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FA, KWOK SAU, and K. G. WANG. "INTEGRO-DIFFERENTIAL EQUATIONS ASSOCIATED WITH CONTINUOUS-TIME RANDOM WALK." International Journal of Modern Physics B 27, no. 12 (2013): 1330006. http://dx.doi.org/10.1142/s0217979213300065.
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The continuous-time random walk (CTRW) model is a useful tool for the description of diffusion in nonequilibrium systems, which is broadly applied in nature and life sciences, e.g., from biophysics to geosciences. In particular, the integro-differential equations for diffusion and diffusion-advection are derived asymptotically from the decoupled CTRW model and a generalized Chapmann–Kolmogorov equation, with generic waiting time probability density function (PDF) and external force. The advantage of the integro-differential equations is that they can be used to investigate the entire diffusion process i.e., covering initial-, intermediate- and long-time ranges of the process. Therefore, this method can distinguish the evolution detail for a system having the same behavior in the long-time limit but with different initial- and intermediate-time behaviors. An integro-differential equation for diffusion-advection is also presented for the description of the subdiffusive and superdiffusive regime. Moreover, the methods of solving the integro-differential equations are developed, and the analytic solutions for PDFs are obtained for the cases of force-free and linear force.
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FA, KWOK SAU. "CONTINUOUS-TIME FINANCE AND THE WAITING TIME DISTRIBUTION: MULTIPLE CHARACTERISTIC TIMES." Modern Physics Letters B 26, no. 23 (2012): 1250151. http://dx.doi.org/10.1142/s0217984912501515.
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In this paper, we model the tick-by-tick dynamics of markets by using the continuous-time random walk (CTRW) model. We employ a sum of products of power law and stretched exponential functions for the waiting time probability distribution function; this function can fit well the waiting time distribution for BUND futures traded at LIFFE in 1997.
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Jara, M., and T. Komorowski. "Limit theorems for some continuous-time random walks." Advances in Applied Probability 43, no. 3 (2011): 782–813. http://dx.doi.org/10.1239/aap/1316792670.
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In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.
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Jara, M., and T. Komorowski. "Limit theorems for some continuous-time random walks." Advances in Applied Probability 43, no. 03 (2011): 782–813. http://dx.doi.org/10.1017/s0001867800005140.
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In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {X n , n ≥ 0} and two observables, τ(∙) and V(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {X n , n ≥ 0} is a sequence of independent and identically distributed random variables.
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AGLIARI, ELENA, OLIVER MÜLKEN, and ALEXANDER BLUMEN. "CONTINUOUS-TIME QUANTUM WALKS AND TRAPPING." International Journal of Bifurcation and Chaos 20, no. 02 (2010): 271–79. http://dx.doi.org/10.1142/s0218127410025715.
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Recent findings suggest that processes such as the excitonic energy transfer through the photosynthetic antenna display quantal features, aspects known from the dynamics of charge carriers along polymer backbones. Hence, in modeling energy transfer one has to leave the classical, master-equation-type formalism and advance towards an increasingly quantum-mechanical picture, while still retaining a local description of the complex network of molecules involved in the transport, say through a tight-binding approach. Interestingly, the continuous time random walk (CTRW) picture, widely employed in describing transport in random environments, can be mathematically reformulated to yield a quantum-mechanical Hamiltonian of tight-binding type; the procedure uses the mathematical analogies between time-evolution operators in statistical and in quantum mechanics: The result are continuous-time quantum walks (CTQWs). However, beyond these formal analogies, CTRWs and CTQWs display vastly different physical properties. In particular, here we focus on trapping processes on a ring and show, both analytically and numerically, that distinct configurations of traps (ranging from periodical to random) yield strongly different behaviors for the quantal mean survival probability, while classically (under ordered conditions) we always find an exponential decay at long times.
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Weron, Karina, Aleksander Stanislavsky, Agnieszka Jurlewicz, Mark M. Meerschaert, and Hans-Peter Scheffler. "Clustered continuous-time random walks: diffusion and relaxation consequences." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2142 (2012): 1615–28. http://dx.doi.org/10.1098/rspa.2011.0697.
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We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.
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MAINARDI, FRANCESCO, ALESSANDRO VIVOLI, and RUDOLF GORENFLO. "CONTINUOUS TIME RANDOM WALK AND TIME FRACTIONAL DIFFUSION: A NUMERICAL COMPARISON BETWEEN THE FUNDAMENTAL SOLUTIONS." Fluctuation and Noise Letters 05, no. 02 (2005): L291—L297. http://dx.doi.org/10.1142/s0219477505002677.
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We consider the basic models for anomalous transport provided by the integral equation for continuous time random walk (CTRW) and by the time fractional diffusion equation to which the previous equation is known to reduce in the diffusion limit. We compare the corresponding fundamental solutions of these equations, in order to investigate numerically the increasing quality of approximation with advancing time.
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Kolokoltsov, Vassili. "CTRW modeling of quantum measurement and fractional equations of quantum stochastic filtering and control." Fractional Calculus and Applied Analysis 25, no. 1 (2022): 128–65. http://dx.doi.org/10.1007/s13540-021-00002-2.
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AbstractInitially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.
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Abdel-Rehim, Enstar A. "From power laws to fractional diffusion processes with and without external forces, the non direct way." Fractional Calculus and Applied Analysis 22, no. 1 (2019): 60–77. http://dx.doi.org/10.1515/fca-2019-0004.
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Abstract In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.
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Klamut, Jarosław, and Tomasz Gubiec. "Continuous Time Random Walk with Correlated Waiting Times. The Crucial Role of Inter-Trade Times in Volatility Clustering." Entropy 23, no. 12 (2021): 1576. http://dx.doi.org/10.3390/e23121576.
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In many physical, social, and economic phenomena, we observe changes in a studied quantity only in discrete, irregularly distributed points in time. The stochastic process usually applied to describe this kind of variable is the continuous-time random walk (CTRW). Despite the popularity of these types of stochastic processes and strong empirical motivation, models with a long-term memory within the sequence of time intervals between observations are rare in the physics literature. Here, we fill this gap by introducing a new family of CTRWs. The memory is introduced to the model by assuming that many consecutive time intervals can be the same. Surprisingly, in this process we can observe a slowly decaying nonlinear autocorrelation function without a fat-tailed distribution of time intervals. Our model, applied to high-frequency stock market data, can successfully describe the slope of decay of the nonlinear autocorrelation function of stock market returns. We achieve this result without imposing any dependence between consecutive price changes. This proves the crucial role of inter-event times in the volatility clustering phenomenon observed in all stock markets.
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Wang, Maomei, Longcang Shu, Gang Zhao, et al. "Simulation of the Riprap Movement Using the Continuous-Time Random Walking Method." Water 13, no. 19 (2021): 2669. http://dx.doi.org/10.3390/w13192669.
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During the implementation of the riprap project, the underwater migration process of the stones is quite uncertain because of its difficulty to observe. The process of stone transportation is discrete, which makes it unsuitable to be described by a continuous differential equation. Therefore, considering the distribution of stone jumping and waiting, a continuous-time random walk (CTRW) model is established. Based on the actual engineering data, five schemes simulate the one-dimensional motion of riprap underwater and further discuss the spatial distribution and particle size of the riprap. The results show that the CTRW model can effectively predict the riverbed elevation change behavior caused by the riprap project. The suitability of the model for the prediction of riprap movement decreases first and then increases with the increase in the selected width. This indicates that the randomness of the motion of the riprap causes the width of the observation zone to have a significant effect on the overall behavior of riprap movement. When the width is large enough, the influence of the randomness of the motion can be reduced by the average movement behavior within the observation zone. While the observation time of riprap movement is from a short to long time scale, the transport behavior changes from subdiffusion to normal diffusion behavior.
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Chen, Zhou, Jin Guo Wang, Wen Zhang Zhang, and Jia Hui Shi. "Experimental Study and Models Comparison for Solute Transport through Riparian Zones." Advanced Materials Research 1073-1076 (December 2014): 1604–8. http://dx.doi.org/10.4028/www.scientific.net/amr.1073-1076.1604.
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Solute transport through riparian zone was studied experimentally and numerically with the consideration of silt layer. The silt layer had markable change on flow field and lead to a significant variation of the breakthrough curves (BTCs). BTCs of solute tracer tests show non-Fickian features as early arrival of peak value and long tailings. BTCs were fitted by advection dispersion equation (ADE), mobile and immobile model (MIM) and the continuous time random walk (CTRW) models. MIM and CTRW can fit BTCs better than ADE and MIM fit better on the capture of the peak value and CTRW fit better in description of the long tailing.
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Wang, Qian, Jianmin Bian, Yihan Li, Chunpeng Zhang, and Fei Ding. "Bimolecular Reactive Transport Experiments and Simulations in Porous Media." Water 12, no. 7 (2020): 1931. http://dx.doi.org/10.3390/w12071931.
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For reactive transport process in porous media, limited mixing and non-Fickian behavior are difficult to understand and predict. To explore the effects of anomalous diffusion and limited mixing, the column-based experiments of bimolecular reactive migration were performed and simulated by the CTRW-FEM model (continuous time random walk-finite element method). Simulated parameters were calibrated and the correlation coefficients between modeled and observed BTCs (breakthrough curves) were greater than 0.9, indicating that CTRW-FEM can solve over-prediction and tailing problems effectively. Porous media with coarser particle size show enhanced mixing and the non-Fickian behavior is not affected by particle size. β (a parameter of CTRW-FEM) and Da (Damköhler number) of CTRW-FEM under different Pe (Péclet number) values showed logarithmic linear relationship. Model sensitivity analysis of the CTRW-FEM model show that the peak concentration is most sensitive to the average pore velocity and the arriving peak time of peak concentration is most sensitive to β. These findings provide a theoretical basis for handling mixing and non-Fickian behavior patterns under actual environmental conditions.
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Dan, Guangyu, Weiguo Li, Zheng Zhong, et al. "Diffusion in Sephadex Gel Structures: Time Dependency Revealed by Multi-Sequence Acquisition over a Broad Diffusion Time Range." Mathematics 9, no. 14 (2021): 1688. http://dx.doi.org/10.3390/math9141688.
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It has been increasingly reported that in biological tissues diffusion-weighted MRI signal attenuation deviates from mono-exponential decay, especially at high b-values. A number of diffusion models have been proposed to characterize this non-Gaussian diffusion behavior. One of these models is the continuous-time random-walk (CTRW) model, which introduces two new parameters: a fractional order time derivative α and a fractional order spatial derivative β. These new parameters have been linked to intravoxel diffusion heterogeneities in time and space, respectively, and are believed to depend on diffusion times. Studies on this time dependency are limited, largely because the diffusion time cannot vary over a board range in a conventional spin-echo echo-planar imaging sequence due to the accompanying T2 decays. In this study, we investigated the time-dependency of the CTRW model in Sephadex gel phantoms across a broad diffusion time range by employing oscillating-gradient spin-echo, pulsed-gradient spin-echo, and pulsed-gradient stimulated echo sequences. We also performed Monte Carlo simulations to help understand our experimental results. It was observed that the diffusion process fell into the Gaussian regime at extremely short diffusion times whereas it exhibited a strong time dependency in the CTRW parameters at longer diffusion times.
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Ma, Weiyuan, Changpin Li, and Jingwei Deng. "Synchronization in Tempered Fractional Complex Networks via Auxiliary System Approach." Complexity 2019 (November 25, 2019): 1–12. http://dx.doi.org/10.1155/2019/6071412.
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In the famous continuous time random walk (CTRW) model, because of the finite lifetime of biological particles, it is sometimes necessary to temper the power law measure such that the waiting time measure has a convergent first moment. The CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. In this article, we introduce the tempered fractional derivative into complex networks to describe the finite life span or bounded physical space of nodes. Some properties of the tempered fractional derivative and tempered fractional systems are discussed. Generalized synchronization in two-layer tempered fractional complex networks via pinning control is addressed based on the auxiliary system approach. The results of the proposed theory are used to derive a sufficient condition for achieving generalized synchronization of tempered fractional networks. Numerical simulations are presented to illustrate the effectiveness of the methods.
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Repetowicz, Przemysław, and Peter Richmond. "Modeling share price evolution as a continuous time random walk (CTRW) with non-independent price changes and waiting times." Physica A: Statistical Mechanics and its Applications 344, no. 1-2 (2004): 108–11. http://dx.doi.org/10.1016/j.physa.2004.06.097.
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Zaheer, Muhammad, Hadayat Ullah, Saad Ahmed Mashwani, Ehsan ul Haq, Syed Husnain Ali Shah, and Fawaz Manzoor. "SOLUTE TRANSPORT MODELLING IN LOW-PERMEABILITY HOMOGENEOUS AND SATURATED SOIL MEDIA." Rudarsko-geološko-naftni zbornik 36, no. 2 (2021): 25–32. http://dx.doi.org/10.17794/rgn.2021.2.3.
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Fickian and non-Fickian behaviors were often detected for contaminant transport activity owed to the preferential flow and heterogeneity of soil media. Therefore, using diverse methods to measure such composite solute transport in soil media has become an important research topic for solute transport modeling in soil media. In this article, the continuous-time random walk (CTRW) model was applied to illustrate the relative concentration of transport in low-permeability homogeneous and saturated soil media. The solute transport development was also demonstrated with the convection-dispersion equation (CDE) and Two Region Model (TRM) for comparison. CXTFIT 2.1 software was used for CDE and TRM, and CTRW Matlab Toolbox v.3.1 for the CTRW simulation of the breakthrough curve. It was found that higher values of determination coefficient (R2) and lower values of root mean square error (RMSE) concerning the best fits of CDE, TRM, and CTRW. It was found that in the comparison of CDE, TRM, and CTRW, we tend to use CTRW to describe the transport behavior well because there are prevailing Fickian and non-Fickian transport. The CTRW gives better fitting results to the breakthrough curves (BTCs) when β has an increasing pattern towards 2.00. In this study, the variation of parameters in three methods was investigated and results showed that the CTRW modeling approach is more effective to determine non-reactive contaminants concentration in low-permeability soil media at small depths.
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Chisaki, Kota, Norio Konno, Etsuo Segawa, and Yutaka Shikano. "Crossovers induced by discrete-time quantum walks." Quantum Information and Computation 11, no. 9&10 (2011): 741–60. http://dx.doi.org/10.26421/qic11.9-10-2.
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We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limits. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability $p \sim 1/n^\beta$ can be evaluated, where $n$ is the final time and $0<\beta<1$. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter $\beta$ shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time $n$. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.
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DENG HUI-FANG. "ASYMPTOTIC BEHAVIORS OF WAITING TIME DISTRIBUTION FUNCTION (WTDF) ψ(t) AND ASYMPTOTIC SOLUTIONS OF CONTINUOUS-TIME RANDOM WALK (CTRW) PROBLEMS". Acta Physica Sinica 35, № 11 (1986): 1436. http://dx.doi.org/10.7498/aps.35.1436.
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Kwaw, Albert Kwame, Zhi Dou, Jinguo Wang, et al. "Influence of Clay on Solute Transport in Saturated Homogeneous Mixed Media." Geofluids 2021 (August 25, 2021): 1–14. http://dx.doi.org/10.1155/2021/1207971.
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In this study, four homogeneous porous media (HPM1-HPM4), consisting of distinct proportions of sand-sized and clay-sized solid beads, were prepared and used as single fracture infills. Flow and nonreactive solute transport experiments in HPM1-HPM4 under three flow rates were conducted, and the measured breakthrough curves (BTCs) were quantified using conventional advection-dispersion equation (ADE), mobile-immobile model (MIM), and continuous time random walk (CTRW) model with truncated power law transition time distribution. The measured BTCs showed stronger non-Fickian behaviour in HPM2-HPM4 (which had clay) than in HPM1 (which had no clay), implying that clay enhanced the non-Fickian transport. As the fraction of clay increased, the global error of ADE fits also increased, affirming the inefficiency of ADE in capturing the clay-induced non-Fickian behaviour. MIM and CTRW performed better in capturing the non-Fickian behaviour. Nonetheless, CTRW’s performance was robust. 12.5% and 25% of clay in HPM2 and HPM3, respectively, decreased the flowing fluid region and increased the solute exchange rate between the flowing and stagnant fluid regions in MIM. For CTRW, the power law exponent ( β CTRW ) values were 1.96, 1.75, and 1.63 in HPM1-HPM3, respectively, implying enhanced non-Fickian behaviour. However, for HPM4, whose clay fraction was 50%, the β CTRW value was 1.87, implying a deviation in the trend of non-Fickian enhancement with increasing clay fraction. This deviation indicated that non-Fickian behaviour enhancement depended on the fraction of clay present. Moreover, increasing flow rate enhanced the non-Fickian transport based on β CTRW .
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Lester, Daniel R., Marco Dentz, Tanguy Le Borgne, and Felipe P. J. de Barros. "Fluid deformation in random steady three-dimensional flow." Journal of Fluid Mechanics 855 (September 19, 2018): 770–803. http://dx.doi.org/10.1017/jfm.2018.654.
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The deformation of elementary fluid volumes by velocity gradients is a key process for scalar mixing, chemical reactions and biological processes in flows. Whilst fluid deformation in unsteady, turbulent flow has gained much attention over the past half-century, deformation in steady random flows with complex structure – such as flow through heterogeneous porous media – has received significantly less attention. In contrast to turbulent flow, the steady nature of these flows constrains fluid deformation to be anisotropic with respect to the fluid velocity, with significant implications for e.g. longitudinal and transverse mixing and dispersion. In this study we derive an ab initio coupled continuous-time random walk (CTRW) model of fluid deformation in random steady three-dimensional flow that is based upon a streamline coordinate transform which renders the velocity gradient and fluid deformation tensors upper triangular. We apply this coupled CTRW model to several model flows and find that these exhibit a remarkably simple deformation structure in the streamline coordinate frame, facilitating solution of the stochastic deformation tensor components. These results show that the evolution of longitudinal and transverse fluid deformation for chaotic flows is governed by both the Lyapunov exponent and power-law exponent of the velocity probability distribution function at small velocities, whereas algebraic deformation in non-chaotic flows arises from the intermittency of shear events following similar dynamics as that for steady two-dimensional flow.
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Zaheer, Muhammad, Zhang Wen, Hongbin Zhan, Xiaolian Chen, and Menggui Jin. "An Experimental Study on Solute Transport in One-Dimensional Clay Soil Columns." Geofluids 2017 (2017): 1–17. http://dx.doi.org/10.1155/2017/6390607.
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Solute transport in low-permeability media such as clay has not been studied carefully up to present, and we are often unclear what the proper governing law is for describing the transport process in such media. In this study, we composed and analyzed the breakthrough curve (BTC) data and the development of leaching in one-dimensional solute transport experiments in low-permeability homogeneous and saturated media at small scale, to identify key parameters controlling the transport process. Sodium chloride (NaCl) was chosen to be the tracer. A number of tracer tests were conducted to inspect the transport process under different conditions. The observed velocity-time behavior for different columns indicated the decline of soil permeability when switching from tracer introducing to tracer flushing. The modeling approaches considered were the Advection-Dispersion Equation (ADE), Two-Region Model (TRM), Continuous Time Random Walk (CTRW), and Fractional Advection-Dispersion Equation (FADE). It was found that all the models can fit the transport process very well; however, ADE and TRM were somewhat unable to characterize the transport behavior in leaching. The CTRW and FADE models were better in capturing the full evaluation of tracer-breakthrough curve and late-time tailing in leaching.
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Kolokoltsov, Vassili N. "The Rates of Convergence for Functional Limit Theorems with Stable Subordinators and for CTRW Approximations to Fractional Evolutions." Fractal and Fractional 7, no. 4 (2023): 335. http://dx.doi.org/10.3390/fractalfract7040335.
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From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes.
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NAKAGAWA, Kei, Yuko HATANO, and Masahiko SAITO. "Reproduction of breakthrough curves for reactive transport experiment in the heterogeneous seepage tank by use of Continuous Time Random Walk (CTRW)." Journal of Groundwater Hydrology 60, no. 3 (2018): 305–15. http://dx.doi.org/10.5917/jagh.60.305.
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Ponprasit, Chaloemporn, Yong Zhang, and Wei Wei. "Backward Location and Travel Time Probabilities for Pollutants Moving in Three-Dimensional Aquifers: Governing Equations and Scale Effect." Water 14, no. 4 (2022): 624. http://dx.doi.org/10.3390/w14040624.
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Backward probabilities have been used for decades to track hydrologic targets such as pollutants in water, but the convenient deviation and scale effect of backward probabilities remain unknown. This study derived backward probabilities for groundwater pollutants and evaluated their scale effect in heterogeneous aquifers. Three particle-moving methods, including the backward-in-time discrete random-walk (DRW), the backward-in-time continuous time random-walk (CTRW), and the particle mass balance, were proposed to derive the governing equation of backward location and travel time probabilities of contaminants. The resultant governing equations verified Kolmogorov’s backward equation and extended it to transient flow fields and aquifers with spatially varying porosity values. An improved backward-in-time random walk particle tracking technique was then applied to approximate the backward probabilities. Next, the scale effect of backward probabilities of contamination was analyzed quantitatively. Numerical results showed that the backward probabilities were sensitive to the vertical location and length of screened intervals in a three-dimensional heterogeneous alluvial aquifer, whereas the variation in borehole diameters did not influence the backward probabilities. The scale effect of backward probabilities was due to different flow paths reaching individual intervals under strong influences of subsurface hydrodynamics and heterogeneity distributions, even when the well screen was as short as ~2 m and surrounded by highly permeable sediments. Further analysis indicated that if the scale effect was ignored, significant errors may appear in applications of backward probabilities of groundwater contamination. This study, therefore, provides convenient methods to build backward probability models and sheds light on applications relying on backward probabilities with a scale effect.
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SALIMI, S., R. RADGOHAR, and M. M. SOLTANZADEH. "SYMMETRY AND QUANTUM TRANSPORT ON NETWORKS." International Journal of Quantum Information 08, no. 08 (2010): 1323–35. http://dx.doi.org/10.1142/s0219749911006661.
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We study the classical and quantum transport processes on some finite networks and model them by continuous-time random walks (CTRW) and continuous-time quantum walks (CTQW), respectively. We calculate the classical and quantum transition probabilities between two nodes of the network. We numerically show that there is a high probability to find the walker at the initial node for CTQWs on the underlying networks due to the interference phenomenon, even for long times. To get global information (independent of the starting node) about the transport efficiency, we average the return probability over all nodes of the network. We apply the decay rate and the asymptotic value of the average of the return probability to evaluate the transport efficiency. Our numerical results prove that the existence of the symmetry in the underlying networks makes quantum transport less efficient than the classical one. In addition, we find that the increasing of the symmetry of these networks decreases the efficiency of quantum transport on them.
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Michelitsch, Thomas M., and Alejandro P. Riascos. "Generalized fractional Poisson process and related stochastic dynamics." Fractional Calculus and Applied Analysis 23, no. 3 (2020): 656–93. http://dx.doi.org/10.1515/fca-2020-0034.
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AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.
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Misiura, Anastasiia, Chayan Dutta, Wesley Leung, Jorge Zepeda O, Tanguy Terlier, and Christy F. Landes. "The competing influence of surface roughness, hydrophobicity, and electrostatics on protein dynamics on a self-assembled monolayer." Journal of Chemical Physics 156, no. 9 (2022): 094707. http://dx.doi.org/10.1063/5.0078797.
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Surface morphology, in addition to hydrophobic and electrostatic effects, can alter how proteins interact with solid surfaces. Understanding the heterogeneous dynamics of protein adsorption on surfaces with varying roughness is experimentally challenging. In this work, we use single-molecule fluorescence microscopy to study the adsorption of α-lactalbumin protein on the glass substrate covered with a self-assembled monolayer (SAM) with varying surface concentrations. Two distinct interaction mechanisms are observed: localized adsorption/desorption and continuous-time random walk (CTRW). We investigate the origin of these two populations by simultaneous single-molecule imaging of substrates with both bare glass and SAM-covered regions. SAM-covered areas of substrates are found to promote CTRW, whereas glass surfaces promote localized motion. Contact angle measurements and atomic force microscopy imaging show that increasing SAM concentration results in both increasing hydrophobicity and surface roughness. These properties lead to two opposing effects: increasing hydrophobicity promotes longer protein flights, but increasing surface roughness suppresses protein dynamics resulting in shorter residence times. Our studies suggest that controlling hydrophobicity and roughness, in addition to electrostatics, as independent parameters could provide a means to tune desirable or undesirable protein interactions with surfaces.
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Frank, Sascha, Thomas Heinze, and Stefan Wohnlich. "Comparison of Surface Roughness and Transport Processes of Sawed, Split and Natural Sandstone Fractures." Water 12, no. 9 (2020): 2530. http://dx.doi.org/10.3390/w12092530.
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In single fractures, dispersion is often linked to the roughness of the fracture surfaces and the resulting local aperture distribution. To experimentally investigate the effects of diverse fracture types and surface morphologies in sandstones, three fractures were considered: those generated by sawing and splitting, and a natural sedimentary fracture. The fracture surface morphologies were digitally analyzed and the hydraulic and transport parameters of the fractures were determined from Darcy and the tracer tests using a fit of a continuous time random walk (CTRW) and a classical advection–dispersion equation (ADE). While the sawed specimen with the smoothest surface had the smallest dispersivity, the natural fracture has the largest dispersivity due to strong anisotropy and non-matching fracture surfaces, although its surface roughness is comparable to the split specimen. The parameterization of the CTRW and of the ADE agree well for β > 4 of the truncated power law. For smaller values of β, non-Fickian transport processes are dominant. Channeling effects are observable in the tracer breakthrough curves. The transport behavior in the fractures is controlled by multiple constraints such as several surface roughness parameters and the equivalent hydraulic aperture.
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Kolokoltsov, Vassili N., and Marianna Troeva. "A New Approach to Fractional Kinetic Evolutions." Fractal and Fractional 6, no. 2 (2022): 49. http://dx.doi.org/10.3390/fractalfract6020049.
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Kinetic equations describe the limiting deterministic evolution of properly scaled systems of interacting particles. A rather universal extension of the classical evolutions, that aims to take into account the effects of memory, suggests the generalization of these evolutions obtained by changing the standard time derivative with a fractional one. In the present paper, extending some previous notes of the authors related to models with a finite state space, we develop systematically the idea of CTRW (continuous time random walk) modelling of the Markovian evolution of interacting particle systems, which leads to a more nontrivial class of fractional kinetic measure-valued evolutions, with the mixed fractional order derivatives varying with the change of the state of the particle system, and with variational derivatives with respect to the measure variable. We rigorously justify the limiting procedure, prove the well-posedness of the new equations, and present a probabilistic formula for their solutions. As the most basic examples we present the fractional versions of the Smoluchovski coagulation and Boltzmann collision models.
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Lester, Daniel R., Marco Dentz, and Tanguy Le Borgne. "Chaotic mixing in three-dimensional porous media." Journal of Fluid Mechanics 803 (August 17, 2016): 144–74. http://dx.doi.org/10.1017/jfm.2016.486.
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Under steady flow conditions, the topological complexity inherent to all random three-dimensional (3D) porous media imparts complicated flow and transport dynamics. It has been established that this complexity generates persistent chaotic advection via a 3D fluid mechanical analogue of the baker’s map which rapidly accelerates scalar mixing in the presence of molecular diffusion. Hence, pore-scale fluid mixing is governed by the interplay between chaotic advection, molecular diffusion and the broad (power-law) distribution of fluid particle travel times which arise from the non-slip condition at pore walls. To understand and quantify mixing in 3D porous media, we consider these processes in a model 3D open porous network and develop a novel stretching continuous time random walk (CTRW), which provides analytic estimates of pore-scale mixing which compare well with direct numerical simulations. We find that the chaotic advection inherent to 3D porous media imparts scalar mixing which scales exponentially with the longitudinal advection, whereas the topological constraints associated with two-dimensional porous media limit the mixing to scale algebraically. These results decipher the role of wide transit time distributions and complex topologies on porous media mixing dynamics, and provide the building blocks for macroscopic models of dilution and mixing which resolve these mechanisms.
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Wang, Hong, and Xiangcheng Zheng. "A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis." Fractional Calculus and Applied Analysis 22, no. 4 (2019): 1014–38. http://dx.doi.org/10.1515/fca-2019-0054.
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Abstract The time-fractional diffusion partial differential equations (tFPDEs) (of order 0 < α < 1) properly model the anomalous diffusive transport or memory effects. Recent work [23] showed that the first-order time derivatives of their solutions have a singularity of O(tα−1) near the initial time t = 0, which makes the error estimates of their numerical approximations in the literature that were proved under full regularity assumptions of the true solutions inappropriate. A sharp error estimate was proved for a finite difference method (FDM) with a graded partition for a one-dimensional tFPDE without artificial regularity assumptions on true solutions, [23]. Motivated by the derivation of the tFPDE from stochastic continuous time random walk (CTRW), we present a modified tFPDE and prove that it has full regularity on the entire time interval (including t = 0) and that its FDM on a uniform time partition has an optimal-order convergence rate only under the assumptions of the regularity of the initial condition and right-hand source term. Numerical experiments show that with the same initial data, the solutions of the modified tFPDE and the classical tFPDE converge to each other as time increases, but the solution of the former does not have the singularity as that to the classical tFPDE near time t = 0.
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Frank, Sascha, Thomas Heinze, Mona Ribbers, and Stefan Wohnlich. "Experimental Reproducibility and Natural Variability of Hydraulic Transport Properties of Fractured Sandstone Samples." Geosciences 10, no. 11 (2020): 458. http://dx.doi.org/10.3390/geosciences10110458.
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Flow and transport processes in fractured systems are not yet fully understood, and it is challenging to determine the respective parameters experimentally. Studies on 10 samples of 2 different sandstones were used to evaluate the reproducibility of tracer tests and the calculation of hydraulic transport properties under identical boundary conditions. The transport parameters were determined using the advection–dispersion equation (ADE) and the continuous time random walk (CTRW) method. In addition, the fracture surface morphology and the effective fracture aperture width was quantified. The hydraulic parameters and their variations were studied for samples within one rock type and between both rock types to quantify the natural variability of transport parameters as well as their experimental reproducibility. Transport processes dominated by the influence of fracture surface morphology experienced a larger spread in the determined transport parameters between repeated measurements. Grain size, effective hydraulic aperture and dispersivity were identified as the most important parameters to evaluate this effect, as with increasing fracture aperture the effect of surface roughness vanishes and the experimental reproducibility increases. Increasing roughness is often associated with the larger effective hydraulic aperture canceling out the expected increased influence of the fracture surface morphology.
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Barrick, Thomas R., Catherine A. Spilling, Matt G. Hall, and Franklyn A. Howe. "The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging." Mathematics 9, no. 15 (2021): 1763. http://dx.doi.org/10.3390/math9151763.
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Quasi-diffusion imaging (QDI) is a novel quantitative diffusion magnetic resonance imaging (dMRI) technique that enables high quality tissue microstructural imaging in a clinically feasible acquisition time. QDI is derived from a special case of the continuous time random walk (CTRW) model of diffusion dynamics and assumes water diffusion is locally Gaussian within tissue microstructure. By assuming a Gaussian scaling relationship between temporal (α) and spatial (β) fractional exponents, the dMRI signal attenuation is expressed according to a diffusion coefficient, D (in mm2 s−1), and a fractional exponent, α. Here we investigate the mathematical properties of the QDI signal and its interpretation within the quasi-diffusion model. Firstly, the QDI equation is derived and its power law behaviour described. Secondly, we derive a probability distribution of underlying Fickian diffusion coefficients via the inverse Laplace transform. We then describe the functional form of the quasi-diffusion propagator, and apply this to dMRI of the human brain to perform mean apparent propagator imaging. QDI is currently unique in tissue microstructural imaging as it provides a simple form for the inverse Laplace transform and diffusion propagator directly from its representation of the dMRI signal. This study shows the potential of QDI as a promising new model-based dMRI technique with significant scope for further development.
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Mitus, Antoni C., Marina Saphiannikova, Wojciech Radosz, Vladimir Toshchevikov, and Grzegorz Pawlik. "Modeling of Nonlinear Optical Phenomena in Host-Guest Systems Using Bond Fluctuation Monte Carlo Model: A Review." Materials 14, no. 6 (2021): 1454. http://dx.doi.org/10.3390/ma14061454.
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We review the results of Monte Carlo studies of chosen nonlinear optical effects in host-guest systems, using methods based on the bond-fluctuation model (BFM) for a polymer matrix. In particular, we simulate the inscription of various types of diffraction gratings in degenerate two wave mixing (DTWM) experiments (surface relief gratings (SRG), gratings in polymers doped with azo-dye molecules and gratings in biopolymers), poling effects (electric field poling of dipolar molecules and all-optical poling) and photomechanical effect. All these processes are characterized in terms of parameters measured in experiments, such as diffraction efficiency, nonlinear susceptibilities, density profiles or loading parameters. Local free volume in the BFM matrix, characterized by probabilistic distributions and correlation functions, displays a complex mosaic-like structure of scale-free clusters, which are thought to be responsible for heterogeneous dynamics of nonlinear optical processes. The photoinduced dynamics of single azopolymer chains, studied in two and three dimensions, displays complex sub-diffusive, diffusive and super-diffusive dynamical regimes. A directly related mathematical model of SRG inscription, based on the continuous time random walk (CTRW) formalism, is formulated and studied. Theoretical part of the review is devoted to the justification of the a priori assumptions made in the BFM modeling of photoinduced motion of the azo-polymer chains.
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Ge, Jianchao, Mark E. Everett, and Chester J. Weiss. "Fractional diffusion analysis of the electromagnetic field in fractured media Part I: 2D approach." GEOPHYSICS 77, no. 4 (2012): WB213—WB218. http://dx.doi.org/10.1190/geo2012-0072.1.
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We address a 2D finite difference (FD) frequency-domain modeling algorithm based on the theory of fractional diffusion of electromagnetic (EM) fields, which is generated by an infinite line source lying above a fractured geological medium. The presence of fractures in the subsurface, usually containing highly conductive pore fluids, gives rise to spatially hierarchical flow paths of induced EM eddy currents. The diffusion of EM eddy currents in such formations is anomalous, generalizing the classical Gaussian process described by the conventional Maxwell equations. Based on the continuous time random walk (CTRW) theory, the diffusion of EM eddy currents in a rough medium is governed by the fractional Maxwell equations. Here, we model the EM response of a 2D subsurface containing fractured zones, based on the fractional Maxwell equations. The governing equation in the frequency domain is discretized using the FD approach. The resulting equation system is solved by the multifrontal massively parallel solver (MUMPS). We find excellent agreement between the FD and analytic solutions for a rough half-space model. Then, FD solutions are calculated for a 2D fault zone model with variable conductivity and roughness. We illustrate a case in which a rough fault zone would not be resolved by classical diffusion modeling, even if its conductivity contrasts with the background.
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Hu, Yingtao, Wenjie Xu, Liangtong Zhan, Zuyang Ye, and Yunmin Chen. "Non-Fickian Solute Transport in Rough-Walled Fractures: The Effect of Contact Area." Water 12, no. 7 (2020): 2049. http://dx.doi.org/10.3390/w12072049.
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The influence of contact area, caused by normal deformation, on fluid flow and solute transport through three-dimensional (3D) rock fractures is investigated. Fracture surfaces with different Hurst exponents (H) were generated numerically using the modified successive random addition (SRA) method. By applying deformations normal to the fracture surface (Δu), a series of fracture models with different aperture distributions and contact area ratios (c) were simulated. The results show that the contact area between the two fracture surfaces increases and more void spaces are reduced as deformation (Δu) increases. The streamlines in the rough-walled fractures show that the contact areas result in preferential flow paths and fingering type transport. The non-Fickian characteristics of the “early arrival” and “long tail” in all of the breakthrough curves (BTCs) for fractures with different deformation (Δu) and Hurst parameters (H) were determined. The solute concentration distribution index (CDI), which quantifies the uniformity of the concentration distribution within the fracture, decreases exponential as deformation (Δu) and/or contact area ratios (c) increase, indicating that increased contact area can result in a larger delay rate of mass exchange between the immobile zone around the contact areas and the main flow channel, thus, resulting in a longer time for the solute to fill the entire fracture. The BTCs were analyzed using the continuous time random walk (CTRW) inverse model. The inverse modeling results show that the dispersion exponent β decreases from 1.92 to 0.81 as c increases and H decreases, suggesting that the increase in contact area and fracture surfaces enhance the magnitude of the non-Fickian transport.
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Raghavan, Rajagopal, and Chih Chen. "An application of a multiindex, time-fractional differential equation to evaluate heterogeneous, fractured rocks." Science and Technology for Energy Transition 78 (2023): 1. http://dx.doi.org/10.2516/stet/2022024.
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A multiindex, distributed fractional differential equation is derived and solved in terms of the Laplace transformation. Potential applications of the proposed model include the study of fluid flow in heterogeneous rocks, the examination of bimodal fluid exchange between mobile-immobile regions in groundwater systems, the incorporation of the existence of liesegang bands in fractured rocks, and addressing the influences of faulted and other skin regions at interfaces, among others. Asymptotic solutions that reveal the structure of the resulting solutions are presented; in addition, they provide for ensuring the accuracy of the numerical computations. Fractional flux laws based on Continuous Time Random Walks (CTRW) serve as a linchpin to account for complex geological considerations that arise in the flow of fluids in heterogeneous rocks. Results are intended to be applied at the Theis scale when combined with geological/geophysical models and production statistics to all aspects of subsurface flow: production of geothermal and hydrocarbon fluids, injection of fluids into aquifers, geologic sequestration and hazardous waste disposal. Results may be extended to study the role of complex wellbores such as horizontal and fractured wells and more complex geological considerations such as faulted systems.
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Favard, Cyril. "Numerical Simulation and FRAP Experiments Show That the Plasma Membrane Binding Protein PH-EFA6 Does Not Exhibit Anomalous Subdiffusion in Cells." Biomolecules 8, no. 3 (2018): 90. http://dx.doi.org/10.3390/biom8030090.
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The fluorescence recovery after photobleaching (FRAP) technique has been used for decades to measure movements of molecules in two-dimension (2D). Data obtained by FRAP experiments in cell plasma membranes are assumed to be described through a means of two parameters, a diffusion coefficient, D (as defined in a pure Brownian model) and a mobile fraction, M. Nevertheless, it has also been shown that recoveries can be nicely fit using anomalous subdiffusion. Fluorescence recovery after photobleaching (FRAP) at variable radii has been developed using the Brownian diffusion model to access geometrical characteristics of the surrounding landscape of the molecule. Here, we performed numerical simulations of continuous time random walk (CTRW) anomalous subdiffusion and interpreted them in the context of variable radii FRAP. These simulations were compared to experimental data obtained at variable radii on living cells using the pleckstrin homology (PH) domain of the membrane binding protein EFA6 (exchange factor for ARF6, a small G protein). This protein domain is an excellent candidate to explore the structure of the interface between cytosol and plasma membrane in cells. By direct comparison of our numerical simulations to the experiments, we show that this protein does not exhibit anomalous diffusion in baby hamster kidney (BHK) cells. The non Brownian PH-EFA6 dynamics observed here are more related to spatial heterogeneities such as cytoskeleton fence effects.
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Magin, Richard L., and Ervin K. Lenzi. "Fractional Calculus Extension of the Kinetic Theory of Fluids: Molecular Models of Transport within and between Phases." Mathematics 10, no. 24 (2022): 4785. http://dx.doi.org/10.3390/math10244785.
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The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases.
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Yan, Xiaosan, Jiazhong Qian, Lei Ma, Mu Wang, and Aofeng Hu. "Non-Fickian Solute Transport in a Single Fracture of Marble Parallel Plate." Geofluids 2018 (June 13, 2018): 1–9. http://dx.doi.org/10.1155/2018/7418140.
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Accurate prediction of solute transport processes in a fracture aquifer is an important task not only for proper management of the groundwater but also for pollution control. A key issue of this task is how to accurately obtain the experimental data and to analyze the solute transport in fracture in subsurface hydrology, which would greatly help us to understand the releasing mechanism and transport of the solute in a fracture. In this study, a fracture experiment is conducted in a laboratory based on previous studies. The fracture used with a length of 60 cm and a width of 10 cm is sealed with glass glue to avoid leakage of tracer due to uneven fracture walls. The sodium chloride (NaCl) solute is injected from the left of the fracture. And an electrical conductivity monitoring system is installed on the right of the fracture. Then breakthrough curves (BTCs) of solute transport are fitted using the classical advection-dispersion equation (ADE) and the truncation power-law function (TPL) model in the package of the continuous time random walk (CTRW). The results show that the flow satisfies non-Darcian law in the experimental conditions, which can be better fitted using the Forchheimer equation and Izbash equation. The solute transport presents non-Fickian phenomena and shows a long tailing. The fitting results of the TPL model are far better than ADE in fitting the long tailing at three different flow velocities. Furthermore, electrical conductivity monitoring method not only is effective but also has an advantage of no disturbance to water and concentration fields in a fracture.
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Li, Xi, Mingyou Wu, Hanwu Chen, and Zhibao Liu. "Algorithms for finding the maximum clique based on continuous time quantum walks." Quantum Information and Computation 21, no. 1&2 (2021): 0059–79. http://dx.doi.org/10.26421/qic21.1-2-4.
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In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.
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Lin, Fang, and Jing-Dong Bao. "Environment-dependent continuous time random walk." Chinese Physics B 20, no. 4 (2011): 040502. http://dx.doi.org/10.1088/1674-1056/20/4/040502.
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Alemany, P. A., R. Vogel, I. M. Sokolov, and A. Blumen. "A dumbbell's random walk in continuous time." Journal of Physics A: Mathematical and General 27, no. 23 (1994): 7733–38. http://dx.doi.org/10.1088/0305-4470/27/23/016.
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Sabhapandit, Sanjib. "Record statistics of continuous time random walk." EPL (Europhysics Letters) 94, no. 2 (2011): 20003. http://dx.doi.org/10.1209/0295-5075/94/20003.
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Lv, Longjin, Fu-Yao Ren, Jun Wang, and Jianbin Xiao. "Correlated continuous time random walk with time averaged waiting time." Physica A: Statistical Mechanics and its Applications 422 (March 2015): 101–6. http://dx.doi.org/10.1016/j.physa.2014.12.010.
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Zhang, Caiyun, Yuhang Hu, and Jian Liu. "Correlated continuous-time random walk with stochastic resetting." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 9 (2022): 093205. http://dx.doi.org/10.1088/1742-5468/ac8c8e.
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Abstract It is known that the introduction of stochastic resetting in an uncorrelated random walk process can lead to the emergence of a stationary state, i.e. the diffusion evolves towards a saturation state, and a steady Laplace distribution is reached. In this paper, we turn to study the anomalous diffusion of the correlated continuous-time random walk considering stochastic resetting. Results reveal that it displays quite different diffusive behaviors from the uncorrelated one. For the weak correlation case, the stochastic resetting mechanism can slow down the diffusion. However, for the strong correlation case, we find that the stochastic resetting cannot compete with the space-time correlation, and the diffusion presents the same behaviors with the one without resetting. Meanwhile, a steady distribution is never reached.
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Briozzo, Carlos B., Carlos E. Budde, and Manuel O. Cáceres. "Continuous-time random-walk model for superionic conductors." Physical Review A 39, no. 11 (1989): 6010–15. http://dx.doi.org/10.1103/physreva.39.6010.
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Denisov, S. I., and H. Kantz. "Continuous-time random walk theory of superslow diffusion." EPL (Europhysics Letters) 92, no. 3 (2010): 30001. http://dx.doi.org/10.1209/0295-5075/92/30001.
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Lv, Longjin, Jianbin Xiao, Liangzhong Fan, and Fuyao Ren. "Correlated continuous time random walk and option pricing." Physica A: Statistical Mechanics and its Applications 447 (April 2016): 100–107. http://dx.doi.org/10.1016/j.physa.2015.12.013.
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