On 1/f-noise of electron in phonon field

TL;DR

This paper derives an exact propagator for an electron interacting with phonons, revealing that energy exchange uncertainties induce flicker noise in the electron's diffusion rate, beyond standard golden rule approximations.

Contribution

It provides an exact propagator for electron-phonon interactions that accounts for flicker noise effects beyond Fermi's golden rule approximation.

Findings

Uncertainties in energy exchanges cause flicker fluctuations in diffusion rates.

The approach extends beyond the simplest Fermi's golden rule approximation.

Reveals fundamental noise characteristics in electron transport due to phonon interactions.

Abstract

Exact propagator of density matrix of particle (electron) under influence of thermal vibrations of its medium (phonons) is treated in simplest approximation beyond the Fermi's golden rule. It is shown that uncertainties \,$\sim\h/t$\, in energy exchanges with the medium give rise to flicker fluctuations in rate of diffusion (diffusivity) of the particle

Equations

$$H_e=\frac{p^2}{2m},H_{ph}=\oint \hslash {\omega }_k{a}_k^{†}{a}_k,$$

$$H_{int}=\oint c_k(e^{ikr}{a}_k+e^{-ikr}{a}_k^{†}),$$

$$\rho (t,X,Y)=⟨X+Y/2|\rho (t)|X-Y/2⟩,$$

$${\mathrm{Tr}}_{ph}\{\exp [{\int }_k{z}_{-}(k)a_k{e}^{ikr{}^{\prime }}]\times$$

$$\times ⟨r{}^{\prime }|{\rho }_{e-ph}(t)|r{}^{\prime \prime }⟩\exp [{\int }_k{z}_{+}(k)a_k^{†}{e}^{-ikr{}^{\prime \prime }}]\},$$

$$\rho (t)=\hat{\mathrm{S}}(t)\rho (0)\equiv$$

$$\equiv [\exp \{t(-\hat{\mathrm{V}}{\nabla }_{\mathrm{X}}+\hat{\mathcal{L}}(z))\}\rho (0)]{|}_{z=0}$$

$$\hat{\mathcal{L}}(z)=i\sum _{+,-}\pm {\int }_k[({\omega }_k-k\hat{\mathrm{V}})z_{\pm }(k)\frac{\delta }{\delta z_{\pm }(k)}-$$

$$-\frac{c_k}{\hslash }{B}_k(Y)z_{\pm }(k)+\frac{c_k}{(2\pi {)}^3\hslash }{A}_k(Y)\frac{\delta }{\delta z_{\pm }(k)}]$$

$$B_k(Y)\equiv e^{ikY/2}{N}_k-e^{-ikY/2}(N_k+1),$$

$$A_k(Y)\equiv e^{ikY/2}-e^{-ikY/2},$$

$$\hat{\mathrm{S}}(t)=\hat{\mathcal{T}}\exp \{{\int }_0^t[-\hat{\mathrm{V}}(t{}^{\prime }){\nabla }_{\mathrm{X}}]dt{}^{\prime }+$$

$$+{\int }_0^t{\int }_0^{t_1}\hat{\mathrm{K}}[t_1,t_2,Y(\cdot ),\hat{\mathrm{V}}(\cdot )]d{t}_{2}d{t}_1\},$$

$$\hat{\mathrm{K}}[t_1,t_2,Y(\cdot ),\hat{\mathrm{V}}(\cdot )]=\oint \frac{|c_k{|}^2}{{\hslash }^2}{A}_k(Y(t_1))\times$$

$$\times 2\cos \{{\int }_{t_2}^{t_1}[{\omega }_k-k\hat{\mathrm{V}}(t{}^{\prime })]dt{}^{\prime }\}{B}_k(Y(t_2)).$$

$$\hat{\mathrm{S}}(t)\Rightarrow {\hat{\mathrm{S}}}_{1/t}(t)\equiv \exp \{t[-\hat{\mathrm{V}}{\nabla }_{\mathrm{X}}+{\hat{\mathcal{K}}}_{1/t}]\},$$

$${\hat{\mathcal{K}}}_{1/t}\equiv \frac{1}{t}{\int }_0^t{\int }_0^{t_1}\hat{\mathrm{K}}[t_1,t_2,Y,\hat{\mathrm{V}}]d{t}_{2}d{t}_1=$$

$$=\oint \frac{|c_k{|}^2}{{\hslash }^2}{A}_k(Y)\frac{2[1-\cos [t({\omega }_k-k\hat{\mathrm{V}})]]}{({\omega }_k-k\hat{\mathrm{V}}{)}^{2}t}{B}_k(Y).$$

$${\delta }_{\nu }(\omega )=\frac{2\nu {\sin }^2(\omega /2\nu )}{\pi {\omega }^2}\approx \frac{1}{\pi }\frac{\nu }{{\nu }^2+{\omega }^2}$$

$${\hat{\mathcal{K}}}_{\nu }=\oint \frac{2\pi |c_k{|}^2}{{\hslash }^2}{A}_k(\hat{\mathrm{Y}}){\delta }_{\nu }({\omega }_k-kV)B_k(\hat{\mathrm{Y}}).$$

$${\hat{\mathcal{K}}}_{\nu }\rho (V)={\int }_{V^{\prime }}[w_{\nu }(V|V^{\prime })\rho (V^{\prime })-w_{\nu }(V^{\prime }|V)\rho (V)]$$

$$w_{\nu }(V|V^{\prime })=\frac{m^3}{(2\pi {)}^2{\hslash }^4}|c_{m(V-V{}^{\prime })/\hslash }{|}^2\times$$

$$\times \{N_{m(V-V^{\prime })/\hslash }{\delta }_{\hslash \nu }(\hslash {\omega }_{m(V-V^{\prime })/\hslash }-E+E^{\prime })+$$

$$+[N_{m(V^{\prime }-V)/\hslash }+1]{\delta }_{\hslash \nu }(\hslash {\omega }_{m(V^{\prime }-V)/\hslash }+E-E^{\prime })\}$$

$$\hat{\mathrm{S}}(t)\Rightarrow {\hat{\mathrm{S}}}_0(t)\equiv \exp \{t[-V{\nabla }_{\mathrm{X}}+{\hat{\mathcal{K}}}_0]\},$$

$${\hat{\mathcal{K}}}_0\equiv \oint \frac{2\pi |c_k{|}^2}{{\hslash }^2}{A}_k(\hat{\mathrm{Y}})\delta ({\omega }_k-kV)B_k(\hat{\mathrm{Y}}),$$

$${\hat{\mathcal{K}}}_0{\rho }_0(V)=0,{\rho }_0(V)\propto \exp (-m{V}^2/2T).$$

$$\delta ({\omega }_k-kV)B_k(\hat{\mathrm{Y}}){\rho }_0(V)=0.$$

$${\hat{\mathcal{K}}}_{\nu }{\rho }_{\nu }(V)=0(\nu =1/t),$$

$${\hat{\mathcal{K}}}_{\nu }{\rho }_0\approx -\nu \oint \frac{4|c_k{|}^2}{{\hslash }^2}{N}_k\exp [\frac{\hslash {\omega }_k}{2T}-\frac{(\hslash k{)}^2}{8mT}]\times$$

$$\times A_k(\hat{\mathrm{Y}})\frac{\sinh [\hslash ({\omega }_k-kV)/2T]}{({\omega }_k-kV{)}^2}{\rho }_0(V)$$

$${\rho }_{1/t}(V)-{\rho }_0(V)\propto \frac{\tau (V)}{t}{\rho }_0(V)$$

$$\displaystyle\begin{array}[]{c}\operatorname{\widehat{S}}_{1/t}(t)\,\rho_{in}(V)-\rho_{0}(V)=\operatorname{\widehat{S}}_{1/t}(t)\,[\rho_{in}(V)-\rho_{0}(V)]\,+\\ +\,[\,1-\operatorname{\widehat{S}}_{1/t}(t)]\,[\rho_{1/t}(V)-\rho_{0}(V)]\end{array}$$

$$\displaystyle\begin{array}[]{c}\operatorname{\widehat{\mathcal{K}}}_{\nu}\,R(V)=-\,\operatorname{\lambda}\,R(V)\,\,\,,\end{array}$$

$$\lesssim {\lambda }_{\ast }(\nu )=\nu \ln \frac{1}{{\tau }_0\nu }=\frac{1}{t}\ln \frac{t}{{\tau }_0}$$

$$\displaystyle\begin{array}[]{c}w_{\nu}(V|V\operatorname{{}^{\prime}})\,\rho_{0}(V\operatorname{{}^{\prime}})\Rightarrow\frac{1}{2}[\,w_{\nu}(V|V\operatorname{{}^{\prime}})\,\rho_{0}(V\operatorname{{}^{\prime}})\,+\\ +\,\rho_{0}(-V)\,w_{\nu}(-V\operatorname{{}^{\prime}}|-V)]\Rightarrow w_{\nu}(V|V\operatorname{{}^{\prime}})\,\rho_{0}(V\operatorname{{}^{\prime}})\end{array}$$

$$\displaystyle\begin{array}[]{c}\operatorname{\widehat{\mathcal{K}}}_{\nu}\,\rho_{0}(V)=\rho_{0}(-V)\,\operatorname{\widehat{\mathcal{K}}}_{\nu}^{\dagger}\,\,,\end{array}$$

Full text

\PACS

05.30.-d, 05.40.-a, 05.60.-k, 71.38.-k

1. One of actual themes of statistical mechanics is effects lost under its standard kinetic coarse-graining, and first of all thermal 1/f noise (see [1]-[12]) thus turned into a mystery like “new phlogiston”. In particular, quantum kinetics essentially exploits the “Fermi’s golden rule” [13] to break continuous interactions of particles and quanta to mutually decoherent events. Corresponding theory’s deficiencies by their nature are insensible to intensity/weakness and details of interactions and therefore may be learned by examples of model mechanical systems of many particle (degrees of freedom).

2. In this respect it is principally interesting to consider  a particle (“electron”) in potential field of vibrations of surrounding medium (“phonons”)  with simple Hamiltonian  $H=H_{e}+H_{ph}+H_{int}$  where

[TABLE]

$\oint\dots\,\equiv\int\dots\,d^{3}k/(2\pi)^{3}$ ,  $c_{k}=c_{k}^{*}$ ,  $|c_{k}|=c(|k|)$  and  $\operatorname{\omega}_{k}\hm=\operatorname{\omega}(|k|)>0$ . There various phonon modes (PM) do not interact one with another, but this is unimportant for the electron (e.) since variety of modes is innumerably infinite and hardly e. could exchange momentum-energy with some of them more than once. Therefore if PM occupation statistics initially (let at  $t=0$) is equilibrium then from e.’s viewpoint it henceforth will be the same.

We are interested in statistics of fluctuations and relaxation of of e.’s velocity  $V=p/m$  and its “Brownian walk” in the space. In [15] it was already pointed out that the walk statistics must be radically non-Gaussian by containing “flicker” fluctuations of e.’s mobility, since their absence would be incompatible with unitarity of evolution of the system (On 1/f-noise of electron in phonon field) that is with that spectrum of its Liouville (von Neumann) operator is purely imaginary. Now we want to go from this “existence theorem” and exact equations for full system’s density matrix (DM)  $\rho_{e-ph}(t)$  and marginal e.’s DM  $\rho(t)\hm=\operatorname{Tr}_{ph}\,\rho_{e-ph}(t)$  to an approximation being as simple as possible but able to catch the flicker noise.

3.  With this purpose it is comfortable to describe e. in the Wigner representation in terms of

[TABLE]

phonons in the coherent states representation, and them together via characteristic functional of electron-phonon (e.-ph.) correlations

[TABLE]

where  $r\operatorname{{}^{\prime}}=X+Y/2,\,r\operatorname{{}^{\prime\!\prime}}=X-Y/2$  and  $\int_{k}\dots\,=\int\dots\,d^{3}k$ . With its means in [14, 15] (our “supplement materials”) electron propagator (EP) was obtained and expressed by

[TABLE]

with e.’s velocity operator  $\operatorname{\widehat{V}}\hm=-i\,(\operatorname{\hbar}/m)\operatorname{\nabla_{\!Y}}$  and the Liouville operator in the form

[TABLE]

with correlations birth-annihilation amplitudes

[TABLE]

where  $N_{k}=[\exp{(\operatorname{\hbar}\omega_{k}/T)}-1]^{-1}$ .

4.  Let us write out EP  $\operatorname{\widehat{S}}(t)$  evidently. Since operator  $\operatorname{\widehat{\mathcal{L}}}(z)$  is of first-order in respect to differentiations  $\delta/\delta z_{\pm}(k)$  the expression (3) would be easy calculable if this was not prevented by non-commutativity of  $\operatorname{\widehat{V}}$  and  $Y$  in  $\operatorname{\widehat{\mathcal{L}}}(z)$ . We can avoid this difficulty by assigning to  $\operatorname{\widehat{V}}$  and  $Y$  auxiliary time arguments and making the exponential chronologically ordered by it. Keeping an eye on it gives possibility to forget the non-commutativity for a time and easily transform the exponential to the normal form in respect to  $\delta/\delta z_{\pm}(k)$  and  $z_{\pm}(k)$ . Thus one finds

[TABLE]

where  $\operatorname{\widehat{\mathcal{T}}}$  symbolizes chronological ordering and

[TABLE]

Of course, after  $\operatorname{\widehat{\mathcal{T}}}$ ’s action the auxiliary arguments are removed.

5.  Ignoring  $\operatorname{\widehat{\mathcal{T}}}$  we come to approximation

[TABLE]

Hear  $\operatorname{\widehat{\mathcal{K}}}_{1/t}$  appears as fictitiously time-local kinetic operator (KO). In fact it is “pseudo-kinetic” (PKO) since it depends on the whole observation time. Introducing  $\nu=1/t$  and fuzzy “quasi-delta-function” (QDF)

[TABLE]

and going from  $\operatorname{\widehat{V}}$  and  $Y$  to Wigner-conjugated pair  $V$  and  $\operatorname{\widehat{Y}}\hm=i\,(\operatorname{\hbar}/m)\operatorname{\nabla_{\!V}}$  let us write shortly

[TABLE]

Unfolding this expression, one can make sure that

[TABLE]

with  $\int_{V}\dots=\int\dots\,d^{3}V$  and

[TABLE]

with  $E\equiv mV^{2}/2$  and  $E\operatorname{{}^{\prime}}\equiv mV^{\prime 2}/2$ . According to (12) our PKO looks like Kolmogorov-Chapman type operators in formalism (“master equations” [13]) for Marcov stochastic processes. However, our transition rates (“probabilities”) (13) are time dependent through the QDF’s width  $\nu=1/t$  in such way that EP (8) is not representable as solution to a first-order time differential equation, which makes the theory strongly non-Marcovian.

6.  If applying the “golden rule”, that is replacing QDF in (11)-(13) by the usual Dirac’s delta-function (DDF)  $\delta(\operatorname{\omega})$  and thus turning PKO into KO,

[TABLE]

one comes to to the conventional (Marcovian) kinetics’ approximation. KO (14) is [14] mere “one-electron version” of the e.-ph. collision integral [16]. It produces the Maxwell distribution as equilibrium one:

[TABLE]

It follows from easy justifiable equalities

[TABLE]

Their validity for any PM separately says about “detail balance” of e.-ph. collisions.

One may believe that at suitable parameters of (On 1/f-noise of electron in phonon field) (may be with several PM branches [14])  $\operatorname{\widehat{\mathcal{K}}}_{0}$  ensures fast relaxation of e.’s velocity distributions and correlators. But this is certainly impossible in the more “high” approximation (8)-(13).

7.  Indeed, after returning QDF to its place instead of DDF the equalities (16) no more are valid, i.e. detail balance (DB) is not completely observed in finite time frameworks. Therefore a distribution  $\rho_{\nu}(V)$  stationary in respect to PKO (11)-(13) and satisfying

[TABLE]

does not coincide with the Maxwellian  $\rho_{0}(V)$ . Their difference is determined by “wings” of QDF. From (11) we have

[TABLE]

with integral in the principal value sense. Then from (17) in the form  $\operatorname{\widehat{\mathcal{K}}}_{\nu}[\rho_{\nu}-\rho_{0}]\hm+\operatorname{\widehat{\mathcal{K}}}_{\nu}\rho_{0}\hm=0$  we see that difference  $\rho_{\nu}-\rho_{0}$  decreases with time not faster than

[TABLE]

with some function  $\tau(V)$ .

It shows that relaxation of arbitrary initial velocity distribution  $\rho_{in}(V)$  to equilibrium  $\rho_{0}(V)$  may go by very slow - time non-integrable - law. However, origin of such behavior is not conservation of some physical quantity but DB’s violation. Hence a slow scenario is not obligatory and there must exist initial conditions  $\rho_{in}(V)$  promoting DB and leading to equilibrium in a fast (integrable) way. In such case in identity

[TABLE]

the first right-hand term compensates “long tail” of the second coming from (19).

This means that among eigen-values of PKO  $\operatorname{\widehat{\mathcal{K}}}_{\nu}\hm=\operatorname{\widehat{\mathcal{K}}}_{1/t}$ , in solutions of problem

[TABLE]

there are small eigen-values (e.v.)

[TABLE]

with some  $\tau_{0}$  (so that  $\exp{[-\operatorname{\lambda}_{*}(\nu)\,t]}\propto 1/t$ ).

Such e.v.s decreasing as QDF’s width just are introducing to e.’s motion statistics “flicker” (unrestrictedly low-frequency) fluctuations. Next consider how it works and then discuss mathematical origin of such e.v.

8.  Being interested namely in flicker noise it is reasonable to neglect DB violation and thus difference between  $\rho_{1/t}(V)$  and  $\rho_{0}(V)$ . For that we may make symmetrization

[TABLE]

thus subjecting PKO to operator equality

[TABLE]

where  ${\dagger}$  is symbol of conjugation - transposition in the Sturm-Liouville sense plus inversion of velocity sign (so that operator  $\rho^{-1/2}_{0}\operatorname{\widehat{\mathcal{K}}}_{\nu}\rho^{1/2}_{0}$  is self-conjugated). This property guarantees time-local DB observance [13, 17, 18]. Now  $\operatorname{\widehat{\mathcal{K}}}_{\nu}\rho_{0}(V)=0$  although QDF keeps its place.

9.  Further recall that EP has argument  $\operatorname{\nabla_{\!X}}$  and

[TABLE]

where  $M_{n}(t,\nu)$  are statistical moments of displacement (path) of e. during time  $t$  under governing by given kinetic operator of velocity [15]. It is taken into account that e.’s walk is spherically symmetric since gas of phonons is isotropic. Therefore eigen-functions (e.f.), or eigen-states (e.s.),  $R(V)$  in (24) can be sorted to odd and even. Let they be enumerated by indices “1” and “’2’ respectively except equilibrium indexed with “0”.

Clearly, operator  $V$  as perturbation of  $\operatorname{\widehat{\mathcal{K}}}_{\nu}$  in (31) induces transitions only between its e.s.s with opposite parities. Hence in the second order of expansion over  $\operatorname{\nabla_{\!X}}$  on the left in (31) we can write, with use of standard notations of perturbation theory (but leaving velocity’s vector indices “in mind”),

[TABLE]

where  $D$  is diffusion coefficient (diffusivity) of e.

We supposed that sets of matrix elements  $V_{01},\,V_{10}$  and e.v.s  $\operatorname{\lambda}_{1}$  are such that  $D$  is finite, i.e. e.’s diffusion is “simple” (not “anomalous”).

10.  Then at correspondingly large  $t$  the fourth order of the expansion tends to form

[TABLE]

The triple sum here represents fourth-order cumulant of the e.’s path,  $C_{4}=M_{4}-3M_{2}^{2}$ , characterizing, in comparison to  $M_{2}^{2}$ , degree of non-Gaussianity of of statistics of the walk [1, 3, 7, 11, 18].

If in (33) sets of even e.s.s of operator  $\operatorname{\widehat{\mathcal{K}}}_{\nu}$  and their e.v.s  $\operatorname{\lambda}_{2}$  were fixed, as for KO (14), then we would expected similar to (32) asymptotic  $C_{4}\propto t$  giving  $C_{4}/M_{2}^{2}\propto 1/t$  and asymptotically Gaussian walk. But our set  $\operatorname{\lambda}_{2}$  is varying with observation time together with parameter  $\nu=1/t$ , moreover, it is essentially varying as far as includes small  $\nu$ -dependent e.v.s (25) (obviously belonging to even e.s.s in view of even parity of functions (18)-(19)). Because of them the fourth-order cumulant grows at large  $t$  not slower than

[TABLE]

while ratio  $C_{4}/M_{2}^{2}$  decreases not faster than logarithmically, and the walk of e. is non-Gaussian even asymptotically. It is convenient to describe such statistics in the language of diffusivity fluctuations with effective correlation function  $K_{D}(t)\hm\approx\operatorname{\partial}_{t}^{2}C_{4}/4!$  [1, 2, 3, 7, 9, 10, 15], that is in our case  $K_{D}(t)\hm\propto D^{2}/\ln{(t/\tau_{0})}$  at  $t\gg\tau_{0}$  if not slower.

Notice that inversely logarithmic tail of  $K_{D}(t)$ , in company with realistic estimate of its weight, naturally appears in statistical phenomenology of e. Brownian motion suggested in [2, 3] (see also [1, 7, 11]). Though generally (33) may produce also other dependencies more close to

[TABLE]

giving “quasi-static” fluctuations of diffusivity.

11.  Let us turn to the question how PKO  $\operatorname{\widehat{\mathcal{K}}}_{\nu}$ ’s small e.v.s of order of  $\nu$  can be explained from pure mathematical point of view. The answer comes with help of the perturbation theory (PT) of linear operators [19] if we treat PKO  $\operatorname{\widehat{\mathcal{K}}}_{\nu}$  as result of perturbation of KO  $\operatorname{\widehat{\mathcal{K}}}_{0}$  and pay attention to those principal circumstance that all e.v.s of operator  $\operatorname{\widehat{\mathcal{K}}}_{0}$  are multiply degenerated. Indeed, DDF in (13) in place of QDF allows only transitions with precise conservation of sum  $H_{e}+H_{ph}$  of own energies of e. and PMs. Therefore one and the same e.v.  $\operatorname{\lambda}$  in (24) with  $\nu=0$  can be associated with different e.f.s corresponding to different values of the “integral of motion”  $H_{e}+H_{ph}$  and specified, for instance, by their dependence on  $E\hm=mV^{2}/2$  on interval  $0\hm\leq E\hm<E_{0}$  where  $E_{0}\hm\equiv\max\,\operatorname{\hbar}\operatorname{\omega}_{k}$  is maximal phonon energy. Visual illustration is presented by special case of “ideally optical” phonons with  $\operatorname{\omega}_{k}\hm=\operatorname{\omega}_{0}\hm\equiv\,$const . There  $E_{0}=\operatorname{\hbar}\operatorname{\omega}_{0}\,$ and  $\operatorname{\widehat{\mathcal{K}}}_{0}$  connects only equidistantly by  $E_{0}$  separated points of  $E$  axis. Hence if  $R(V)$  is a solution of (24) with  $\nu=0$  then  $\Pi(E/E_{0})\,R(V)$ , where  $\Pi(x)=\Pi(x\pm 1)$  is arbitrary periodic function, also is solution of the same equation with same e.v., so that the degeneration is at least countably-multiple.

Perturbation  $\operatorname{\widehat{\mathcal{K}}}_{\nu}-\operatorname{\widehat{\mathcal{K}}}_{0}$  owing to QDF introduces “non-quantized” transitions from any point of e.’s energy axis to all other its points and thus transitions between different  $H_{e}+H_{ph}$  values. Physically it approaches the e.’s motion picture to reality where not  $H_{e}+H_{ph}$  but the full energy  $H_{e}+H_{ph}+H_{int}$  is conserved. Formally according to PT it must lead to splitting of each e.v. to many different values  $\propto\nu$ . As the result, even a discrete e.v.s spectrum of parent operator transforms into arbitrarily dense one in whole permissible region,  $\,0\hm\leq\operatorname{\lambda}\hm<\operatorname{\lambda}_{max}\hm\approx\int_{V}\int_{V\operatorname{{}^{\prime}}}w_{0}(V\operatorname{{}^{\prime}}|V)\,\rho_{0}(V)$  in our case. At that the zero e.v. connected to equilibrium serves as source of most interesting for us e.v.s  $\sim\nu$ .

12.  We can ascertain the said by the example of optical phonons. For complete description of perturbed spectrum at lowest order of PT it would be necessary, for some full set of linearly independent functions  $\Pi(x)$ , to compose and solve corresponding secular equation. But for estimates of character and value of perturbation it is sufficient to apply simple formula of first-order PT for shift  $\operatorname{\varDelta}\operatorname{\lambda}\hm=\operatorname{\lambda}\hm-\operatorname{\lambda}_{0}$  of e.v.  $\operatorname{\lambda}_{0}$  as if it was non-degenerated:

[TABLE]

where  $\Pi=\Pi(E/E_{0})$ , and  $F_{0}(V)\,\rho_{0}(V)=R_{0}(V)$  is most smooth of all e.f.s of  $\operatorname{\widehat{\mathcal{K}}}_{0}$  connected to  $\operatorname{\lambda}_{0}$  (in view of (30)  $\operatorname{\widehat{\mathcal{K}}}_{0}^{\dagger}F_{0}\hm=-\operatorname{\lambda}_{0}F_{0}$ ). Choosing here  $\Pi(x)\hm=\exp{(2\pi inx)}$  (with integer $n$ ) and treating QDF as sharp energy function as compared to  $R_{0}(V)$ , one can obtain

[TABLE]

where  $\tau_{0}\equiv 2\pi\operatorname{\hbar}/E_{0}$ . Taking alternatively

[TABLE]

with  $x_{0}\leq 1$ , assuming for simplicity that  $\operatorname{\hbar}\nu/E_{0}\ll x_{0}$  and  $\operatorname{\lambda}_{0}$  lies at beginning of spectrum, and rejecting insignificant details, one finds

[TABLE]

These test calculations quite highlight tendency of the perturbation to expand  $\operatorname{\lambda}_{0}$  into ascending from  $\operatorname{\lambda}_{0}$  sequence of e.v.s with step  $\sim\operatorname{\lambda}_{max}\tau_{0}\,\nu\,\ln\,({\tt const}/\tau_{0}\nu)$  (non-analiticity of expressions (37)-(41) in respect to variable  $\nu$  is caused by that  $\nu=0$  is peculiar point of  $\operatorname{\widehat{\mathcal{K}}}_{\nu}$ ). For us of most importance is perturbation of equilibrium state where  $F_{0}(V)\hm=$ const ,  $\operatorname{\lambda}_{0}=0$  and  $\operatorname{\lambda}\hm=\operatorname{\varDelta}\operatorname{\lambda}$ . Formulae (37)-(41) show detachment of many small e.v.s from it including e.v.s  $\approx\operatorname{\lambda}_{*}(\nu)$  with  $\operatorname{\lambda}_{*}(\nu)$  from (25).

13.  Consequently, among characteristic relaxation times of correlators and distributions of e.’s velocity there appear unbounded large ones  $\sim\nu^{-1}=t$ . Of course, in fact they only enter in definite types of correlators, since the perturbation does not destroy statistical isotropy of phonon field, and all e.s.s related to the equilibrium’s degeneration and its removing are spherically symmetric along with  $\rho_{0}(V)$  (and with (18) and (19)). Therefore they do not contribute to (32) or to sums over  $1$  and  $1\operatorname{{}^{\prime}}$  in (33) but contribute to sum over  $2$  in (33), moreover, this contribution becomes dominating with time. In the light of our estimates it is clear that the result is in between (34) and (35).

However, determination of numeric details in asymptotic like (34) or (35) requires more investigation of properties of  $\operatorname{\widehat{\mathcal{K}}}_{0}$  and similar operators (not familiar, to the best of our knowledge, even to mathematicians). In the past the physical kinetics had no need in it but novadays it becomes practically significant problem.

14.  To resume, we suggested relatively simple approximation, (8)-(9), of quantum dynamics of electron in phonon surroundings in terms of pseudo-kinetic operator (PKO) (9)-(13) which differs from operators of convenient kinetics by evident taking account of finiteness of duration of real experiments. The PKO neglects many-phonon collisions but instead visually reveals uncertainty of electron’s rate (coefficient) of diffusion (mobility) in the form of its flicker fluctuations (1/f-noise).

Common sense prompts [1, 2, 3, 11] that related unbounded from above correlation times in essence signify absence of correlations in these fluctuations. The seeming paradox is resolved, - as was underlined in [4, 5] (see also [7, 18, 11]), - at the expense of unusual statistics with Cauchy type probability distributions.

Curiously, in the framework of PKO approximation the Cauchy distribution (CD) appears already in PKO itself in the form of quasi-delta-function (on the right in (10)). Due to it the PKO acts as if phonon energy undergoes random deviations from dispersion law obeying CD with width  $\operatorname{\hbar}\nu\hm=\operatorname{\hbar}/t$ . At that those CD’s peculiarity caused by its inversely quadratic tail is important that under its convolutions (reproducing CD) summation of widths of distributions takes place, i.e. independent (in the sense of probability theory) random quantities, each subject to CD, are summed like fully commonly dependent ones, which means their full indifference to averaging. Hence in the course of many collisions the random deviations of phonon energies do not suppress one another but produce effective deviation with amplitude (CD’ width)  $\approx n(t)\,\operatorname{\hbar}/t\hm\approx\operatorname{\hbar}\operatorname{\lambda}_{max}$  which does not decrease with time ( $n(t)\hm\approx\operatorname{\lambda}_{max}t$  is typical number of collisions during time  $t$ ). This view onto PKO can be useful for numeric estimates and models of 1/f-noise, including generalizations of the PKO approximation to other systems.