Dissipation in Laplacian fields across irregular boundaries

PHYSICAL REVIEW E, VOLUME 64, 011115

Dissipation in Laplacian fields across irregular boundaries K. Karamanos and G. Nicolis Centre for Nonlinear Phenomena and Complex Systems, Universite´ Libre de Bruxelles, Campus Plaine, Code Postal 231, Boulevard du Triomphe, B-1050 Brussels, Belgium

T. Massart and P. Bouillard Continuum Mechanics Department, Universite´ Libre de Bruxelles, Code Postal 194/5, Avenue F. D. Roosevelt 50, B-1050 Brussels, Belgium ~Received 20 November 2000; published 26 June 2001! The entropy production associated to a Laplacian field distributed across irregular boundaries is studied. In the context of the active zone approximation an explicit expression is given for the entropy production in terms of geometry, whose relation to the variational formulation is discussed. It is shown that the entropy production diminishes for successive prefractal generations of the same fractal generator, so that the final fractal object is expected to dissipate less than all previous ones. The relevance of this result in the abundance of fractal surfaces or interfaces observed in nature is discussed. DOI: 10.1103/PhysRevE.64.011115

PACS number~s!: 65.40.Gr, 05.70.Ln, 05.45.Df, 47.53.1n

I. INTRODUCTION

Recently, a number of studies on the role of the irregularity of a boundary in the spatial distribution of a field obeying Laplace’s equation and of its associated flux has been reported. In particular, the case of fractal boundaries has been analyzed in depth using results from harmonic analysis along with extensive numerical computations based on finite element techniques @1–4#. One of the principal conclusions emerging from these studies is the existence of universal scaling laws culminating in the derivation of interesting expressions for the impedance describing the system’s linear response. Many of these laws find their origin in Makarov’s theorem stating that, whatever the shape of an irregular ~simply connected! boundary in two dimensions might be, the active zone in which most of the flux generated by a Laplacian field is concentrated, scales as a length @5–7#. The relevance of the above results stems mainly from two factors. First, under ordinary conditions many familiar transport phenomena such as diffusion and heat conduction are described in the steady state by Laplacian fields @8#. And second, in nature as well as in technology the space in which these fields are distributed is far from regular. The terminal part of the respiratory system of mammals, biological membranes, porous electrodes or catalysts, provide some characteristic examples @8–14#. Our objective in the present study is to explore an alternative way to characterize the complexity of Laplacian transport across irregular boundaries, based on irreversible thermodynamics. More specifically we will be interested in the behavior of the dissipation generated by the underlying process, as the irregularity of the boundary is increased. Dissipation is here measured by the entropy production, arguably the central quantity of irreversible thermodynamics @15,16#. It has been shown that under the assumption of constant phenomenological coefficients linking the fluxes to the constraints this quantity satisfies a variational principle, thereby playing in nonequilibrium a role analogous to thermody1063-651X/2001/64~1!/011115~10!/$20.00

namic potentials in equilibrium. Curiously, as it will turn out, the range of validity of this result is incompatible with the Laplacian character of the associated field and, conversely, when this field is Laplacian entropy production no longer follows a variational principle. Nevertheless, it still provides an interesting characterization of nonequilibrium states, both locally in space and globally for the system as a whole. We shall see how the ‘‘active zone’’ concept allows one to incorporate in this characterization information pertaining to the geometry of the boundaries. The general formulation is laid down in Sec. II. In Sec. III the entropy production associated to diffusion is evaluated within the active zone approximation for boundaries corresponding to the first two generations of an eventually fractal boundary. It is found that fragmentation tends to decrease both the total dissipation and the dissipation per unit surface. In Sec. IV the results are confronted with, and complemented by, those of numerical simulations. The main conclusions are summarized in Sec. V. II. FORMULATION

Let f be a scalar field associated to a conserved quantity. It is supposed that under the action of a nonequilibrium constraint this field gives rise to a single irreversible process whose flux J and the associated force X are vectors. While, typically, X is the space derivative of some function f of f related to the derivative of a thermodynamic potential, J can be related to f only through an appropriate phenomenological, or constitutive relation, J5L ~ f ! “ f ~ f ! ,

L.0,

~1!

where L is the ~generally state-dependent! phenomenological or Onsanger coefficient. The field f obeys, then, to a closed evolution equation of the form

64 011115-1

]f 52div J52div L ~ f ! “ f ~ f ! , ]t

~2!

©2001 The American Physical Society

KARAMANOS, NICOLIS, MASSART, AND BOUILLARD

PHYSICAL REVIEW E 64 011115

it being understood that any extra factors in the left-hand side arising from thermodynamic derivatives ~such as specific heat, etc.! have been absorbed in a suitably rescaled time. The local entropy production associated with ~1! and ~2! is

s 5J•X5L ~ f ! „“ f ~ f ! …2 >0,

~ Fick’s law! ,

J52D“c

~10! kL D5 'const, c

~3!

the overall dissipation being P5

slowly in a certain relevant range of values of the state variables. Taking up our two examples of diffusion and heat conduction one has, for instance,

and

Es

~4!

dr,

~ Fourier’s law! ,

J52l“T

V

~11!

where V is the volume occupied by the system. Let us assume first that L is strictly constant ~stateindependent!. Equation ~2! implies, then, that in the steady state f ( f ) obeys Laplace’s equation, div “ f ~ f ! 5“ 2 f ~ f ! 50.

~5!

Typically, the dependence of f on f is nonlinear. For instance, in the diffusion of a solute in a solvent, say in an electrolytic cell, and in heat conduction in a slab one has, respectively,

m T

f 52

L l5 2 'const T It follows immediately from Eq. ~2! that c and T are, then, Laplacian fields. As a counterpart, the entropy production loses its variational character. Below, we illustrate this in the case of diffusion, but the arguments apply to heat conduction as well. We have P5

E

V

~ 2D“c ! •“

~ heat conduction! ,

P5Dk

m being the chemical potential and T the temperature. In an ideal mixture m 5 m * 1kT ln c, c being the solute concentration, k the Boltzmann constant, and m * the standard chemical potential. It follows that the Laplacian fields associated to diffusion and heat conduction when L is constant are respectively, ln c and T 21 rather than c and T themselves. As shown in irreversible thermodynamics under the same assumptions, and provided that the boundary conditions are fixed or zero-flux ones, the total entropy production P is minimum at the steady state ~Prigogine’s theorem! @15,16#, P5L

E

„“ f ~ f ! …2 dr.

To see this we differentiate both sides of Eq. ~7! with respect to time,

E

“ f ~ f ! •“

V

]f dr ]t

]P 5Dk ]t

~8!

E S DS D V

]f ]f

]f ]t

2

dr