Anomalies of Lévy-based thermal transport from the Lévy-Fokker-Planck equation

Lévy-type behaviors are widely involved in anomalous thermal transport, yet generic investigations based on the mathematical descriptions of the confined Lévy flights are still lacking. In the frameworks of classical irreversible thermodynamics and Boltzmann-Gibbs statistical mechanics, the Lévy-Fokker-Planck equation is connected to near-equilibrium thermal transport. In this work, we show that thermal transport dominated by the confined Lévy flights will be paired with an anomaly, namely that the local effective thermal conductivity is nonlocal. It is demonstrated that the near-equilibrium assumption is not unconditionally valid, which relies on several thermodynamic restrictions expressed by the probability density function (PDF). It is illustrated that the Lévy-Fokker-Planck equation based on the Caputo operator will give rise to two signatures of anomalous thermal transport, the power-law size-dependence of the global effective thermal conductivity and nonlinear boundary asymptotics of the stationary temperature profile. These anomalies are interrelated with each other, and their quantitative relations can be considered as criteria for Lévy-based thermal transport.


Introduction
The Lé vy process [1] is commonly defined by the characteristic function wherein  denotes the Lé vy index, k is the variable in the Fourier space, and c  is a constant.
In recent years, Lé vy-type behaviors have been widely used to interpret signatures of anomalous thermal transport in low-dimensional systems [2][3][4][5][6][7]. A typical example is the power-law size-dependence of the effective thermal conductivity eff  [4,[8][9][10][11], namely, with L denoting the system size. Based on the Monte Carlo technique for solving the phonon Boltzmann transport equation, Upadhyaya and Aksamija [5] have observed a Lé vy-type (or heavy-tailed) distribution of the phonon mean free paths in Si-Ge alloy nanowires, which gives rise to a divergent exponent 13   . Denisov and co-authors [12] connected the size-dependence exponent to the Lé vy index   1, 2   for one-dimensional dynamical channels, 2   . This relation is supported by a recent investigation on the long-range interacting Fermi-Pasta-Ulam chains [7]. Furthermore, the results in Si-Ge alloy nanowires and one-dimensional dynamical channels also show that the Lé vy processes will be paired with another signature of anomalous thermal transport, the superdiffusive growth of the mean-square energy displacement [5], . The coexistence of the Lé vy-type regimes and superdiffusive thermal transport has also been acquired in semiconductor alloys [6] and two-dimensional nonlinear lattices [8].
There is another conceptual connection between the Lé vy processes and anomalous thermal transport in low-dimensional systems, the spatial fractional-order operators [13][14][15]. For instance, the energy perturbation   , e x t  in the one-dimensional harmonic chains is commonly governed by a 3/4-fractional diffusion equation [14,15] [18][19][20][21][22]. In these studies, the Lé vy-type behaviors are observed based on the specific physical regimes of the heat carriers, which differ from model to model, yet generic mathematical descriptions are not much involved with signatures of anomalous thermal transport. In mathematics, spatial fractional-order governing equations are widely applied to the Lé vy processes [23][24][25] , P x t denotes the probability density function (PDF), K  is the noise intensity with the dimension 1 xt   , RL Lx D  and 0 RL x D  stand for the right-hand and left-hand Riemann-Liouville operators respectively. For engineering or experimental problems, the boundary points must be attained, which will give rise to infinite Lé vy measure. In this work, we apply Eq (1.5) to one-dimensional thermal transport, wherein the PDF is defined in terms of the correlation function of the energy fluctuations [3], namely, xx  , which indicates that the nonlocality is symmetric. Based on the entropic functionals, a connection between the evolution of the PDF and thermal transport is established. Anomalous features of thermal transport thereafter arise from the entropic connection, including the nonlocality of the local effective thermal conductivity, power-law size-dependence of the global effective thermal conductivity, and nonlinear boundary asymptotics of the stationary temperature profile. Thermal transport and confined Lé vy flights.

Thermal transport and confined Lé vy flights
The Lé vy-Fokker-Planck equation describes the evolution of the PDF, while thermal transport focuses on thermodynamic quantities, i.e., the heat flux   is the density of the entropy production rate.
Besides the entropy balance equation, there is another restriction termed as continuity equation, and we thereafter arrive at Then,

 
, J x t and   , P x t can be connected to thermal transport via the relationship between For thermal transport not far from local equilibrium, Boltzmann-Gibbs statistical mechanics typically coincides with classical irreversible thermodynamics [26], which gives the following expressions for the above entropic functionals, where eq s is the entropy density independent of thermal transport, and c is the specific heat capacity per volume. Upon combining Eqs (2.8) and (2.9) with Eqs (2.5) and (2.6) respectively, one can derive the relations between The two relations do not rely on specific constitutive models between   given by [27] and substituting it into Eqs (2.10) and (2.11) leads to x but also all states in   0, L . In other words, any points in   0, L will contribute to the heat flux at 0 x . Such nonlocality will vanish in the limit 2   , which leads to a degeneration into the standard diffusion equation. In this degenerate case, Eq (2.13) becomes which illustrates that the diffusive heat flux is proportional to the PDF gradient. Note that the gradient of the entropy density is written as Combining Eqs (2.15) and (2.16) yields Here,  is the so-called thermal conductivity, which is an intrinsic material property and independent of geometric parameters such as the system size.  Furthermore, as a thermodynamically irreversible process, non-vanishing thermal transport (   ,0 q J x t  ) must be paired with a strictly positive value of the entropy production rate. Conversely, if the total entropy production rate of a system is zero, this system must be in thermal equilibrium, which indicates that . In the framework of classical irreversible thermodynamics, the thermodynamic restriction stated above corresponds to the following corollary  According to the result in [23], the equilibrium solution of the Lé vy-Fokker-Planck equation is given by Non-uniform   eq Px will give rise to a non-uniform temperature distribution, namely, From a physical perspective, it is non-trivial that the non-vanishing temperature gradient coexists with the thermal equilibrium state, which means absolute thermal insulation, 0 Px is singular at the boundary, which will induces infinite boundary temperatures. These non-trivial behaviors have not been observed in existing studies on anomalous thermal transport [2][3][4].

Modification based on Caputo operator
If the temperature distribution is uniform in the absence of thermal transport, the equilibrium PDF should be written as For the Lé vy-Fokker-Planck equation, this equilibrium solution can be acquired via replacing the Riemann-Liouville operator by the Caputo operator [31], and the constitutive relation between   , J x t and   , The corresponding local effective thermal conductivity reads which is still nonlocal. The thermodynamic restrictions for q J x t T x t remain unchanged, and the restrictions on the PDF take the following forms We now consider stationary thermal transport in the presence of a small temperature difference, namely, In this case, the solution of the modified Lé vy-Fokker-Planck equation is written as The power-law size-dependence of the effective thermal conductivity presently occurs, while the size-dependence exponent is 2   . This relation between  and  formally coincides with Ref. [12], but it is derived from the confined Lé vy flight rather than the Lé vy walk model. In existing numerical and experimental investigations [2][3][4], the range of the size-dependence exponent is observed as 1   . This range will not allow the case of 01   , that is why the Lé vy exponent is restricted as 12  . In the following, the local effective thermal conductivity will be discussed. which depends on not only the system size but also the location. Eq (3.14) also exhibits another signature of anomalous thermal transport, the nonlinear boundary asymptotics of the stationary temperature profile [4], namely, 16) In the diffusive limit 2   , loc eff  will be independent of the system size and location, and meanwhile, the asymptotic exponent 2   becomes linear. All of these degenerate behaviors agree with Fourier's law, which is physically reasonable. It should be underlined that the expanding approach stated above is inapplicable to the standard Lé vy-Fokker-Planck equation based on the Riemann-Liouville operator. That is because the assumption of sufficiently small temperature difference ( 0 TT  ) is invalid for the Riemann-Liouville operator.

Concluding remarks
The symmetric Lé vy-Fokker-Planck equation is applied to investigating anomalous thermal transport in a one-dimensional confined domain. Based on the frameworks of classical irreversible thermodynamics and Boltzmann-Gibbs statistical mechanics, we establish a connection between the evolution of the probability density function and thermal transport dominated by the confined Lé vy flights. The expression of the local effective thermal conductivity is derived as a nonlocal formula, which depends on all states in the domain. The thermal transport process therefore becomes anomalous. It is demonstrated that the diffusive limit 2   will lead to the degeneration into conventional Fourier's law of heat conduction as if the thermal conductivity and specific heat capacity possess the same temperature-dependence. The thermodynamic connection between the Lé vy-Fokker-Planck equation and anomalous thermal transport relies on the near-equilibrium assumption, which needs certain physical restrictions on the evolution of the probability density function. It is found that the Riemann-Liouville operator will be paired with thermodynamically non-trivial behaviors, namely that the equilibrium state corresponds to the non-uniform temperature distribution and infinite boundary temperature. In order to avoid the non-uniform equilibrium state, the Lé vy-Fokker-Planck equation is modified in terms of the Caputo operator. It is shown that the modified Lé vy-Fokker-Planck equation will give rise to two signatures of anomalous thermal transport, the power-law size-dependence of the global effective thermal conductivity and nonlinear boundary asymptotics of the stationary temperature profile. The results illustrate that the anomalies of Lé vy-based thermal transport are not independent of each other, and should fulfill certain quantitative relations. For instance, the size-dependence exponent of the global effective thermal conductivity and asymptotic exponent of the stationary temperature profile are constrained by 22   . The quantitative relations can be used to test whether a specific thermal transport process is dominated by the confined Lé vy flights.