Anomalous kinetics and transport from 1D self-consistent mode-coupling theory

Abstract

Abstract. We study the dynamics of long-wavelength fluctuations in onedimensional (1D) many-particle systems as described by self-consistent modecoupling theory. The corresponding nonlinear integro-differential equations for the relevant correlators are solved analytically and checked numerically. In particular, we find that the memory functions exhibit a power-law decay accompanied by relatively fast oscillations. Furthermore, the scaling behaviour and, correspondingly, the universality class depends on the order of the leading nonlinear term. In the cubic case, both viscosity and thermal conductivity diverge in the thermodynamic limit. In the quartic case, a faster decay of the memory functions leads to a finite viscosity, while thermal conductivity exhibits an even faster divergence. Finally, our analysis puts on a more firm basis the previously conjectured connection between anomalous heat conductivity and anomalous diffusion.

Thermal Transport in Low Dimensions

“…It can thus be shown [21,22] that, for small q-values and long times C(q,t) = g( √ εtq 3/2 ) i.e. z = 3/2 in agreement with the above mentioned numerics.…”

“…A reasonable argument, that can be invoked to delimit the KPZ universality class, is the symmetry of the interaction potential with respect to the equilibrium position. With reference to the MCT, one realizes that the symmetry of the fluctuations implies that the quadratic kernel in (36) should be replaced by a cubic one 4 , thus yielding different values of the exponents [22]. In the language of KPZ interfaces, whenever the coefficient of the nonlinear term vanishes, the evolution equation reduces to the Edwards-Wilkinson equation that is indeed characterized by different scaling exponents.…”

Crossover between different universality classes: Scaling for thermal transport in one dimension

“…The underlying picture is suggested to be understood by phonons performing various kinds of continuous-time random walks (in most cases, be the Lévy walks but not always), probably induced by the peculiar phonon dispersions along with nonlinearity. The results and suggested mechanisms may provide insights into controlling the transport of heat in some 1D materials.Transport in one dimension has, for a long time, been realized to be anomalous in most cases [1,2], with signatures of a universal power-law scaling of transport coefficients, among which the heat transport has been extensively investigated in the recent decades, both by various theoretical techniques, such as the renormalization group [3], mode coupling [4,5] or cascade [6][7][8], nonlinear fluctuating hydrodynamics [9][10][11], and Lévy walks [12][13][14][15]; and also by computer simulations [16][17][18][19][20][21][22][23][24][25][26]. For all studied cases two main scaling exponents have been given the most focus, i.e., α describing the divergence of heat conductivity with space size L as L α and γ characterizing the space(x)-time(t) scaling of heat spreading density ρ(x, t) as t −1/γ ρ(t −1/γ x, t).…”

“…Unfortunately, however, depending on the focused system's different parameter regimes, different theoretical models have been employed, and different predictions have been suggested. Thus, the universality classes of both scaling exponents and their relationship [27,28] remain controversial: (i) for α, two classes: α = 1/3 [3,9,[14][15][16]19] and α = 1/2 [4,5,9] have been reported; however, the universality has been doubted [25,26] and a Fibonacci sequence of α values converging on α * = (3 − √ 5)/2 (≃ 0.382) [6][7][8] has been suggested; (ii) for γ, (a) E-mail: phyxiongdx@fzu.edu.cn two universality classes, γ = 5/3 [9-11, 13-15, 20-23] and γ = 3/2 [9-11, 22, 23] have been predicted recently. The discussion of the latter scaling exponent γ is currently very hot [9-15, 20-23, 29, 30] because it involves more detailed space and time information [31], thus it can present a very detailed prediction for heat transport.…”

Boundary-Induced Instabilities in Coupled Oscillators

“…We conjecture that, as in the context of (anomalous) heat conduction in one-dimensional systems, the discriminating factor is given by the presence of symmetries [24,32]. In the XY case, the average value of the torque is immaterial, since it can be removed by selecting a suitably rotating frame.…”