Diffusion transitions induced by shear-thinning viscosity: application to laser-cooled atomic gases

We study the diffusive dynamics of a system in a nonlinear velocity-dependent frictional environment within a continuous time random walk model. In this model, the motion is governed by a shear-thinning frictional force, −γ0v/[1+(v2/vc2)]μ ( 0<μ⩽1 ), where γ 0 represents the coefficient of static friction and µ is the scaling index. Through analytical and numerical results, we construct a diffusion phase diagram that encompasses different regimes upon variations in parameters γ 0 and µ: normal diffusion; superdiffusion; and hyperdiffusion. These transitions occur because the induced weaker friction enhances the diffusion. With a decrease in the scaling index, we find that the γ 0-dependent exponent of diffusion converges towards the experimental findings for ultracold 87Rb atoms because the strong effective friction arises. The discrepancies between the fractional Lévy kinetics and the experimental findings may be potentially reconciled. We believe that these findings are helpful for analyzing experimental observations of cold atoms diffusing in optical lattices.

To reconcile the discrepancies between the theory of fractional Lévy kinetics and experimental findings of ultracold 87 Rb atoms, several physical effects were considered.A non-negligible physical effect is that, the escape of significant numbers of particles from the optical traps in the experiment.This induces a serious problem that the tails of the Lévy probability density function (PDF) might be truncated so as to fit the central region of the atomic cloud accurately [27].There are few quantitative comparisons between theoretical results considering the effects of many particles escaping from the optical lattice and experimental findings.Within the standard semiclassical picture [29][30][31], two competing mechanisms determine the cooling force, f(v ), where γ 0 denotes the coefficient of static friction and v c the critical velocity.To avoid the truncation of tails of Lévy PDF, or alternatively, many particles escape from the optical lattices in the experiment, particles should experience the strong friction force.In this work, a modification to the cooling force is assumed, where 0 < µ ⩽ 1 is the range for the scaling index.Aided by Green-Kubo relation [13,32], we illustrate that the strong effective dynamical friction arises by modifying the scaling index of the cooling force, which weakens the diffusion.The diffusion exponent gradually approaches to the experimental fitting.Therefore, the discrepancies of γ 0 -dependent diffusion exponent between the fractional Lévy kinetics and the experimental findings for ultracold 87 Rb atoms may be potentially reconciled.
The frictional force described by equation ( 1) has been used in a popular model of shear-thinning viscosity in nonlinear rheology [11,33].For µ = 0, the frictional force f(v) = −γ 0 v reduces to viscous Stokes friction.For µ = 0.5, f (v) provides the basis of an important model corresponding to a plateau in viscoelastic stress often observed in viscoelastic materials [8,[34][35][36][37].For µ = 1.0, the friction force f ) has been previously employed for cold atoms diffusing in optical lattices [23-25, 38, 39].In order to investigate anomalous diffusion in complex systems, various phenomenological models, which include continuous time random walks (CTRWs) [40][41][42][43], generalized Lévy walks [44,45], and other related stochastic models [46,47], have been extensively developed.These models combining with generalized Montrol-Weiss equation have been successfully applied in disordered, biological, and economic systems [48][49][50][51].In this paper, we extend the CTRW approach by considering each step as a dynamical process, and therefore the effects of environment-damping can be introduced naturally [52,53].Within dynamical CTRW model, we explore the diffusive dynamics of system with the shear-thinning viscous frictional force.From analytical and numerical results, we construct the whole diffusion phase diagram comprising the normal diffusion, superdiffusion, and hyperdiffusion regimes by varying parameters γ 0 and µ for this model.When µ = 1.0, the results of dynamical CTRW model are consistent with those of fractional Lévy kinetics.From the perspective of strong friction through a decrease in µ, we reconcile the discrepancies in the γ 0 -dependent diffusion exponent for ultracold 87 Rb atoms between the fractional Lévy kinetics [27] and the experimental findings [28].
Our paper is structured as follows.In section 2, we provide a generalized discussion of nonlinear viscoelastic dynamics based on the dynamical CTRW model.In section 3, we apply the dynamical CTRW model to calculate analytically and numerically the diffusive behaviors induced by the shear-thinning frictional force.We compare the γ 0 -dependent diffusion exponent for ultracold 87 Rb atoms between the dynamical CTRW model with various scaling index and the experimental findings in section 4. The last section presents our conclusions.

Nonlinear viscoelastic dynamics within the dynamical CTRW model
Before focusing on the shear-thinning frictional force given by equation (1), we begin by discussing the diffusive dynamics of a system in a nonlinear viscoelastic environment.We use the dynamical CTRW model, which allows the effects of environment-damping to be consider [52,53], whereas standard CTRWs are phenomenological and do not capture the complex microscopic dynamics or the influence of environmental damping on diffusive behaviors.The velocity of the walkers sampled from a broad distribution is random and dissipative in each jump.The key factors in this dynamical CTRWs approach are dissipative launch velocity and hopping time cost, which are expressed through the PDFs λ(v) and ω(t), respectively.In its simple one-dimensional realization, a walk is performed by a particle jumping with a launch velocity.Once the particle of mass m undergoes renewal processes, each step obeys dynamical equations where g(x) = − dU(x) dx , and U(x) denotes an external potential.We set m = 1 hereafter for simplicity.Indeed, for standard CTRW, the particle waits for a random duration at each site before making a jump to another site.Thus, the waiting time can be interpreted to the jump length.However, for the dynamical CTRW, the waiting time is replaced by the random time cost in each jump called the hopping time, which represents a different approach to considering waiting time compared to the standard CTRW.Notably, the length of each jump will be decided by the contribution of both the hopping time and the launch velocity [54,55] instead of relying on the distribution of waiting times as in standard CTRWs [56].Because the jump trajectory of the particle is based on dynamical equations, dynamical CTRW is a well-established approach to investigating the diffusive dynamics of complex systems in the presence of various frictional environments.2) and ( 1).In each jump, the random time cost is the hopping time T sampled from (4), while the random launch velocity v samples from equation (3).The details of the numerical scheme are given in appendix B. Parameter values m = 1.0, τ0 = 0.5, vc = 1.0, σ = 0.5, µ = 1.0, and γ0 = 3.0 are used.
We assume that a particle, initially located at x 0 , undergoes jumps with launch velocity v and hopping time T. The PDF of the launch velocity is a Gaussian function with variance 2σ 2 , which ensures the principle of causality constraining finite jump speeds [57].The hopping time in each jump follows a exponential distribution given by where τ 0 denotes the characteristic hopping time.After this jump, the particle arrives at x with velocity v T , and therefore the jump length is X = x − x 0 and the time just arrived at position x is t = t 0 + T, where T sampled from equation ( 4) is the random time cost in this jump.In figure 1, we provide a single trajectory of the dynamical CTRW.The particle starts a jump driven by a random launch velocity and a random hopping time.Since each jump step is governed by equation ( 2), the PDF of jump length Q(x, t) is determined by both λ(v) and ω(t).Substituting Q(x, t) into the generalized Montrol-Weiss equation, we obtain the PDF W(x, t), which represents the probability of a particle being at position x at time t.Details of derivations are given below.
The PDF W(x, t) is given by [58-60] where W 0 (x) ≡ W(x, 0) denotes the initial condition and Φ(t) = 1 − ´t 0 ω(t 1 )dt 1 the survival probability.The second term of the right side is the probability that the particle remains at (x, 0) until time t.The PDF of jump length , which indicates the probability density for jumping to (x, t), conditional on arrival at (x 0 , t 0 ) previously.The expression for the conditional jump length distribution ψ(x − x 0 |t − t 0 ) is obtained from [61]: We assume v can be expressed in terms of x − x 0 and t − t 0 through the time integration of equation ( 2), yielding: The form of function B(•) depends upon velocity-dependent frictional force f (v).Substituting equations ( 6) and ( 7) into equation ( 5) yields New J. Phys.26 (2024) 023055 By performing Fourier F (x → k) and Laplace L (t → s) transformations respectively, we obtain Simplifying equation ( 9) with W 0 (k) ≡ 1, we get where From equation (11), it becomes evident that the PDF of jump length is decided by the contribution of both hopping time distribution and launch velocity distribution.The function B(x; t), which is derived through the time integration of equation ( 2) governing each jump step, connects the dynamical CTRW with the standard CTRW.
Based on equations ( 10) and ( 11), the mean squared displacement (MSD) is calculated as where ⟨x(t)⟩ and ⟨x 2 (t)⟩ are given by where L −1 denotes the inverse Laplace transformation.The exponent of diffusion α is determined as the growth in the spread of the average particle trajectory, which helps to identify the diffusion anomalies for the system [41].

Analysis of the shear-thinning frictional force
Up to this point, the modeling framework and analysis of diffusive dynamics are general.We here concern ourselves with shear-thinning frictional force, given by f We focus on the diffusive properties of the system in the present of shear-thinning friction and the external potential is absent (U(x) = 0) therein and hereafter.To investigate the diffusive properties of the system in the present of shear-thinning friction based on dynamical CTRW model, we first analyze the asymptotic behaviors of f (v).In the low-velocity regime (|v| ≪ v c ), a frictional force f(v) ≃ −γ 0 v results.In contrast, if the velocity is high (v ≫ v c ), the frictional force behaves as a power-law function, f(v) ≃ −γ 0 v 1−2µ /v 2µ c .In figure 2, we depict the frictional force f (v) for various µ, and the asymptotic behaviors in the low-and high-velocity regimes are clearly visible.The shearing-thinning friction given by equation (1) introduces two parameters, i.e. γ 0 and µ.It can infer that, for a diffusive system, the growth of MSD over time and the exponent of diffusion α depend upon these two parameters, specifically To gain an intuitive understanding of the diffusive dynamics of the system, one needs to construct a diffusion phase diagram.Unfortunately, obtaining closed solutions for W(k, s) has proven challenging because it is difficult to obtain a unified expression for Q(k, s) with equation ( 1).However, the diffusive dynamics of the system can be inferred from two limiting cases.

Dynamical phase diagram 3.2.1. Normal diffusion in large γ 0 limit
In the limit of large γ 0 (γ 0 ≫ 1), the particle dominates the low-velocity regime, and thus f (v) is reduced to −γ 0 v for all instances 0 < µ ⩽ 1.By substituting f(v) = −γ 0 v into equation ( 2) and performing time integration, we obtain By substituting equation ( 17) into equation ( 11), we find that in the long-time limit where λ(k) = exp(−k 2 σ 2 ) and ω(s) = 1/(1 + τ 0 s).Combining equations ( 10) and ( 18), the MSD emerges as where K 1 = σ 2 /(γ 2 0 τ 0 ).Details of the derivation of equation ( 19) are given in appendix A. Therefore, Equation ( 20) indicates that the system falls into normal diffusion phase for all instances 0 < µ ⩽ 1 in the large γ 0 limit.According to equation ( 18) derived under the condition of large γ 0 limit, the considered model represents decoupled CTRW and the normal diffusive process naturally occurs.However, we can infer from equation ( 11) that the PDF of jump length Q(k, s) behaves as different forms upon variations in γ 0 and µ.The resulting model would correspond to coupled CTRW and diffusion transitions emerge, which is discussed below.

Diffusion transitions in small γ 0 limit
In the limit of small γ 0 (γ 0 ≪ 1), the particle possesses high velocity and experiences shear-thinning frictional force f(v) ≃ −γ 0 v 1−2µ , which contributes to a rich phase diagram of dynamics.For simplicity, we have set v c = 1 therein and hereafter.As the parameter µ gradually increases, three distinct diffusion phases emerge.
(i) Normal diffusion persists for small µ values.Upon inspection of figure 2, it is evident that when 0 < µ ⩽ µ 0 , where µ 0 is some positive value, |f(v)| ≫ γ 0 in high velocity regime.Consequently, the frictional force is so strong that the inertial term in equation ( 2) becomes negligible and can be set to zero (v = 0), leading to the expression: and subsequently, New J. Phys.26 (2024) 023055

M-G Li et al
Substituting equation ( 22) into (10), we arrive at where K 2 = 2σ 2 τ 0 .Details of the derivation of equation ( 23) are provided in appendix A. The corresponding exponent of diffusion is For 0 < µ ⩽ µ 0 , the system exhibits normal diffusion in the small γ 0 limit.The specific value of µ 0 can be determined by numerical simulations.As µ gradually increases, shear-thinning frictional force weakens and the system transitions towards a superdiffusion phase (1 < α < 2), for which stronger diffusion is induced by weaker friction.(ii) Ballistic diffusion occurs for µ = 0.5.In the small γ 0 limit, when µ = 0.5, the frictional force acting on the particle is approximately f(v) ≃ −γ 0 .By substituting this expression into equation ( 2) and performing time integration, we have Equation ( 11) with equation ( 25) is then given by where the shift and similarity theorems of Fourier transforms are used.Combining equations ( 10) and ( 26), one obtains which expresses the MSD for a force-free particle subjected to shear-thinning friction scales with t 2 at long times.The derivation of equation ( 27) is given in appendix A. The exponent of diffusion is calculated as which indicates ballistic diffusion of the particle.As µ gradually increases from µ = 0.5, friction becomes weaker and the system enters a hyperdiffusion phase, for which 2 < α(γ 0 ≪ 1; 0.5 in the small γ 0 limit.This scenario corresponds to the Obukhov model for a tracer particle path in turbulence, in which the velocity follows simple Brownian motion ⟨v 2 (t)⟩ ∼ t [62,63].In this instance, the time integral over the velocity scales as ⟨∆x 2 (t)⟩ ∼ t 3 and α(γ 0 ≪ 1; µ = 1.0) = 3 [64][65][66].Therefore, as friction weakens with increasing µ, the system exhibits three distinct diffusion phases: normal diffusion, superdiffusion, and hyperdiffusion.
To obtain the whole diffusion phase diagram, we performed numerical simulations for dynamical CTRW model described by equation (2) with equation (1).The details of the numerical scheme are given in appendix B. In figure 3, we reveal the three distinct phases-normal diffusion, superdiffusion, and hyperdiffusion-obtained by varying the coefficient of static friction γ 0 and scaling index µ: in the large γ 0 limit (γ 0 ⩾ 2.5), the diffusion behavior is Gaussian for 0 < µ ⩽ 1.In the small γ 0 limit (γ 0 = 0.1), normal diffusion is observed for 0 < µ ⩽ µ 0 with µ 0 = 0.3 determined through numerical simulations.As µ increases gradually, the system enters a superdiffusion phase, and at µ = 0.5, ballistic diffusion occurs.For 0.5 < µ ⩽ 1 in small γ 0 limit, hyperdiffusion is observed.By gradually increasing the coefficient of static friction γ 0 , the system undergoes diffusion transitions from hyperdiffusion to superdiffusion, and finally to normal diffusion.

Application to the experimental findings of ultracold 87 Rb atoms
For µ = 1.0, the frictional force f(v) = −γ 0 v/(1 + v 2 ) represents the cooling force in the semiclassical theory of fractional Lévy kinetics for cold atoms diffusing in optical lattices [27].In this theory, a key parameter D = 2σ 2 /γ 0 was defined as so to analyze the diffusion transitions, for which 2σ 2 is the velocity variance.For deep wells D < 1/5 (specifically, γ 0 ⩾ 2.5 with σ = 0.5), the diffusion follows a Gaussian distribution, whereas for 1/5 < D < 1 (0.5 ⩽ γ 0 ⩽ 2.5), Lévy statistics is obeyed and the exponent of diffusion is given by α = 7/2 − 1/(2D).When D > 1 (γ 0 < 0.5), the diffusion behavior follows x ∼ t 3/2 and α = 3. Notably, the results obtained from the dynamical CTRW in a shear-thinning frictional environment with µ = 1.0 are consistent with the results within the standard semiclassical picture (see figure 4).Nevertheless, experiments have so far not detected transitions from Richardson's diffusion (α = 3) to ballistic diffusion (α = 2) derived from the theory of fractional Lévy kinetics.Sagi and collaborators, in their experimental study of anomalous diffusion for ultracold 87 Rb atoms in a one-dimensional polarization optical lattice [28], found that the anomalous exponents depend on the depth of the optical potential U 0 , and the system has shown at most ballistic behavior.
The main reason for the discrepancies between the theory of fractional Lévy kinetics and the experiment findings of ultracold 87 Rb atoms is that the tails of Lévy PDF should be truncated because many particles escape from the optical traps in the experiment.In previous studies on cold atoms, a cooling force that decays as v −1 in the high velocity regime had been used.To fit the central region of the atomic cloud accurately, we assume the particles experience a strong frictional force to prevent them from reaching higher velocities.This consideration might reconcile discrepancies between the fractional Lévy kinetics and the experimental findings of ultracold 87 Rb atoms.In a shearing-thinning frictional environment, the exponent of diffusion α was found to depend upon the scaling parameter µ besides γ 0 .Here we illustrate that decreasing µ while keeping γ 0 constant induces stronger friction, which result in a smaller diffusion exponent.Figure 4(a) compares the experimental results of the exponent of diffusion with a dynamical CTRW model incorporating frictional force f(v) = −γ 0 v/(1 + v 2 ) µ for various µ.The experimental exponent of diffusion as a function of γ 0 is determined based on microscopical considerations with γ 0 = 2σ 2 U 0 /(cE R ) [17,29], where E R denotes the recoil energy, and dimensionless parameter c depends on the atomic transition involved.For ultracold 87 Rb atoms in a one-dimensional polarization lattice with angular momentum J g = 1/2 → J e = J g + 1 = 3/2 transitions, c = 12.3 [17].Upon inspection of figure 4(a), we find that the exponent of diffusion gradually approaches the value of experimental results of ultracold 87 Rb atoms as µ decreases for a fixed value of γ 0 .The behavior of the MSD of a system subjected to the shear-thinning frictional force with various µ and γ 0 is shown in figure 4(b), where the diffusion transitions can be observed intuitively.
To understand further the effect of modifying the scaling index of the cooling force on particle diffusive dynamics, we compared the effective dynamical friction for various values of µ, while keeping γ 0 fixed.Aided by Green-Kubo relation [13,67], one obtains the diffusion coefficient D eff = lim t→∞ ⟨v 2 (t)⟩ ´∞ 0 dτ ⟨v(t)v(t + τ )⟩/⟨v 2 (t)⟩.Assuming the diffusion coefficient is given by D eff = lim t→∞ k B T eff (t)/mγ eff (t), where k B T eff (t) = m⟨v 2 (t)⟩.Thus, the effective dynamical friction is expressed as The influence of modifying the scaling index of the cooling force is reflected in the effective dynamical friction.Although it is difficult to obtain analytical expressions for ⟨v 2 (t)⟩ and ⟨v(t)v(t + τ )⟩, simulation results are obtained from equation (2) with equation (1).Covering the transition regime from hyperdiffusion to normal diffusion, Figure 5(a) plots the effective dynamical friction γ eff vs scaling index µ for several γ 0 , obtained from the time-dependent of γ eff (t) in the long time limit shown in the inset of figure 5(a).We find that the effective dynamical friction becomes stronger as the value of µ increases.The stronger effective friction resulting from a decrease in µ yields a smaller value of the exponent of anomalous diffusion.The decrease in µ prevents the escape of many particles from the optical trap in the experiment, which is illustrated in figure 5(b).As the scaling index µ decreases, large velocities are suppressed considerably and particles predominantly reside in the low-velocity regime for small µ.Therefore, modifying the scaling index of the cooling force could potentially reconcile the deviation in the diffusion exponent between the theory of fractional Lévy kinetics and experimental fitting.However, achieving full agreement requires further investigation.

Conclusion
We conducted a comprehensive study on the anomalous diffusion transitions of a system in a shear-thinning frictional environment using a dynamical CTRW approach.Through analytical derivations and numerical simulations, a diffusion phase diagram that encompasses the entire range of variations in the parameters γ 0 and µ was constructed.The system exists three distinct diffusive phases: normal diffusion, superdiffusion, and hyperdiffusion.By applying the results of dynamical CTRW in shear-thinning frictional environments with experimental findings for ultracold 87 Rb atoms, discrepancies between experiment and fractional Lévy kinetics may be reconciled with µ decreasing.This is because the stronger effective friction induced by decreasing µ yields a smaller value of the exponent of anomalous diffusion.We believe that the present study provides useful information about systems in nonlinear viscoelastic environments that are related to Sisyphus cooling force.Statistical mechanics of cold atoms in optical lattices remains an open topic, a further study of which may reveal other surprising findings when these systems are extended to quantum scenarios.

Figure 1 .
Figure 1.Numerical realization of a single trajectory for x and v of the dynamical CTRW with equations (2) and (1).In each jump, the random time cost is the hopping time T sampled from (4), while the random launch velocity v samples from equation(3).The details of the numerical scheme are given in appendix B. Parameter values m = 1.0, τ0 = 0.5, vc = 1.0, σ = 0.5, µ = 1.0, and γ0 = 3.0 are used.

Figure 3 .
Figure 3.The diffusion phase diagram of system subjected to the shear-thinning frictional force obtained by varying the coefficient of static friction γ0 and scaling index µ.Data points are calculated numerically using equation (2) with equation (1) with the absence of an external potential.Parameter values m = 1.0, τ0 = 0.5, vc = 1.0, and σ = 0.5 are used.

Figure 4 .
Figure 4. (a) The γ0-dependent exponent of diffusion α for cold atoms with various evaluated methods: experimental fitting, fractional Lévy kinetics, and dynamical CTRW with various scaling index µ.(b) Time-dependent MSD ⟨∆x 2 (t)⟩ for a system subjected to the shear-thinning frictional force with various µ and γ0.Date points are calculated numerically from equation (2) with equation (1).Parameter settings are those of figure 3.

Figure 5 .
Figure 5. (a) Effective dynamical friction γ eff vs scaling index µ for several values of γ0.The inset shows the behavior of the time-dependent effective dynamical friction γ eff (t) for several values of µ with γ0 = 0.5.(b) Velocity scatter of particles subjected to shear-thinning frictional force for various values of µ and γ0.Scattering is performed with a sample size of 10 5 particles for asymptotic times.Data points are obtained through numerical calculations from equation (2) with equation (1).Parameter settings are those of figure 3.