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arXiv:cond-mat/0109018 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum thermodynamics of systems with anomalous dissipative coupling

Alessandro Cuccoli, Andrea Fubini, V. Tognetti, Ruggero Vaia

Published Nov 26, 2001• 34 citations
arXivHTMLPDF

Abstract

The standard system-plus-reservoir approach used in the study of dissipative systems can be meaningfully generalized to a dissipative coupling involving the momentum, instead of the coordinate: the corresponding equation of motion differs from the Langevin equation, so this is called anomalous dissipation. It occurs for systems where such coupling can indeed be derived from the physical analysis of the degrees of freedom that can be treated as a dissipation bath. Starting from the influence functional corresponding to anomalous dissipation, it is shown how to derive the effective classical potential that gives the quantum thermal averages for the dissipative system in terms of classical-like calculations; the generalization to many degrees of freedom is given. The formalism is applied to a single particle in a double well and to the discrete phi(4) model. At variance with the standard case, the fluctuations of the coordinate are enhanced by anomalous dissipative coupling.

DOI
10.1103/physreve.64.066124

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The standard system-plus-reservoir approach used in the study of dissipative systems can be meaningfully generalized to a dissipative coupling involving the momentum, instead of the coordinate: the corresponding equation of motion differs from the Langevin equation, so this is called anomalous dissipation. It occurs for systems where such coupling can indeed be derived from the physical analysis of the degrees of freedom that can be treated as a dissipation bath. Starting from the influence functional corresponding to anomalous dissipation, it is shown how to derive the effective classical potential that gives the quantum thermal averages for the dissipative system in terms of classical-like calculations; the generalization to many degrees of freedom is given. The formalism is applied to a single particle in a double well and to the discrete phi(4) model. At variance with the standard case, the fluctuations of the coordinate are enhanced by anomalous dissipative coupling.
The study of dissipative systems holds significant interest in physics.This thesis aims to explore these phenomena in both classical and quantum regimes.In the first part, we utilize the system-reservoir approach to investigate the dynamics of two Brownian particles immersed in the same bath.Two methods are employed to address this problem: one with bilinear coupling and the other with nonlinear coupling between the particles and the bath.The extension of the system to include two particles with bilinear coupling yields unphysical results, such as free-particle motion for the relative coordinate and a lack of interaction between closely spaced particles.To address this issue, authors have been introduced an exponential coupling.In this work, we propose a method to reconcile both linear and nonlinear couplings.We demonstrate how to derive the same nonlinear dissipation rates starting from the bilinear Lagrangian, achieved through a modified spectral function that explicitly depends on the distance between the particles.Additionally, we implement a modified spectral function to mitigate anomalous diffusion observed in the standard nonlinear model, along with a phenomenological model describing hydrodynamic interaction between a pair of Brownian particles in a viscous fluid.In the quantum regime, we adopt the same system-reservoir approach to investigate the dissipative adaptation hypothesis proposed by Jeremy England.This hypothesis proposes a general thermodynamic mechanism that explains the self-organization of systems through the dissipation of work absorbed from an external drive.In the second part of this thesis, we explore the quantum dynamics of systems subjected to an external drive, evaluating the thermodynamic quantities of a self-organization process.To do so, we utilize a time-dependent spin-boson Hamiltonian that characterizes a particle subject to a metastable double potential with time-dependent parameters controlling the asymmetry of the wells.Our objective is to demonstrate that the asymmetric potential can localize the particle in the unstable side of the well and verify that this transition results in the highest energy absorption.In conclusion, we propose further investigations into the driven spin-boson model to establish a comprehensive theory of the system's evolution and its thermodynamic implications.
We consider quantum nonlinear many-body systems with dissipation described within the Caldeira-Leggett model, i.e., by a nonlocal action in the path integral for the density matrix. Approximate classical-like formulas for thermodynamic quantities are derived for the case of many degrees of freedom, with general kinetic and dissipative quadratic forms. The underlying scheme is the pure-quantum self-consistent harmonic approximation (PQSCHA), equivalent to the variational approach by the Feynman-Jensen inequality with a suitable quadratic nonlocal trial action. A low-coupling approximation permits us to get manageable PQSCHA expressions for quantum thermal averages with a classical Boltzmann factor involving an effective potential and an inner Gaussian average that describes the fluctuations originating from the interplay of quanticity and dissipation. The application of the PQSCHA to a quantum phi(4) chain with Drude-like dissipation shows nontrivial effects of dissipation, depending upon its strength and bandwidth.
Abstract This review provides a brief and quick introduction to the quantum Langevin equation for an oscillator, while focusing on the steady-state thermodynamic aspects. A derivation of the quantum Langevin equation is carefully outlined based on the microscopic model of the heat bath as a collection of a large number of independent quantum oscillators, the so-called independent-oscillator model. This is followed by a discussion on the relevant ‘weak-coupling’ limit. In the steady state, we analyze the quantum counterpart of energy equipartition theorem which has generated a considerable amount of interest in recent literature. The free energy, entropy, specific heat, and third law of thermodynamics are discussed for one-dimensional quantum Brownian motion in a harmonic well. Following this, we explore some aspects of dissipative diamagnetism in the context of quantum Brownian oscillators, emphasizing upon the role of confining potentials and also upon the environment-induced classical-quantum crossover. We discuss situations where the system-bath coupling is via the momentum variables by focusing on a gauge-invariant model of momentum-momentum coupling in the presence of a vector potential; for this problem, we derive the quantum Langevin equation and discuss quantum thermodynamic functions. Finally, the topic of fluctuation theorems is discussed (albeit, briefly) in the context of classical and quantum cyclotron motion of a particle coupled to a heat bath.
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We develop a perturbation scheme to treat the dynamics of a quantum Brownian particle (coordinate xxx) coupled nonlinearly to a bath of oscillators (coordinates qn{q}_{n}qn​) with an interaction action in the form $\ensuremath{\lambda}f(x){q}_{n}^{k}$. We derive the influence functionals for the k=2,3,4k=2, 3, 4k=2,3,4 cases to order ${\ensuremath{\lambda}}^{2}$ and derive the master equations for the special cases of local dissipation and white noise, as well as the general cases of nonlocal dissipation and colored noise for f(x)=xandx2f(x)=x \mathrm{and} {x}^{2}f(x)=xandx2. We show that a generalized fluctuation-dissipation relation exists between the lthl\mathrm{th}lth-order noise kernels of the kthk\mathrm{th}kth-order coupling and their corresponding dissipation kernels.
A derivation of the equations of motion for a mixed quantum-classical system in a dissipative environment is presented. The classical environment of the quantum subsystem is taken to be composed of two parts: the first part, termed the classical subsystem, comprises those degrees of freedom that are coupled directly to the quantum subsystem, whereas the second part is a large bath that is coupled only to the classical subsystem. Projection operator methods are used to eliminate the bath degrees of freedom yielding a dissipative equation of motion for the coupled quantum and classical subsystems.