Quantum surface diffusion in Bohmian mechanics

Surface diffusion of small adsorbates is analyzed in terms of the so-called intermediate scattering function and dynamic structure factor, observables in experiments using the well-known quasielastic Helium atom scattering and Helium spin echo techniques. The linear theory applied is an extension of the neutron scattering due to van Hove and considers the time evolution of the position of the adsorbates in the surface. This approach allows us to use a stochastic trajectory description following the classical, quantum and Bohmian frameworks. Three different regimes of motion are clearly identified in the diffusion process: ballistic, Brownian and intermediate which are well characterized, for the first two regimes, through the mean square displacements and Einstein relation for the diffusion constant. The Langevin formalism is used by considering Ohmic friction, moderate surface temperatures and small coverages. In the Bohmian framework, analyzed here, the starting point is the so-called Schrödinger-Langevin equation which is a nonlinear, logarithmic differential equation. By assuming a Gaussian function for the probability density, the corresponding quantum stochastic trajectories are given by a dressing scheme consisting of a classical stochastic trajectory followed by the center of the Gaussian wave packet, and issued from solving the Langevin equation (particle property), plus the time evolution of its width governed by the damped Pinney differential equation (wave property). The Bohmian velocity autocorrelation function is the same as the classical one when the initial spread rate is assumed to be zero. If not, in the diffusion regime, the Brownian-Bohmian motion shows a weak anomalous diffusion.


I. INTRODUCTION
Surface diffusion is one of the most elementary dynamical process occurring on surfaces and a preliminary step to more complex surface phenomena.It is a very active field of surface science from fundamental as well as technological (catalysis, crystal growth, energy storage, etc.) points of view.Typically, this diffusion process is analyzed as in spectroscopic experiments where a probe particle is interacting perturbatively with a given system at thermal equilibrium with a reservoir (or thermal bath) and measuring its response.According to van Hove's theory for neutron scattering by crystal and liquids [1][2][3], the nature of particles (photons, neutrons, electrons or atoms) probing systems of moving and interacting particles (adsorbates) is largely irrelevant when the Born approximation is assumed, reducing this scattering event to a typical statistical mechanics problem.The corresponding linear response is then determined by the spectrum of the spontaneous fluctuations of the reservoir as established by the very well-known fluctuation-dissipation theorem [4].Information provided by the experiment together with a theoretical support or theory behind can allow us to better understand the dynamics as well as extract valuable information for molecular interactions (adsorbate-substrate and adsorbate-adsorbate interactions) within the general framework of stochastic processes.A very large amount of information about the diffusion process in surfaces has been gathered along the last twenty eight years from the well-known review paper by Gomer [5].For fast diffusion motions, we are going to focus on He atoms as nondestructive probe particles used in two types of experiments, quasielastic He atom scattering (QHAS) [6] and He spin echo spectroscopy (HeSE) [7].These time of flight techniques are sensitive to surface processes on the length and time scales on which single atoms diffusion occurs (length scales between around 10 −10 up to 10 −8 meters and time scales going from around 10 −12 up to 10 −8 seconds).Time of flight spectra are usually converted to energy transfer scale allowing a frequency analysis of the surface phonons as well as slow motions of the adsorbates.Angular (around 0.3 0 ) as well as velocity (around 1 %) resolutions are very small covering a large dynamical range in intensity; much better for the HeSE technique.Typical He velocities are less than 3 × 10 3 m/sec.The practical limit of these techniques lies in the velocity spread in the beam but, with the spin-echo method, one measures velocity changes of individual atoms rather than the velocity change with respect to the mean incident velocity.The major challenges facing these techniques are to analyze and extract valuable information from the observed line shapes as well as time behavior.
Van Hove's theory of neutrons was generalized to atom surface scattering within the transition matrix formalism [8] and the Chudley-Elliott aproximation [9,10].In surface diffusion problems, most of work is based on the Langevin equation formalism which is widely used when dealing with stochastic processes, as the diffusion one.Thanks to Caldeira and Leggett, this formalism can be derived from a Hamiltonian which is split into three parts describing the dynamics of the system, the thermal bath or reservoir and their mutual interaction [11,12].The surface is usually considered to be corrugated and at a given temperature, being replaced by an infinite number of harmonic oscillators and mimicking the phonon dynamics as well as the mechanism of dissipation.An Ohmic friction is typically assumed and the damping mechanism is mainly due to acoustic phonons.For barriers greater than 3k B T (k B is the Boltzmann factor and T the surface temperature), the diffusion process is activated and the instantaneous jump picture works quite well.Activation barrier heights are extracted from an Arrhenius plot of the diffusion coefficient.Large discrepancies are obtained when comparing the experimental or theoretical results to the classical transtiton state theory [13] due to the existence of long jumps at high surface temperatures, multiple jumps where the Chudley-Elliott model does not apply.A quantum and classical Kramer's theory was developed to overcome such discrepancies [14][15][16][17][18], leading to analytic expressions for diffusion coefficients, escape rates and hopping distributions within the Langevin formalism.Whenever the diffusing atoms are light such as hydrogen or deuterium, quantum effects are present.It is known that quantum diffusion coefficients can be smaller or greater than the classical ones [15].For example, if the substrate is Pt(111), Arrhenius plots of the diffusion constant and overall hoping rate show clearly a region where deviations from the linearity are observed, which is characteristic of the classical transition state theory (TST) [19].This deviation starts occuring at low temperatures (below 90 K) and the theory of dissipative tunneling [20], based on the quantum TST, is sufficient to be applied.The flattening of the Arrhenius plot at the crossover temperature is however not observed which is a feature of deep tunneling [12].In this regime, Grabert and Weiss accounted for quantum diffusion in periodic potentials [21,22] by using the so-called bounce technique together with the Chudley-Elliott model, leading to analyticl expressions for transition rates and diffusion constants in an incoherent tunneling regime.This theoretical framework was successfully applied to this diffusion problem [23] for low coverages.In any case, as far as we know, this interesting and particular quantum dynamics has not been analyzed in the Langeving formalism, that is, by using quantum stochastic trajectories.
A natural theoretical approach considering quantum trajectories is Bohmian mechanics which is being more and more applied to many conservative problems [24][25][26].Recently, an extension to open quantum systems (see, stochastic processes), within the nonlinear, logarithmic Schrödinger-Langevin (SL) equation framework derived by Kostin [27], has been proposed under the presence or not of a continuous measurement [28,29] and for nonlinear dissipation [30].The resulting quantum stochastic trajecories have been applied to simple systems such as the damped free particle, linear potential, and harmonic oscillator [31] and dissipative quantum tunnelling through an inverted parabolic barrier under the presence of an electric field [32] when analysing the classical-quantum transition of trajectories in the gradual decoherence process.These works introduced the so-called scaled trajectories having as a particular case the Bohmian ones.By assuming a time-dependent Gaussian ansatz for the probability density, theses scaled trajectories are written as a sum of a classical trajectory (a particle property) plus a term containing the width of the corresponding wave packet (a wave property) within of what has been called dressing scheme [29].
The organization of this work is as follows.In Section II, the general theory for neutron scattering due to van Hove is briefly reviewed to better understand the extension to atom scattering.Two main observable functions the socalled dynamic structure factor and intermediate scattering function are introduced and written in terms of adsorbate trajectories.These trajectories are analyzed within the Langevin formalism starting from the so-called Caldeira-Leggett Hamiltonian in the classical, quantum and Bohmian frameworks.This last framework is developed in terms of the Schrödinger-Langevin equation.The adsorbate coverage is introduced by a collisional friction.In Section III, three main regimes in the diffusion process are characterized and analyzed in the classical and quantum domains: the ballistic, Brownian (or diffusion) and intermediate regimes.For each case, the corresponding trajectories are analyzed in terms of the mean square displacements and velocity autocorrelation functions leading to analytical expressions for the observable lines shapes.In the second regime, the Brownian-Bohmian motion shows a weak anomalous diffusive behavior.

A. Observables
In 1954, van Hove [1] established the differential cross section of the scattering of slow neutrons by a system of interacting particles in terms of the generalized pair distribution function, the so-called G(r, t) function of van Hove (with r being a position vector and t a time interval).This G function is a natural extension of the standard pair distribution function g(r) well known, for example, in liquids with G(r, 0) = g(r).Moreover, G describes the correlation between a particle in position r + r ′ at t + t ′ and a particle in position r ′ at time t ′ .In the Born approximation or first order perturbation theory, the scattering problem is reduced essentially to a problem in statistical mechanics [1][2][3] where the nature of the scattered particles (neutrons, light, atoms, etc.) and details of the interaction potential are irrelevant.In this formalism, the linear response of the system implies that it is determined entirely by the properties exhibited by the system in the absence of probe particles.This differential cross section can also be written in terms of the independent variables associated with the momentum transfer, k, and energy transfer ω as providing the probability that the probe particles scattered from the diffusing system reach a certain solid angle Ω in an interval of outgoing energy ω.The response function or line shape S(k, ω) is also termed the scattering law or dynamic structure factor (DSF) where N is introduced for convenience and represents the number of interacting particles in the system under study.The spatial Fourier transform of the G-function is called intermediate scattering function (ISF) and therefore S and I are related by the inverse Fourier transform in time.These functions are easily showed to be expressed in terms of the density-density correlation function where the particle density operator is defined as In this work, we are going mainly to focus on the QHAS technique probing the dynamics of adsorbates or adparticles on surfaces [13].With this technique, at thermal energies, time-of-flight measurements of the probe particles are converted to energy transfer spectra given by the dynamics structure factor.In this scattering, He atoms presents an energy exchange ω = E f inal − E initial and a parallel (to the surface) momentum transfer K = K f inal − K initial (it is standard to express variables projected on the surface as capital letters for position R = (x, y) and parallel momentum K).The prominent peak around the zero energy transfer, the so-called quasi-elastic peak (Q-peak), provides direct information of adsorbate diffusion.Additional weaker peaks at low energy transfers around the Q-peak are also observed and attributed to the parallel frustrated translational motion of some adsorbates (the so-called T-mode) and to surface phonons excitations.Long distance and time correlations are extracted from the scattering law when considering small values of K and ω, respectively.The nature of the adsorbate-substrate and adsorbate-adsorbate interactions can also be known from the scattering law.In this context, the dynamic structure factor is usually expressed as with where the brackets denote an ensemble average and R j (t) the position vector of the j adparticle at time t on the surface.This intermediate scattering function is precisely what is directly observed from the HeSE technique [7] which is quite similar to the well known neutron spin echo one.At this point, it is important to stress the main difference between neutron and Helium scattering.The G-function can naturally be split into a part describing the correlations between the same particle, G s , and distinct particles, G d , where the crossing terms are taken into account.Thus, the full pair correlation function can then be expressed as According to its definition, G s (R, 0) = δ(R) which the Dirac delta function gives the presence of the particle at that position and G d (R, 0) = g(R).At low adparticle concentrations (coverage, θ ≪ 1), when interactions among adsorbates can be neglected because they are far apart from each other, the main contribution to (6) is G s (particleparticle correlations are negligible and G d ≈ 0).On the contrary, at high coverages, G d is expected to have a significant contribution to (6).As a result of this splitting, the intermediate scattering function can also be expressed as a sum of distinct (I d ) and self (I s ) functions.Following neutron scattering language, the corresponding Fourier transforms of I and I s give the so-called coherent scattering law, S(K, ω) and incoherent scattering law S s (K, ω), respectively.In QHAS and HeSE experiments, only coherent scattering is observed.After Eq. ( 5), the ISF contains information about the dynamics of the adsorbates through R j (t).This dynamics is open since the surface can be seen as a reservoir or thermal bath at a given temperature, leading to dissipation and stochasticity within a classical or quantum framework.In the following, we are going to focus on the nature of the adsorbate-adsorbate and adsorbate-substrate interactions.In any case, a proper comparison between the experimental and theoretical observables (issued from any theoretical method) has to be carried out through a convolution integral which takes into account the response of the apparatus which is usually assumed a Gaussian function.

B. Classical stochastic trajectories
For heavy adsorbates, the time-dependent position vectors can be obtained from classical stochastic trajectories.As mentioned above, if the coverage is very small, the adsorbate-adsorbate interaction is negligible and the dynamics can be well described only by the self part of the G-function, G s .The main interaction is then the adsorbate-substrate interaction as well as the thermal fluctuations of the surface through a random force or noise.In the literature, the standard Hamiltonian used is that proposed by Magalinskij [33] and Caldeira and Leggett [11] written in this context as [18,34] where (p x , p y ) and (x, y) are the adparticle momenta and positions with mass m, (p xi , x i ) and (p yi , y i ) with i = 1, • • • , N are the momenta and positions of the bath oscillators (phonons) for each degree of freedom, with mass and frequency given by m i and ω i , respectively.Phonons with polarization along the z-direction are not considered.The adsorbate-substrate interaction V (x, y) is a periodic function describing the surface corrugation at zero temperature.The Hamiltonian (7) is not translational invariance since the term coupling the parallel motions to the phonon bath in both directions is not periodic but linear [35].However, this Hamiltonian is still used because it leads to the correct generalized Langevin equation once the bath degrees of freedom are eliminated where the friction coefficients are defined through the cosine Fourier transform of the spectral densities, with i = x, y and The nonhomogeneity of (8) represents a fluctuating or random force ξ for each degree of freedom which depends on the initial position of the system and initial positions and momenta of the oscillators of each bath according to [12] and If Ohmic friction is assumed, γ i (t) = 2γ i δ(t), where γ i is a constant and δ(t) is Dirac's δ-function.Eqs. ( 8) then reduce to two coupled standard Langevin equations (the δ-function counts only one half when the integration is carried out from zero to infinity) within the Markov approximation.The properties of noise are: (i) ξ i (t) = 0 (zero mean) and (ii) The corresponding classical stochastic trajectories are given by R(t) = (x(t), y(t)).When a flat surface is considered, V (x, y) = 0 and the standard Brownian motion takes place.At higher coverages, adsorbate-adsorbate interactions can no longer be neglected and typically pairwise interaction potentials are usually introduced in Langevin molecular dynamics simulations [36].These simulations always result in a relatively high computational cost due to the time spent by the codes in the evaluation of the forces among particles.This problem is even worse when working with long-range interactions, since a priori they imply that one should consider a relatively large number of particles to numerical convergency.An alternative approach is to consider a purely stochastic description for these interactions [37][38][39] through what is called the interacting single adsorbate (ISA) approximation in a two-bath model.The motion of a single adsorbate is then modelled by a series of random pulses within a Markovian regime (i.e., pulses of relatively short duration in comparison with the system relaxation and acting during a long period of time).These pulses simulate the collisions among adsorbates and are described by means of a white shot noise.In this way, a typical molecular dynamical simulation problem involving N adsorbates is substituted by the dynamics of a single adsorbate where the action of the remaining N − 1 adparticles is replaced by a random force given by the white shot noise.The surface coverage is related to a collisional friction providing the average number of collisions per unit time, γ c .The probability of observing a given number of collisions, after an elapsed time, follows closely a Poisson distribution.The adsorbate is then subject to two uncorrelated white noises, one coming from the substrate and the other one from the surrounding adsorbates.Thus, the total friction coefficient η in the ISA approximation is a sum of two friction coefficients, η = γ + γ c and the total noise is given by ξ = ξ G + ξ S (where G stands for Gaussian and S for shot) for each degree of freedom of the surface (x, y).In this way, differences between self and distinct time-dependent pair correlation function do not exist but Eqs. ( 1) and (2) still hold.The ISF can now be rewritten as Within the so-called Gaussian approximation [40], which is exact when the velocity correlations at more than two different times are negligible, Eq. ( 14) is expressed again as a second order cumulant expansion in K with being the velocity autocorrelation function (VAF) projected onto the direction of the parallel momentum transfer.The velocity is considered to be a stationary stochastic process.This autocorrelation function decays with time, allowing us to define a characteristic time, the so-called correlation time, as where v 2 0 = k B T /m is the average thermal velocity in one dimension, along the direction given by K, m, T and k B being the adsorbate mass, surface temperature and Boltzmann constant, respectively.
The advantage of this approximation consists in providing a direct expression for the coherent scattering which is observed when He atoms are used as probe particles.The dynamical structure factor depends on the VAF through the intermediate scattering function if the Gaussian approximation is also assumed.Two extreme regimes are well characterized in this context, the ballistic diffusion, at very small times (ηt << 1), which is a frictionless motion and the diffusion regime, at very long times (ηt >> 1), when the thermal equilibrium is already reached.Analytical expressions for line shapes in these two extreme regimes are easily derived due to have simple velocity autocorrelations functions [34].

C. Quantum Langevin equation
When considering light adsorbates, quantum mechanics in the Heisenberg picture should be applied.Quantum vector positions in Eqs. ( 5) and ( 14) are then seen as operators.At two different times, they do not commute.However, it is possible to factorize the ISF in two factors due to the disentangling theorem according to e A e B = e A+B e [A,B]/2 which holds when the corresponding commutator is a c-number.Thus, if A = iK.R(0) and B = −iK.R(t) then [34] with the I 2 -factor is given by Eq. ( 15).The I 1 factor can be readily obtained from the formal solution of the corresponding Langevin equation (if the Ohmic friction is assumed) given by Eq. ( 13) where is the random force including the Gaussian and shot noises and R(0) and P(0) are the initial conditions for the position and momentum, respectively.The commutator involved in I 1 is i since [R(0), P(0)] = i , [R(0), F] = i ∂F/∂P(0) = 0 and [R(0), δF r ] = 0 if the noise is assumed to be classical (moderate surface temperatures).Thus, where E r = 2 K 2 /2m is the adsorbate recoil energy.I 1 is a time dependent phase factor which is less and less important when the adsorbate mass and the friction coefficient increase.

D. Bohmian stochastic trajectories
An alternative way to describe the quantum diffusion motion is through Bohmian (or quantum) stochastic trajectories.For this goal, we start from the so-called SL o Kostin equation [27].In 1972, Kostin derived heuristically this equation from the standard Langevin equation.In this context, from Eq. ( 13), the corresponding nonlinear Schrödinger equation is written as where the random potential is given by the damping potential by and is a time dependent function resulting from the average value of V D by integration with respect to the position variable.The norm of the wave function is conserved and the expectation value of the corresponding nonlinear Hamiltonian is, as usual, the sum of the kinetic and potential energies at any time.The SL equation does not fulfill the superposition principle.
If the wave function is written in polar form as where φ(R, t) and S(R, t) are real valued functions and then is substituted into Equation ( 21), the resulting Schrödinger-Langevin-Bohm (SLB) equation reads as [29] i ∂φ ∂t Now, by writing the real and imaginary parts separately, we readily reach the continuity equation with ρ = φ 2 , and the velocity field defined by v = ∇S/m, and the quantum dissipative Hamilton-Jacobi equation given by Q being the quantum potential defined in terms of ρ as follows Let us assume that the probability density is a Gaussian function where δ(t) and q(t) are the width and the center of mass of the wave packet, respectively.From Equation ( 27), the velocity field turns out to be where the dot on the variable means time derivation.The time integration of this velocity field is straightforward leading to the equation for the Bohmian stochastic trajectories Eq. ( 32) is given by a sum of a particle property through a classical trajectory plus a wave property involving the time evolution of the wave packet width.This scheme is known as dressing scheme [29] which is issued only from the continuity equation (27).Now, substitution of Eq. ( 31) into Eq.( 28) and after lengthy but straightforward calculations, we reach leading to the standard Langevin equation for the center of the Gaussian wave packet and the so-called dissipative or damped Pinney equation for its width The solution of this nonlinear differential equation was given by Pinney for the conservative case [41] (η = 0) when is replaced by an arbitrary constant.The commutation rule for the positions at different times does not work in this context.Moreover, the ISF given by Eq. ( 5) can be replaced by Eq. ( 14) within the ISA approximation and Eq. ( 15) when assuming the Gaussian approximation.Within this approximation, the VAF is the key function to be known or evaluated.The velocity of the quantum stochastic trajectories (32) is readily obtained to be and the VAF along the K direction is then where cross correlations are zero due to the statistical independence.It should be noticed that if the initial spread rate is assumed to be zero, δ(0) = 0, the VAF behaves classically.The quantum stochastic dynamics involved in the surface diffusion process within the Bohmian framework is thus reduced to solve Eqs. ( 34) and (35).

III. APPLICATIONS
In surface diffusion, three regimes of motion can be clearly distinguished.First, at very short times, ηt << 1, the motion is frictionless giving place to the so-called ballistic regime.Second, at very long times, ηt >> 1, the thermodynamical equilibrium has already been reached and we speak about the Brownian or diffusion regime.And, finally, we have the intermediate regime where the thermodynamical equilibrium is still far to be reached.We pass now to analyze these three different regimes within the classical, quantum and Bohmian frameworks and provide analytical expressions (if possible) of the lines shapes within the Gaussian and ISA approximations.

A. The ballistic regime
Due to the frictionless motion taking place at very short times (less than the mean free time), the corrugation of the surface plays no role in the surface dynamics.In the classical framework, the VAF is expected to be constant with time and given by the thermal velocity along the K direction as From Eq. ( 15), we have and from Eq. ( 4) which are the Gaussian behaviors predicted for both observables, the ISF and DSF or line shape.This regime has been observed for a two dimensional free gas of Xe atoms on Pt(111) [42,43].Thus, for times much shorter than the mean collision time, the adsorbate displays a free motion showing a dynamical coherence since no memory lost of its velocity takes place.Furthermore, the full width at half maximum (FWHM) of the line shape is linearly dependent on the wave vector transfer, Γ ∝ v 2 K (0) |K|.In this ballistic regime, the mean square displacement (MSD) of the classical stochastic trajectories is known to be characterized by showing a quadratic behavior with time.
In the quantum Langevin framework, the ISF is given by Eq. ( 18) together with Eqs. ( 15) and ( 20).As mentioned above, I 1 is a time dependent phase factor.In the limit of small times, Φ(ηt) ∼ ηt and The second factor I 2 is similar to the classical case and therefore and The Gaussian lineshape is thus shifted by the recoil energy whereas the FWHM is the same as before.
In the Bohmian framework, the starting point is Eq.(37).At very short times, the adsorbate represented by a Gaussian function follows a free motion whose center is ruled by the simple differential equation q(t) = 0 (45) and its width is governed by the nondissipative Pinney equation The solution of this nonlinear differential equation is [44] δ which gives the standard time behavior for the width of a free Gaussian wavepacket when δ(0) = 0, In order to have the width contribution in Eq. ( 37), we can assume, for example, that < δ(0) leading, in the so-called Fresnel or short time regime [26], to where the spreading increases quadratically with time.Thus, in the ballistic regime, and after Eq. ( 32), the Bohmian stochastic trajectories projected on K have the expression where v is the constant velocity of the adsorbate, q(0) gives the initial condition for the center of the Gaussian wave packet and R(0) is generated from the initial Gaussian function.
On the other hand, the VAF along the K direction is expressed according to Eq. (37) as and from Eq. ( 15), we have and from Eq. ( 4) where where only the first term in Eq. ( 49) has been considered in order to keep constant the velocity autocorrelation function which is the key argument in the ballistic regime.The Gaussian functions thus obtained are different from those of the classical case except, as mentioned before, for the case where δ(0) = 0.The commutation rule for the positions at different times is replaced, in this formalism, by the statitical choice of R(0).Finally, in this regime, the MSD of the Bohmian or quantum stochastic trajectories is characterized by showing as expected a quadratic behavior with time.

B. The Brownian or diffusion regime
In this regime, as mentioned above, the thermodynamical equilibrium is already reached.This takes place at long times, that is, when ηt >> 1.Here, again, the surface corrugation plays no role in the surface dynamics.After Doob's theorem [45], the classical VAF is now given by which tell us that the corresponding correlation is decreasing exponentially with time.The ISF and DSF in this classical framework are well known and given by [34] I(K, t) = e −χ 2 (e −ηt +ηt−1) (57) and respectively, where the so-called shape parameter χ is defined by which governs the dynamical coherence of the diffusion process.In this expression, l is the mean free path.It is well known that the time asymptotic behavior of the MSD gives the diffusion coefficient through Einstein's relation The information about D can also be extracted from the observable ISF and DSF.In this diffusion regime, we have that χ << 1 and the ISF is given by a time exponential function and the DSF by a single Lorentzian function which its FWHM is Γ = 2DK 2 .Interestingly enough, in the extreme opposite case, χ >> 1, we approach the ballistic regime already discussed previously.In general, the continuous variation of the χ-parameter can also be seen as a simple way to define the surface dynamical regime.When decreasing χ, the corresponding DSF or line shape becomes narrower and narrower.This gradual change of line shape is known as the motinal narrowing effect [4], going from a Gaussian to a Lorentzian line shape for the two extreme cases studied so far.
In the quantum Langevin framework, it has been shown [34] that the VAF is given by with ν n = 2πn/ β (with β = (k B T ) −1 ) being the so-called Matsubara frequencies.Quantum effects are important at low temperatures, the long time behavior being mainly determined by the first term of the Matsubara series.Thus, relaxation is no longer governed only by the damping constant.The ISF in this quantum framework is then given by [34] I(K, t) = e −χ 2 (ηt−Φ(ηt))−iEr t−K 2 g(t) (64) whith and where it is clearly seen that the extra term K 2 g(t) in the argument of the ISF exponential is the difference with respect to the classical result.The DSF is now much more involved and can not be reduced to a simple analytical function.The diffusion coefficient is a complex number given by whose real part is Einstein's law.The same result can be obtained from the MSD by considering only the symmetric part of the autocorrealtion function.In any case, the limit to very small temperatures is questionable since we are not taking into account the quantum noise correlation.
In the Bohmian framework, from Eqs. ( 37) and (56), the VAF along the K direction is expressed as and the dynamical equations governing this regime are given by Eqs.(34) and (35).As mentioned before, the damped Pinney equation has not an analytical solution but it is possible to look for an approximate one [44].Eq. ( 35) can be rewritten as The expression inside brackets is essentially a positive definite quantity; the first term could be seen as the kinetic energy of the spreading and the second one as a potential function.At long times, due to the negative derivative (decreasing function with time), both terms tend to be negligible at different rates.In this regime, the spreading acceleration is expected to be much smaller than the the damping term η δ, leading to a simple solution for Eq. ( 35) to be It is then straightforward to have that δ justifying the assumption made when ηt >> 1.Thus, Eq. (67) becomes showing the time dependence of the Bohmian VAF.The time depedendent extra contribution goes with t −3/4 , typical from a dissipative behavior for the spreading of the Gaussian distribution function [44].The Bohmian stochastic trajectories, after Eq. ( 32), are then given by The ISF in this framework is then given by Now, at very long times, the argument of the first and second factors contributes linearly with t and then where ā is an average time value of the extremely slow varying function t 1/4 .The DSF or line shape is now expressed as which again a single Lorentzian function is obtained but with a higher FWHM given by Γ = 2(D + α)K 2 .The parameter α is zero at least when δ(0) = 0.In the diffusion regime, the corresponding MSD is no longer linear with time since the crossing term goes to zero at long times.This MSD also keeps the same dressing scheme of the stochastic trajectories, the first contribution is a particle contribution given in terms of the diffusion coefficient D (behaving as in the classical case) and the second one comes from the wave spreading but with different time dependent behaviors.This dressing scheme for the MSD and its time dependence characterize the so-called Bohmian-Brownian motion [29].This slight deviation from the linearity could be seen as a weak anomalous diffusion process [46] C. The intermediate regime In this intermediate regime, analytical results for ISF and DSF are only obtained if the classical VAF is assumed to follow simple functional forms; in any case, numerical simulations of the Langevin equations have to be carried out to extract the parameters involving the particular functional form assumed.
In the classical framework, it is acceptable [34] to assume that the VAF is well described by where a temporary trapping of the adsorbate is expected to occur inside the wells of the corrugated surface interaction potential.The ω-parameter gives the frequency of these intrawell oscillations with a certain dephase δ.Physically, this expression has the correct time behavior corresponding to the ballistic and Brownian regimes analyzed previously.
As has been shown elsewhere, the ISF issue from Eq. (79) has a more or less simple analytical expression leading to a DSF taking into account the intrawell motions which are of low energy quite close to the main quasielastic peak due to zero energy transfer.
For massive particles, the mean interparticle distance is most of the time greater than the thermal de Broglie wavelength λ B = / √ 2mk B T and quantum effects are only considered to be a correction [47].The I 2 factor could be replaced by the classical Eq. ( 15) but this approximation is not good at small times.However, due to the fact the diffusion regime is reached at long times, the only quantum correction comes from the I 1 factor.Obviously, for light particles, where tunnelling can be present, the approach is radically different.
The nice thing about the Bohmian framework with respect to the quantum one is that information of the classical motion can still be used.Thus, the Bohmian VAF is now expressed as k B T m e −ηt cos(ωt + δ) + (R(0) − q(0)) 2 K δ(0) As mentioned before, the Pinney equation governing the width of the Gaussian density can not be solved analytically.
A numerical solution has been obtained by Tsekov [48] and a first-order perturbation solution by Haas et al. [44] where the acceleration term is assumed to be small, reproducing very well the asymptotic behavior.For initially rapidly expanding wave packets, the damping term becomes dominant after a long period of time.With this in mind, the ISF and DSF are only known by numerical calculations.Again, for an initial spreading velocity zero, the standard classical stochastic trajectories and VAF are recovered as well as the ISF and DSF.As a word of conclusion, in this work we have put in evidence that Bohmian stochastic trajectories are also able to describe surface diffusion processes when Ohmic friction, moderate surface temperatures and small coverages are assumed.An important difference can be seen when the initial spreading rate of the Gaussian wave packet is considered zero or not.In particular, when this initial rate is not zero, the diffusion process described in terms of Bohmian stochastic trajectories displays a weak anomalous diffusive behavior.Within this approach, the incoherent tunnelling regime sould be carried out with success after our experience of applying it to the dissipative tunneling by a parabolic barrier [32].It is true that the corresponding formalism should be extended to include the surface periodicity.At the same time, this diffusion process could also be extended and described by scaled trajectories, recently proposed to study dissipative dynamics [31,32] providing a smooth classical-quantum transition.When the Ohmic friction is not a good asumption then the generalized Langevin equation formalism is the appropriate dynamical equation together with colored noise [49].These interesting topics as well as to analyse the diffusion process in terms of continuous measurement [29] are hopefully to be considered in the near future.