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arXiv:1803.09250

Kinetics, pseudo-kinetics, uncertainty principle and quantum 1/f noise

Yu. E. Kuzovlev

Published Mar 25, 2018• 0 citations
arXivHTMLPDF

Abstract

1/f noise at arbitrary low frequences is the way of existence of irreversibility in thermal motion governed by reversible laws of mechanics. This statement not once was confirmed in statistical mechanics beyond its traditional kinetical roughenings. Here we point out that in case of quantum statistical mechanics in principle it is sufficient to avoid such the roughening as the "Fermi golden rule". This means taking into account the time-energy uncertainty principle (time-frequency one in classical limit) and thus uncertainties in characteristics of real collisions and scatterings of particles and/or quanta. We consider the resulting "pseudo-kinetics" and demonstrate how it produces quantum 1/f-noise

DOI
10.48550/arxiv.1803.09250

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1/f noise at arbitrary low frequences is the way of existence of irreversibility in thermal motion governed by reversible laws of mechanics. This statement not once was confirmed in statistical mechanics beyond its traditional kinetical roughenings. Here we point out that in case of quantum statistical mechanics in principle it is sufficient to avoid such the roughening as the "Fermi golden rule". This means taking into account the time-energy uncertainty principle (time-frequency one in classical limit) and thus uncertainties in characteristics of real collisions and scatterings of particles and/or quanta. We consider the resulting "pseudo-kinetics" and demonstrate how it produces quantum 1/f-noise
Arguments are presented for the viewpoint that irreversibility is inherent in quanturn mechanics, irrespective of the number of particles participating. This fact is related to the notion of transition–ramifying evolution–which is characteristic of quantum mechanics. It is in contrast to classical mechanics based on the notion of orbit, where the irreversibility arises only as the result of a large number of particles or a certain complexity of the system. The statistical assumption needed for the derivation of irreversibility in classical mechanics is absorbed into the uncertainty principle of quantum mechanics, and is revived as inherent irreversibility. The statements conform with the parallelism of quantum mechanics and classical statistical mechanics emphasized in a previous report.
Heisenberg’s uncertainty principle for position-momentum delta x delta p >= hbar/2 is considered as an important statement regarding quantum mechanics i.e. one cannot know both p and x exactly. Delta x and delta p stand out because in classical mechanics one has an eact p and x at each t. Mathematically, however, delta x and delta p are standard deviations which may also be calculated in classical statistical mechanics, for example with the Maxwell-Boltzmann distribution. In particular, if one considers the ground state of a simple harmonic oscillator, the quantum P(x) proportional to exp(-pp/a) and P(p) proportional to exp(-xxa) match the classical statistical probabilities P(x) and P(p) for a fixed value of T proportional to sqrt(k/m) where k is the spring constant. Thus for this special T, delta x delta p for a classical statistical mechanical gas is identical to delta x delta p for the ground state quantum oscillator. The probability is normalized the same way for both. The physics of the two situations, however, is completely different. Thus the Heisenberg uncertainty principle does not necessarily describe quantum mechanical behaviour alone. If one uses entropic uncertainty, i.e. Sp + Sx where S is Shannon’s entropy, once again one uses P(p)=a*(p)a(p) and P(x)=W*(x)W(x) where W(x)=wavefunction = Sum over p a(p)exp(ipx). In principle, one could also calculate such an entropy for a classical gas i.e. use P(x)P(p) in Shannon’s entropy. The result Sp+Sx does not imply one does not know about p if one knows x i.e. the usual quantum statement.
We address the paradoxical transformation of a classical-mechanical particle motion when the space and time scales of observation pass below the uncertainty principle threshold. This is analyzed in the language of classical statistical mechanics, considering specifically many-particle systems inhomogeneous along one spatial direction. We employ the density functional formalism in its square-gradient form and find: (i) The macroscopic solution is analogous to the classical trajectory of a particle under a potential of force given by (minus) the free energy density. Whereas, (ii) fluctuations around the solution in (i) are equal to the quantum-mechanical wave functions of a particle under a potential given by the curvature of the free energy density. We illustrate this situation with three textbook examples: A particle in a box, the harmonic oscillator, and the hydrogen atom. We show that their time-independent Schrödinger equation wave functions describe, respectively, the fluctuations of a fluid interface, of critical point fluctuations, and of a confined ideal gas. At large scales, sharp probability distributions make fluctuations irrelevant; the vanishing of the first variation yields the macroscopically observable statistical-mechanical non-uniformity, equivalent to the classical particle trajectory. But at sufficiently small scales, with necessarily very few particles, distributions appear much wider, fluctuations dominate, and one obtains the Schrödinger equation (for the microscopic potential).
This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. It is in part a continuation of an earlier paper on the uncertainty principle for quantum-mechanical histories. We consider histories characterized by position samplings at n moments of time. This includes the case of position sampling continuous in time in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, F=mx\ifmmode\ddot\else\textasciidieresis\fi{}+V'(x), at n-2 times. We find that it is peaked around F=0; thus classical correlations are exhibited. We show that the width of the peak \ensuremath{\Delta}F is largely independent of the initial state. We show that the uncertainty principle takes the form 2${\mathrm{\ensuremath{\sigma}}}^{2}$(\ensuremath{\Delta}F)2{)}^{2})2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/t2{\mathit{t}}^{2}t2, where \ensuremath{\sigma} is the width of the position samplings, and t is the time scale between projections. We determine the modifications to this result when the system is coupled to a thermal environment, thus obtaining a measure of the comparative sizes of quantum and thermal fluctuations. We show that the thermal fluctuations become comparable to the quantum fluctuations under the same conditions that decoherence effects come into play, in agreement with earlier work. We also study an alternative measure of classical correlations, namely, the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred.We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. Under appropriate conditions, which we describe, they admit an interpretation as a statistical ensemble of classical solutions, with the probability of each individual classical solution given by a smeared Wigner function of its initial data. We study the decoherence properties of histories characterized by the value of the field equations, F, at a sequence of times. We argue that they will be decoherent if their initial data are decoherent.
We investigate the large-time asymptotical behavior of a mobile impurity immersed in a degenerate Tonks-Girardeau gas. We derive a correct weak-coupling kinetic equation valid for N arbitrary ratio of masses of gas and impurity particles. When gas particles are either lighter or heavier than the impurity we find that our theory is equivalent to the Boltzmann theory with the collision integral calculated via the Fermi Golden Rule. On the contrary, in the equal-mass case, Fermi Golden Rule treatment gives false results due to not accounting for multiple coherent scattering events. The latter are treated by the resummation of ladder diagrams, which leads to a new kinetic equation. The asymptotic momentum of the impurity produced from this equation coincides with the result obtained by means of the Bethe ansatz.