arXiv subjects

Alex V. Plyukhin

Publications and source records attributed to Alex V. Plyukhin.

12 recordsLinked to original sources

arXiv2026

Nonergodic Brownian oscillator

We consider an open (Brownian) classical harmonic oscillator in contact with a non-Markovian thermal bath and described by the generalized Langevin equation. When the bath's spectrum has a finite upper cutoff frequency, the oscillator may have ergodic and nonergodic configurations. In ergodic configurations (when exist, they correspond to lower oscillator frequencies) the oscillator demonstrates conventional relaxation to thermal equilibrium with the bath. In nonergodic configurations (which correspond to higher oscillator frequencies) the oscillator in general does not thermalize, but relaxes to periodically correlated (cyclostationary) states whose statistics vary periodically in time. For a specific dissipation kernel in the Langevin equation, we evaluate explicitly relevant relaxation functions, which describe the evolution of mean values and time correlations. When the oscillator frequency is switched from a lower value to higher one, the oscillator may show parametric ergodic to nonergodic transitions with equilibrium initial and cyclostationary final states. These transitions are shown to resemble phase transitions of the second kind.

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arXiv2026

Langevin equations with non-Gaussian thermal noise: Valid but superfluous

We discuss the statistics of additive thermal (internal) noise in systems governed by the generalized Langevin equation with linear dissipation. To assess the equation's validity, it is common to assume that the system is ergodic and to verify that solutions approach correct equilibrium values at asymptotically long times. In this paper, we instead consider the consistency of the generalized Langevin equation with the Jarzynski equality at finite times and do not assume the system's ergodicity. Specifically, we consider a classical Brownian oscillator whose initial stiffness, or frequency, is perturbed by a rectangular pulse of duration $τ$. We find that the Jarzynski equality is satisfied unconditionally only up to the seventh order in $τ$; in higher orders, the Jarzynski equality holds if and only if the noise is Gaussian. These results imply that, unless it is exact, the Langevin equation can only be used to evaluate properties that are linear or quadratic in noise and its derivatives. Such properties are insensitive to the noise statistics, so the Langevin equation with linear dissipation and non-Gaussian noise (though not inconsistent by itself) is superfluous.

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arXiv2026

Langevin equation with potential of mean force: The case of anchored bath

The potential of mean force (PMF) is an effective average potential acting on an open system, renormalized due to the interaction with the surrounding thermal bath. The PMF determines the correction to the equilibrium Gibbs distribution, but it is generally unclear how to implement the concept for time-dependent phenomena described by a (generalized) Langevin equation. We study a model where the system is a single particle (so there are no complications related to internal forces) and a non-trivial PMF is due to the presence of on-site (anchor) potentials applied to the bath particles. We found that the PMF does not merely replace the external potential, but also makes the dissipation kernel and statistical properties of noise dependent on the system's position. That dependence is determined by the internal bath and system-bath interactions and is a priori unknown. Therefore, in the general case the Langevin equation with the PMF is not closed and thus inoperable. However, for systems with linear forces the aforementioned dependence on the system's position is canceled. As an example, we consider a model where the bath is formed by the Klein-Gordon chain, i. e. a harmonic chain with on-site harmonic potentials. In that case, the generalized Langevin equation has the standard form with an external potential replaced by a quadratic PMF.

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arXiv2025

Heat exchange for oscillator strongly coupled to thermal bath

The heat exchange fluctuation theorem (XFT) by Jarzynski and Wójcik [Phys. Rev. Lett. 92, 230602 (2004)] addresses the setting where two systems with different temperatures are brought in thermal contact at time $t=0$ and then disconnected at later time $τ$. The theorem asserts that the probability of an anomalous heat flux (from cold to hot), while nonzero, is exponentially smaller than the probability of the corresponding normal flux (from hot to cold). As a result, the average heat flux is always normal. In that way, the theorem demonstrates how irreversible heat transfer, observed on the macroscopic scale, emerges from the underlying reversible dynamics. The XFT was proved under the assumption that the coupling work required to connect and then disconnect the systems is small compared to the change of the internal energies of the systems. That condition is often valid for macroscopic systems, but may be violated for microscopic ones. We examine the validity of the XFT's assumption for a specific model of the Caldeira-Leggett type, where one system is a classical harmonic oscillator and the other is a thermal bath comprised of a large number of oscillators. The coupling between the system and the bath, which is bilinear, is instantaneously turned on at $t=0$ and off at $t=τ$. For that model, we found that the assumption of the XFT can be satisfied only for a rather restricted range of parameters. In general, the work involved in the process is not negligible and the energy exchange may be anomalous in the sense that the internal energy of the system, which is initially hotter than the bath, may further increase.

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arXiv2024

Brownian oscillator with time-dependent strength: a delta function protocol

We consider a classical Brownian oscillator of mass $m$ driven from an arbitrary initial state by varying the stiffness $k(t)$ of the harmonic potential according to the protocol $k(t)=k_0+a\,δ(t)$, involving the Dirac delta function. The microscopic work performed on the oscillator is shown to be $W=(a^2/2m)\,q^2-a q v$, where $q$ and $v$ are the coordinate and velocity in the initial state. If the initial distribution of $q$ and $v$ is the equilibrium one with temperature $T$, the average work is $\langle W \rangle=a^2T/(2m\,k_0)$ and the distribution $f(W)$ has the form of the product of exponential and modified Bessel functions. The distribution is asymmetric and diverges as $W\to 0$. The system's response for $t>0$ is evaluated for specific models.

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arXiv2023

Nonergodic Brownian oscillator: Low-frequency response

An undisturbed Brownian oscillator may not reach thermal equilibrium with the thermal bath due to the formation of a localized normal mode. The latter may emerge when the spectrum of the thermal bath has a finite upper bound $ω_0$ and the oscillator natural frequency exceeds a critical value $ω_c$, which depends on the specific form of the bath spectrum. We consider the response of the oscillator with and without a localized mode to the external periodic force with frequency $Ω$ lower than $ω_0$. The results complement those obtained earlier for the high-frequency response at $Ω\ge ω_0$ and require a different mathematical approach. The signature property of the high-frequency response is resonance when the external force frequency $Ω$ coincides with the frequency of the localized mode $ω_*$. In the low-frequency domain $Ω<ω_0$ the condition of resonance $Ω=ω_*$ cannot be met (since $ω_*>ω_0$). Yet, in the limits $ω\toω_c$ and $Ω\toω_0^-$, the oscillator shows a peculiar quasi-resonance response with an amplitude increasing with time sublinearly.

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arXiv2023

Nonergodic Brownian oscillator: High-frequency response

We consider a Brownian oscillator whose coupling to the environment may lead to the formation of a localized normal mode. For lower values of the oscillator's natural frequency, $ω\leω_c$, the localized mode is absent and the unperturbed oscillator reaches thermal equilibrium. For higher values of $ω>ω_c$, when the localized mode is formed, the unperturbed oscillator does not thermalize but rather evolves into a nonequilibrium cyclostationary state. We consider the response of such an oscillator to an external periodic force. Despite the coupling to the environment, the oscillator shows the unbounded resonance (with the response linearly increasing with time) when the frequency of the external force coincides with the frequency of the localized mode. An unusual resonance (``quasi-resonance") occurs for the oscillator with the critical value of the natural frequency $ω=ω_c$, which separates thermalizing (ergodic) and non-thermalizing (nonergodic) configurations. In that case the resonance response increases with time sublinearly, which can be interpreted as a resonance between the external force and the incipient localized mode.

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arXiv2020

Non-Clausius heat transfer: The method of the nonstationary Langevin equation

Compared to other formulations of the second law of thermodynamics, the Clausius statement that heat does not spontaneously flow from cold to hot concerns a system in non-equilibrium states, and in that respect is more ambitious but also more ambiguous. We discuss two scenarios when the Clausius statement in its plain form does not hold. First, for ergodic systems, the energy transfer may be consistent with the statement on a coarse-grained time scale, but be anomalously directed during time intervals shorter than the thermalization time. In particular, when an initially colder system is brought in contact to a hotter bath, the internal energy of the former increases with time in a long run but not monotonically. Second, the heat transfer may not respect the Clausius statement on any time-scale in non-ergodic systems due to the formation of localized vibrational modes. We illustrate the two scenarios with a familiar model of an isotope atom attached to a semi-infinite harmonic atomic chain. Technically, the discussion is based on a Langevin equation for the isotope, using the initial condition when the isotope and chain are initially prepared in uncorrelated canonical states under the constraint that the boundary atom between the isotope and chain is initially fixed and later released. In such setting, the noise in the Langevin equation is non-stationary, and the fluctuation-dissipation relation has a non-standard form.

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arXiv2020

Non-Clausius heat transfer: the example of harmonic chain with an impurity

Motivated by recent discussion about the possibility of non-Clausius (from cold to hot) heat flow, we revisit the familiar model of an impurity atom of mass $M$ embedded in an otherwise uniform one-dimensional harmonic lattice of host atoms of mass $m$. Assuming that the initial distributions for the impurity and the rest of the lattice have canonical forms with given temperatures, we show that the average kinetic energy of the impurity may increase with time even if its initial temperature is higher than or equal to that of the lattice. Such an increase is only temporary in uniform lattices and in lattices with a heavy impurity ($M\ge m$), but may be permanent in lattices with a localized vibrational mode generated by a light impurity ($M<m$). Thus the model shows a non-Clausius spontaneous heat flow directed from a colder lattice to a hotter impurity.

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arXiv2019

Random walks on uniform and non-uniform combs and brushes

We consider random walks on comb- and brush-like graphs consisting of a base (of fractal dimension $D$) decorated with attached side-groups. The graphs are also characterized by the fractal dimension $D_a$ of a set of anchor points where side-groups are attached to the base. Two types of graphs are considered. Graphs of the first type are uniform in the sense that anchor points are distributed periodically over the base, and thus form a subset of the base with dimension $D_a=D$. Graphs of the second type are decorated with side-groups in a regular yet non-uniform way: the set of anchor points has fractal dimension smaller than that of the base, $D_a 1$) and numerically tested for the Sierpinski brush (with the base and anchor set built on the same Sierpinski gasket). As an example of nonuniform graphs we consider the Cantor comb composed of a one-dimensional base and side-groups, the latter attached to the former at anchor points forming the Cantor set. A peculiar feature of this and other nonuniform systems is a long-lived regime of super-diffusive transport when side-groups are of a finite size.

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arXiv2016

Random walks with fractally correlated traps: Stretched exponential and power law survival kinetics

We consider the survival probability $f(t)$ of a random walk with a constant hopping rate $w$ on a host lattice of fractal dimension $d$ and spectral dimension $d_s\le 2$, with spatially correlated traps. The traps form a sublattice with fractal dimension $d_a w$, including the limit of perfect traps $w_a\to \infty$, the stretched exponential regime is absent and the decay of $f(t)$ follows, after a short transient, the aforementioned power law for all times.

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arXiv2015

Correlations of correlations: Secondary autocorrelations in finite harmonic systems

The momentum or velocity autocorrelation function C(t) for a tagged oscillator in a finite harmonic system decays like that of an infinite system for short times, but exhibits erratic behavior at longer time scales. We introduce the autocorrelation function of the long-time noisy tail of C(t) ("a correlation of the correlation"), which characterizes the distribution of recurrence times. Remarkably, for harmonic systems with same-mass particles this secondary correlation may coincide with the primary correlation C(t) (when both functions are normalized) either exactly, or over a significant initial time interval. When the tagged particle is heavier than the rest, the equality does not hold, correlations shows non-random long-time scale pattern, and higher order correlations converge to the lowest normal mode.

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