aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Area Distribution of Elastic Brownian Motion
M. A. Rajabpour a ∗ a Dip. di Fisica Teorica and INFN, Universit`a di Torino, Via P. Giuria 1, 10125 Torino,ItalyDecember 2, 2018
Abstract
We calculate the excursion and meander area distributions of the elastic Brownianmotion by using the self adjoint extension of the Hamiltonian of the free quantum particleon the half line. We also give some comments on the area of the Brownian motionbridge on the real line with the origin removed. We will stress on the power of selfadjoint extension to investigate different possible boundary conditions for the stochasticprocesses. We discuss also some possible physical applications.
Keywords : Elastic Brownian Motion, Quantum Mechanics, Self Adjoint ExtensionPACS number(s): 02.50.FZ and 03.65 Db and 05.40.Jc
In this paper we study some area distributions of elastic Brownian motion [1], such as elas-tic Brownian excursion and elastic Brownian meander. The goal apart from the calculationof the area distribution of elastic Brownian motion is to give a unified framework to studydifferent area distributions for Brownian motion with generic boundary conditions. Thisgeneralization is in close connection with the concept of the self adjoint extension in quan-tum mechanics. The self adjoint extension gives a reasonable classification for the possibleboundary conditions of the quantum particle and so for the stochastic process. ∗ e-mail: [email protected] The elastic Brownian motion is the natural generalization of the Brownian motion in thehalf line with special boundary condition on the origin. To define the process we need firstto introduce the local time. The definition of the local time of the path ω at the point a is2s follows t l ( a ) := 12 lim ǫ → ε Z T x + ε ( B s ) ds, (2.1)where x + ε ( B s ) is the indicator for the time that the process is in the interval [0 , x + ǫ ].One can naively write the above equation as an integral over a delta function as t l ( ω,
0) = R T δ ( x ( t )) dt . The local time has dimensions of the inverse of velocity and can be written as t l ( a ) = n ∆ t ∆ x for the discrete random walk, where n is the number of times the path hits theorigin. Then one can write the exponential of the local time asexp ( − πη t l ) ≈ (1 − πη ∆ x ) n . (2.2)To go to the discrete level we multiplied t l with (∆ x ) ∆ t that comes from the central limittheorem. The equation (2.2) means that by considering 1 − πη t l as the probability for asingle reflection from the origin it is possible to interpret the exponential of local time asthe probability that a particle on a given path is reflected from the origin . Then the Greenfunction of the elastic Brownian motion is just by the following expectation value G ( x, y, T ) = < exp ( − πη t l ) > . (2.3)The elastic Brownian motion is in close connection with the quantum particle on the half line.The Hamiltonian operator of the quantum particle on the half line has self adjoint extensionwith the following boundary condition ψ (0) = − η π dψdx | x =0 . (2.4)The above boundary condition is called Robin boundary condition. The energy of the particleis E k = k and the wave functions are ψ k = r π cos( kx + δ k ) , (2.5)where tan( δ k ) = πηk and δ k is the phase shift corresponding to the s -wave. The solutions arenormalizable and complete.The Hamiltonian is self adjoint for all of the real values of η but to avoid the cases withbound states we will consider just non-positive η . η = 0 is the Dirichlet boundary condition The name elastic Brownian motion comes from this property of the process. η = ∞ is the reflectingbarrier, the particle will be reflected from the boundary with probability one. η = 0 is the absorbing barrier,the particle will be absorbed after hitting the boundary. Other cases between two extreme cases called elasticbarrier. η → −∞ is the Neumann boundary condition. The Green function with respect to thesolutions of the Hamiltonian has the following form G η ( x, y, t ) = Z ∞ dke − iE k t ϕ ( y ) ϕ ∗ ( x ) . (2.6)Using the above equation the Green function has the following form for arbitrary values ofthe self adjoint extension [11, 12, 13] G η ( x, y, t ) = G F ( x − y, t ) + G F ( x + y, t ) + 4 πη Z ∞ dwe πη w G F ( x + y + w ) η ≤
0; (2.7) G η ( x, y, t ) = G F ( x − y, t ) + G F ( x + y, t ) − πη Z ∞ dwe − πη w G F ( x + y − w )+ 4 πη e i π tη e − πη ( x + y ) η ≥
0; (2.8) G F ( x − y, t ) = 1 √ πit e i ( x − y ) / t . (2.9)For the special cases, Dirichlet and Neumann the results are as follows G η =0 ( x, y, t ) = G F ( x − y, t ) − G F ( x + y, t ); (2.10) G η →−∞ ( x, y, t ) = G F ( x − y, t ) + G F ( x + y, t ) . (2.11)One can use the above equations to get the Green function of the elastic Brownian motion byjust Wick rotation. The important point of this section is the possibility of using quantummechanics language to describe the elastic Brownian motion. The other interesting point isthe possibility of extending this equality in to the level of path integral representation [11, 12,13]. In the next section we use this correspondence to calculate different area distributionsof the elastic Brownian motion. In this section we will solve the problem of the area for the restricted Brownian motion,in particular we will solve the problem of the area distribution of elastic Brownian excursionand elastic Brownian meander. In the extreme limits we will get the well known results.
In this subsection we calculate excursion area of the elastic Brownian motion. The definitionof the excursion area is as follows: take an elastic Brownian process starts at x (0) = ǫ T to x ( T ) = ǫ , without crossing the origin between. We areinterested to the probability density P ( A, T, ǫ ) of the area A = R T x ( τ ) dτ for a fixed ǫ andthen finally take the limit ǫ →
0, it plays the role of the regulator and can be treatedindependent of the limit in the local time process. This regularization is just necessary forthe Dirichlet boundary condition but we are happy to keep it in our calculation to see itsrelevance in the calculation of Dirichlet boundary condition. To do this calculation we mapthe problem to the quantum mechanical problem. Now we use the method of Comtet andMajumdar [9] to calculate the excursion area distribution. To satisfy the constraint thatprocess stays non-negative between 0 and T one can multiply the above measure with theindicator function Q Tτ =0 θ [ x ( τ )] exp ( − πη t l ) which θ is the step function. The distribution P ( A, T ) of the area under the elastic Brownian excursion can then be expressed as thefollowing quantum mechanical problem P η ( A, T ) = 1 Z ηE Z x ( τ )= ǫx (0)= ǫ D x ( τ ) e − R T ( ( dxdτ ) + πη δ ( x ( t )) T Y τ =0 θ [ x ( τ )] δ ( Z T x ( τ ) dτ − A ) , (3.1)where Z ηE is the normalization and corresponds to the quantum mechanics of a particle withinfinite wall at the origin and zero potential at the positive real line Z ηE = < ǫ | e − H T | ǫ > . (3.2)The Hamiltonian, H is equal to the self adjoint Hamiltonian that we discussed in the previoussection. After integration, for small ǫ we have Z ηE ≃ η − πǫ ) ( 1 √ πT η − πη e π Tη Erfc( √ T πη )) . (3.3)The integral for the Dirichlet and Neumann cases are Z E ≃ √ π ǫ T − / + O ( ǫ ); η = 0 ,Z ∞ E ≃ r πT + O ( ǫ ); η → −∞ . (3.4)The above partition functions are just the probability that an elastic Brownian motion goesfrom x (0) = ǫ to x ( τ ) = ǫ in time T without crossing the origin. Taking the Laplace transform P ( λ, T ) = R ∞ P ( A, T ) e − λA dA of both sides of the the equation (3.1) gives P η ( λ, T ) = 1 Z E Z x ( τ )= ǫx (0)= ǫ D x ( τ ) e − R T ( ( dxdτ ) + πη δ ( x ( t ))+ λx ( τ )) T Y τ =0 θ [ x ( τ )] . (3.5)5n the numerator we have the propagator < ǫ | e − H T | ǫ > where H = − ( dxdτ ) + V ( x )and V ( x ) = λx for x > x ≤
0. We absorb the Dirac delta function in tothe boundary condition as the case of quantum particle on the half line, in the other wordswe consider the self adjoint extension of this operator. The boundary condition of the wavefunction after self adjoint extension is the same as (2.4). The solution of the Schr¨odingerequation is given by the Airy function as follows ψ ηi ( x ) = Ai ((2 λ ) / ( x − E/λ )) qR ∞ Ai ((2 λ ) / ( y − E/λ )) dy . (3.6)Using the boundary conditions one can determine the discrete eigenvalues as follows E ηi = 2 − / λ / c ηi , Ai ( − c ηi ) Ai ′ ( − c ηi ) = − η (2 λ ) / π . (3.7)Unfortunately since the above equation is transcendental the exact form of c ηi for the genericboundary condition is not available. However, for the Dirichlet and Neuman boundary con-ditions the solutions are just the magnitude of the zeros of Ai ( z ) and Ai ′ ( z ) on the negativereal axes respectively, we show them by − c i and − c ∞ i . The first few real roots of Ai ( z ) areapproximately -2.33811, -4.08795, -5.52056 and -6.78671, etc. The first few real roots of Ai ′ ( z ) are approximately -1.018792, - 3.248197, -4.820099 and -6.163307, etc. Then thewave functions in these two cases are ψ i ( x ) = (2 λ ) / Ai ((2 λ ) / x − c i ) Ai ′ ( − c i ) , (3.8) ψ ∞ i ( x ) = (2 λ ) / Ai ((2 λ ) / x − c ∞ i ) p c ∞ i Ai ( − c ∞ i ) . (3.9)To get the above results we used the identity R ∞ x Ai ( x ) dx = − xAi ( x ) + Ai ′ ( x ). Using theenergy eigenvalues and wave functions one can write the equation (3.5) as follows P η ( λ, T ) = < ǫ | e − H T | ǫ >Z ηE = 1 Z ηE ∞ X i =1 | ψ ( ǫ ) | e − − / λ / c ηi T . (3.10)For the Dirichlet and Neumann cases after considering small ǫ we have P ( λ, T ) = √ πλT / ∞ X i =1 e − − / λ / c i T , (3.11) P ∞ ( λ, T ) = r πT λ ) / ∞ X i =1 c ∞ i e − − / λ / c ∞ i T . (3.12)6ne can write the equations (3.11) and (3 .
12) in the scaling form as follows P ( λ, T ) = s √ π ∞ X i =1 e − − / s / c i , (3.13) P ∞ ( λ, T ) = u r π / ∞ X i =1 c ∞ i e − − / u c ∞ i , (3.14)where s = λT / and u = T / λ / . It is possible to do the inverse Laplace transform of thefunctions (3.11) and (3 .
12) explicitly and find the distribution of the excursion area. To doso we need the formula L − [exp( − sλ a ); A ] = asA a +1 M a ( sA − a ) (3.15)where M β ( z ) is the well known Wright function given by the following series M β ( z ) = 1 π ∞ X k =0 ( − z ) k k ! Γ( β ( k + 1)2 ) sin( β ( k + 1)2 ) . (3.16)For the Dirichlet boundary condition the inverse Laplace transform for T = 1 gives P ( A ) = √ π ∞ X k =1 c i ∂ A ( A − M ( c i A − ,
23 )) =2 √ A ∞ X k =1 e − c i )327 A ( 2( c i )
27 ) / U ( − , , c i ) A ) . (3.17)where U ( a, b, z ) is the confluent hypergeometric function.To calculate the moments of the area one can work in the Laplace space and then use theequality Γ( n ) < A − n > = R ∞ P ( λ, T ) λ n − dλ . To do the calculations we need to first definethe generalized Riemann function Λ η ( s ) = P ∞ i =1 1( c ηi ) s where c ηi comes from the equation (3.7).The above relations help us to calculate the moments explicitly as follows < A n > = √ π − n n Γ(1 + − n )2 )Γ(2 − n ) Λ ( 3( − n + 1)2 ) . (3.18)It was shown in [14] that the moments after regularization are < A n > = √ π − n Γ( n + 1)Γ( n − ) K n (3.19)where K n is by the following recursion relation K n = 3 n − K n − + n − X j =1 K j K n − j , s ≥ , (3.20)7he first few values are K = − , K = , K = and K = . Then the first fewmoments of the excursion area are < A > = 1 , < A > = √ π , < A > = 512 , < A > = 15 √ π , .... (3.21)There is also a nice relation between the Airy zeta function and K n as followsΛ ( 3 − n −
43 cos( πn )sin( πn ) K ( n ) . (3.22)For example we have the following limitslim n → n Λ ( 32 ( n − π , Λ (0) = 14 . (3.23)For the Neumann boundary condition we need another Laplace transform pair L − [ λ − α exp( − sλ − a ); A ] = 1 A − α φ ( a, α ; − sA a ); − < a < , s > , < α < , (3.24)where φ ( a, α ; z ) = P ∞ k =0 z k k !Γ( ak + α ) is the generalized Wright function defined for the a > − P ∞ ( A ) = r πT / ∞ X i =1 c ∞ i ∂∂A ( 1 A / φ ( − ,
23 ; − − / c ∞ i T A − )) . (3.25)The moments of the area for T = 1 can be written as < A n > = 3 √ π n +12 Γ( − n )Γ( − n ) Λ ∞ ( 3 − n . (3.26)These moments are the same as the moments of the Brownian bridge and can be regularizedin the same way [8]. Then the moments are < A n > = √ π − n/ Γ(1 + n )Γ( n ) D n , n ≥ , (3.27)where D n is by the following recursion relation D n = 3 n − D n − − n − X j =1 D j D n − j , s ≥ . (3.28)The first few values are D = 1, D = , D = and D = . Then the first few momentsof the area are < A > = 1 , < A > = 14 r π , < A > = 760 , < A > = 21512 r π , .... (3.29)8t is not possible to find exact probability distribution for the generic η because for Robinboundary condition c ηi is not independent of λ and since they are related by non-algebraicrelation it is not possible to get c ηi with respect to λ explicitly. However one can follow thecalculations in the level of Laplace space. The wave function has the following form ψ ηi ( x ) = (2 λ ) / Ai ((2 λ ) / x − c ηi ) p c ηi Ai ( − c ηi ) + Ai ′ ( − c ηi ) . (3.30)The Laplace transform of the distribution of the area is P η ( λ, T ) = (2 λ ) / Z E ( ǫ = 0) ∞ X i =1 ( η (2 λ ) / π )( 11 + ( η (2 λ ) / π ) c ηi ) e − − / λ / c ηi T . (3.31)To pursuit the calculation let us consider small η s. One can write c ηi = c i + δc ηi where δc ηi is the small perturbation around the zeros of the Airy function. After expansion of (3.7) theperturbation is δc ηi ≈ η (2 λ ) / π . (3.32)For small η one can also expand Z E as follows Z ηE ≈ η / π / + O ( η ) . (3.33)Then the P ( λ, T ) after expansion is P η ( λ, T ) ≈ √ πλT ∞ X i =1 e − − / λ / c i T − ηT π λ . (3.34)The inverse Laplace transform of the function after using the equation (3.15) is P η ( A ) ≈ √ π / ∞ X k =1 c i ∂ A (cid:18) ( A − η π ) − M ( c i ( A − η π ) − ,
23 ) (cid:19) . (3.35)The moments of the area can be find by the same method as before by just replacing A inthe equation (3.19) by A − η π .For the large η s the same calculation can be done. The partition function of the elasticBrownian motion for large η is Z ηE ≈ r πT − πη + O ( 1 η ) . (3.36)The perturbation of c ηi after expansion of the equation (3.7) is δc ηi ≈ − πc ∞ i η (2 λ ) / . (3.37)Unfortunately since the perturbation of the energy is dependent on the energy level we arenot able to find simple equation for the distribution of the area in this case.9 .2 The area under the elastic Brownian meander The definition of the elastic Brownian meander is similar to the elastic Brownian excursion,the only difference is for the elastic Brownian meander we do not need to force the processto come back to the starting point. In this case the partition function is Z ηM = Z ∞ db < b | e − H T | ǫ > . (3.38)One can show that < b | e − H T | ǫ > = r πT (cid:18) e −
12 ( ǫ − b )2 T + e −
12 ( ǫ + b )2 T (cid:19) + πη e π η T − π ( ǫ + b ) η Erf c [( 2 π Tη ) / − π ( ǫ + b ) η ( 2 T π η ) − / ] . (3.39)For small η one can expand the error function up to the second order < b | e − H T | ǫ > ≈ √ πT (cid:18) e −
12 ( ǫ − b )2 T + e −
12 ( ǫ + b )2 T (cid:19) − √ πT e −
12 ( ǫ + b )2 T (1 − η ǫ + b πT ) . (3.40)Taking the first order correction with respect to the η and integrating over b gives Z ηM ≈ Erf ( ǫ √ T ) − ηπ √ πT e − ǫ T . (3.41)For η → ∞ it is easy to get Z ∞ M ≈ . (3.42)Similar to the calculation that we did in the elastic Brownian excursion case one can writethe Laplace transform of the distribution of area as P η ( λ, T ) = 1 Z ηM Z ∞ db < b | e − H T | ǫ > . (3.43)Using the wave function (3.30) one can get P η ( λ, T ) = 1 Z ηM ∞ X i =1 Ai ((2 λ ) / ǫ − c ηi ) R ∞− c ηi Ai ( y ) dyc ηi Ai ( − c ηi ) + Ai ′ ( − c ηi ) e − − / λ / c ηi T . (3.44)After expansion of the function with respect to the ǫ and η the first correction appears in thespectrum as follows P η ( λ, T ) = √ π − / ( λT / ) / ∞ X i =1 B ( c i ) e − − / λ / c ηi T . (3.45)10here B ( c i ) = R ∞− c i Ai ( y ) dyAi ′ ( − c i ) and c ηi = c i + δc ηi . The distribution of the area after inverseLaplace transform is P η ( A, T ) = √ π − / T / ∞ X i =1 B ( c i ) ∂∂A ( 1( A − η π ) / φ ( − ,
23 ; − − / c ηi T ( A − η π ) − )) . (3.46)The continuation of calculation is now straightforward we just need to use the well knownresults for the Brownian meander. The moments of the area for Brownian meander, i.e. η = 0, for T = 1 is < A n > = √ π − n/ Γ( n + 1)Γ( n +12 ) Q n . (3.47) Q n satisfies the following recursion relations Q n = β n − n X j =1 α j Q n − j ,β n = α n + 34 (2 n − β n − , (3.48) α n = 6 − n Γ( n + 1) Γ(3 n + 1 / n + 1 / . The first few values are Q = 1, Q = , Q = and Q = . Then the first few values ofthe moments of the area are < A > = 1 , < A > = 34 r π , < A > = 5960 , < A > = 465512 r π , .... (3.49)To get the results for the small η one needs to replace A with A − η π in the (3.49).The next interesting case is the Neumann boundary with η → ∞ . The Laplace transformof the distribution of the area after a little algebra is P ∞ ( λ, T ) = ∞ X i =1 κ i e − − / λ / c ∞ i T , (3.50)where κ i = AI ( c ∞ i ) c ∞ i Ai ( c ∞ i ) with AI ( z ) = R ∞ z Ai ( y ) dy . This distribution is exactly the same as thedistribution of the area of the Brownian motion, i.e. R T | B t | dt . This is not surprising becausethe absolute value of Brownian motion is like considering the area in the presence of totallyreflecting boundary condition. Using the inverse Laplace transform one can get P ∞ ( A, T ) = 2 − / TA / ∞ X i =1 κ i c ∞ i M ( 2 − / c ∞ i TA / ) (3.51)Using the well known results for the area of the absolute value of the Brownian motion [8]the moments of the area can be written as < A n > = 2 − n/ Γ(1 + n )Γ( n +22 ) L n , (3.52)11here L n satisfies the following recursion relation L n = β n + n X i =1 j + 16 j − α j L n − j , (3.53)and α j and β j are the same as in the equation (3.48). The first few values are L = 1, L = 1, L = and L = . Then the first few values of the moments of the area are < A > = 1 , < A > = 23 r π , < A > = 38 , < A > = 263630 r π , .... (3.54) Different applications of the area distributions of Brownian motion in computer science,graph theory, fluctuating one-dimensional interfaces and localization in electronic systemswere already discussed in length in many papers, see [16, 17] and references therein. TheAiry distribution function and its derivative appear extensively in all applications. In thissubsection we will summarize some immediate simple applications of our extended Airy dis-tribution. We will discuss the distribution of the average distance of a particle from a disorderwith point interaction in three dimensions and the distribution of the average relative heightdistribution of interacting interfaces in two dimensions. We will also discuss one more elasticBrownian functional distribution related to vicious walkers interacting with the boundary.The first immediate application comes from the equality of the norm of the 3-dimensionalBrownian motion (called three dimensional Bessel process) and Brownian motion on the halfline [1, 10]. This is very easy to show by considering the radial part of the Fokker-Planckequation of three dimensional Brownian motion. Now consider a Brownian motion movingin three dimensions in the presence of the disorder at the origin , one can map the systemto the problem of one particle on the half line. The most generic point interaction betweendisorder and particle comes from the self adjoint extension of the Hamiltonian of the freeparticle in punctured three dimensional space [10] which is equal to the free particle on thehalf line with the boundary condition that we discussed in section 2. Then it is easy to seethat the area distribution that we calculated is just the average distance distribution in theperiod T between the disorder and the free particle with the generic point interaction.Another simple application, which is in the close connection with the previous example,is the interacting interfaces. A path of Brownian motion in the x − t space is just like aninterface with zero roughness exponent. One can also look at this interface as an ensemble of12rowth models such as Edwards-Wilkinson model. Consider now two non-crossing interfacesin the region [0 , L ] with the similar boundary conditions. This problem is equivalent to theproblem of two free particles on the real line. Consider, like the previous example, pointinteractions between the particles. The interaction between the particles is equivalent to theinteraction between the interfaces. Then the area distribution that we calculated for theelastic Brownian excursion is just the distribution of the average distance of the interfaces inthe interval [0 , L ]. One can also relax the boundary condition in one of end points and considerdifferent boundary conditions for the different interfaces and use the results corresponding toelastic Brownian meander area.Since the elastic Brownian motion is the generalization of Brownian motion in the presenceof the boundary we believe that all the possible applications should deal with the boundaryinteraction or point interaction between two particles. In this paper we just calculated onepossible functional of the elastic Brownian motion, the area. However, there are many otherfunctionals that can have interesting physical applications in the study of the interactingnon-crossing walkers or interacting interfaces. We will discuss some of these functionals inthe conclusion of the paper and give here one more example with more detail.Consider the problem of p non-crossing walkers, which has application in describing do-main walls of elementary topological excitations in the commensurate adsorbed phases closeto the commensurate-incommensurate transition [18], in the presence of the boundary. Oneinteresting quantity is the maximal height distribution of the top walker that was alreadycalculated exactly for the vicious walkers in [19]. Walkers are vicious if they annihilate eachother when they meet. For simplicity we will discuss the simplest case p = 1. Consider H as the maximal height of the walker in [0 ,
1] then one can define the cumulative distributionas P ( M ) = P rob [ H ≤ M ]. We define N ( ǫ, M ) as the probability that the walker do anexcursion in the period [0 ,
1] starting at ǫ and coming back to the same point staying withinthe interval [0 , M ]. The cut-off ǫ is just necessary for the Dirichlet boundary condition as wediscussed before. Since for this case the result is already known [20] we will put ǫ = 0 here-after. Then it is easy to show that N (0 , M ) = P E | ψ E | e − E . The wave function is as (2.5) ,i.e. ψ E = q π cos( kx + δ k ), with E = k and k is the solution of the equation cot( kM ) = πηk .Finally one can write the cumulative distribution as P ( M ) = 1 Z η X k πηk ) )( M − k sin(2 kM )) e − k / , (3.1)where Z η comes from the equation (3.3). For small η and small M with M < η one can13implify the equation as P ( M ) ≈ √ πM k e − k / , where k = q − πMη M .Generalization of the above results to arbitrary p is straightforward, however, it is rathercumbersome. From the perspective of our study in this paper one can generalize this problemin two directions: the first possibility is considering non-vicious walkers, interacting domain-walls, with a Dirichlet boundary condition on the wall. The second possibility is consideringvicious walkers with non-trivial interaction with the boundary. In this section we will summarize some results come from the area calculation for theBrownian bridge on the line with a point defect. The quantum mechanical counterpart wasdiscussed lengthly in the literature and it is called the free particle on a line with a hole [15].The functional integral for this problem was discussed in [13] and it is based on the differentlocal times of the particle in the two sides of the origin. The most general boundary conditionthat respects the time reversal symmetry for the defect on the origin, after using self adjointextension theory, is ψ ′ + (0) ψ + (0) = − α − β − δ − γ ψ ′− (0) ψ − (0) , (4.1)with a constraint αγ − βδ = 1. For simplicity we will consider some special cases. It is easyto see that for δ → ∞ the boundary condition decouples and we will have two decoupled halflines and so the area distribution is as the previous section.The next simple case is α = γ = − δ = 0 which is equal to the free particle on theline with the following delta function potential V ( x ) = − β δ ( x ) . (4.2)To avoid the bound state we consider non-positive β . To calculate the area of this kind ofBrownian bridge one can use the method of previous section. Interestingly the results arevery similar to the previous section. The energy of the particle is E k = k and the wavefunctions are ψ k ( x ) = r π cos( k | x | + δ k ) , (4.3)14here tan( δ k ) = β k . Then Z βE is the same as (3.3) after replacing η with πβ . The wavefunctions in the presence of the potential λ | x | can have different parities, they are ψ βi ( x ) = 2 − / (2 λ ) / Ai ((2 λ ) / | x | − c βi ) p c βi Ai ( − c βi ) + Ai ′ ( − c βi ) , even parity , (4.4) ψ i ( x ) = sgn( x )2 − / (2 λ ) / Ai ((2 λ ) / | x | − c −∞ i ) Ai ′ ( − c −∞ i ) , odd parity , (4.5)where c β is the same as c η after replacing η with πβ . Odd wave functions do not contributein the distribution of the area. The result for the even part is the same as the result for theelastic Brownian excursion. It is easy to see that β = 0 is like the free Brownian motion andso the distribution of the area is like Brownian bridge or like elastic Brownian excursion withNeumann boundary condition, i.e. (3.25). The β = −∞ is like the the Dirichlet boundarycondition and the distribution of the area is (3.17) after considering η = 0.The other simple boundary condition comes from α = γ = − β = 0 and δ = 0, in theother words ψ ′ + (0) − ψ ′− (0) = 0 , (4.6) ψ + (0) − ψ − (0) = − δψ ′− (0) . (4.7)The energy of the particle is E k = k and the wave functions are ψ k ( x ) = sgn( x ) r π cos( k | x | + δ k ) , (4.8)where tan( δ k ) = δk . Then Z δE is the same as (3.3) after replacing η with πδ .The wave functions in the presence of the potential λ | x | can have different parities, theyare ψ δi ( x ) = sgn( x ) 2 − / (2 λ ) / Ai ((2 λ ) / | x | − c πδi ) q c πδi Ai ( − c πδi ) + Ai ′ ( − c πδi ) , odd parity , (4.9) ψ i ( x ) = (2 λ ) / Ai ((2 λ ) / | x | − c ∞ i ) p c ∞ i Ai ( − c ∞ i ) , even parity . (4.10)The above calculation shows that in this case the distribution of the area can be calculatedby adding two terms. In this case the even parity has contribution for the distribution ofthe area which is equal to the Neumann boundary condition. Replacing η with πδ in theformula of elastic Brownian excursion will give the contribution of the odd part. The caseof δ → ∞ is equal to the Neumann boundary condition that separates the system in to thetwo regions, positive and negative part of the real line. Of course having two solutions is an15ndicator of degeneracy coming from the parity invariancy of the system in this limit. δ = 0is the free particle case and one can see that only the even parity has contribution in the areadistribution. In this paper we found the area distribution of the elastic Brownian motion in some limits.Our method was based on the equality of this process with the self adjoint Hamiltonian of thequantum particle on the half line. The corresponding Hamiltonian for the area distributionis just the Hamiltonian with linear potential. The eigenvalues of this Hamiltonian after selfadjoint extension satisfy a transcendental equation and so for the generic case the distributionof the area is not available, however, in some limits the calculation is tractable. We foundperturbatively the area distribution of the Brownian excursion and the Brownian meanderin the presence of the weekly reflecting barrier. By using self adjoint extension we found aunified method to classify different possible area distribution for the Brownian motion in thepresence of the boundary. Some possible applications in diffusion phenomena in the presenceof disorder and interacting interfaces were also discussed.We did similar calculations for the Brownian motion on the pointed line. The self ad-joint Hamiltonian in this case has three independent parameters and the eigenvalues of theHamiltonian satisfy the same transcendental equation in some interesting limits. Similarcalculations can be useful in describing different distributions of diffusing particles in thepresence of point disorder.There are plenty of questions remained to be answered in the study of the elastic Brow-nian motion by using quantum mechanical technics. Some of them are: The case of themaximal height of p non-intersecting Brownian excursions and Brownian bridges is also in-teresting to be calculated [19], the possible connection of this study to the interacting domainwalls of elementary topological excitations in the commensurate adsorbed phases close to thecommensurate-incommensurate transition can be interesting. The other example is the dis-tribution of the time to reach the maximum [21]. Unfortunately the eigenvalue equations forthe above cases are transcendental as we faced for one example in the end of section 3 and soit is impossible to get a closed formula for the distributions, however, the exact calculationsin some limits are may be possible. The other example is the distribution of the time spent16y the particle on the positive side of the origin out of the total time t . This distributionwas first calculated by Levy in the case of the Brownian motion [22]. For the pointed linethe equations are again transcendental and need to be solved by numerical calculation. Wemostly focused on the distributions in one dimension however one can also try to calculatethe different distributions like the algebraic area distribution, winding number distributionin the pointed two dimensional space, the number of defects could be finite or infinite .We believe that the method of self adjoint extension in quantum mechanics can be usefulto calculate such kind of distributions. It is also interesting to check our results with thenumerical calculations. References [1] D. Revuz, M. Yor:
Continuous Martingales and Brownian Motion , Springer, New York,1999[2] M. Kac, Trans. Amer. Math. Soc, 59(1946), 401-414. MR0016570[3] D. A. Darling, Ann, Prbab, 11(1983)803-806[4] G. Louchard, J. App. Prob, 21 (1984)479-499[5] M. Perman and J. A. Wellner, Ann. Appl. Prob, 6(1996)1091-1111[6] P. Flajolet, P.Poblete and A. Viola, Algorithmica, 22(1998)490-515[7] P. Flajolet and G. Louchard, Algorithmica, 31(2001)361-377[8] S. Janson, Probability Surveys, 4(2007)80-145[9] S. N. Majumdar and A. Comtet, Phys. Rev. Lett. 92, 225501 (2004)and Journal ofStatistical Physics, 119( 2005) 777-826[10] M. A. Rajabpour,Journal of Statistical Physics 136 (2009) 785 [arXiv:0906.1728][11] T. E. Clark, R. Menikoff, and D. H. Sharp, Phys. Rev. D 22, 3012 - 3016 (1980)[12] E. Farhi, S. C. Gutmann, Int. J. Mod. Phys. A5 (1990) 3029. 84 For one defect point we don’t expect significant change in the winding number distribution in two dimen-sions.
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