Time-energy uncertainty as cause of thermal flicker noise

TL;DR

This paper proposes that quantum energy uncertainty in transition states causes persistent correlations and flicker noise in thermal systems, affecting particle mobility and transition rates.

Contribution

It introduces a quantum-based explanation for flicker noise by linking energy uncertainty to non-decaying correlations in thermal processes.

Findings

Quantum energy uncertainty leads to non-decaying correlations.

Flicker noise arises from energy fluctuations in intermediate states.

Particle mobility can be suppressed due to these quantum effects.

Abstract

It is shown that if kinetics of quantum transitions takes account of energy uncertainty of intermediate states, then it creates non-decaying correlations and non-averagable (flicker) fluctuations in the energy as well as in rates of transitions-induced irreversible processes, in particular, flicker noise or maybe suppression of mobility (rate of wandering) of particle interacting with thermally equilibrium medium

Equations

$$\frac{2[1-\cos (\tau E_{21}/\hslash )]}{E_{21}^2}|{\Phi }_{21}{|}^2\Rightarrow \frac{2\pi \tau }{\hslash }|{\Phi }_{21}{|}^2\delta (E_{21}).$$

$$2\sum _2[1-\cos (\tau E_{21}/\hslash )]|{\Phi }_{21}{|}^2$$

$$\frac{\partial \rho (t)}{\partial t}={\int }_0^t\hat{\mathcal{Q}}(t-t^{\prime })\rho (t^{\prime })d{t}^{\prime },$$

$$(\hat{\mathcal{Q}}(\tau )\rho {)}_1=\sum _2\frac{2|{\Phi }_{12}{|}^2}{{\hslash }^2}\cos \frac{\tau E_{12}}{\hslash }[{\rho }_2-{\rho }_1].$$

$$\tilde{\rho }(z)=[z-{\tilde{\mathcal{Q}}}_z{]}^{-1}\rho (0)$$

$$({\tilde{\mathcal{Q}}}_{z}f{)}_1=\sum _2\frac{2\pi }{\hslash }|{\Phi }_{12}{|}^2{\Delta }_z(E_{12})[f_2-f_1],$$

$${\Delta }_z(ϵ)=\frac{1}{\pi }\frac{\hslash z}{h^2{z}^2+{ϵ}^2}.$$

$$({\tilde{\mathcal{Q}}}_{z}f{)}_1\to z\sum _2\frac{2\pi }{\hslash }|{\Phi }_{12}{|}^2\frac{f(E_2)-f(E_1)}{(E_2-E_1{)}^2}\propto z$$

$${\tilde{\mathcal{Q}}}_z\to {\tilde{\mathcal{Q}}}_{+0}+\hat{\mathrm{\Pi }}{\tilde{\mathcal{Q}}}_z\hat{\mathrm{\Pi }},$$

$$(\hat{\mathrm{\Pi }}f{)}_1\text{hm}=N^{-1}(E_1)\sum _2\delta (E_1-E_2)f_2,$$

$$W(\infty ,E)=N(E)\underset{z\to 0}{\lim }z\tilde{\rho }(z)=\frac{1}{1+\hat{\chi }}\delta (E-E_0),$$

$$\hat{\chi }f(E)=-\frac{\hslash }{\pi }\int \frac{G(E,E{}^{\prime })}{N(E{}^{\prime })}\frac{f(E{}^{\prime })-f(E)}{(E{}^{\prime }-E{)}^2}dE{}^{\prime }$$

$$G(E,E{}^{\prime })=\frac{2\pi }{\hslash }\sum _{1,2}\delta (E_1-E)|{\Phi }_{12}{|}^2\delta (E_2-E{}^{\prime }).$$

$$\Xi (t,ik)=⟨e^{ikR(t)}⟩=\mathrm{Tr}{e}^{t(\hat{\mathcal{L}}+ik{\hat{\mathcal{J}}}_V)}\varrho (0),$$

$$\frac{\partial \Xi (t,ik)}{\partial t}=ikV\Xi (t,ik)+{\int }_0^t\hat{\mathcal{Q}}(t-t^{\prime },ik)\Xi (t{}^{\prime },ik)d{t}^{\prime }$$

$$\tilde{\Xi }(z,ik)=\mathrm{Tr}[z-ikV-{\tilde{\mathcal{Q}}}_z(ik){]}^{-1}\rho (0),$$

$${\Delta }_z(E_{12})\Rightarrow {\Delta }_{z-ik(V_1+V_2)/2}(E_{12})$$

$${\tilde{\mathcal{Q}}}_z(ik)-{\tilde{\mathcal{Q}}}_z(0)\text{hm}\equiv ik{\tilde{\mathcal{V}}}_z(ik)$$

$${\tilde{\mathcal{D}}}_z(ik)=\hat{\mathrm{\Pi }}(V+{\tilde{\mathcal{V}}}_z(ik))\hat{\mathrm{\Pi }}{}^{\prime }\times$$

$$\times \{z-ikV-\hat{\mathrm{\Pi }}{}^{\prime }{\tilde{\mathcal{Q}}}_z(ik)\hat{\mathrm{\Pi }}{}^{\prime }{\}}^{-1}\hat{\mathrm{\Pi }}{}^{\prime }(V+{\tilde{\mathcal{V}}}_z(ik))\hat{\mathrm{\Pi }}$$

$$\tilde{\Xi }(z,ik)\to \mathrm{Tr}\{z+k^2\tilde{\mathcal{D}}(z,ik)+O(z^2){\}}^{-1}\hat{\mathrm{\Pi }}\rho (0)$$

$$\tilde{\Xi }(z,ik)\to \frac{1}{z+k^2\mathrm{Tr}{\tilde{\mathcal{D}}}_z(ik)\rho (0)}\equiv \frac{1}{z+k^{2}D(z,ik)}$$

$$D(z,ik)\to {\tau }_0\mathrm{Tr}V[V+{\tilde{\mathcal{V}}}_z(ik)\hat{\mathrm{\Pi }}]\rho (0),$$

$$({\tilde{\mathcal{V}}}_z(ik)f{)}_1=\sum _{2,\pm }\frac{V_{12}|{\Phi }_{12}{|}^2[f_2-f_1]}{(\hslash z\pm i{E}_{12}{)}^2+(\hslash k{V}_{12}{)}^2}$$

$$\sum _1{\rho }_{V1}(0)=\sum _1\frac{\delta ({\Sigma }_1+E(V)-E_0)}{N(E_0)}=W_0(V)\equiv$$

$$\equiv \frac{N_{th}(E_0-E(V))}{N(E_0)}=\frac{N_{th}(E_0)}{N(E_0)}\exp [-E(V)/T],$$

$$D(z,ik)\to D_0+{\tau }_0\iint \int \frac{\hslash V(V+V{}^{\prime })}{8\pi }\times$$

$$\times \sum _{\pm }\frac{W(V,V{}^{\prime };ϵ)-W(V{}^{\prime },V;-ϵ)}{(\hslash z\pm iϵ{)}^2+(\hslash k(V+V{}^{\prime })/2{)}^2}$$

$$W(V,V{}^{\prime };ϵ)=\frac{2\pi }{\hslash }\sum _{1,2}\delta (E_0+ϵ-{\Sigma }_1-E(V))\times$$

$$\times |{\Phi }_{V1V{}^{\prime }2}{|}^2\frac{\delta (E_0-{\Sigma }_2-E(V{}^{\prime }))}{N(E_0)}$$

$${\tilde{\mathcal{Q}}}_{+0}f(V)=\int W(V,V{}^{\prime };0)\left[\frac{f(V{}^{\prime })}{W_0(V{}^{\prime })}-\frac{f(V)}{W_0(V)}\right]dV{}^{\prime }.$$

$$W(V,V{}^{\prime };ϵ)=w(V,V{}^{\prime };|E-ϵ-E{}^{\prime }|)\times$$

Full text

\PACS

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

1.  Effects of interactions in systems of many particles (degrees of freedom) usually are thought in terms of such random interaction events as quantum transitions. Then one has to prescribe them unambiguously definite personal probabilities. But this is impossible without applying of violence to exact equations of statistical mechanics. Frequently the violence is bringing in the non-stationary perturbation theory (NPT) [1] so called “Fermi golden rule” (GR) [2, 3], that is replacement

[TABLE]

Here, for a system with Hamiltonian  $H=H_{0}+\Phi$ ,  $\Phi_{21}$  are matrix elements (ME) of interections from viewpoint of orthogonal basis formed by eigen-states of unperturbed, i.e. interactionless, Hamiltonian  $H_{0}$  with eigen-values  $\operatorname{\mathcal{E}}_{1}$  and  $\operatorname{\mathcal{E}}_{2}$ , and  $\operatorname{\mathcal{E}}_{21}=\operatorname{\mathcal{E}}_{2}-\operatorname{\mathcal{E}}_{1}$ . The GR makes transitions’ probabilities proportional to time of their expectation and thus encourages their treatment in the spirit of Marcovian stochastic processes [2]. At that one ignores fluctuations of “unperturbed” energy (UE)  $\operatorname{\mathcal{E}}$  of states what are passed in transitions’ sequences.

Meanwhile, the uncertainty principle based estimate of steps of these fluctuations,  $\operatorname{\mathcal{E}}_{21}\sim\operatorname{\hbar}/\tau$ , not at all points at their “vanishing smallness”, for their mean square

[TABLE]

has non-zero limit t  $\tau\rightarrow\infty$  and even may be infinitely large. This circumstance prompts tbat UE fluctuations neglected by kinetics in fact are flicker ones (i.e. possessing time-non-integrable correlations), which agrees [4, 5] with other results on statistics of mechanical thermal motion [6]-[12].

In reality, of course, that are fluctuations in transitions’ probabilities and rates of relaxation, friction, dissipation, diffusion and any other irreversible processes caused by interactions.

Let us show how one can reveal such fluctuations, taking in mind interactions between a “small subsystem” and large “thermostat”, so that  $H_{0}\hm=E(V)+H_{th}$ , for instance, between “Brownian” particle (BP) with  $E(V)=MV^{2}/2$  and thermodynamically equilibrium medium.

2.  In statistical mechanics all interactions are governed by the von Neumann equation for system’s density matrix (DM)  $\operatorname{\varrho}$ , while kinetics deals with DM’s diagonal, that is distribution  $\rho$  of probabilities  $\rho_{1}=\operatorname{\varrho}_{11}$  of the “unperturbed” states. If at  $t=0$  all non-diagonal MEs were zeros, then later

[TABLE]

where operator  $\operatorname{\widehat{\mathcal{Q}}}$  is presented by formula [5] which for a weak interaction (in the framework of second order of NPT) reduces to

[TABLE]

Using the Laplace transformation and marking its resultants and related objects with tilde, we have

[TABLE]

with operator  $\operatorname{\widetilde{\mathcal{Q}}}_{z}$  acting by formula

[TABLE]

where

[TABLE]

3.  Factor  $\Delta_{z}(\operatorname{\epsilon})$  determines  $\operatorname{\widetilde{\mathcal{Q}}}_{z}$ ’s properties at small  $z\rightarrow+0$  and hence  $\rho(t)$ ’s behavior at large time. For the first look,  $\Delta_{z\rightarrow+0\,}(\operatorname{\mathcal{E}}_{12})$  must do the same work as the delta-function  $\delta(\operatorname{\mathcal{E}}_{12})$  in GR and allows to write  $\operatorname{\widetilde{\mathcal{Q}}}_{+0}$  in (3) in place of  $\operatorname{\widetilde{\mathcal{Q}}}_{z}$ , with  $\operatorname{\widetilde{\mathcal{Q}}}_{+0}$  being the Marcovian probability evolution generator from usual kinetics. But a truth is more complex. If distribution  $f$  in (4) is only depending on UE of states, i.e.  $f_{1}\hm=f(\operatorname{\mathcal{E}}_{1})$ , then  $\operatorname{\widetilde{\mathcal{Q}}}_{+0}\,f=0$ , that is GR artificially “pins” UE to be constant. In fact, however,

[TABLE]

(treating the sun in the sense of principal value), thus showing that during a finite time UE always changes by a finite value, although in a slow way.

Consequently, it will be more right to write

[TABLE]

where  $\operatorname{\widehat{\Pi}}$  is operator of projecting onto functional space of quasi-equilibrium distributions, that is ones uniform on any constant UE hyper-surface:

[TABLE]

where   $\operatorname{\mathcal{N}}(\operatorname{\mathcal{E}})\hm=\sum_{1}\,\delta(\operatorname{\mathcal{E}}-\operatorname{\mathcal{E}}_{1})\,$  is density of states with given UE. The first term of (5) is responsible for fast relaxation to quasi-equilibrium, while second term represents slow relaxation, like diffusion, over UE axis. Because of the latter, if starting from  $\operatorname{\mathcal{E}}_{0}$ , UE with time achieves distribution,  $W(t,\operatorname{\mathcal{E}})\hm=\sum_{1}\,\delta(\operatorname{\mathcal{E}}-\operatorname{\mathcal{E}}_{1})\,\rho_{1}(t)$ , which “freezes”, or “solidifies”, at

[TABLE]

where operator  $\widehat{\chi}$  acts according to

[TABLE]

with density of transitions between different UEs

[TABLE]

Such the freezing means that UE’s fluctuations include infinitely long living correlations, even with initial conditions, that is [5] these are flicker fluctuations.

4.  To consider influence of these fluctuations onto the small subsystem, let it be BP, we will exploit characteristic function (ChF) of integral  $R(t)=\int_{0}^{t}V(t\operatorname{{}^{\prime}})\,dt\operatorname{{}^{\prime}}$ , i.e. BP’s path passed during the observation time. We define this ChF [13] basing on the orrespondence principle, in analogy with classical statistical mechanics, as

[TABLE]

where  $\operatorname{\widehat{\mathcal{L}}}$  and  $\operatorname{\widehat{\mathcal{J}}}_{V}$  are quantum Liouville super-operator and Jordan super-operator of symmetrized multiplication, respectively, $\operatorname{\widehat{\mathcal{J}}}_{V}\,A\,\hm\equiv(VA+AV)/2$ . Then, quite similarly to derivation [5] of (1)-(2) and (3), in the second-order NPT one finds

[TABLE]

with operator  $\operatorname{\widehat{\mathcal{Q}}}(\tau,ik)$  which differs from above  $\operatorname{\widehat{\mathcal{Q}}}(\tau)$  by product  $\cos{(\tau\operatorname{\mathcal{E}}_{12}/\operatorname{\hbar})}\,\exp{[ik\,(V_{1}+V_{2})\,\tau/2]}$  in place of cosine  $\cos{(\tau\operatorname{\mathcal{E}}_{12}/\operatorname{\hbar})}$ , with  $V_{1,\,2}$  denoting BP’s velocity values in  $H_{0}$ ’s eigen-states  $1,\,2$ . Thus

[TABLE]

where operator  $\operatorname{\widetilde{\mathcal{Q}}}_{z}(ik)$  differs from  $\operatorname{\widetilde{\mathcal{Q}}}_{z}$  by replacement

[TABLE]

in the sum inside it (so that  $\operatorname{\widetilde{\mathcal{Q}}}_{z}(0)\hm=\operatorname{\widetilde{\mathcal{Q}}}_{z}$ ). This is expression of interference between BP-medium interaction and BP’s motion.

For description of this interference let us write

[TABLE]

and introduce operator of (temp of) BP’s diffusion:

[TABLE]

with  $\operatorname{\widehat{\Pi}}\operatorname{{}^{\prime}}\equiv 1-\operatorname{\widehat{\Pi}}$ . The ChF reduces to it, when initial distribution is (quasi-) equilibrium, i.e.  $\operatorname{\widehat{\Pi}}\rho(0)\hm=\rho(0)$ , for example,  $\rho(0)\hm=\delta(\operatorname{\mathcal{E}}-\operatorname{\mathcal{E}}_{0})/\operatorname{\mathcal{N}}(\operatorname{\mathcal{E}}_{0})$ . Then

[TABLE]

under  $z\rightarrow 0$  with  $k$ -independent  $O(z^{2})$ . If, in addition, the interaction is uniform (invariant in respect to shifts) on UE axis, so that the above quantities  $G(\operatorname{\mathcal{E}},\operatorname{\mathcal{E}}\operatorname{{}^{\prime}})/\operatorname{\mathcal{N}}(\operatorname{\mathcal{E}}\operatorname{{}^{\prime}})$  and  $W(t,\operatorname{\mathcal{E}})$  in fact depend only on differences  $\operatorname{\mathcal{E}}\!-\!\operatorname{\mathcal{E}}\operatorname{{}^{\prime}}$  and  $\operatorname{\mathcal{E}}\!-\!\operatorname{\mathcal{E}}_{0}$ , then it is not hard to guess and prove that

[TABLE]

with  $\operatorname{\mathcal{D}}(z,ik)$  insensible to (quas-equilibrium)  $\rho(0)$ .

5.  Now we have to discuss function   $\operatorname{\mathcal{D}}(z,ik)\hm=\operatorname{Tr}\,\operatorname{\widetilde{\mathcal{D}}}_{z}(ik)\,\rho(0)$ . Operator  $\operatorname{\widehat{\Pi}}\operatorname{{}^{\prime}}\operatorname{\widetilde{\mathcal{Q}}}_{z}\operatorname{\widehat{\Pi}}\operatorname{{}^{\prime}}$  in (8) represents fast relaxation of BP’s velocity distribution to equilibrium one (balanced with medium). Presuming  $k$  and  $z$  to be infinitesimally small, it seems much reasonable to approximate contents of the braces in (8) by  $-\operatorname{\widetilde{\mathcal{Q}}}_{+0}$  and the latter, for simplicity, by characteristic eigen-value of  $-\operatorname{\widetilde{\mathcal{Q}}}_{+0}$ , i.e. characteristic rate (inverse time)  $g_{0}=1/\tau_{0}$  of the velocity relaxation. This roughening helps us to focus at much more important role of operator  $\operatorname{\widetilde{\mathcal{V}}}_{z}(ik)$  which mixes UE fluctuations to BP’s wandering. With taking into account that  $\operatorname{Tr}\,\operatorname{\widetilde{\mathcal{V}}}_{z}(ik)\,\dots\hm=0$  we have

[TABLE]

where action of  $\operatorname{\widetilde{\mathcal{V}}}_{z}(ik)$  can be presented, in view of obvious symmetries of interaction MEs, by formula

[TABLE]

with   $V_{12}=(V_{1}+V_{2})/2$ .

Next, it will be comfortable to separate BP’s velocity from the full system states’ indices  $1,\,2,\,\dots$  and instead of  $V_{1},\,V_{2},\,\dots$  write  $V,\,V\operatorname{{}^{\prime}},\,\dots$  while giving ciphers  $1,\,2,\,\dots$  to medium’s states (eigen-states of  $H_{th}$ ). Then  $\rho_{1}$  turns to  $\rho_{V1}$  and

[TABLE]

where  $\Sigma_{1}$  are energies of medium states,  $\operatorname{\mathcal{N}}_{th}(\Sigma)$  is their density,  $T\hm=[d\ln{\operatorname{\mathcal{N}}_{th}(\operatorname{\mathcal{E}})}/d\operatorname{\mathcal{E}}]^{-1}$  is temperature of micro-canonical ensemble [14] of medium states, and  $W_{0}(V)$  is Maxwellian equilibrium distribution of BP’s velocity. Using these designations and mentioned interaction uniformity and symmetries, we can write

[TABLE]

with integrals over velocities  $V,\,V\operatorname{{}^{\prime}}$  and over UE deviation from conservation,  $\operatorname{\epsilon}$ , with function

[TABLE]

and with   $D_{0}\hm=\tau_{0}\int V^{2}\,W_{0}(V)\,dV$ . Simultaneously, in terms of only velocity of BP (under “manually kept” constant UE value), operator  $\operatorname{\widetilde{\mathcal{Q}}}_{+0}$  becomes

[TABLE]

6.  Now, consider the integrals in (10), for brevity dealing with  $V$  like velocity of one-dimensional motion along  $k$ ’s direction. Notice that function (11) by very its definition is proportional to (relative) density of medium states at lowest of its “left” and “right” energies,  $\Sigma_{1}\hm=\operatorname{\mathcal{E}}_{0}+\operatorname{\epsilon}\!-\!E(V)$  and  $\Sigma_{2}\hm=\operatorname{\mathcal{E}}_{0}\!-\!E(V\operatorname{{}^{\prime}})$ . It means that

[TABLE]

with  $E=E(V)$  and  $E\operatorname{{}^{\prime}}=E(V\operatorname{{}^{\prime}})$ , where, again due to the definition (11), symmetry  $w(V\operatorname{{}^{\prime}},V;\sigma)\hm=w(V,V\operatorname{{}^{\prime}}\sigma)$  takes place, and factor  $\sigma\hm\equiv|E-\operatorname{\epsilon}-E\operatorname{{}^{\prime}}|\hm=|\Sigma_{1}-\Sigma_{2}|$  represents energy donated or obtained by medium during a transion. We also took into account that multiplier  $w(V\operatorname{{}^{\prime}},V;\dots)$  characterizes excitations of medium in itself, therefore it may involve the energy “discrepancy” (deviation),  $\operatorname{\epsilon}$ , only just through medium energy change  $\sigma$ . Besides, if medium along with its interactions is spatially uniform and isotropic, then  $w(\cdot)$  must have only two scalar arguments:   $w(V,V\operatorname{{}^{\prime}};\sigma)\hm\Rightarrow w(|V-V\operatorname{{}^{\prime}}|;\sigma)$ . Consequenntly,

[TABLE]

where

[TABLE]

and, evidently, only odd in respect at once to  $\operatorname{\epsilon}$  and  $E-E\operatorname{{}^{\prime}}$  component of  $F(V,V\operatorname{{}^{\prime}};|E-\operatorname{\epsilon}-E\operatorname{{}^{\prime}}|)$  in fact contrbutes to the integrals.

Formula (13) is main result of our communication. In its rest we point out some of consequences from (13).

7.  Consider expansion

[TABLE]

First of all we want to know about behavior of coefficient  $\operatorname{\mathcal{D}}_{2}(z)$  when $z\rightarrow+0$ , since it is coonected to long-time asymptotics of fourth-order cumulant of the BP’s path:

[TABLE]

and thus says about large-scale deviations of BP’s wandering statistics from the Gaussian one.

From (13) it follows that

[TABLE]

with second of velocity integrals

[TABLE]

where   $v\equiv(V\!+\!V\operatorname{{}^{\prime}})/2$ . All they are odd functions of  $\operatorname{\epsilon}$ . It is visible from here that if  $I_{2}(\operatorname{\epsilon}\rightarrow 0)\propto\operatorname{\epsilon}$ , then

[TABLE]

This means that absolute value of the fourth-order path cumulant grows with observation time proportionally to its square. In other words, non-Gaussianity of the wandering does not decrease with time at all.

At that, seemingly, both positive and negative signs of the non-Gaussianity are possible, dependently on sign of  $[dI_{2}(\operatorname{\epsilon})/d\operatorname{\epsilon}]_{\operatorname{\epsilon}=0}$ . Positive case allows natural interpretation as the result of smooth flicker fluctuations of BP’s diffusivity (diffusion coefficient) [15]-[18]. Then their effective correlation function  $C_{D}(\tau)$  (asymptotically) is determined by relation

[TABLE]

So, asymptotics (14) implies  $C_{D}(\infty)=$ const $>0$ , that is, verbally, “quasi-static” fluctuations.

Negative sign in (14), of course, also reflects flicker fluctuations of BP’s wandering, but of some different type, probably discontinuos like “telegraph signals”.

In particular, both signs can realize when variety of medium’s energy changes  $\sigma$  (irradiated or absorbed quanta) possible in one transition consists of single number (like in the simple phonon medium in [4]):   $w(|u|;\sigma)\hm=w(|u|)\,\delta(\sigma-q(|u|))$ , where  $u\equiv V\!-\!V\operatorname{{}^{\prime}}$ . At that, positive and negative contributions to the fourth cumulant are coming mainly from quanta  $q(|u|)$  smaller and greater than medium temperature, respectively.

8.  Complete enough analysis of  $\operatorname{\mathcal{D}}(z,ik)$  as a whole we leave for the future. Just here we have only to comment BP’s diffusivity (diffusio coefficient) as such, i.e.  $\operatorname{\mathcal{D}}_{2}(0)\hm=\operatorname{\mathcal{D}}(0,0)$ , which is given by

[TABLE]

There, second (integral) term also, like (14), is able for any sign, that is it can both increase and decrease the seed value  $D_{0}$  given by the usual kinetics.

It is necessary to underline insensitivity of the second term to degree of weakness of the interaction:  since  $w(\cdot)\hm\propto g_{0}=1/\tau_{0}$  by physical meaning as well as formal definitions of these objects, the product  $\tau_{0}I_{0}(\operatorname{\epsilon})$  is indifferent to  $\tau_{0}$ . At the same time  $D_{0}\propto\tau_{0}$ , therefore, in case of positivity of integral in (15), too strong diminution of  $\tau_{0}$  would turn  $\operatorname{\mathcal{D}}_{0}(0)$  to zero or less. But, clearly, it would be mere artifact of too unwary application of low-order NPT (in strict all-orders NPT  $\tau_{0}$  hardly can turn to zero).

9.  In conclusion, one more principal remark. Our above reduction of formalism from full micro-states to the pair of variables  $\operatorname{\mathcal{E}},\,V$  has cut off possibilities to include effects of medium’s memory. But, instead, we have concentrated on most fundamental effects of unavoidable time-energy uncertainty in real-life, finite-duration, interactions and observations. We hope that it is a noticable progress in quantum microscopic theory of “pure” flicker (1/f-) noise, that is one which, - as already was shown [7, 9, 12] in classical statistical mechanics, - is created by interaction even with such memoryless media as ideal gas.