ANALYSIS OF STATISTICAL CHARACTERISTICS OF QUASI-BREATHER IN MONOATOMIC FCC METALS Au , Cu , Ni , Pd AND Pt © 2018

The molecular dynamics method is used to calculate and analyze the statistical characteristics of a quasi-breather with a hard type of nonlinearity in monoatomic FCC metals, for example, Cu, Au, Pt, Ni and Pd. Within the framework of this model, the following statistical characteristics and dependencies were calculated for quasi-breathers: a grouped statistical series of absolute and relative frequencies, a polygon of absolute and relative frequencies, a histogram of relative frequencies, an empirical distribution function, an estimate of the mathematical expectation and variance of the original sample. The densities of phonon states are calculated for all crystals. Statistics allow you to understand the causes of the destruction of breathers and more fully describe the process of their dissipation of energy.


INTRODUCTION
One of the most interesting and important objects of nonlinear physics for practical application are solitontype waves (solitary waves) [1][2].Despite the fact that solitons are known to science for more than 180 years, they have been little studied in many fi elds of knowledge.So, recently interest to discrete nonlinear systems in which the existence of dynamic solitons is possible increases.An example of a dynamic soliton can serve as discrete breathers (DB) -loca lized in space and periodic in time high-amplitude excitations in nonlinear discrete structures with translational symmetry [3].
It is assumed that the DBs participate in various solid-state processes.In particular, DB can increase the catalytic properties of nanoparticles with a disordered structure, lead to radiation-stimulated growth of pores in metals, contribute to diffusion, transport electric charge, lead to annealing of defects, reduce the energy barrier of chemical reactions in crystalline solids, etc. [4][5].
Discrete breathers can be divided into two types according to the nature of their frequency dependence on the amplitude [6].In discrete breathers of a soft type, the frequency decreases with increasing amplitude (such discrete breathers can exist only in crystals having a gap in the phonon spectrum: their frequency lies in the slit of the phonon spectrum and therefore they are called slotted), and in discrete hard type breather the opposite occurs (they can have frequencies, both in the gap and above the phonon spectrum).Discrete breathers with a soft type of nonlinearity can be excited in biatomic crystals, for example, in NaCl [6], Pt 3 Al [7][8][9][10][11][12][13][14], as well as in graphene and graphane [15].Breather with a hard type of nonlinearity exist in pure metals with FCC-, BCC-, and HCP-structures.For pure metals or ordered alloys with a small difference in mass, the conditions for excitation of a DB with a hard type of nonlinearity are more specifi c than when exciting slit DBs with a soft type of nonlinearity.
It is necessary to make a terminological reservation.In mathematical physics, DB means strictly periodic, non-local oscillations localized in space, continuous in time, at one frequency, but in real systems where the presence of all possible perturbations is inevitable, one should consider quasi-breathers having a non-strict periodicity of oscillations with frequencies in a certain range and fi nite life time [16].Unlike idealized DB, quasi-breathers have an infi nite, but rather long, lifetime.Quasi-breathers arise in the presence of small deviations from accurate breather solutions in the multidimensional space of all possible initial conditions in the solution of the Cauchy problem for nonlinear differential equations, since in this case there is no complete suppression of the contributions from oscillations of peripheral particles with their own frequencies.Thus, weakening of the leading vibration of the breather core (in our calculations, the core of a symmetric breather forms one particle, and in the case of an antisymmetric breather, two of its central particles) leads to the presence of additional vibration frequencies in the breather solution.These small contributions can be found in the vibrations of all quasi-breather particles, in particular, central ones.If the frequencies of oscillations of all quasi-breather particles computed at a certain time interval near t = t k are determined suffi ciently accurately, then they will not be strictly identical.Further, the terms breather, discrete breather and quasi-breather will be used as synonyms.
The monoatomic FCC metals Cu, Au, Pt, Ni and Pd are considered in this paper.The main goal of the paper is to calculate and analyze the statistical characteristics of quasi-breathers with a hard type of nonlinearity in the specifi ed materials.The data obtained will make it possible to characterize the evolution of a quasibreather over time.

COMPUTER MODEL AND DESCRIPTION
OF THE EXPERIMENT The models we are considering are bulk crystals containing 10 5 to 3•10 5 particles interacting via a potential obtained by the immersed atom method (EAM potential).The simulation was carried out using the LAMMPS package [17].
In computational chemistry, the immersed atom model is used to approximate the interaction energy between two atoms, taking into account the presence of neighboring atoms.The choice of the potential and the validity of its use for a specifi c task is an important stage in the modeling.
The total energy E of the crystal can be expressed , where j ij represents the pair interaction energy of atoms i and j, located at a distance r ij from each other, and F i is the embedding energy associated with placing the atom i in a location with electron density r i .The electron density takes into account the position of the surrounding atoms and can be calculated from the formula r i ( )is the electron density at the site of the atom i located at a distance r ij from the atom j.
The EAM potential of a pure element is determined by three functions: the pair energy φ, the electron density ρ, and the embedding energy F. For the alloy, the EAM potential contains not only the three functions φ, ρ, and F for each of the constituent elements, but also the pair energies j ab between the different elements a and b (a≠b).As a result, the functions j, r, and F calculated for pure metals can not be directly applied to the alloy or multilayer systems.Nevertheless, the procedure for generalizing EAM potentials and their trimming distance by normalizing EAM potentials and introducing an alloy model was proposed by the author of [18].This procedure enables the construction of EAM potentials of alloys from EAM-potentials for individual elements.Such potentials of alloys were used in molecular modeling and gave good results in experiments [18]; we used this potential for the CuAu crystal [19].
The main factor determining the lifetime of the DB in real crystals is the remoteness of its frequency from the frequencies of the phonon spectrum, and therefore dispersion curves and phonon-state densities for the crystals under study were calculated (see Fig. 1).The calculations used the software package LAMMPS, which includes the procedures necessary for these purposes, based on the Fourier transform of the autocorrelation functions of atomic displacements versus time.
Next, the statistical characteristics of DB in monoatomic crystals Cu, Au, Pt, Ni and Pd will be calculated and analyzed.

RESULTS AND DISCUSSIONS
The main statistical characteristics of the quasibreather are the standard deviation η(t k ) (Fig. 2) and the mean value of the frequency v of atomic vibrations, where t k is the quasi-breather lifetime [20][21]: The lifetime of these quasi-breathers was divided into fi ve equal parts.Thus, fi ve points were obtained for analyzing the statistical characteristics of breathers (see Fig. 2).That is, there was a sample of fi ve elementsfrequencies of quasi-breathers, see Table 1.
Next, we constructed the statistical series of absolute frequencies for this sample, i. the extreme values for each model in Table 1.Find the length of the grouping interval by the formula: h = (w max -w min ) / m.
(3) We find the right boundaries of the grouping intervals: (4) We fi nd the centers w * k intervals of the grouping by the formula: w * k = w k -h / 2 (к = 1, ..., 5).
(5) For each grouping interval (w k-1 , w k ) we fi nd the number n k * of sample elements that fall in this interval.It is important that each sample element is assigned to one and only to one interval, and if the value of the element falls on the interval boundary, then we will refer it to the interval with the lowest number.The minimum element is always referred to the fi rst interval, the maximum to the last.The results are shown in Table 2.
We build a grouped statistical series of relative frequencies, which is a sequence of pairs of numbers  3).
Based on Table 3, we construct the relative frequency polygons for each of the crystal models (see Fig. 3).
To complete the statistical picture of the cha-racteristics of quasi-breathers, we estimate the mathematical expectation and variance, and also construct empirical distribution functions.
Estimate of the mathematical expectation (sample mean) of a grouped sample is calculated using the formula: КОНДЕНСИРОВАННЫЕ СРЕДЫ И МЕЖФАЗНЫЕ ГРАНИЦЫ, ТОМ The estimation of variance, not grouped sample, is carried out according to the formula: For the models we are considering, we have obtained the values given in Table 4.
For clarity, we construct empirical distribution functions F(щ) (see Fig. 4).The obtained statistical data show the process of energy dissipation by breathers on the whole interval of their lifetime.The destruction of quasi-breathers occurs at a time when the root-mean-square deviation exceeds the difference between the average frequency of the breather and the nearest boundary of the phonon spectrum of the crystal.In this case, this process may not be uniform, which is primarily due to the properties of the crystals, as well as the method of exciting the breathers.

CONCLUSION
In the work of molecular dynamics using the statistical approach, quasi-breathers in monoatomic FCC crystals Cu, Au, Pt, Ni, and Pd are considered.Dispersion curves and densities of phonon states are calculated for all crystals.All the basic statistical characteristics of quasi-breather frequencies are calculated: the standard deviation of the frequencies of the atoms, the mean frequencies of the quasibreather, the polygons of the relative frequencies, the mathematical expectation, the variance, and the empirical distribution functions.It is established that the root-mean-square deviation of vibration frequencies of quasi-breather atoms, that is, the degree of their quasi-breathing, increases with time (see Fig. 2), and the average frequency of their oscillations decreases, approaching the upper boundary of the phonon spectrum (see Table 1).Quasi-breathers are destroyed when the root-mean-square deviation of the vibration frequencies exceeds the difference between the average frequency of the breather and the nearest boundary of the phonon spectrum of the crystal.The obtained statistical data allow describing the process of degradation of DB with the passage of time.It is important that the described approaches make it possible to establish that quasi-breathers having a shorter lifetime dissipate energy at the initial stages of existence, which is caused both by the method of exciting the breathers and by the properties of model crystals.

Table 3 .
Grouped statistical series of relative frequencies