The following question is addressed: under what conditions can a strange diffusive process, defined by a semi-dynamical V-Langevin equation or its associated Hybrid kinetic equation (HKE), be described by an equivalent purely stochastic process, defined by a Continuous Time Random Walk (CTRW) or by a Fractional Differential Equation (FDE)? More specifically, does there exist a class of V-Langevin equations with long-range (algebraic) velocity temporal correlation, that leads to a time-fractional superdiffusive process? The answer is always affirmative in one dimension. It is always negative in two dimensions: any algebraically decaying temporal velocity correlation (with a Gaussian spatial correlation) produces a normal diffusive process. General conditions relating the diffusive nature of the process to the temporal exponent of the Lagrangian velocity correlation (in Corrsin approximation) are derived.
Deep Dive into V-Langevin Equations, Continuous Time Random Walks and Fractional Diffusion.
The following question is addressed: under what conditions can a strange diffusive process, defined by a semi-dynamical V-Langevin equation or its associated Hybrid kinetic equation (HKE), be described by an equivalent purely stochastic process, defined by a Continuous Time Random Walk (CTRW) or by a Fractional Differential Equation (FDE)? More specifically, does there exist a class of V-Langevin equations with long-range (algebraic) velocity temporal correlation, that leads to a time-fractional superdiffusive process? The answer is always affirmative in one dimension. It is always negative in two dimensions: any algebraically decaying temporal velocity correlation (with a Gaussian spatial correlation) produces a normal diffusive process. General conditions relating the diffusive nature of the process to the temporal exponent of the Lagrangian velocity correlation (in Corrsin approximation) are derived.
Strange Transport has been the object of intense studies in recent years (A very recent qualitative review is [1]). (We use the terminology "Strange transport" [2] rather than "Anomalous transport" which is customary in Dynamical Systems theory, but has a different meaning in Plasma transport theory). The concept first appeared in the theory of stochastic processes, especially the theory of Continuous Time Random Walks (CTRW) [3], [4]. Consider a disordered system (e.g. a turbulent fluid or plasma). The position x(t) at time t of one of its particles is determined by its interactions with the other particles and/or with external sources. In a strongly disordered system these interactions are modelled by a random field. The statistical description of the system thus involves the definition of an ensemble of realizations. One of the important quantities describing the system is the mean square deviation (MSD) of the random variable x(t). In many cases, this is represented asymptotically (t → ∞) by a simple increasing function of time:
where δx(t) = x(t) -x(t) . The value of the “diffusion exponent” µ characterizes the diffusion regime of the system1 :
The subdiffusive and the superdiffusive regimes are the ones called “STRANGE”. It will be seen in the forthcoming sections that the concept of “strange transport” involves much more than a simple statement about the behaviour of the MSD.
All present theories of strange transport are of stochastic nature. From the very voluminous literature we only cite here a few of the more comprehensive, physically oriented review papers and books: [3] - [13], where additional references will be found.
As stated in [2], transport theories (e.g., for fluids or plasmas) can be constructed on three levels. α) A purely statistical mechanical theory would be based on the kinetic equation for a set of interacting charged particles, combined with Maxwell’s equations. This would be the most fundamental description, but becomes impossibly complicated in practice.
β) The next level of description would be a compromise, based on a “semidynamical” model of particles moving according to the laws of mechanics (Newton’s equation), but under the action of a fluctuating field, representing the action of the turbulent environment. This leads to stochastic ordinary differential equations of the Langevin type. More specifically, we consider V-Langevin equations [2], i.e. equations for the position of a “particle”:
x(0) = 0 (3) where the right hand side represents a fluctuating velocity field, defined by its statistical properties. We always consider a divergenceless velocity field:
Associated with ( 3) is a kinetic-type first-order partial differential equation for the fluctuating particle distribution function f (x, t) (whose average is the density profile), called a hybrid kinetic equation (HKE) [2]. By definition, the characteristics of this equation are precisely the V-Langevin equations (3):
Due to the property (4) this equation is equivalent to the following:
This equation ensures the conservation of the number of particles, and is rightly interpreted as a kinetic equation.
γ) The last level of description is the level of continuous time random walk (CTRW) theories and fractional diffusion equations (FDE), in which the deterministic dynamical laws are completely given up, and replaced by a purely random process.
All these concepts will be defined and used in the forthcoming sections. These three levels are, of course, interrelated: each one of them should be justifiable as an approximation of the more fundamental one.
The strange transport theories associated with CTRW and with FDE have been very successful in recent years in modelling many peculiar behaviors observed, among other fields, in plasma and fusion physics. There is, however, still a serious gap in justifying these purely stochastic models on the basis of a molecular theory, i.e., theories of the type α), or at least β).
In [2] we treated several models exhibiting strange transport, showing that a stochastic CTRW can be associated with a semi-dynamical V-Langevin equation under certain limiting conditions. All these models (diffusion in a fluctuating electrostatic field or in a fluctuating magnetic field) were either diffusive, or subdiffusive. One of the purposes of the present work is to try to determine whether there exist V-Langevin equations leading to temporal superdiffusive behavior and associating them with a CTRW or with a fractional diffusion equation.
A first work following the same philosophy was done by West et al. [14]. In that paper a specially simplified model of a one-dimensional, two-state CTRW was used. This means that a “particle” performs a CTRW in one dimension, moving with a velocity that is constant in absolute value: ±V . The particle moves during a given (random) time, after which it suddenly jumps and reverses its velocity. The velocity autocorrelation function is given as an algebraically
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