Localization of Defects Via Residence Time Measures

Localization of defects via residence time measures A. Ciallella E mail: alessandro.ciallella@univaq.it arXiv:2011.08624v1 [cond-mat.stat-mech] 17 Nov 2020 Dipartimento di Ingegneria Civile, Edile – Architettura e Ambientale, Centro di Ricerca M&MoCS Università degli Studi dell’Aquila, via Giovanni Gronchi 18, 67100 L’Aquila, Italy. E.N.M. Cirillo E mail: emilio.cirillo@uniroma.it Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sapienza Università di Roma, via A. Scarpa 16, 00161, Roma, Italy. B. Vantaggi E mail: barbara.vantaggi@uniroma1.it Dipartimento Metodi e Modelli per l’Economia, il Territorio e la Finanza, Sapienza Università di Roma, via Del Castro Laurenziano 9, 00161, Roma, Italy. Abstract. We show that residence time measure can be used to identify the geometrical and transmission properties of a defect along a path. The model we study is based on a one–dimensional simple random walk. The sites of the lattice are regular, i.e., the jumping probabilities are the same in each site, except for a site, called defect, where the jumping probabilities are different. At each side of the lattice an absorbing site is present. We show that by measuring the fraction of particles crossing the channel and/or the typical time they need to cross it, it is possible to identify the main features of the lattice and of the defect site, namely, the jumping probabilities at regular and at the defect sites and the position of the defect in the lattice. Keywords: residence time, random walk, defect localization 1. Introduction The effect of obstacles in the transport of moving agents has been widely studied in different contexts [1]. Macromolecules, playing the role of obstacles for the diffusing smaller molecules [2–5], tend to slow down the dynamics, while in granular systems [6–9] and in the framework of pedestrian dynamics [10–15], the correct positioning of physical obstacles can accelerate the exit of the agents from a bounded region hindering clogging patterns [16–19]. ccv-defect˙v˙nov.tex – 18 novembre 2020 1 1:49 Geometrical and dynamical properties of obstacles, such as position, shape, and attitude to transfer mass, influence the transport properties of the whole system. We refer the reader to some old and more recent studies where the effect of obstacles has been studied with different techniques and for many different stochastic dynamics, such as the simple exclusion 1D model [20], a Probabilistic Cellular Automata [21] the 1D and 2D symmetric and asymmetric random walk [22, 23], the linear Boltzmann dynamics [24], the zero range process [25–27], and the simple exclusion 2D process [28, 29]. As it is clearly demonstrated by those studies, in the applications it can be of the outmost importance the knowledge of the geometrical and transport properties of obstacles which limit and affect the dynamics of the agents in a region of space, since the currents observed in the system strongly depend on them. On the other hand obstacles can be often not easily accessible to instruments for experimental measures: imagine an obstacle placed inside a biological channel or in a long pipe. Hence, it is rather natural to ask ourselves if it is possible to deduce the properties of the obstacle measuring observable quantities connected to the motion of agents. In this paper we shall consider a model of particles flowing on a lane and the observables that will be used to deduce the properties of the lane and of the obstacle are the fraction of particles crossing the lane and the mean time they take to cross it. We shall consider two different experiments. In experiment 1 we imagine to start a known number of particles at one end point of the lane and compute the number of particles which will exit the lane through the second end point. Thus, in this experiment we have access to the fraction of particles that succeed to cross the lane. In experiment 2 we imagine to start, all at the same time, an unknown number of particles at one end point of the lane and to measure the time that the particles that succeed to cross the lane take to exit it through the second end point. Thus, in this experiment we are not able to measure the fraction of particles that actually cross the lane, but we have access to the mean value of the time that the particles that cross the lane need to cross it. The typical time needed for a particle to cross the channel will be called residence time. We mention that the idea of residence time as average crossing time conditioned to crossing a region was firstly introduced in [28, 29] for a 2D simple exclusion walk: a thorough study of its properties in absence of obstacle is reported in [29], whereas the presence of obstacles was considered in [28]. We notice that in the 1D version of the simple exclusion model with Langmuir kinetics the idea of residence time was exploited in [30]. The problem introduced above is faced in this paper via a one–dimensional model based on a simple random walk. At each side of the lattice an absorbing site is present. We deal with the problem of identify the features of the lattice, once we measured some observable ccv-defect˙v˙nov.tex – 18 novembre 2020 2 1:49 such as the fraction of particles reaching the right absorbing site, and the average time needed for a walker to reach it. More precisely, the model we consider in this paper is a one–dimensional simple random walk on {0, 1, . . . , L} possibly with a defect site. The sites 0 and L ≥ 3 are absorbing, so that when the particle reaches one of these two sites the walk is stopped. All the sites 1, . . . , L − 1 are regular excepted for one site called defect [22, 23]. The defect site is the site d. At each unit of time the walker jumps to a neighboring site according to the following rule: if it is on a regular site, then it jumps to the right with probability p or to the left with probability q = 1 − p, so that it cannot stay still at the site. If it is at the defect site it jumps to the right with probability p0 or to the left with probability q 0 = 1 − p0 . The array 1, . . . , L − 1 will be called the lane. The sites 0 and L will be, respectively, called the left and right exit of the lane. In this framework the residence time is defined by starting the walk at site 1 and computing the typical time that the particle takes to reach the site L provided the walker reaches L before 0. More precisely, we let xt be the position of the walker at time t and denote by P and E the probability associated to the process and the related expected operator for the walk started at x0 = 1. We let τ be the first hitting time to L or crossing time, namely, τ = inf{t > 0 : xt = L} , (1.1) with the convention that τ = ∞ if the trajectory reaches 0 before reaching L. Note that the trajectories cannot be trapped for an infinite time in the lane because L is finite. Note also that, since the particle starts at site 1, the values that τ can assume are of the form L + 2k − 1, for any integer k ≥ 0. We thus define the residence time as R = E(τ |R.E.) , (1.2) where we conditioned to the event R.E. meaning that the particle exits the lane through the right exit in L. In absence of defect, namely, the sole parameters that one wants to deduce is the jumping probability p which accounts for the transport properties of the lane. In case of experiment 1, namely, when the fraction of particles crossing the lane is available, a standard maximum likelihood argument provides a correct estimate for the parameter, which is fully identified. On the other hand, in case of experiment 2, again an estimate is provided by a maximum likelihood argument, but a complete identification of the parameter p is lacking since the residence time is symmetric when p is exchanged with 1 − p. In presence of defects the situation is much more complicated, since one wants to deduce the three parameters p, d, and p0 , which respectively account for the transport properties of ccv-defect˙v˙nov.tex – 18 novembre 2020 3 1:49 the lane (p), the geometry of the defect (its position d), and the transmission property of the obstacle (p0 ). In our discussion we shall assume p to be known and we shall show that both in case of experiment 1 and 2 it will not possible to estimate the values of d and p0 , but it will possible to find an equation connecting the two. In other words, in the parameter space d–p0 , a level curve on which the pair (d, p0 ) has to lay will be identified. If the two experiments, 1 and 2, are performed simultaneously, namely, provided the fraction of particles crossing the lane and the residence time are both measured, then in many case the identifiability problem will be fully solved and an estimate for the parameters d and p0 will be found. The paper is organized as follows: in Section 2 we consider the case in which the lane does not present any defect. The presence of a defect site is then discussed in Section 3. Finally, the Section 4 is devoted to some brief conclusions. 2. Simple random walk: no defect We first consider the simple random walk case, namely, here we assume that p0 = p. For a given p, considering a single particle, the joint probability of the event that it exits through the right side starting from site 1 and the first hitting time to L is n = L + 2k − 1 is P(τ = n, R.E.) = αn p L+n−1 2 q n−(L−1) 2 = αL+2k−1 pL+k−1 q k , (2.3) where αn is the number of different possible paths that starts in 1 and exits the lane reaching the site L for the first time at time n, without hitting the site 0. Moreover, in each of these paths the walker performs (n − (L − 1))/2 steps to the left and (n − (L − 1))/2 + L − 1 steps to the right. We stress that αn is smaller that the total number of paths made of (n − (L − 1))/2 steps to the left and (n − (L − 1))/2 + L − 1 steps to the right, since the constraint that the particles reaches L before touching 0 must be taken into account. It is well known, see, e.g., [22, 31–33], that the probability of the event R.E. is given by the classic gambler’s ruin probability P(R.E.) = 1 − (q/p) 1 if p 6= q and P(R.E..) = if p = q. L 1 − (q/p) L (2.4) Thus, for p 6= q P(τ = n|R.E.) = αn p L−1 L X −1 q L − pL q = αn p k q k = α n pk q k pj q L−1−j , (q/p) − 1 q−p j=0 L−1+k k (q/p) (2.5) where, we recall, the time n is in the form L + 2k − 1. Note the last expression in (2.5) is valid also for p = q, indeed P(τ = n|R.E.) = αn pL+2k−1 L. ccv-defect˙v˙nov.tex – 18 novembre 2020 4 1:49 Now, we consider M independent identically distributed walkers starting at 1 and denote by τi the crossing time of the particle i, for i = 1, . . . , M . The probability that the M particles exit the lane through the right exit with crossing times ni = L + 2ki − 1 is P(τ1 = n1 , τ2 = n2 , . . . , τM = nM , R.E.1 , R.E.2 , . . . , R.E.M ) = M Y αni pL+ki −1 q ki , (2.6) i=1 where with R.E.i we denote the event that the i particle exits the lane through the right exit. Using, as above, the gambler’s ruin result, we get P(τ1 = n1 , τ2 = n2 , . . . , τM = nM |R.E.1 , R.E.2 , . . . , R.E.M ) = ! !M  M  M L−1 L L Y Y X PM PM q − p = αni pki q ki = αni p i=1 ki q i=1 ki pj q L−1−j q − p i=1 i=1 j=0 (2.7) We remark that conditioning to exiting through the right side do not influence the independence of the observed crossing times for given p. Consider, for simplicity, the case M = 2: P(τ1 = n1 , R.E.1 , τ2 = n2 , R.E.2 ) (2.8) P(τ1 = n1 , τ2 = n2 |R.E.1 , R.E.2 ) = P(R.E.1 , R.E.2 ) P(τ1 = n1 , R.E.1 ) P(τ2 = n2 , R.E.2 ) P(τ1 = n1 |R.E.1 )P(τ2 = n2 |R.E.2 ) = · . (2.9) P(R.E.1 ) P(R.E.2 ) Since the crossing times are independent and identically distributed random variables it follows immediately that the left–hand sides of (2.8)–(2.9) are equal. As we discussed in the Introduction, the problem we address in this Section is the estimate of the parameter p: we assume to perform an experiment and after we measure some observables we want to trace back the value of p. We discuss, now, two different experiments: in the first case we measure the fraction of particles which succeed to cross the lane, namely, the fraction of particles which exit the lane through the right exit in L. In the second experiment we shall assume to measure only the average time spent by the particles in the lane before exiting through the right end without having any information on the fraction of successful particles. In these experiments the role of identifiability of models is discussed. In the following a statistical model is said to be identifiable whenever the likelihood has no flat region, so it is theoretically possible to estimate the underlying parameters. This means that different values of the parameters must generate a different value for the likelihood over the observable variables, i.e. L(θ, (x)) 6= L(θ0 , (x)) for any θ 6= θ0 . For non-identifiable models: two or more values of the parameters are observationally equivalent. In these cases, it is relevant to determine the non-identifiable regions. Furthermore a ccv-defect˙v˙nov.tex – 18 novembre 2020 5 1:49 model is said locally identifiable whenever for any θ in the parameter space Θ there exists an open neighborhood Nθ of θ in Θ such that for any θ0 ∈ Nθ it holds L(θ, (x)) 6= L(θ0 , (x)) for any θ 6= θ0 . 2.1. Experiment 1: measuring the fraction of successful particles The knowledge of the fraction of the number of particles which exit through the right side is sufficient to find a maximum likelihood estimate of p using the probability of success in the gambler’s ruin problem (2.4). This probability as a function of p is strictly monotonic, hence, if we prepare N particles at site 1 and denote by m the number of those particles that exits the lane through the right exit in L, the equation in the unknown p m = P(R.E.) N (2.10) has an unique solution p̂m . Note that m/N is the maximum likelihood estimate of P(R.E.). Indeed, the random variables χi taking value 1 if the particle i reaches the right side of the interval and 0 otherwise are i.i.d. Bernoulli random variables, so that we recall here that in P this case the maximum likelihood estimator N i=1 χi /N is not biased. The classical computation follows for completeness: we can estimate p by finding the   N arg max (P(R.E.))m (1 − P(R.E.))N −m . p∈[0,1] m By considering the log–likelihood we write   N `(p) = ln + m ln P(R.E.) + (N − m) ln(1 − P(R.E.)) . m (2.11) To search for its critical point we derive with respect to p: d 1 d 1 d `(P(R.E.)) = m P(R.E.) − (N − m) P(R.E.) = dp P(R.E.) dp 1 − P(R.E.) dp m N −m d P(R.E.)[ − ]=0 = dp P(R.E.) 1 − P(R.E.) (2.12) Since dP(R.E.)/dp is strictly positive, the former has a unique solution P(R.E.) = m/N , as we wrote in (2.10). 2.2. Experiment 2: measuring the crossing time In case the fraction of particles crossing the lattice is not know, it is possible to setup an estimate of p by measuring the average crossing time. Even in this case we will be able to perform a maximum likelihood estimate. Suppose we measure the crossing time ni of m particles, for i = 1, . . . , m, we evaluate the parameter p by comparing the experimental mean ccv-defect˙v˙nov.tex – 18 novembre 2020 6 1:49 
Pm crossing time n /m to the theoretical residence time R. The equation providing our i i=1 estimate for p will be found via a maximum likelihood argument. We note that the residence time presents a symmetry in the role of p and q = 1 − p that suggests the presence of identifiability problems. That is to say, if the residence time of walkers is the sole information we have access to, it is not possible to uniquely deduce the jumping parameter p that characterize the walk. Indeed, looking at (2.7), we notice that the crossing time observed for particles that eventually exit through the site L starting from site 1, is symmetric1 in p ∈ [0, 1] with respect to 1/2 (recall that q = 1 − p). The existence of this symmetry is also evident if one looks at the dependence of the residence time on p, see, e.g., Fig. 2.1. 3500 3000 2500 R 2000 1500 1000 500 0 0.0 0.2 0.4 0.6 0.8 1.0 p Figure 2.1: Residence time R as a function of p ∈ [0, 1] for L = 100 in absence of defect. In other words, looking at the symmetry residence time, we can expect that the experimental information on the crossing time won’t be sufficient to distinguish different estimates of p. More precisely, we find two possible estimates p̄ and 1 − p̄ for the parameter p. These two solutions will reduce to a single one in the case in which p̄ = 1 − p̄ = 1/2. This will be, indeed, the sole case in which we will have complete identifiability for the parameter p. Note, however, that, if the value of p is not very close to 1/2, a partial information on the fraction of particles exiting from the right side (e.g., the knowledge of the order of magnitude of the number of particles started from 1) is sufficient to distinguish between the two cases and identify the correct estimate p̄ or 1 − p̄. To construct the estimate of p based on the residence time measure we, now, setup a maximum likelihood argument: we focus on m particles that we know to have reached the site L and observe their crossing times ni . We know, from (2.5) and (2.7) the probability of observing τ1 = n1 , τ2 = n2 ,. . ., τm = nm conditioned to R.E.i , for i = 1, . . . , m. Maximum 1 We refer the interested reader to [33] and [34] where other classical results from Stern and Samuels for walks starting from the middle point are discussed. ccv-defect˙v˙nov.tex – 18 novembre 2020 7 1:49 likelihood gives the estimation pm of p as pm = arg max m Y p∈[0,1] = arg max ! αni p P p P i ki P q i ki i=1 m Y p∈[0,1] ! αni i=1 i ki P q i ki  qL − pL q−p L−1 X m (2.13) !m pj qL−1−j , j=0 where ni = L + 2ki − 1. Notice that the last expression shows that the function is symmetric in the exchange p into 1 − p. The log–likelihood is `(p) = m X ln αni + m X i=1 ki ln p + i=1 m X  ki ln(1 − p) + m ln i=1 (1 − p)L − pL 1 − 2p  (2.14) and, looking for the arg max of `(p), we find the critical points of `(p) as the solutions of the equation ! m X ki 1 − 2p 2 (1 − p)L−1 + pL−1 = + L . (2.15) 2 L − pL m p − p 2p − 1 (1 − p) i=1 To discuss the structure of the solutions of (2.15), due to the invariance under the exchange p in 1 − p, we restrict the discussion to the interval [1/2, 1] and denote by f (p) and g(p) its left–hand and right–hand sides. It is possible to check that the following statements hold: – lim f (p) = 0, lim− f (p) = −∞, lim g(p) = 0, lim− g(p) = −L + 2; p→1/2+ – lim f 0 (p) = −8 p→1/2+ p→1/2+ p→1 m X ki i=1 m p→1 , lim g 0 (p) = −8/3 + 4L − 4L2 /3; p→1/2+ – f (p) is a concave monotonically decreasing function in (1/2, 1). Moreover, we checekd numerically that there exists p∗ ∈ (1/2, 1] that depends on L such that g(p) is convex and strictly monotonically decreasing in (1/2, p∗ ), while it is monotonically P ki 2 increasing in (p∗ , 1). Hence, if −8 m i=1 m ≤ −8/3 + 4L − 4L /3 the sole solution of (2.15) is p = 1/2, and it is a maximum point. Otherwise, there exists one more solution of (2.15) in (1/2, 1). It is easy to see that such a solution is a maximum point, while 1/2 becomes in this case a minimum point. P ki 2 The above remarks yield the following conclusions: If −8 m i=1 m ≤ −8/3+4L−4L /3 the model is globally identifiable and the maximum likelihood estimate is p̄m = 1/2. Otherwise, the model is locally (not globally) identifiable and there exists two maximum likelihood estimates p̄m and 1 − p̄m . ccv-defect˙v˙nov.tex – 18 novembre 2020 8 1:49 4000 150 200 5000 100 2000 100 50 0 0 0 0 -50 -2000 -100 -100 -5000 -150 -4000 0.2 0.3 0.4 0.5 0.6 0.7 0.8 p -200 0.2 0.3 0.4 0.5 0.6 0.7 0.8 p 0.48 0.49 0.50 0.51 0.52 p 0.48 0.49 0.50 0.51 0.52 p Figure 2.2: Graphical solution of equation (2.15) for L = 100: the orange and the blue curves (color online) are, respectively, the plots of the right and left hand sides of the equation. The second and the fourth panel are a magnification of the first and the third, respectively. P The experimental measure i ki /m is equal to 895 in the first two panels and 1623 in the last two. The experimental data used in the first two plot have been obtained by simulating the process with p = 0.48, whereas for the last two plots the simulation has been run with p = 0.5. The two curves intersect each others at 0.4802, 0.5 and 0.5198 in the first two plots and at 0.5 in the last two. We conclude this section testing the above procedure performing a numerical experiment. We first run the model with L = 100 and p = 0.48. We find 1889.17 as numerical estimate P for the residence time, so that the experimental value of the quantity m i=1 ki /m appearing in (2.8) is 895.085. The graphical solution of (2.15) is depicted in the two leftmost panels of Fig. 2.2. The two curves intersect in 1/2 and in the two symmetric points 0.4802 and 0.5198. These two last values are the maximum likelihood estimate for p and, as we discussed above, the information provided by this experiment is not sufficient to identify correctly the parameter. On the other hand, by running the simulation with p = 1/2, we found 3344.65 as numerical estimate for the residence time, so that the experimental value of the quantity Pm i=1 ki /m appearing in (2.8) is 1622.83. The graphical solution of (2.15) is, now, depicted in the two rightmost panels of Fig. 2.2. The two curves intersect only in p = 1/2 which is the maximum likelihood estimate for p in this case. 3. Identification of a defect We consider now the case of presence of a defect in the lane. The defect is the site d, where the probabilities are p0 to jump to the right and q 0 = 1 − p0 to the left. We can have different situations: p, d, p0 can be known or unknown. In this section2 we suppose that p is 2 In case we suppose that the geometric and transport properties of the defect which, in our model, are represented by d and p0 , are known, then we face the problem of estimating p. Suppose we know N and we measure the number m of crossing particles: since for fixed p0 and d the probability of reaching the site L is monotonically increasing as a function of p, as we did in the case of no defect (see eq.(2.10)), the maximum ccv-defect˙v˙nov.tex – 18 novembre 2020 9 1:49 known, namely, the transport properties of the lane are known, and we want to estimate the geometric and transport properties of the defect which, in our model, are represented by d and p0 . Note first of all that this case is more complex than the previous one without the defect site. The probability for a walker to jump out from the right–side exit in a time exactly n, starting from 1 is now P(τ = n, R.E.) = j k X X αi,j,d pL−1+k−i (p0 )i q k−(j−i) (q 0 )j−i , (3.16) j=1 i=1 where j counts the total number of times that the particles jumped on the defect d before reaching the site L, while i is the number of times that the particle starting from d jumped to d + 1. Moreover, αi,j,d is the total number of different paths that the particle can perform in the line for each choice of i, j, d. Calculation of αi,j,d is in principle possible but it is not easy and not necessary for our purpose. 3.1. Measuring the fraction of successful particle in presence of a defect It is possible to produce explicit formulas for the probability of exiting through the right end point even in presence of a defect (see [22]). Indeed, the probability to cross the lane is given by (3.17) and (3.18) and can be considered (for p fixed and known) as a function of p0 and d: p0 (p − q)q P(R.E.)(p0 , d) = , p 6= q; (3.17) p(−p0 q(−1 + (q/p)d ) + pq 0 (−(q/p)L + (q/p)d )) 1 , p = q. (3.18) d + (L − d)q 0 /p0 Using the expressions of the probability to reach the right side (3.17) and (3.18), the observation of how many particles reached the site L allows to estimate (p0 , d) through a maximum likelihood procedure. Indeed, if m of N independent particles started at 1 exit the lane through the right end point, similarly to what we did in absence of defect we can write   N 0 ˆ (p̂ , d) = arg 0 max (P(R.E.))m (1 − P(R.E.))N −m , (3.19) p ∈[0,1],d∈{2,...,L−2} m P(R.E.)(p0 , d) = likelihood estimation gives m/N = P(R.E.). Hence, solving such an equation we get an estimate for p and the parameter p is completely identifiable. On the other hand, in case N is not known, but it is possible to measure the mean crossing time we find a lack of complete identifiability. Indeed, as in the case of no defect, the identification of p as the most likely value producing the observed residence time in general do not allow to distinguish between two different possible values of p, since the residence time as a function of p is monotonically increasing up to a value p∗ and then it decreases (see, for instance, [22, right panel of Fig. 9]). The effect of the presence of the defect is that of modifying the numerical value of the measured residence time and to shift the value of p∗ , that in the case of no defect is 0.5. ccv-defect˙v˙nov.tex – 18 novembre 2020 10 1:49 under the assumption that the Bernoulli random variables taking values 1 if R.E. is true and zero otherwise are independent and identically distributed. We consider d to be a continuous variable, to simplify the evaluation, but we should remember that only discrete values d = {2, 3, . . . , L − 2} for the defect are significant in our problem. Considering the log–likelihood we write   N 0 `(d, p ) = ln + m ln P(R.E.) + (N − m) ln(1 − P(R.E.)) (3.20) m and we derive it with respect to p0 and d to search for critical points   ∂` = m ∂ P(R.E.) − N −m ∂ P(R.E.) =  m − N −m  ∂ P(R.E.) = 0 ∂p0 P(R.E.) ∂p0 1−P(R.E.) ∂p0 P(R.E.) 1−P(R.E.) ∂p0  ∂ m ∂ N −m ∂ m ∂`  = P(R.E.) − P(R.E.) = − N −m P(R.E.) = 0 . ∂d P(R.E.) ∂d 1−P(R.E.) ∂d P(R.E.) 1−P(R.E.) ∂d Hence, all the points (d, p0 ) that verify P(R.E.) = m N (3.21) are critical, where, we recall, P(R.E.) is given in (3.17) and (3.18). To prove that the solutions of (3.21) are the sole critical point for the maximum likelihood function, we show that ∂P(R.E.)/∂p0 and ∂P(R.E.)/∂d are different from zero. We first discuss in detail the case p = 1/2: from (3.18) we have ∂ (L − d)/p02 P(R.E.) =  2 ∂p0 d + (L − d)(1 − p0 )/p0 and ∂ 1 − (1 − p0 )/p0 P(R.E.) =  2 . ∂d d + (L − d)(1 − p0 )/p0 The quantity on the left is always positive for d < L and it is equal to 0 if and only if d = L, while the quantity on the right is positive for p0 > 1/2 and negative for p0 < 1/2, and it is equal to 0 if and only if p0 = 1/2, that is to say in absence of defect (p = q = p0 = 1/2). The case p 6= q can be discussed analogously and, again, we find that the partial derivative with respect to p0 is equal to zero if and only if d = L, while the partial derivative with respect to d is equal to zero if and only if p0 = p, that is to say in the case where no defect is present. Given m and N , the solutions of the equation (3.21) determine a curve in the plane p0 –d. These curves are depicted in Fig. 3.3 for different values of p. The contour plots in Fig. 3.3 show that, given m and N , the geometric locus made of the solutions of (3.21) provides d as a function of p0 which is decreasing if p0 < p, increasing for p0 > p, and constant for p0 = p. Note that equation (3.21) is easily explicitly solvable, finding p0 as a function of d and taking into account only the values of d that produce p0 ∈ [0, 1]. Summarizing, dealing with the experiment 1 we are not able to fully estimate the transport and geometric properties of the obstacle, p0 and d. Indeed, we are able to find only a ccv-defect˙v˙nov.tex – 18 novembre 2020 11 1:49 d 60 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 100 80 d 60 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0.00400 0.00300 0.00200 0.00100 0.00076 0.00050 0.00025 0.00010 0.14000 0.13000 0.12000 0.11500 0.11321 0.11000 0.10000 0.09000 80 60 d 80 100 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 100 80 60 d 100 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0.030 0.026 0.022 0.018 0.014 0.010 0.006 0.002 0.18400 0.18300 0.18200 0.18182 0.18100 0.18000 0.17900 0.17800 Figure 3.3: Contour plot of P(R.E.) in the plane p0 –d for L = 100. Top left p = 0.49, top right p = 0.5, bottom left p = 0.53, bottom right p = 0.55. relation between the two parameters, that is to say we can determine a curve in the plane p0 –d where the parameters have to lay. Hence, the model is not anymore locally identifiable. 3.2. Residence time in presence of a defect In presence of a defect we have just shown that experiment 1, namely, the measure of the fraction of particle crossing the lane, is not sufficient to achieve an estimate for the model parameters p0 and d. In this section we approach the problem also from the point of view of the residence time estimate: we shall see that using both the fraction and the residence time measure, in some cases, it will be possible to identify completely the two parameters. Due to the complicated structure of the problem, we do not rely on a pure maximum likelihood estimate, but we simply identify the theoretical residence time with the experimental mean crossing time by referring the method of moments. In order to derive a theoretical expression for the residence time we follow the strategy ccv-defect˙v˙nov.tex – 18 novembre 2020 12 1:49 outlined in [22, 31–33] and based on a generating function computation. We let un be the probability that the walk started at 1 exits through the right end after n steps. We construct the generating function of the probability of exiting from the right side starting P from 1, U (s) = n≥0 un sn . The generating function can be explicitly calculated but has not a simple and concise expression; we report the computation in the Appendix A for completeness, see also [22]. Now, we note that P(R.E.) = U (1). Since the derivative of the generating function is P 0 U (s) = n≥1 nun sn−1 , following [32] we have that X X X lim− U 0 (s) = nun = nP(τ = n, R.E.) = nP(τ = n|R.E.)P(R.E.) = R P(R.E.), s→1 n≥1 n≥1 n≥1 (3.22) where R is the residence time as defined in (1.2). We note that R is finite for any fixed p, q, p0 , and q 0 in (0, 1). By deriving two times U (s) we find the following expression, that is useful to calculate the variance of the residence time: " # X X X X lim U 00 (s) = n(n − 1)un = n2 un − nun = P(R.E.) n2 P(τ = n|R.E.) − R s→1− n≥2 n≥1 n≥1 n≥1   = E(τ 2 |R.E.) − R P(R.E.) . Thus, it is possible to calculate the variance of the residence time as Var(τ |R.E.) = E(τ 2 |R.E.) − [E(τ |R.E.)]2 = 1 [U 00 (1) + U 0 (1)] − R2 . P(R.E.) The residence time as a function of p0 and d can be computed using the equations shown above. Data for L = 100 are reported in Fig. 3.4 as a scatter plot. We first notice that the picture in the case p = 0.49 is specular to that one for p = 0.51, and this is always true for any choice of p and 1 − p. In the symmetric case p = 1/2 the residence time has a peculiar behavior: for p0 = p = 1/2 (no real defect is indeed present) and in the case d = L/2 the residence time in presence of the defect is not influenced by the defect, it is the same as the symmetric case with no defect, see the yellow region in the plot (panel top right). These two straight lines divide the rectangle [0, 1] × Z ∩ {2, 3, . . . , L − 2} into four zones symmetric with respect to the center: two specular regions where the residence time is smaller with respect to the case of no defect (blue zone in the plot) and two specular regions where the presence of the defect produce a larger residence time (from red to green in the plot). In particular, a defect with p0 < p = 1/2 close to the entrance favors a faster egress of (a smaller number of) particles, as well as the presence of a defect with p0 > p = 1/2 close to the exit. The defects close to the ccv-defect˙v˙nov.tex – 18 novembre 2020 13 1:49 25 0.5 25 0.5 3000 0.25 0.5 0.75 p' 1 16000 14000 residence time d 50 4000 0 22000 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 75 5000 2000 0.75 p' 0.25 50 25 75 12000 10000 d 0.25 6000 residence time 50 7000 75 residence time residence time d 75 8000 d 22000 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 50 8000 6000 4000 25 2000 0 0.75 0.25 p' 0.5 0.75 p' Figure 3.4: Contour plot of the residence time R for L = 100, p = 0.49 (top left), p = 0.50 (top right), p = 0.51 (bottom left), p = 0.53 (bottom right). exit with p0 < p = 1/2 tend to trap particles that reached the second half of the interval, as well as defects with p0 > p = 1/2 close to the entrance that tend to favor a long permanence of the walkers in the first half of the interval. We remark that every observation about the residence time takes into account only the particles that eventually reach the right exit site L, ignoring those that come out of the interval at the 0 site. In the case p > 1/2, the behavior is a perturbation of the case p = 1/2: the four zones are still present if p is close to 1/2, but the two regions in the top of the plot tend to invade the whole rectangle [0, 1] × Z ∩ {2, 3, . . . , L − 2} when p is increased. It is still present a cross made of points (yellow in the plots in Fig. 3.4) where the residence time is the same as in the case of no defect, the vertical arm of the cross is at p0 = p, while the horizontal one moves downward when p is increased (in the plot is visible at about d = 27 for p = 0.51, at about d = 10 for p = 0.53). The regions below the horizontal arm are not the specularly ccv-defect˙v˙nov.tex – 18 novembre 2020 14 1:49 symmetric of the top regions, indeed, the values of the residence time vary more consistently in the top regions. When p is close to 1 in the residence time plot only two regions are left: for p0 < p the residence time is larger than in the no defect case, while for p0 > p it is the opposite. The case p < 1/2 is specular to the case p > 1/2. 3.3. Numerical Experiments We consider, now, a numerical experiment in which we melt both experiments 1 and 2, namely, considered N particles started at site 1 of the lane, we count the number of particles m which exit the lane through the right end and we measure the crossing time ni that each of them takes to reach the exit. Hence, we get both a numerical measure of the crossing P fraction m/N and of the mean crossing time m i=1 ni /m. The question we pose is the following: supposing to know the transport properties of the lane expressed by the parameter p, can we estimate the geometry and the transport property of the obstacle represented respectively by d and p0 ? Recalling the definition of the i.i.d. Bernoulli random variables χi introduced in Section 2.1, we note that m is the value assumed in the experiment by the random variable (χ1 + . . . + χN )/N and remark that, by the law of large numbers, such a variable converges to P(R.E.) as N → ∞. We now estimate the rate of convergence: for any given ε > 0, thanks to the Central Limit Theorem, we have that √  (χ + . . . + χ )    Nε 1 N P − P(R.E.)) ≤ ε ' 2Φ −1 (3.23) N P(R.E.)(1 − P(R.E.)) √ where Φ is the cumulative Gaussian distribution. Given ε > 0, for N ε/[P(R.E.)(1 − P(R.E.))] ≥ 3 we can estimate this probability to be very close to 1 (about 0.9974). Note that we do not know the correct value of P(R.E.) since we do not know (p0 , d) that identify the defect. By equation (3.21) the probability P(R.E.) is estimated by m/N . Considering the unbiased sample variance estimator of the i.i.d. χi ! N  N 2  m 2 X X 1 m N 1 ŝ2 = χi − = χ2i − , N − 1 i=1 N N − 1 N i=1 N we estimate ŝ2 as   m N −m N m m2 − 2 = . N −1 N N N −1 N The standard error of the mean is now computed as the observed realization of the sample standard deviation divided by the square root of the sample size: r ŝ m N −m σ̂χ = √ = . N − 1 N2 N ccv-defect˙v˙nov.tex – 18 novembre 2020 15 1:49 label p p0 d N m rex 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.53 0.53 0.53 0.53 0.53 0.53 0.55 0.55 0.55 0.130 0.130 0.065 0.130 0.750 0.800 0.800 0.800 0.050 0.800 0.250 0.250 0.250 0.800 0.250 0.100 0.700 19 8 92 70 10 85 85 85 43 50 75 75 75 15 78 43 25 1.0 · 108 2.0 · 108 1.0 · 106 2.0 · 107 2.0 · 107 1.0 · 106 1.0 · 107 1.0 · 108 5.0 · 106 1.0 · 107 1.0 · 107 5.0 · 107 2.0 · 108 1.0 · 107 1.0 · 107 5.0 · 106 5.0 · 106 1069462 1505795 36345 634906 1412009 40614 404194 4040196 507651 1134138 1132454 5660505 22631148 1285723 1817491 908079 914097 2441.266 2460.133 6366.541 5440.954 2930.654 2358.224 2349.55 2349.153 3751.445 1292.222 1701.426 1702.267 1701.527 1361.100 1030.056 1392.074 877.913 Table 1: List of experiments discussed in Section 3.3. A progressive number is associated to each experiment and the parameters p, p0 , d, and N are listed. In the last two columns we list the measured values for m and rex . Given an experimental observation of the fraction of the particles exiting the lane through L, we can consider the interval [m/N − 3σ̂χ , m/N + 3σ̂χ ] where P(R.E.) lies with very high probability. In the same way we study the standard error of the residence time, i.e., of the average crossing time, producing the confidence interval at level 0.9974. Then, we can find the regions of points of the plane p0 –d to which corresponds a residence time in that interval and plot those regions. Note that in this second case the particles producing the experimental estimate of the residence time, and so producing the corresponding confidence interval, are only those m which exit the lane through the right end point at L. We can thus conclude that, given the numerical measures fex = m/N for the fraction Pm of crossing particles and rex = i=1 ni /m for the mean crossing time, the estimate for the parameters d and p0 will be constructed by intersecting the regions of the plane p0 –d corresponding to the confidence intervals found for the crossing fraction and for the crossing ccv-defect˙v˙nov.tex – 18 novembre 2020 16 1:49 time. In the sequel of this section we discuss several experiments performed with different values of the jump probability p = 0.51, 0.53, 0.55 (see the list of the experiments in Table 1). d 60 40 20 80 0.01072548 0.01069462 0.01066376 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 100 80 2444.968 60 2441.266 d 80 100 40 2437.564 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 60 d 100 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.5: Experiment 1. Left: region associated to confidence interval for fex . Center: region associated to confidence interval for rex . Right: intersection of the two regions. Beside the real value of the obstacle parameters identified by the cyan point, a second intersection region around (0.1645, 11) is found (orange point). In Fig. 3.5 we show results for experiment 1: the regions associated to the confidence interval for fex and rex are shown together with their intersection. The confidence intervals for fex and rex are, respectively, [0.01066376, 0.01072548] and [2437.564, 2444.968]. The intersection is made by two small regions, from which we select as candidates only pairs (p0 , d) with integer d. The candidates we find are p0 ∈ [0.129445, 0.130208] for integer values of d = 19, and p0 ∈ [0.163927, 0.164804] for d = 11. For these values of the parameters, the residence time lies in the interval [2441.69, 2442.21] when p0 ∈ [0.129445, 0.130208] and d = 19 and it lies in the interval [2440.63, 2441.15] for p0 ∈ [0.163927, 0.164804] and d = 11. Thus, all these candidate pairs (p0 , d) give a residence time in the confidence interval for rex , so it is not possible to reject any of them. Hence, a complete identification is not possible. Nevertheless, it is clear that all candidate pairs (p0 , d) correspond to a defect close to the entrance which tends to obstruct the passage of walkers (p0 < p). No more pairs (p0 , d) with integer d lie on the intersection region. In Fig. 3.6 we show the results of experiment 2. The setting is similar to the experiment 1, but we are here able to identify the defect parameter d and restrict the possible values of p0 to a small interval. The regions associated to the confidence interval for fex and rex are shown together with a magnification of their intersection. The confidence intervals for fex and rex are, respectively, [0.00751064, 0.0754731] and [2456.74, 2465.526]. The intersection is made by two small regions. As we did in the case of experiment 1, we look for the pairs (p0 , d) with integer d which lie in this regions. The first region ccv-defect˙v˙nov.tex – 18 novembre 2020 17 1:49 100 d 60 40 20 80 0.00754731 0.00752898 60 d 80 25 40 0.00751064 20 2460.133 15 10 2456.740 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 2463.526 d 100 5 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0.08 0.10 0.12 0.14 p' Figure 3.6: Experiment 2. Left: region associated to confidence interval for fex . Center: region associated to confidence interval for rex . Right: magnification of the intersection of the two regions. Points identify the two intersections: the first around the actual defect (in cyan) and a second intersection at about (0.0835, 21.5) (in orange). let us identify as possible parameters of the defect p0 ∈ [0.129606, 0.130195] for the integer value of d = 8. No more admissible pairs can be found. In fact, considering here the two closest integer values (d1 = 21 and d2 = 22) to points in the second region. Taking d = 21, one can verify that the values of p0 that give a crossing probability laying in the confidence interval for fex are p0 ∈ [0.0846366, 0.0850611] but the residence times associated to these pairs are in the interval [2449.79, 2450.06], out of the confidence interval for rex . In the same way, taking d = 22, the candidate pairs associated to the confidence interval for fex are p0 ∈ [0.0817259, 0.0821386] whose associated residence times lie in the interval [2472.9, 2473.15], out of the confidence interval for rex . Hence, we identify the defect to have p0 ∈ [0.129606, 0.130195] and d = 8. 100 40 20 80 0.0369064 0.0363450 0.0357836 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 80 6442.95 60 6366.54 40 6290.13 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 60 d d 60 100 d 80 100 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.7: Experiment 3. Left, center and right panel as in Fig. 3.5. The cyan point identifies the defect. ccv-defect˙v˙nov.tex – 18 novembre 2020 18 1:49 In experiments 3 and 4, a defect with p0 < p, close to the exit site, can be identified as shown in Figs. 3.7 and 3.8. The intersection produces a unique connected region whose dimension is related to the number of starting particles N . In experiment 3, see Fig. 3.7, the regions associated to the confidence interval for fex and rex are shown together with their intersection. The confidence intervals for fex and rex are, respectively, [0.0357836, 0.0369064] and [6290.13, 6442.95]. The number of starting particles is N = 106 , that is sufficient to produce an intersection region that clearly identifies some peculiarities of the defect (d is close to L, so the defect is close to the exit site, and p0 is small, so that the defect hinder the passage) but not to identify a single possible value of d. 100 100 80 0.0318629 d 0.0317453 40 0.0316277 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 60 5440.95 d 60 80 5456.87 40 60 d 80 100 5425.03 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.8: Experiment 4. See caption of Fig. 3.7. In the case of experiment 4, instead, the confidence intervals for fex and rex are, respectively, [0.0316277, 0.0318629] and [5425.03, 5456.87]. The large value of N = 2 · 107 allow to individuate a smaller intersection region, so that the defect is completely identifiable in this case. The only pairs with integer d laying in the intersection region are those with p0 ∈ [0.129184, 0.130592] and d = 70. Defects that favor the passage (p0 > p) and close to the entrance site 1 can produce a lack of complete identifiability as shown in Fig. 3.9. There, experiment 5 is depicted. The intersections between the two regions associated to the confidence interval for fex and rex produce two connected regions. The confidence intervals for fex and rex are, respectively, [0.0704286, 0.0707723] and [2926.287, 2935.022]. In both regions of intersection pairs with integer d are present, so that a complete identification is not possible. The estimated values of the defect parameters are p0 ∈ [0.749253, 0.751496] when d = 10, or p0 ∈ [0.986593, 0.992496] when d = 20. In Fig. 3.10 the results of experiments 6, 7 and 8 are presented. Here, the defect parameters are still the same in the three cases, but the different number of starting particles N allows to find different intersection regions. In this case, it is interesting to show that even ccv-defect˙v˙nov.tex – 18 novembre 2020 19 1:49 d 60 40 20 100 80 0.0707723 60 0.0706004 2930.654 40 0.0704286 2926.287 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 80 2935.022 d 80 100 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 60 d 100 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.9: Experiment 5. Left, center and right panel as in Fig. 3.5. Beside the real value of the obstacle parameters identified by the cyan point, a second intersection region around (0.99, 20) is found (orange point). 100 100 2377.54 60 2338.91 60 40 2343.463 40 0.0406140 0.0406062 20 0.0404194 0.0400218 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 60 2351.074 2349.153 2347.232 40 0.0412062 20 80 2349.550 d 80 d 2358.22 d 80 100 2355.637 0.04046103 20 0.0402326 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0.04040196 0.04034289 Figure 3.10: From left to right: experiments 6 – 7 – 8 respectively. The cyan point identifies the actual defect, that is the same in the three experiments. The number of starting particles is different: N = 106 in experiment 6, N = 107 in experiment 7, N = 108 in experiment 8. for a relatively small number of starting particles (N = 106 , experiment 6) a unique connected region is found. Nevertheless, there are many different possible values of d. When N considered is larger, the intersection region dimension decreases, and the defect is identified to be in the last quarter of the lane, and to favor the passage of the particle (p0 > p), but still a large number N = 108 is not sufficient to identifies exactly the position of the defect. A further increasing of the number of starting particles N is needed to enhance the estimate. It is interesting to notice that in all the considered experiments the knowledge of the numbers of crossing particles n and starting particles N allows to deduce if p0 is larger or smaller than p, but on its own it does not give any clue where the defect is, except for the experiment 5. On the other hand, one can note that the only knowledge of the residence time ccv-defect˙v˙nov.tex – 18 novembre 2020 20 1:49 is generally not sufficient by itself to deduce if the defect is pushing forward or backward the walkers, since two unconnected regions with both possible behaviors are identified by the confidence interval found (see experiments 1–2 and 5–8). The lack of identifiability tends to disappear when p is larger, i.e. a single connected region is usually found as the intersection of the regions associated to the confidence interval for fex and rex . However, it is common that the intersection identifies a unique, connected but large region, so that many different values of d are possible. This happens due to the fact that in presence of a consistent drift (p far enough from 1/2) a single defect do not modify very consistently the probability to cross the interval if it is far from the entrance site 1 or if p0 is very close to p. Hence, this phenomenon was more rare in the case of p = 0.51 previously considered since the drift was sufficiently small, so that the region associated to the confidence interval of fex was usually quite small. The cases in which the defect is close to site 1 while p0 is not close to p are instead the cases in which it is easier to fully identify the defect. Note that in principle, increasing sufficiently the number of starting particle N , it is possible to let the intersection region become as small as wanted, but the required N can be very large. However, the larger the drift is, the more is common to deduce some information about the defect from the knowledge of the residence time alone. For instance, in many of the following experiments we can notice that the residence time allows to individuate if p0 is smaller or larger than p. Indeed, we recall that as shown in Fig. 3.4 the scatter plot of the residence time tends to reduce, when p move away from 1/2, to a first region where p0 < p and a second region (p0 > p), one of them in which the residence time is larger than the case of no defect, and the opposite in the other one. To clarify what we said we present now some experiments in which the different situations are encountered. Let us now consider the case of p = 0.53. 100 80 60 d 0.1015302 40 0.1011252 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 80 3763.11 80 60 3751.44 60 d 0.1019352 100 40 d 100 3739.78 20 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.11: Experiment 9. See caption of Fig. 3.7. ccv-defect˙v˙nov.tex – 18 novembre 2020 21 1:49 The experiment 9 is shown in Fig. 3.11. The actual defect is in the first half of the interval with a value of p0 that is very small. We are now in a case in which it is possible to fully identify the defect. The confidence intervals for fex and rex are, respectively, [0.1011252, 0.1019352] and [3739.78, 3763.11]. The defect is correctly identify to have p0 ∈ [0.049815, 0.050315] and d = 43, that are the pairs (p0 , d) with integer d that lay in the intersection region. 100 80 80 0.1137146 d 0.1134138 40 60 20 1292.222 40 0.1131130 60 40 1290.837 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 80 1293.608 d 60 100 d 100 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.12: Experiment 10. In Fig. 3.12 we plot the results of experiment 10. Here, as in the case of experiments 11–13, the complete identification is made difficult by the fact that the defect is far from the entrance site (the actual defect is in the middle of the lane), and the parameter p0 is not sufficiently high or low (p0 = 0.8 in the experiment). In this case the information that one can easily get is that p0 > 0.75 from the residence time, and that d > 40. Any refinement of this information would require a consistently larger number of starting particles N . 100 80 80 0.1135454 d 0.1132454 40 0.1129454 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 80 1703.673 60 1701.426 d 60 100 40 1699.179 20 60 d 100 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.13: Experiment 11. See caption of Fig. 3.7. In Figs. 3.13–3.14 we show that if the defect is not so influential on the probability to cross, the full identification of the defect will require very large N . The results of experiments ccv-defect˙v˙nov.tex – 18 novembre 2020 22 1:49 100 100 1703.673 80 80 1701.426 1699.179 40 0.1132454 60 1701.0557 d 40 40 0.1135454 20 1701.5270 1701.259 60 1701.9983 80 1702.267 d d 60 100 1703.274 0.1133445 20 0.1132101 0.1129454 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0.11322294 20 0.11315574 0.1130757 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0.11308854 Figure 3.14: From left to right: experiments 11 – 12 – 13 respectively. The cyan point identifies the actual defect, that is the same in the three experiments. The number of starting particles is different: N = 107 in experiment 11, N = 5 · 107 in experiment 12, N = 2 · 108 in experiment 13. 11, 12 and 13 are shown in these plots. The three experiments refer to the case of the same defect, with a different number of starting particles N . It is evident that in this case the increasing of N produces a very slow shrinkage of the intersection region. Nevertheless, it soon appears clear that the defect is in the second half of the interval, and p0 is smaller than p. 100 80 80 0.1288899 d 0.1285723 40 0.1282548 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 80 1362.549 60 1361.100 d 60 100 40 1359.651 20 60 d 100 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.15: Experiment 14. See caption of Fig. 3.7. Instead, in Fig. 3.15 we show that a defect with the same p0 = 0.8 of the case of the experiment 10 is fully identifiable if it is sufficiently close to the starting point (d = 15 in the experiment). Notice that in this experiment the information about the confidence interval of the residence time alone is not sufficient to determine whether the defect is pushing the walker forward or backward, nor where the defect is located. When p increases, the possible different behaviors further reduces. We propose here some ccv-defect˙v˙nov.tex – 18 novembre 2020 23 1:49 experiment where p = 0.55. 100 100 80 60 d 0.1817491 40 0.1813833 20 80 60 1030.8688 40 1030.0559 d 0.1821149 40 1029.2430 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 60 d 80 100 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.16: Experiment 15. See caption of Fig. 3.7. 100 80 80 0.1821318 d 0.1816158 40 60 20 1392.074 40 0.1810998 60 40 1389.923 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 80 1394.226 d 60 100 d 100 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.17: Experiment 16. See caption of Fig. 3.7. 100 80 80 0.1833384 d 0.1828194 40 0.1823004 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 80 878.7650 60 877.9133 d 60 100 40 877.0617 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' 60 d 100 40 20 0 0.0 0.2 0.4 0.6 0.8 1.0 p' Figure 3.18: Experiment 17. See caption of Fig. 3.7. In the experiments 15, 16 and 17 illustrated by Figs. 3.16, 3.17, 3.18 one can notice that the study of the region associated to the confidence interval of the experimental residence ccv-defect˙v˙nov.tex – 18 novembre 2020 24 1:49 time is always useful to identify whether the defect is favoring the walker to pass or not. In experiments 15 and 16, the situation is similar to that one of experiments 10–13. The defect is not sufficiently close to the starting site 1, so that the study of the confidence interval associated to the experimental fraction of particles crossing is not sufficient to fully identify the parameters of the defect, since many different couples (p0 , d) are still in the intersection region. The experiment 17 shows instead that a defect located at d = 25 (with p0 = 0.7 in the experiment) produces an intersection region that restrict the possible values of the estimated d to a small number, with a quite accurate estimate of the actual value of p0 . 4. Conclusions In the framework of the 1D simple random walk, we have studied the effect of obstacles in the transport of moving agents on a lane. We have shown how to setup experiments to measure the transport properties of a lane and the geometrical and transport properties of possibly present obstacles. In absence of obstacles, we have shown that measuring the fraction of particles that cross the lane (experiment 1) or the average time taken by the crossing particles to cross the lane (experiment 2) is sufficient to deduce the transport properties of the system, that in the case of study is the probability for a particle to jump to its right. To be precise, in the case of experiment 2, the problem is not completely identifiably, since the two symmetric estimates p and 1 − p are found. In presence of defects, supposing to know p, we are interested to estimate the geometric and transport properties of the obstacles, that is to say, its position d and the probability p0 that a particle sitting on the obstacle jumps to its right. We have shown that performing just experiment 1 or 2 is not sufficient to do the job, but knowing the results of both experiments allows to solve the problem and find an estimate of the two parameters d and p0 , apart from some particular case in which identifiability issues arise. We have discussed experiments for different values of the total number of moving particles N and we have shown that the identifiability problems are weakened when N is increased. A. Generating function For the sake of completeness, we report here the construction of the generating function U introduced in Section 3.2 following [22] (and references therein) and adapting, where necessary, the notation to the present setting. We call un,i the probability to exit the walk through the right side, starting from i and after n steps. We construct the generating function of the probability of exiting through the ccv-defect˙v˙nov.tex – 18 novembre 2020 25 1:49 P right side starting from i, Ui (s) = n≥0 un,i sn , see [31–33]. We define ti = Pi (R.E.), where the index i denotes that the walk is started at the site i. Note that ti = Ui (1). Therefore, the series defining Ui (s) is totally and thus uniformly convergent for s ∈ [0, 1]. Since the P derivative of the generating function is Ui0 (s) = n≥1 nun,i sn−1 , following [32] we have that lim− Ui0 (s) = s→1 X nun,i = ti Ri , (A.24) n≥1 where Ri is the conditional expectation of the duration of the game given that the random walk ends in L and it is finite for any fixed p, q, p0 , and q 0 . Following the approach of [31, 33], we find the generating function as the solution of a system combining the equations for the generating function in the bulks (regular sites) and on the singular site. Recalling that on the regular sites p and q are, respectively, the probabilities to jump to the right and to the left, we find that un+1,i = pun,i+1 + qun,i−1 (A.25) on the regular sites, namely, for i = 1, . . . , d − 1 and i = d + 1, . . . , L − 1, and boundary values un,0 = un,L = 0 when n ≥ 1, u0,L = 1, u0,i = 0 when i ≤ L. Thus, multiplying (A.25) by sn+1 and summing for n = 0, 1, 2, . . . we find the following equation in the bulk, i.e., i = 1, . . . , d − 1 and i = d + 1, . . . , L − 1, Ui (s) = psUi+1 (s) + qsUi−1 (s), (A.26) to be solved with boundary conditions UL (s) = 1 and U0 (s) = 0. The equation for Ud (s) on the defect site is Ud (s) = p0 sUd+1 (s) + q 0 sUd−1 (s), (A.27) since the probabilities on the singular site are p0 to jump to the right and q 0 to the left. It is known [31] that in the bulk of regular sites the generating function Ui for fixed s can be searched in the form λi (s), except for the case p = q and s = 1 at the same moment, where the generating function is linear, see later. Substituting in (A.26) it is found p 1 ± 1 − 4pqs2 . (A.28) λ± (s) = 2ps The generating function in the bulk can be written as a linear combination of terms in the form λi− and λi+ for i = 0, . . . , L. We consider two different linear combinations in the ccv-defect˙v˙nov.tex – 18 novembre 2020 26 1:49 bulk on the left and on the right of the defect site, namely, we introduce two coefficients on the left and two (possibly) different coefficients on the right. Thus we find two different representation of Ud (s). Therefore, we will be able to find the unknown coefficients by requiring that these two representation are equal for i = d, that is to say the equation at the defect (A.27) and the boundary conditions are satisfied. More precisely, the generating function reads Ui (s) = G(s)λi+ (s) + H(s)λi− (s), Uj (s) = U (s)λj+ (s) +V (s)λj− (s), i = 0, 1, . . . , d; (A.29) j = d, d + 1, . . . , L, and the coefficients G(s), H(s), U (s), and V (s) solve the system   Gλd+ + Hλd− = U λd+ + V λd−      Gλd + Hλd = q 0 s(Gλd−1 + Hλd−1 ) + p0 s(U λd+1 + V λd+1 ) + − + − + −  0=G+H      1 = U λL+ + V λL− . (A.30) The system (A.30) has a unique solution (G, H, U, V ) that can be explicitly expressed in terms of jump probabilities, λ± and s. Thus, by substituting λ± given by the (A.28) in the formulas (A.29), we find the explicit expressions of the generating function Ui (s). However, due to the length and complexity of these expressions, we prefer not to report here the solutions of the system in the general case. Note that in the easiest case, i.e., if the site d is regular, namely p0 = p and q 0 = q, the solutions are G = U and H = V which reduce to the classical ones in Feller [31, equation (4.10) in Paragraph XIV.4]. The conditional expectation Ri of the duration of the game starting from i and ending in L can be computed using equation (A.24). The limit allows us to include in this formula even the symmetric case p = q, where the generating function has not the form of a combination of powers λi anymore for s = 1. References [1] E. Cristiani and D. Peri. Applied Mathematical Modelling, 45:285 – 302, 2017. [2] M.J. Saxton. Biophysical Journal, 66:394–401, Feb 1994. [3] F. Höfling and T. Franosch. Reports on Progress in Physics, 76(4), 2013. [4] M.A. Mourão, J.B. Hakim, and S. Schnell. Biophysical Journal, 107:2761–2766, Jun 2017. ccv-defect˙v˙nov.tex – 18 novembre 2020 27 1:49 [5] A.J. Ellery, M.J. Simpson, S.W. McCue, and R.E. Baker. The Journal of Chemical Physics, 140(5):054108, 2014. [6] K. To, P. Lai, and H.K. Pak. Phys. Rev. Lett., 86:71–74, Jan 2001. [7] I. Zuriguel, A. Garcimartı́n, D. Maza, L.A. Pugnaloni, and J.M. Pastor. Phys. Rev. E, 71:051303, May 2005. [8] F. Alonso–Marroquin, S.I. Azeezullah, S.A. Galindo–Torres, and L.M. Olsen-Kettle. Phys. Rev. E, 85:020301, Feb 2012. [9] I. Zuriguel, A. Janda, A. Garcimartı́n, C. Lozano, R. Arévalo, and D. Maza. Phys. Rev. Lett., 107:278001, Dec 2011. [10] D. Helbing. Rev. Mod. Phys., 73:1067–1141, Dec 2001. [11] D. Helbing, I. Farkas, P. Molnàr, and T. Vicsek. In M. Schreckenberg and S. D. Sharma, editors, Pedestrian and Evacuation Dynamics, pages 21–58, Berlin, 2002. Springer. [12] E.N.M. Cirillo and A. Muntean. Physica A: Statistical Mechanics and its Applications, 392(17):3578 – 3588, 2013. [13] A. Muntean, E.N.M. Cirillo, O. Krehel, M. Bohm, In “Collective Dynamics from Bacteria to Crowds”, An Excursion Through Modeling, Analysis and Simulation Series: CISM International Centre for Mechanical Sciences, Vol. 553 Muntean, Adrian, Toschi, Federico (Eds.) 2014, VII, 177 p. 29 illus, Springer, 2014. [14] E.N.M. Cirillo, A. Muntean, Comptes Rendus Macanique 340, 626–628, 2012. [15] A. Ciallella, E.N.M. Cirillo, P.L. Curseu, A. Muntean, Mathematical Models and Methods in Applied Sciences, https://doi.org/10.1142/S0218202518400079. [16] G. Albi, M. Bongini, E. Cristiani, and D. Kalise. SIAM Journal on Applied Mathematics, 76(4):1683–1710, 2016. [17] D. Helbing, I. Farkas, and T. Vicsek. Nature, 407:487–490, Sep 2000. [18] D. Helbing, L. Buzna, A. Johansson, and T. Werner. Transportation Science, 39(1):1–24, 2005. ccv-defect˙v˙nov.tex – 18 novembre 2020 28 1:49 [19] R. Escobar and A. De La Rosa. In W. Banzhaf, J. Ziegler, T. Christaller, P. Dittrich, and J.T. Kim, editors, Advances in Artificial Life, Proceedings of the 7th European Conference, ECAL, 2003, Dortmund, germany, September 14–17, 2003, Proceedings. Lecture Notes in Computer Science, vol. 2801., pages 97–106, Berlin, 2003. Springer. [20] S.A. Janowsky, J.L. Lebowitz, J. Stat. Phys. 77, 35–51 (1994). [21] B. Scoppola, C. Lancia, R. Mariani, J. Stat. Phys. 161, 843–858 (2015). [22] A. Ciallella, E.N.M. Cirillo, Conditional expectation of the duration of the classical gambler problem with defects European Physical Journal Special Topics 228 111-128, 2019 [23] A. Ciallella, E.N.M. Cirillo, J. Sohier. Physical Review E, 97:052116, 2018. [24] A. Ciallella and E.N.M. Cirillo. Kinetic & Related Models 11, 1475–1501 (2018). [25] E.N.M. Cirillo, M. Colangeli, and A. Muntean. Physical Review E 94, 042116 (2016). [26] E.N.M. Cirillo, M. Colangeli, Phys. Rev. E 96, 052137, 2017. [27] E.N.M. Cirillo, M. Colangeli, A. Muntean, Physica A 488, 30–38 (2017) [28] E.N.M. Cirillo, O. Krehel, A. Muntean, and R. van Santen. Phys. Rev. E, 94:042115, Oct 2016. [29] E.N.M. Cirillo, O. Krehel, A. Muntean, R. van Santen, and Aditya S. Physica A: Statistical Mechanics and its Applications, 442:436 – 457, 2016. [30] J. Messelink, R. Rens, M. Vahabi, F.C. MacKintosh, A. Sharma, Physical Review E 93, 01211 (2016) [31] W. Feller. An Introduction to Probability Theory and its Applications, volume 1. John wiley & Sons, Inc, New York – London – Sidney, 1968. [32] F. Stern, Math. Mag. 48, 200–203 (1975). [33] T. Lengyel, Applied Mathematics Letters 22, 351–355 (2009). [34] W.A. Beyer, M.S. Waterman, Symmetries for conditioned ruin problems, Math. Mag. 50 (1) (1977) 42–45. ccv-defect˙v˙nov.tex – 18 novembre 2020 29 1:49