenkov, CNRS Ecole Polytechnique p4/35
9 Heterogeneous catalysis Description C = 0 C = C 0 C n = C Λ Λ = D/K in the volume on the source on the surface characteristic length of the problem D diffusion coefficient K catalyst s reactivity Denis S Grebenkov, CNRS Ecole Polytechnique p4/35
10 Electric transport through an irregular electrode Denis S Grebenkov, CNRS Ecole Polytechnique p5/35
11 Electric transport through an irregular electrode Microroughness (of metallic surface) Irregular geometry on different scales Impedance? 1µm Denis S Grebenkov, CNRS Ecole Polytechnique p5/35
12 Electric transport through an irregular electrode Microroughness (of metallic surface) Irregular geometry on different scales Impedance? 1µm Denis S Grebenkov, CNRS Ecole Polytechnique p5/35
13 Electric transport through an irregular electrode Microroughness (of metallic surface) Irregular geometry on different scales Impedance? 1µm Denis S Grebenkov, CNRS Ecole Polytechnique p5/35
14 Electric transport through an irregular electrode Electric transport: (direct constant potential) V = 0 in the volume V = V 0 on the counter-electrode V n = 1 Λ V on the working electrode 1µm Λ = r/ρ characteristic length of the problem r resistance of the working electrode ρ resistivity of the electrolyte Denis S Grebenkov, CNRS Ecole Polytechnique p5/35
15 Laplacian transport source u = 1 u = 0 Λ u n = u working surface Denis S Grebenkov, CNRS Ecole Polytechnique p6/35
16 Laplacian transport source u = 1 u = 0 Λ u n = u working surface How does a geometrical irregularity influence the transport properties of the system? Denis S Grebenkov, CNRS Ecole Polytechnique p6/35
17 Spectroscopic impedance Mathematical model: mixed boundary value problem for Laplace equation source Diffusive transport C = C 0 C = 0 Λ C n = C working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
18 Spectroscopic impedance Mathematical model: mixed boundary value problem for Laplace equation We are interested in the total flux Φ Λ through the working surface: source Φ Λ = D Diffusive transport C = C 0 Ω C = 0 ds C Λ n Λ C n = C Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
19 Spectroscopic impedance Mathematical model: mixed boundary value problem for Laplace equation We are interested in the total flux Φ Λ through the working surface: Cell impedance: source Φ Λ = D Diffusive transport C = C 0 Ω C = 0 ds C Λ n Z cell (Λ) = C 0 Φ Λ Λ C n = C Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
20 Spectroscopic impedance Mathematical model: mixed boundary value problem for Laplace equation We are interested in the total flux Φ Λ through the working surface: Spectroscopic impedance: source Φ Λ = D Diffusive transport C = C 0 Ω C = 0 ds C Λ n Λ C n = C Λ = D W Z(Λ) = Z cell (Λ) Z cell (0) working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
21 Spectroscopic impedance Dirichlet mode: Λ 0 source Φ Λ = D Diffusive transport C = C 0 Ω C = 0 ds C Λ n Λ C n = C Z(Λ) = C 0 Φ Λ C 0 Φ 0 Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
22 Spectroscopic impedance Dirichlet mode: Λ 0 Absorbing surface source Φ Λ = D Diffusive transport C = C 0 Ω C = 0 ds C Λ n Z(Λ) = C 0 Φ Λ C 0 Φ 0 C 0 Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
23 Spectroscopic impedance Dirichlet mode: Λ 0 Absorbing surface Z(Λ) Λ source C = C 0 C = 0 DL dir Φ Λ = D Diffusive transport Ω ds C Λ n Z(Λ) = C 0 Φ Λ C 0 Φ 0 L dir Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
24 Spectroscopic impedance Neumann mode: Λ source Φ Λ = D Diffusive transport C = C 0 Ω C = 0 ds C Λ n Λ C n = C Z(Λ) = C 0 Φ Λ C 0 Φ 0 Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
25 Spectroscopic impedance Neumann mode: Λ Reflecting surface Z(Λ) source C = C 0 C = 0 Λ DL p Φ Λ = D Diffusive transport Ω ds C Λ n Z(Λ) = C 0 Φ Λ C 0 Φ 0 C n 0 Λ = D W L p working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
26 Spectroscopic impedance Intermediate mode: Constant phase angle behavior Z(Λ) Λ β source Φ Λ = D Diffusive transport C = C 0 Ω C = 0 ds C Λ n Λ C n = C Z(Λ) = C 0 Φ Λ C 0 Φ 0 Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
27 Spectroscopic impedance Intermediate mode: Constant phase angle behavior Z(Λ) Λ β Fractal boundary β = 1 D f β = τ(2) D f source Φ Λ = D Diffusive transport C = C 0 Ω C = 0 ds C Λ n Λ C n = C Z(Λ) = C 0 Φ Λ C 0 Φ 0 Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
28 Spectroscopic impedance Intermediate mode: Arbitrary surface How does the geometrical irregularity influence C = 0 transport proper- ties? source Φ Λ = D Diffusive transport C = C 0 Ω ds C Λ n Λ C n = C Z(Λ) = C 0 Φ Λ C 0 Φ 0 Λ = D W working surface Denis S Grebenkov, CNRS Ecole Polytechnique p7/35
29 Plan Laplacian transport phenomena Theoretical approach Numerical and experimental study Conclusion, questions and perspectives Denis S Grebenkov, CNRS Ecole Polytechnique p8/35
30 Theoretical description Laplace equation + mixed condition: Description in average Denis S Grebenkov, CNRS Ecole Polytechnique p9/35
31 Theoretical description Laplace equation + mixed condition: Description in average Partially reflected Brownian motion: Random trajectory of each particle Denis S Grebenkov, CNRS Ecole Polytechnique p9/35
32 Theoretical description Laplace equation + mixed condition: Description in average Partially reflected Brownian motion: Random trajectory of each particle Dirichlet-to-Neumann operator: Spectral analysis of the problem Denis S Grebenkov, CNRS Ecole Polytechnique p9/35
33 Definition of the Dirichlet-to-Neumann operator For a given domain Ω, Ω Ω Denis S Grebenkov, CNRS Ecole Polytechnique p10/35
34 Definition of the Dirichlet-to-Neumann operator For a given domain Ω, one takes a function f H 1 ( Ω) on the boundary Ω, Ω Ω f Denis S Grebenkov, CNRS Ecole Polytechnique p10/35
35 Definition of the Dirichlet-to-Neumann operator For a given domain Ω, one takes a function f H 1 ( Ω) on the boundary Ω, one solves the Dirichlet problem in this domain Ω Ω u = f u = 0 Denis S Grebenkov, CNRS Ecole Polytechnique p10/35
36 Definition of the Dirichlet-to-Neumann operator For a given domain Ω, one takes a function f H 1 ( Ω) on the boundary Ω, one solves the Dirichlet problem in this domain, and one applies the normal derivative to the solution Ω Ω u = f u = 0 f g = u n [Mf] Denis S Grebenkov, CNRS Ecole Polytechnique p10/35
37 Properties of the Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator M is dependent solely on the geometry; Denis S Grebenkov, CNRS Ecole Polytechnique p11/35
38 Properties of the Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator M is dependent solely on the geometry; self-adjoint; Denis S Grebenkov, CNRS Ecole Polytechnique p11/35
39 Properties of the Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator M is dependent solely on the geometry; self-adjoint; with discrete spectrum {µ α } bounded below; Denis S Grebenkov, CNRS Ecole Polytechnique p11/35
40 Properties of the Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator M is dependent solely on the geometry; self-adjoint; with discrete spectrum {µ α } bounded below; with smooth eigenfunctions V α (s) (defined on the boundary Ω) which form a complete basis in L 2 ( Ω) Denis S Grebenkov, CNRS Ecole Polytechnique p11/35
41 Representation of the spectroscopic impedance Z(Λ) = Z cell (Λ) Z cell (0) = C 0 Φ Λ C 0 Φ 0 Denis S Grebenkov, CNRS Ecole Polytechnique p12/35
42 Representation of the spectroscopic impedance Z(Λ) = Z cell (Λ) Z cell (0) = C 0 Φ Λ C 0 Φ 0 Z(Λ) = Λ D α=0 F α 1 + µ α Λ Denis S Grebenkov, CNRS Ecole Polytechnique p12/35
43 Representation of the spectroscopic impedance Z(Λ) = Z cell (Λ) Z cell (0) = C 0 Φ Λ C 0 Φ 0 Physics Λ, D Z(Λ) = Λ D Λ = D W, D K, r ρ α=0 F α 1 + µ α Λ Geometry µ α, F α F α = ( V α ω 0 ) 2 Denis S Grebenkov, CNRS Ecole Polytechnique p12/35
44 Plan Laplacian transport phenomena Theoretical approach Numerical and experimental study Conclusion, questions and perspectives Denis S Grebenkov, CNRS Ecole Polytechnique p13/35
45 Koch interface Denis S Grebenkov, CNRS Ecole Polytechnique p14/35
46 Koch interface Denis S Grebenkov, CNRS Ecole Polytechnique p14/35
47 Koch interface Denis S Grebenkov, CNRS Ecole Polytechnique p14/35
48 Koch interface D f = ln 5/ ln 3 Denis S Grebenkov, CNRS Ecole Polytechnique p14/35
49 Localization of the spectrum µ α L tot Denis S Grebenkov, CNRS Ecole Polytechnique p15/35
50 Localization of the spectrum µ α L tot Denis S Grebenkov, CNRS Ecole Polytechnique p15/35
51 Localization of the spectrum µ α L tot Denis S Grebenkov, CNRS Ecole Polytechnique p15/35
52 Localization of the spectrum µ α L tot Denis S Grebenkov, CNRS Ecole Polytechnique p15/35
53 Reduction of eigenmodes µ α L p Z(Λ) = Λ D α=0 F α 1 + µ α Λ Denis S Grebenkov, CNRS Ecole Polytechnique p16/35
54 Reduction of eigenmodes 05 F α L Z(Λ) Λ D ( F0 1 + µ 0 Λ + F µ 1 Λ ) µ α L p Denis S Grebenkov, CNRS Ecole Polytechnique p16/35
55 Eigenvectors First generation Plane surface (line) V 0 01 V F 0 L tot = 1 F 0 L tot = 1 Denis S Grebenkov, CNRS Ecole Polytechnique p17/35
56 Eigenvectors First generation Plane surface (line) V 1 01 V F 1 L tot = 067 F 1 L tot = 0 Denis S Grebenkov, CNRS Ecole Polytechnique p17/35
57 Eigenvectors First generation Plane surface (line) V 2 01 V F 2 L tot = 0 F 2 L tot = 0 Denis S Grebenkov, CNRS Ecole Polytechnique p17/35
58 Eigenvectors First generation Plane surface (line) V 3 01 V F 3 L tot = 0 F 3 L tot = 0 Denis S Grebenkov, CNRS Ecole Polytechnique p17/35
59 Eigenvectors First generation Plane surface (line) V 4 01 V F 4 L tot = 0017 F 4 L tot = 0 Denis S Grebenkov, CNRS Ecole Polytechnique p17/35
60 Eigenvectors First generation Plane surface (line) V 9 01 V F 9 L tot = 0021 F 9 L tot = 0 Denis S Grebenkov, CNRS Ecole Polytechnique p17/35
61 Eigenvectors First generation Plane surface (line) V V F 50 L tot = 0 F 50 L tot = 0 Denis S Grebenkov, CNRS Ecole Polytechnique p17/35
62 Reduction of eigenmodes 05 0 g=1 0 S µ α L p Denis S Grebenkov, CNRS Ecole Polytechnique p18/35
63 Reduction of eigenmodes 05 g=1 0 g=2 0 S µ α L p Denis S Grebenkov, CNRS Ecole Polytechnique p18/35
64 Reduction of eigenmodes g=1 05 g=2 0 g=3 0 S µ α L p Denis S Grebenkov, CNRS Ecole Polytechnique p18/35
65 Reduction of eigenmodes g=1 g=2 g= S g= µ α L p Denis S Grebenkov, CNRS Ecole Polytechnique p18/35
66 Reduction of eigenmodes g=1 g=2 g= S g= ˆµ k L p 5 k µ α L p Denis S Grebenkov, CNRS Ecole Polytechnique p18/35
67 Reduction of eigenmodes g=1 g=2 g= S g= ˆµ k L p 5 k ˆFk L p (5/3) k µ α L p Denis S Grebenkov, CNRS Ecole Polytechnique p18/35
68 Spectroscopic impedance: model and calculations Fourth generation: 100% 50% (Z Z mod )/Z Z mod (Λ) = Λ D 4 k=0 ˆF k 1 + Λˆµ k 0 50% 100% Λ/L Denis S Grebenkov, CNRS Ecole Polytechnique p19/35
69 Analytical model: anomalous behavior Tenth generation: Z mod (Λ) = Λ D 10 k=0 ˆF k 1 + Λˆµ k Denis S Grebenkov, CNRS Ecole Polytechnique p20/35
70 Analytical model: anomalous behavior Tenth generation: 10 5 Z mod (Λ) = Λ D 10 k=0 DZ mod (Λ) Λ/L p ˆF k 1 + Λˆµ k DZ mod (Λ) Λ/L Z mod (Λ) Λ β β Λ/L Denis S Grebenkov, CNRS Ecole Polytechnique p20/35
71 Analytical model: anomalous behavior Tenth generation: β(λ) Z mod (Λ) = Λ D 10 k=0 ˆF k 1 + Λˆµ k g= β 068 1/D f = Λ Denis S Grebenkov, CNRS Ecole Polytechnique p20/35
72 Analytical model: anomalous behavior Tenth generation: β(λ) Z mod (Λ) = Λ D 10 k=0 ˆF k 1 + Λˆµ k g=10 g=5 07 β 068 1/D f = Λ Denis S Grebenkov, CNRS Ecole Polytechnique p20/35
73 Analytical model: anomalous behavior Tenth generation: β(λ) g=10 g=5 Z mod (Λ) = Λ D g=3 10 k=0 ˆF k 1 + Λˆµ k 07 β 068 1/D f = Λ Denis S Grebenkov, CNRS Ecole Polytechnique p20/35
74 Irregular boundaries Random Koch curves Denis S Grebenkov, CNRS Ecole Polytechnique p21/35
75 Irregular boundaries Random Koch curves Koch curves of different dimensions Denis S Grebenkov, CNRS Ecole Polytechnique p21/35
76 Irregular boundaries Random Koch curves Koch curves of different dimensions Koch surfaces Denis S Grebenkov, CNRS Ecole Polytechnique p21/35
77 Irregular boundaries Random Koch curves Koch curves of different dimensions Koch surfaces Denis S Grebenkov, CNRS Ecole Polytechnique p21/35
78 Irregular boundaries Random Koch curves Koch curves of different dimensions Koch surfaces Effect of eigenmodes reduction Denis S Grebenkov, CNRS Ecole Polytechnique p21/35
79 Irregular boundaries Random Koch curves Koch curves of different dimensions Koch surfaces Effect of eigenmodes reduction Hierarchy of eigenvalues Denis S Grebenkov, CNRS Ecole Polytechnique p21/35
80 Irregular boundaries Random Koch curves Koch curves of different dimensions Koch surfaces Effect of eigenmodes reduction Hierarchy of eigenvalues Analytical model of the impedance Denis S Grebenkov, CNRS Ecole Polytechnique p21/35
81 Experimental study How does the geometrical irregularity influence impedance behavior of real electrodes? Denis S Grebenkov, CNRS Ecole Polytechnique p22/35
82 Experimental study How does the geometrical irregularity influence impedance behavior of real electrodes? Denis S Grebenkov, CNRS Ecole Polytechnique p22/35
83 Classical theory of double layer Electric transport: V = V 0 e iωt V = 0 V = 0 V = V 0 V n = 1 Λ V Λ = ζ(ω) ρ in the volume on the counter-electrode on the working electrode where ζ(ω) is the surface impedance Z(Λ) = ρλ α= F α 1 + µ α Λ Denis S Grebenkov, CNRS Ecole Polytechnique p23/35
84 Practical implementation One measures the physical characteristics, ρ and ζ(ω), to obtain Λ = ζ(ω)/ρ; Denis S Grebenkov, CNRS Ecole Polytechnique p24/35
85 Practical implementation One measures the physical characteristics, ρ and ζ(ω), to obtain Λ = ζ(ω)/ρ; One finds the geometrical characteristics, µ α and F α ; Denis S Grebenkov, CNRS Ecole Polytechnique p24/35
86 Practical implementation One measures the physical characteristics, ρ and ζ(ω), to obtain Λ = ζ(ω)/ρ; One finds the geometrical characteristics, µ α and F α ; One substitutes them into the spectral decomposition Z(Λ) = ρλ α=0 F α 1 + µ α Λ Denis S Grebenkov, CNRS Ecole Polytechnique p24/35
87 Second generation: experiment and theory Re[Z(ω)] R (Ohm) Im[Z(ω)] (Ohm) ω (Hz) ω (Hz) Denis S Grebenkov, CNRS Ecole Polytechnique p25/35
88 Second generation: experiment and theory Re[Z(ω)] R (Ohm) Im[Z(ω)] (Ohm) ω (Hz) ω (Hz) Denis S Grebenkov, CNRS Ecole Polytechnique p25/35
89 Second generation: experiment and theory Re[Z(ω)] R (Ohm) Im[Z(ω)] (Ohm) ω (Hz) ω (Hz) In conclusion, without knowledge of the microscopic transport mechanism, the geometrical irregularity can be taken into account by our theoretical approach Denis S Grebenkov, CNRS Ecole Polytechnique p25/35
90 Theoretical results Theoretical description of Laplacian transport towards irregular interfaces based on the Dirichlet-to-Neumann operator and study of the partially reflected Brownian motion Implementation of the (discrete) boundary element method for contruction of the Dirichlet-to-Neumann operator Denis S Grebenkov, CNRS Ecole Polytechnique p26/35
91 Numerical results Effective contribution of a small number of eigenmodes to linear response of the system Hierarchy of these significative eigenmodes (characteristic scales of the boundary) Analytical model of the impedance Denis S Grebenkov, CNRS Ecole Polytechnique p27/35
92 Experimental results Measurement of the impedance of brass and nickel electrodes with boundaries reproducing two generations of the convex Von Koch surface Observation of anomalous behavior of the impedance in the case of planar electrodes Accounting for geometrical irregularity by theoretical approach Denis S Grebenkov, CNRS Ecole Polytechnique p28/35
93 Construction of the Dirichlet-to- Neumann operator Present approach: A kind of the boundary element method based on potential theory Denis S Grebenkov, CNRS Ecole Polytechnique p29/35
94 Construction of the Dirichlet-to- Neumann operator Present approach: A kind of the boundary element method based on potential theory Main problem: one has to calculate all the elements of the matrix representing this operator in a discrete form Denis S Grebenkov, CNRS Ecole Polytechnique p29/35
95 Construction of the Dirichlet-to- Neumann operator Present approach: A kind of the boundary element method based on potential theory Main problem: one has to calculate all the elements of the matrix representing this operator in a discrete form Reality: Only a small fraction of the eigenmodes is in fact needed Denis S Grebenkov, CNRS Ecole Polytechnique p29/35
96 Construction of the Dirichlet-to- Neumann operator We are looking for a numerical method allowing to compute the eigenmodes of the operator M successively, without constructing the whole operator Denis S Grebenkov, CNRS Ecole Polytechnique p30/35
97 Construction of the Dirichlet-to- Neumann operator Denis S Grebenkov, CNRS Ecole Polytechnique p30/35
98 Perspectives Extensions of the theoretical approach: Denis S Grebenkov, CNRS Ecole Polytechnique p31/35
99 Perspectives Extensions of the theoretical approach: other equations (time-dependent diffusion) Denis S Grebenkov, CNRS Ecole Polytechnique p31/35
100 Perspectives Extensions of the theoretical approach: other equations (time-dependent diffusion) other conditions (non linear) Denis S Grebenkov, CNRS Ecole Polytechnique p31/35
101 Perspectives Extensions of the theoretical approach Numerical study of higher orders generations, especially in 3 dimensions Denis S Grebenkov, CNRS Ecole Polytechnique p32/35
102 Perspectives Extensions of the theoretical approach Numerical study of higher orders generations, especially in 3 dimensions Analytical model of the impedance for random interfaces? Denis S Grebenkov, CNRS Ecole Polytechnique p32/35
103 Perspectives Extensions of the theoretical approach Numerical study of higher orders generations, especially in 3 dimensions Analytical model of the impedance for random interfaces? Inverse problem: knowing the spectroscopic impedance, what can one say about the geometry? Denis S Grebenkov, CNRS Ecole Polytechnique p32/35
104 Perspectives Extensions of the theoretical approach Numerical study of higher orders generations, especially in 3 dimensions Analytical model of the impedance for random interfaces? Inverse problem: knowing the spectroscopic impedance, what can one say about the geometry? Experimental study of irregular electrodes Denis S Grebenkov, CNRS Ecole Polytechnique p32/35
105 Perspectives Denis S Grebenkov, CNRS Ecole Polytechnique p33/35
106 Open problems Justification of the approach for really irregular interfaces Denis S Grebenkov, CNRS Ecole Polytechnique p34/35
107 Open problems Justification of the approach for really irregular interfaces Reduction of the number of contributing eigenmodes Denis S Grebenkov, CNRS Ecole Polytechnique p34/35
108 Open problems Justification of the approach for really irregular interfaces Reduction of the number of contributing eigenmodes Relation between the eigenvalues and the characteristic length scales of the boundary Denis S Grebenkov, CNRS Ecole Polytechnique p34/35
109 Open problems Justification of the approach for really irregular interfaces Reduction of the number of contributing eigenmodes Relation between the eigenvalues and the characteristic length scales of the boundary Correction, generalization and mathematical justification for the simplified impedance model β = 1 = β = τ(2) D f D f Denis S Grebenkov, CNRS Ecole Polytechnique p34/35
110 Related papers 1 B Sapoval, Transport Across Irregular Interfaces: Fractal Electrodes, Membranes and Catalysts, in Fractals and Disordered Systems, Eds A Bunde, S Havlin, pp (Springer, 1996) 2 B Sapoval, General Formulation of Laplacian Transfer Across Irregular Surfaces, Phys Rev Lett 73, 3314 (1994) 3 M Filoche and B Sapoval, Can One Hear the Shape of an Electrode? II Theoretical Study of the Laplacian Transfer, Eur Phys J B 9, 755 (1999) 4 D S Grebenkov, M Filoche, B Sapoval, Spectral Properties of the Brownian Self-Transport Operator, Eur Phys J B 36, (2003) 5 D S Grebenkov, M Filoche, B Sapoval, Mathematical Basis for a General Theory of Laplacian Transport towards Irregular Interfaces, Phys Rev E 73, (2006) 6 D S Grebenkov, M Filoche, B Sapoval, A Simplified Analytical Model for Laplacian Transfer Across Deterministic Prefractal Interfaces, Fractals (2007, in press) 7 D S Grebenkov, Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena, in Focus on Probability Theory, Ed L R Velle, pp (Nova Science Publishers, 2006) [Electronic version is available on Denis S Grebenkov, CNRS Ecole Polytechnique p35/35
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