Diffusion of Randomly Driven Solitons in Molecular Chains (original) (raw)
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Thermal diffusion of solitons on anharmonic chains with long-range coupling
Physical Review E, 2007
We extend our studies of thermal diffusion of non-topological solitons to anharmonic FPU-type chains with additional long-range couplings. The observed superdiffusive behavior in the case of nearest neighbor interaction (NNI) turns out to be the dominating mechanism for the soliton diffusion on chains with long-range interactions (LRI). Using a collective variable technique in the framework of a variational analysis for the continuum approximation of the chain, we derive a set of stochastic integro-differential equations for the collective variables (CV) soliton position and the inverse soliton width. This set can be reduced to a statistically equivalent set of Langevin-type equations for the CV, which shares the same Fokker-Planck equation. The solution of the Langevin set and the Langevin dynamics simulations of the discrete system agree well and demonstrate that the variance of the soliton increases stronger than linearly with time (superdiffusion). This result for the soliton diffusion on anharmonic chains with long-range interactions reinforces the conjecture that superdiffusion is a generic feature of non-topological solitons.
Anomalous diffusion of a quantum Brownian particle in a one-dimensional molecular chain
Physical Review E, 2020
We discuss anomalous relaxation processes of a quantum Brownian particle which interacts with an acoustic phonon field as a thermal reservoir in one-dimensional chain molecule. We derive a kinetic equation for the particle using the complex spectral representation of the Liouville-von Neumann operator. Due to the onedimensionality, the momentum space separates into infinite sets of disjoint irreducible subspaces dynamically independent of one another. Hence, momentum relaxation occurs only within each subspace toward the Maxwell distribution. We obtain a hydrodynamic mode with transport coefficients, a sound velocity, and a diffusion coefficient, defined in each subspace. Moreover, because the sound velocity has momentum dependence, phase mixing affects the broadening of the spatial distribution of the particle in addition to the diffusion process. Due to the phase mixing, the increase rate of the mean-square displacement of the particle increases linearly with time and diverges in the long-time limit.
Soliton diffusion on the classical, isotropic Heisenberg chain
2001
Abstract. We investigate the diffusive motion of a solitary wave on a classical, isotropic, ferromagnetic Heisenberg spin chain with nearest-neighbour exchange interaction. The spins are coupled magnetically to Gaussian white noise and are subject to Gilbert damping. The noise induces a collective, stochastic time evolution of the solitary wave. Within a continuum version of the model we employ implicit collective variables to describe this stochastic behaviour. Thermally excited magnons are disregarded.
Real-time dynamics of soliton diffusion
Physical Review B, 1998
We study the non-equilibrium dynamics of solitons in model Hamiltonians for Peierls dimerized quasi-one dimensional conducting polymers and commensurate charge density wave systems. The real time equation of motion for the collective coordinate of the soliton and the associated Langevin equation is found in a consistent adiabatic expansion in terms of the ratio of the optical phonon or phason frequency to the soliton mass. The equation of motion for the soliton collective coordinate allows to obtain the frequency dependent soliton conductivity. In lowest order we find that although the coefficient of static friction vanishes, there is dynamical dissipation represented by a non-Markovian dissipative kernel associated with two-phonon processes. The correlation function of the noise in the quantum Langevin equation and the dissipative kernel are related by a generalized quantum fluctuation dissipation relation. To lowest adiabatic order we find that the noise is gaussian, additive and colored. We numerically solve the equations of motion in lowest adiabatic order and compare to the Markovian approximation which is shown to fail both in the φ 4 and the Sine Gordon models even at large temperatures.
Soliton diffusion on chains of coupled nonlinear oscillators
Physica A: Statistical Mechanics and its Applications, 2004
We study the di usivity of solitons in noisy-coupled nonlinear oscillators. We consider an anharmonic atomic chain and a network of coupled RLC circuits with nonlinear capacitance and linear inductances and resistances. The solitons propagate in the presence of either thermal noise and/or shot noise. We use a Langevin formalism in order to model the noisy systems, for that reason noise and damping are added to the discrete equations of motion. In the long-wave approximation both systems can be described by identical noisy and damped Korteweg-de Vries (KdV) equations. By using the results of the well-known adiabatic perturbation theory of the forced KdV equation we derive ordinary di erential equations (ODEs) for the relevant variables of the soliton, namely position and inverse of the width. We solve these ODEs by using standard perturbation theory to obtain analytical expressions of the variance and average of the soliton position. We perform Langevin dynamics simulations of the full discrete systems which conÿrm our analytical results, namely superdi usivity of the solitons depending on the initial velocity.
Diffusion of solitons in anisotropic Heisenberg models
The European Physical Journal B, 2004
We are interested in the thermal diffusion of a solitary wave in the anisotropic Heisenberg spin chain (HSC) with nearest-neighbor exchange interactions. The shape of the solitary wave is approximated by soliton solutions of the continuum HSC with on-site anisotropy, restricting ourselves to large width excitations. Temperature is simulated by white noise coupled to the system. The noise affects the shape and position of the solitary wave and produces magnons. Using implicit collective variables we describe the former effects and neglect magnons (i.e. we use the so-called adiabatic approximation). We derive stochastic equations of motion for the collective variables which we treat both analytically and numerically. Predictions for the mean values and the variances of the variables obtained from these equations are compared with the corresponding results from spin dynamics simulations. For the soliton position we find reasonable agreement between spin dynamics and the results of the collective variable treatment, whereas we observe deviations for the other collective variables. The stochastic dynamics of the position shows both a standard Brownian and a super-diffusive component. These results are analogous to results for the isotropic case, previously studied by some of the authors. In the present article we discuss in particular how the anisotropy enters the stochastic equations of motion and the quantitative changes it causes to the diffusion.
The European Physical Journal B, 2009
We study the excitation of solitons in lattices with Morse interactions in a wide temperature range and their influence on (free) electrons moving in the lattice. The lattice units (considered as "atoms" or "screened ion cores") are treated by classical Langevin equations. For visualizations the densities of the core (valence) electrons are in a first estimate represented by Gaussian densities, thus permitting to visualize lattice compressions. The evolution of the (free) electrons is modelled in the tight binding approximation first using Schrödinger equation and, subsequently, a stochastic description of the evolution as a Markov process. We investigate electron transfer assisted by solitons and solitonic influences on macroscopic transport in particular on diffusion. Then we consider the electron-lattice interaction and obtain numerical solutions of the simultaneously evolving Langevin and Pauli master equations. We show that the proposed mechanism of riding on thermal solitons is relatively fast (of the order of the sound velocity).
Kinetic properties of multiquanta Davydov-like solitons in molecular chains
Physical Review E, 1999
The Fokker-Planck equation for multivibron solitons interacting with lattice vibrations in a molecular chain has been derived by means of the nonequilibrium statistical operator method. It was shown that a soliton undergoes diffusive motion characterized by two substantially different diffusion coefficients. The first one corresponds to the ordinary ͑Einsteinian or dissipative͒ diffusion and characterizes the soliton Brownian motion, while the second one corresponds to the anomalous diffusion connected with frictionless displacement of the soliton center of mass coordinate due to the interaction with phonons. Both processes are the consequence of the Cherenkov-like radiation of phonon quanta arising when soliton velocity approaches the phase speed of sound. ͓S1063-651X͑99͒01707-9͔
Thermal diffusion of supersonic solitons in an anharmonic chain of atoms
Physical Review E, 2003
We study the nonequilibrium diffusion dynamics of supersonic lattice solitons in a classical chain of atoms with nearest-neighbor interactions coupled to a heat bath. As a specific example we choose an interaction with cubic anharmonicity. The coupling between the system and a thermal bath with a given temperature is made by adding noise, ␦ correlated in time and space, and damping to the set of discrete equations of motion. Working in the continuum limit and changing to the sound velocity frame we derive a Korteweg-de Vries equation with noise and damping. We apply a collective coordinate approach which yields two stochastic ODEs which are solved approximately by a perturbation analysis. This finally yields analytical expressions for the variances of the soliton position and velocity. We perform Langevin dynamics simulations for the original discrete system which confirm the predictions of our analytical calculations, namely, noise-induced superdiffusive behavior which scales with the temperature and depends strongly on the initial soliton velocity. A normal diffusion behavior is observed for solitons with very low energy, where the noise-induced phonons also make a significant contribution to the soliton diffusion.