ENTROPY PRODUCTION OF REACTION-DIFFUSION SYSTEMS UNDER CONFINEMENT

Diffusion processes under confinement within a channel in which one coordinate is longer than the others have been studied by projecting the diffusion equation into one dimension. This results in the so-called Fick-Jacobs equation that introduces an effective diffusion coefficient dependent on the position. Several approaches have been used to propose position-dependent diffusion coefficients, and it has been found that it depends on the channel's width function as well as the geometric properties of the midline, such as its curvature and torsion. Within this approach we study the entropy production for a reaction-diffusion process of two species on a two dimensional channel. Recently, it has been seen that the Turing instability conditions, the range of unstable modes for patterns formation, as well as the spatial structure of the patterns themselves, can be modified through the geometric parameters of the confinement. In this contribution, the effect of the confinement on entropy production is analyzed and characterized in terms of the geometry of the corresponding channel.


Effective 1D projection method
For narrow channels, there is a method that projects the 2D motion to an effective one-dimensional motion. From 2D diffusion equations: C(x, y, t) = Dx ∂ 2 ∂x 2 C(x, y, t) + Dy ∂ 2 ∂y 2 C(x, y, t), we introduce the marginal distribution where f i (x) are the upper and lower boundaries of the channel, such that the width function is w(x) = f 2 (x) − f 1 (x).
After integration under reflecting boundary conditions, the so-called Fick-Jacobs equation is obtained: The entropic potential U (x) = − ln w(x) which contains effects of the shape of the boundaries, is introduced. Zwanzig propose an adjustment due to variations of the diffusion coefficient: Diffusion on narrow channels D(x) for 2D and 3D symmetric channels García-Chung, Chacón-Acosta, Dagdug
G. Chacón Acosta (UAM-C) Entropy of RD systems on confinement 7 / 23 Diffusion on narrow channels D(x) for 2D and 3D asymmetric channels Cuadro: D(x) for asymmetric channels, the midline in 2D is y0(x) and in 3D is r0(x). Width function in 2D is w(x) and in 3D πR(x) 2 .
G. Chacón Acosta (UAM-C) Entropy of RD systems on confinement 8 / 23 Diffusion on narrow channels D(s) from the differential geometrical method Different geometry methods show that when a tube has a constant cross-section, the diffusion coefficient takes the following form in the corresponding limit cases.
For a variable cross-section R = const., and when the channel has a twisted midline, one has b The Reaction-Diffusion system: where u y v are the species' concentrations that diffuse and react according to the kinetics f and g. Here d = Du/Dv and γ is the relative strength of the reactions. The problem may have Neumann, Dirichlet, Robin, or periodic boundary conditions.
The stationary state of the system is (u 0 , v 0 ), such that in the absence of diffusion f = g = 0.
Under certain conditions of the values of the parameters, the system's state becomes unstable in the presence of diffusion.
The system is linearized around (u 0 , v 0 ) and the polynomial λ(k) is the dispersion relation whose roots give the range of unstable wavenumbers Let us consider two chemical species (u, v) confined in a channel whose longitudinal coordinate is larger than the transversal one. This system is then described by the Fick-Jacob-Zwanzig operator Eq. (1), which is rewritten as a Fokker-Planck operator: with the fluxes Ju, Jv and the advection coefficient By doing an expansion of the concentrations around the steady-state and a rescaling of the coordinates as a function of D(x), it is then possible to obtain a linearized set of equations for the perturbations that gives us a dispersion relation from where it is possible to analyze the stability of the system 1 .
Harmonic oscillator potential, with the shape parameter 1.5 (red) ≤ σ ≤ 6 (blue). The entropy production rate of a reaction-diffusion system is given by the sum of two components: chemical reactions and by diffusion Chemical contribution. The entropy production rate per unit length due to chemical reactions is as follows 2 where S ij are the stoichiometric coefficient for the reaction, J i is the i-th net chemical reaction current, and c j , c 0 j are the concentration and its equilibrium value respectively. The reversible part of the reactions is not considered to obtain S ij , which avoids the thermodynamic study of pattern formation. However, we consider it as an approximation to study the geometric effects in these types of expressions.
For the present reaction σ R is the total entropy production rate is the spatial integration of this term along the entire length of the channel.

Summary
Confinement influences the diffusivity of the system. Turing's mechanism gives us the range of unstable modes where the patterns can form for specific values of the parameters.
When the system is under confinement, both the range of unstable modes and the pattern itself, change depending on the geometry of the channel. Also, the rate of entropy production grows faster as the width of the channel increases.