Abstract
We consider a model of surface-mediated diffusion with alternating phases of bulk and surface diffusion for two geometries: the disk and rectangles. We develop a spectral approach to derive an exact formula for the mean exit time of a particle through a hole on the boundary. The spectral representation of the mean exit time through the eigenvalues of an appropriate self-adjoint operator is particularly well-suited to investigate the asymptotic behavior in the limit of large desorption rate \(\lambda \). For a point-like target, we show that the mean exit time diverges as \(\sqrt{\lambda }\). For extended targets, we establish the asymptotic approach to a finite limit. In both cases, the mean exit time is shown to asymptotically increase as \(\lambda \) tends to infinity. That implies that the pure bulk diffusion is never an optimal search strategy. We also investigate the influence of rectangle elongation onto the mean exit time, in particular, the dependence of the critical ratio of bulk and surface diffusion coefficients on the rectangle aspect ratio. We show that the intermittent search strategy can significantly outperform pure surface diffusion for elongated rectangles.
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Appendices
Appendix A: Asymptotic behavior of the eigenvalues of the operator \(V\tilde{T}V\)
Theorem 5.1
Let \(\lambda _n\) be the decreasing sequence of eigenvalues of the operator \(V\tilde{T}V\), where the operators \(\tilde{T}\) and \(V\) are defined in Eqs. (2.9) and (2.8). We have
where \(A_{\epsilon }\) depends only on \(\epsilon \).
Proof
In order to prove this statement, we first investigate the following problem:
Let \(A\) and \(B\) are two compact, positive, self-adjoint operators. We assume that the eigenvalues of the operator \(A\) are ordered in a decreasing sequence: \(\lambda _1(A)\ge \lambda _2(A)\ge \ldots \ge \lambda _n(A)\ge \ldots \ge 0\). We recall the variational principle as following \(\square \)
Theorem 5.2
The \(\max \) is taken over \(F\), the subspace associated with the first \(n\) eigenvectors of \(A\).
We state two following lemmas which will be needed to prove Theorem 5.1.
Lemma 5.3
Let \(\lambda _n(A)\) and \(\lambda _n(A+B)\) be the \(n^{th}\) eigenvalues of the operators \(A\) and \(A+B\). Then, we have
where \(\Vert .\Vert \) defines the norm of an operator in \(L^2[a,b]\) space.
Proof
Let \(F(A)\) be the subspace of \(L^2[a,b]\) associated with the first \(n\) eigenvectors of \(A\). For all \(x\in F(A)\), we have
According to the variational principle, we have
Besides, we have
It follows from Eqs. (5.4) and (5.5) that
This gives
Again, according to the variational principle, we thus get
In the same manner, if we take \(F(A+B)\) be associated to the first \(n\) eigenvectors of \(A+B\), we can get
and the Lemma 5.3 follows. \(\square \)
Lemma 5.4
With the notations used in Lemma 5.3, if \(\mathrm{rank}(B) < n < \infty \), then
Proof
We call \(F\) the subspace of \(L^2[a,b]\) associated with the first \(n+\mathrm{rank}(B)\) eigenvectors of \(A\).
By the variational principle, we have
Consequently,
Since
we have
So, we conclude that
The second inequality in (5.6) of this lemma is obtained when we put \(A'=A+B\), \(B'=-B\), \(n'=n-\mathrm{rank}(B)\) and apply the inequality (5.7) for \(A'\), \(B'\) and \(n'\) instead of \(A\), \(B\) and \(n\). \(\square \)
We now call \(\pi _N\) be the orthogonal projection on the first \(N\) eigenvectors of \(B\).
By the property of an orthogonal projection, we can rewrite
then
We note that
and
By applying Lemma 5.3, we get \(\forall ~ n\ge N\),
From (5.8), we obtain
According to Lemma 5.4,
We can thus conclude that
Lemma 5.5
Let \(\{\lambda _n(A)\}\) and \(\{\lambda _n(B)\}\) be the eigenvalues of two self-adjoint operators \(A\) and \(B\). If \(\lambda _n(A)\sim c n^{-s}\) and \(\lambda _N(B)=\rho ^N\) where \(A\) and \(\rho \) are some constants, \(0 \le \rho \le 1\), then
Proof
Indeed, if we choose \(N(n)=n^{\delta }\), with \(\delta < 1\), then the inequalities (5.9) imply
As \(n \rightarrow \infty \), we have
Hence, we conclude that \(\lambda _n(A+B)\sim c n^{-s}\). \(\square \)
We now turn back to prove Theorem 5.1.
We consider the eigenpairs of the operator \(V\tilde{T}V\), where \(\tilde{T}\) is defined in Eq. (2.9), and \(V\) is defined by
where \(\{e_n\}\) are the eigenvectors of the operator \(V\tilde{T}V\), \(0 \le a_n \le 1\) (the last approximate equality is valid for \(a_n \rightarrow 0\) as \(n\rightarrow \infty \)). Setting
then,
Let us denote by \(K_N\) the image of the orthonormal projection on the first \(N^\mathrm{th}\) eigenvectors of \(R\) and by \(R_N\) the image of the orthonormal projection on the remaining eigenvectors of \(R\). By definition, \(R=K_N+R_N\). Then,
We note that \(\mathrm{rank}(K_N)=N\) and in formula (5.10), whenever there is a \(K_N\), we have an operator of rank \(N\). Moreover, \(-R_N\tilde{T}-\tilde{T}R_N+R_N\tilde{T}R_N\) has the norm dominated by the \(N^\mathrm{th}\) eigenvalue of \(R\):
Since the operator \(\tilde{T}\) is the solution of the Sturm–Liouville problem, \(\lambda _n(\tilde{T})\sim A_{\epsilon } n^{-2}\). Hence, according to Lemma 5.5, we get
Appendix B: Numerical computation of spectral characteristics
We briefly present a numerical algorithm to compute the spectral characteristics \(\lambda _n\) and \(\psi _n\). In order to compute the eigenvalues \(\lambda _n\) and the eigenvectors \(e_n\) of the operator \(V\tilde{T}V\), we get an explicit representation of this operator in the basis \(\cos n\theta \). First, we find
and
from which the expansion of \(\tilde{T}(\cos n\theta )\) (\(n\ge 0\)) in the basis \(\{\cos n\theta \}\) of \(L_{\text {even}}^{2}{[0,\pi ]}\) is
where the coefficients \(\mathbf{T}_{mn}\) are defined by
In turn, the operator \(V\) has a diagonal representation (we set \(R = 1\)):
Combining these results, the operator \(V\tilde{T}V\) is represented by the infinite-dimensional matrix \(\mathbf{V}\mathbf{T}\mathbf{V}\) whose elements are
and \([\mathbf{V}\mathbf{T}\mathbf{V}]_{m,n} = 0\) if \(m = 0\) or \(n = 0\). Finding the eigenvalues \(\{\lambda _n\}\) and the eigenvectors \(\{e_n\}\) of the operator \(V\tilde{T}V\) is equivalent to finding the eigenpairs of the associated matrix \(\mathbf{V}\mathbf{T}\mathbf{V}\). Note that this matrix is symmetric.
The matrix \(\mathbf{V}\mathbf{T}\mathbf{V}\) is diagonalized in Matlab that finds the eigenvalues \(\lambda _n\) and the coefficients \(v_{mn}\) determining the orthonormal basis \(\{e_n\}_{n \ge 0}\) as
The spectral weights \(\psi _n\) are then given as
where
and \(\langle \psi ,1 \rangle = \langle V\tilde{T}(1), 1 \rangle = \langle \tilde{T}(1), V1 \rangle = 0\).
Appendix C: Pure bulk diffusion phase
In [24], the mean exit time for pure bulk diffusion phase (\(\lambda = \infty \), \(a = 0\)) was found to be
where
where \(P_n(z)\) are Legendre polynomials. In the limit \(r\rightarrow R\), one gets
The average of the uniformly distributed starting point \(\theta \) yields
For small \(\epsilon \), the first term dominates yielding \(\langle t_1\rangle _b \simeq \frac{R^2}{D_2} \ln (2/\epsilon ) (1 + O(\epsilon ))\).
Figure 11 shows the mean exit times \(\langle t_1\rangle _{\lambda =0}\) and \(\langle t_1\rangle _{\lambda = \infty }\) from Eqs. (2.40) and (7.5) for surface diffusion and pure bulk diffusion, as a function of \(\epsilon \).
Appendix D: Transportation case (\(a=R\))
As we earlier discussed, one typically considers small values of the reflection distance \(a\). Nevertheless, the results of this paper are applicable to any value of \(a\) from \(0\) to \(R\). The so-called transportation case \(a = R\) when the particle is reflected to the origin of the disk, was studied by Bénichou et al. [14]. In this case, successive explorations between any two reflections are independent that allows one to get much simpler formulas. For instance, in the limit \(\lambda \rightarrow \infty \), the Laplace-transformed probability density of the exit time, \({\mathcal L}[P(t)](s)\) (with a uniformly chosen initial point on the circle), has a simple expression: \({\mathcal L}[P(t)](s) = q\bigl [1 - (1-q)/I_0\bigl (\sqrt{s R^2/D_2}\bigr )\bigr ]^{-1}\), where \(q = \epsilon /\pi \), and \(1/I_0(\sqrt{s R^2/D_2})\) is the Laplace-transformed probability density for the first passage time to the circle when started from the origin (with \(I_0(z)\) being the modified Bessel function of the first kind). As a consequence, the mean exit is simply
This limit is clearly seen on Fig. 8.
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Bénichou, O., Grebenkov, D.S., Hillairet, L. et al. Mean exit time for surface-mediated diffusion: spectral analysis and asymptotic behavior. Anal.Math.Phys. 5, 321–362 (2015). https://doi.org/10.1007/s13324-015-0098-0
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DOI: https://doi.org/10.1007/s13324-015-0098-0